John Wallis: Writings on Music [1 ed.] 0754668703, 9780754668701

John Wallis (1616-1703), was one of the foremost British mathematicians of the seventeenth century, and is also remember

632 266 3MB

English Pages 250 [254] Year 2014

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

John Wallis: Writings on Music [1 ed.]
 0754668703, 9780754668701

Table of contents :
Contents
Series Editor’s Preface
List of Figures
Acknowledgements
List of Abbreviations
Introduction
1 Letters to Henry Oldenburg, May 1664
2 Letters to Henry Oldenburg, March 1677
‘3 The Harmonics of the Ancients compared with Today’s’: Appendix to Ptolemy’s Harmonics, 1682
4 Notice of Wallis’s Edition of Ptolemy’s Harmonics in the Philosophical Transactions, January 1683
5 ‘A Question in Musick’, Philosophical Transactions, March 1698
March 1698
6 Letter to Samuel Pepys, June 1698
7 Letters to Andrew Fletcher, August 1698
Select Bibliography
Index

Citation preview

John Wallis: Writings on Music

Music Theory in Britain, 1500–1700: Critical Editions Series Editor Jessie Ann Owens, University of California, Davis, USA

Also published in this series: The Music Treatises of Thomas Ravenscroft ‘Treatise of Practicall Musicke’ and A Briefe Discourse Ross W. Duffin Thomas Salmon: Writings on Music Volume I: An Essay to the Advancement of Musick and the Ensuing Controversy, 1672–3 Benjamin Wardhaugh Thomas Salmon: Writings on Music Volume II: A Proposal to Perform Musick and Related Writings, 1685–1706 Benjamin Wardhaugh ‘The Temple of Music’ by Robert Fludd Peter Hauge John Birchensha: Writings on Music Christopher D.S. Field and Benjamin Wardhaugh A Briefe and Short Instruction of the Art of Musicke by Elway Bevin Edited by Denis Collins Synopsis of Vocal Musick by A.B. Philo-Mus. Edited by Rebecca Herissone A Briefe Introduction to the Skill of Song by William Bathe Edited by Kevin C. Karnes A New Way of Making Fowre Parts in Counterpoint by Thomas Campion and Rules how to Compose by Giovanni Coprario Edited by Christopher R. Wilson

John Wallis: Writings on Music

Edited by David Cram Jesus College, University of Oxford, UK Benjamin Wardhaugh All Souls College, University of Oxford, UK

First published 2014 by Ashgate Publishing Published 2016 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN 711 Third Avenue, New York, NY 10017, USA Routledge is an imprint of the Taylor & Francis Group, an informa business Copyright © David Cram and Benjamin Wardhaugh David Cram and Benjamin Wardhaugh have asserted their right under the Copyright, Designs and Patents Act, 1988, to be identified as the editors of this work. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Bach musicological font ©Yo Tomita British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library The Library of Congress has cataloged the printed edition as follows: Wallis, John, 1616–1703. [Prose works. Selections] John Wallis : writings on music / edited by David Cram and Benjamin Wardhaugh. pages cm. – (Music theory in Britain, 1500-1700. Critical editions) Includes bibliographical references and index. ISBN 978-0-7546-6870-1 (hardcover) 1. Wallis, John, 1616-1703 – Correspondence. 2. Oldenburg, Henry, approximately 1615–1677 – Correspondence. 3. Pepys, Samuel, 1633–1703 – Correspondence. 4. Fletcher, Andrew, 1655–1716 – Correspondence. 5. Music theory – Early works to 1800. I. Cram, David, editor of compilation. II. Wardhaugh, Benjamin, 1979–, editor of compilation. III. Title. ML174.W15 2014 781–dc23 2014005013 ISBN 9780754668701 (hbk)

Contents Series Editor’s Preface   List of Figures   Acknowledgements   List of Abbreviations   Introduction   John Wallis, 1616–1703 ‘My present un polished thoughts’: The letters of 1664 ‘A new Musical Discovery’: The letters of 1677 ‘The Harmonics of the ancients compared with today’s’: Wallis’s appendix to Ptolemy’s Harmonics, 1682 ‘Your Most Obliged & Faithfull Servant’: Thomas Salmon ‘A thing wherein I am so little acquainted’: musical publications in 1698 ‘Very elaborate and judicious’: Wallis’s legacy in musical scholarship Editorial policy

vii ix xi xiii 1 1 2 8 12 17 19 27 30

1 Letters to Henry Oldenburg, May 1664  

41

2 Letters to Henry Oldenburg, March 1677  

69

3 ‘The Harmonics of the Ancients compared with Today’s’: Appendix to Ptolemy’s Harmonics, 1682  

75

4 Notice of Wallis’s Edition of Ptolemy’s Harmonics in the Philosophical Transactions, January 1683  

205

5 ‘A Question in Musick’, Philosophical Transactions, March 1698   207 6 Letter to Samuel Pepys, June 1698  

213

7 Letters to Andrew Fletcher, August 1698  

223

Select Bibliography   231 Index237

Series Editor’s Preface The purpose of this series is to provide critical editions of music theory in Britain (primarily England, but Scotland, Ireland and Wales also) from 1500 to 1700. By ‘theory’ is meant all sorts of writing about music, from textbooks aimed at the beginner to treatises written for a more sophisticated audience. These foundational texts have immense value in revealing attitudes, ways of thinking and even vocabulary crucial for understanding and analysing music. They reveal beliefs about the power of music, its function in society and its role in education, and they furnish valuable information about performance practice and about the context of performance. They are a window into musical culture every bit as important as the music itself. The editions in this series present the text in its original form. That is, they retain original spelling, capitalization and punctuation, as well as certain salient features of the type, for example the choice of font. A textual commentary in each volume offers an explication of difficult or unfamiliar terminology as well as suggested corrections of printing errors; the introduction situates the work and its author in a larger historical context. Jessie Ann Owens Professor of Music Department of Music University of California, Davis, USA

List of Figures 1.1 The opening of John Wallis’s letter to Henry Oldenburg of 14 May 1664, in the copy (C) made for Narcissus Marsh. Dublin, Marsh’s Library, Z3.4.24, fol. 2r. By kind permission of the Governors and Guardians of Marsh’s Library in whom copyright remains vested. 3.1

3.2

44

John Wallis’s diagrammatic representation of the ancient and modern scales. Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium (Oxford: E Theatro Sheldoniano, 1699), p. 157 (Oxford, Bodleian Library, shelfmark: Savile B 12). By permission of the Bodleian Libraries, University of Oxford.

88

John Wallis’s diagrammatic representation of the genera according to Aristoxenus. Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium (Oxford: E Theatro Sheldoniano, 1699), p. 164 (Oxford, Bodleian Library, shelfmark: Savile B 12). By permission of the Bodleian Libraries, University of Oxford.

104

Acknowledgements It is a pleasure once again to acknowledge the kindness and support of Jessie Ann Owens, the editor of this series of volumes, who took an interest in this project from its inception. It is a matter of regret that the pressure of other work prevented her from being, as originally planned, one of the editors named on the title page. She and her assistant Minji Kim have provided invaluable assistance with the later stages of this project. The editors also acknowledge the support of the Arts and Humanities Research Council for the John Wallis Project, under which work on this volume began. Particular thanks are due to the staff of Ashgate Publishing, in particular Heidi Bishop and Pam Bertram, for their work. All Souls College, Jesus College, and Wolfson College, Oxford supported the editors during periods of work on this book, and we have pleasure in recording our gratitude to those institutions. Philip Beeley has been of particular help in matters connected with Wallis’s correspondence. The editors of this volume are specialists in early modern mathematics (Wardhaugh) and linguistics (Cram). The publication of Chapter 3 would not have been possible without the very generous help of Richard Ashdowne and Armand d’Angour with the translation of Wallis’s Latin. Our English text owes its merits to them, its remaining defects to us. We are grateful to Marsh’s Library, Dublin and the Bodleian Library, Oxford, for permission to reproduce the illustrations in this volume.

List of Abbreviations DNB Leslie Stephen and Sidney Lee (eds), The Dictionary of National Biography (63 vols; London, 1885–1900) ECCO Eighteenth-Century Collections Online: galenet.galegroup. com/servlet/ECCO EEBO Early English Books Online: eebo.chadwyck.com EMLO Early Modern Letters Online: emlo.bodleian.ox.ac.uk ESTC The Electronic Short-Title Catalogue: estc.bl.uk GMW Andrew Barker (ed.), Greek Musical Writings II: harmonic and acoustic theory (Cambridge, 1989) GMO Stanley Sadie (ed.), The New Grove Dictionary of Music and Musicians (29 vols; 2nd edn; London, 2001; online edn; 2007–10: www.oxfordmusiconline.com) OCD Simon Hornblower and Antony Spawforth (eds), The Oxford Classical Dictionary (3rd edn; Oxford, 1996) ODNB H.C.G. Matthew and Brian Harrison (eds), Oxford Dictionary of National Biography: from the earliest times to the year 2000 (60 vols; Oxford, 2004; online edn; 2008: www.oxforddnb.com).

Introduction This volume presents the main writings on music of John Wallis (1616–1703), doctor of divinity, Savilian Professor of Geometry, and fellow of the Royal Society. Chief among these is ‘The Harmonics of the Ancients compared with Today’s’, the appendix to his 1682 edition of Ptolemy’s Harmonics; also included are three letters to Henry Oldenburg, the secretary of the Royal Society, written in 1664, and five letters published in the Philosophical Transactions of the Royal Society: two in 1677 and three in 1698. These writings are concerned with several aspects of the theory of music, including the mechanical production of musical sound and the effects of music on the human person. A principal and recurring concern is the mathematical foundation of musical harmony and the relationship between ancient thinking on that subject – as represented in particular by Ptolemy (c. a.d. 90–168) – and the music of seventeenth-century England. The sources presented here enable the evolution of Wallis’s ideas on this subject to be discerned and provide evidence for the place which music held in his intellectual life. The purpose of this volume is to situate these texts within Wallis’s life and intellectual biography, and within the wider discussion of musical theory taking place in England at the time. It is not to situate Wallis himself within a wider intellectual or political milieu, nor to examine more widely the Greek musical tradition on which he at times reflected, or its reception.1 This introduction comprises a discussion, roughly in chronological order, of the various pieces of evidence for Wallis’s interest in music, including the texts edited in this volume. It outlines the contexts for the production of those texts and sketches the intellectual backgrounds to Wallis’s work on the musical topics he addressed in them: his sources, his relationship with Thomas Salmon, his interest in experimentalism, and his views on the nature and human effects of music. The editors thus attempt to provide the materials for a fuller appreciation for this substantial and significant body of musical writings from Restoration England. John Wallis, 1616–1703 Most of John Wallis’s published works have yet to receive modern editions; the edition of his correspondence, which will be a vital resource for the study of his intellectual biography, so far covers only the first part of his life.2 Biographical work on him is therefore necessarily somewhat provisional, and this introduction will present only the brief sketch necessary for understanding when and how his main writings on music were produced.3

2

John Wallis: Writings on Music

John Wallis was born in 1616 in Ashford, Kent, and educated at schools in Kent and Essex and at Emmanuel College, Cambridge. During the 1640s he became increasingly prominent as a public intellectual; he also became known for his cryptographical work for the government, which would continue after the Restoration. In 1649 he was appointed to the Savilian Chair of Geometry at Oxford in place of Peter Turner. Wallis remained in that position for more than 50 years, until his death in 1703, and became one of the most important English mathematicians of his generation. He was a fellow of the Royal Society from 1661, although his continuing residence in Oxford resulted in a relatively slight visibility at meetings compared with other early fellows of comparable intellectual range and seriousness.4 Nevertheless, he published frequently in the Philosophical Transactions (more than 60 papers). Mathematics had no exclusive claim on his attention. He became keeper of the university archives in 1658, having been ordained in 1640 and become a doctor of divinity in 1654, and published quite frequently on theological subjects. In 1660 he became a royal chaplain; several of his sermons were printed.5 Wallis’s publications during his lifetime numbered several dozen,6 and they ranged very widely: grammar, logic, the mechanics of speech, mathematics of all kinds, mathematical mechanics, theology. He was eclectic rather than, perhaps, encyclopaedic in his interests, but there were very few even among the most versatile of his contemporaries who could have claimed to have made real contributions to so many fields. Those contributions were not made without controversy, and indeed Wallis’s pattern of publication suggests that at times he sought out topics on which he could engage in robust refutation. Individuals including the Danish philologist and mathematician Marcus Meibom, the English clergyman and natural philosopher William Holder, and most famously the English political thinker and natural scientist (and mathematician) Thomas Hobbes all came in for attack.7 A combative and vigorous manner of writing, and possibly a sometimes cavalier attitude to giving credit, seem to have marked his interventions in musical topics too, as we will see. It will be evident from all of this, and from the contents of the present volume, that Wallis considered his responsibilities as Savilian Professor of Geometry in no narrow sense; the Savilian statues, indeed, required him to do so, specifying that the professor of geometry should teach not just the classics of ancient Greek geometry but also arithmetic, surveying (with practical demonstrations ‘in the fields or spots adjacent to the University’), mechanics, and ‘canonics or music’.8 Although some of Wallis’s lectures on Euclid’s Elements survive,9 no lectures on ‘canonics’ by him are known. It is not clear just when he began to take a deeper interest in that particular subject. ‘My present un polished thoughts’: the letters of 1664 Our earliest direct evidence for Wallis’s interest in music is contained in correspondence with Henry Oldenburg (c. 1619–77) in 1664. On 27 April John

Introduction

3

Birchensha (c. 1605–?1681), the subject of an earlier volume in this series, had appeared before the Royal Society, to receive the society’s thanks for a letter he had submitted setting out in brief his views on music theory.10 The musical committee (containing William Brouncker, Robert Boyle, and others) which the society set up to examine Birchensha’s ideas does not seem to have come to any conclusion – indeed, we have no evidence that it met at all – but the incident produced some resonance in the correspondence of Oldenburg, the secretary of the society. He wrote to Wallis on 4 May, in a letter – now lost – which apparently gave a ‘large account’ of the Royal Society’s recent activities and mentioned Birchensha and his paper without giving any details. (The letter also seems to have discussed the handling of the papers of the astronomer Jeremiah Horrox and to have asked Wallis to press Thomas Hyde concerning work on the Latin translation of works by Ulugh Beg.) Birchensha undertook ‘to bring the art of music to that perfection, that even those, who could neither sing nor play, should be able, by his rules, to make good airs, and compose two, three, four, or more parts artificially’.11 It is clear from what followed that Oldenburg was correct in his belief that the information about Birchensha would be of particular interest to Wallis. Wallis wrote three letters to Oldenburg during the remainder of May 1664, which represent three stages in his response (Chapter 1 below). In the first, dated 7 May, he remarked briefly that his limited reading had shown him that the modern theory of music was defective, but said little more. There possibly followed another letter from Oldenburg, also now lost, in which he invited or seemed to Wallis to invite further comment. On 14 May Wallis obliged, providing a treatise of a little under 5,000 words dealing with the mathematics of music: in particular with the construction of a satisfactory mathematical description of the modern musical scale. Hedged though it was with disclaimers about his scanty expertise and scantier reading, this text in fact made it clear that Wallis had given some thought to questions of musical tuning. What seems to be draft material for it, indeed, survives in the Bodleian Library, in a single opening of one of Wallis’s notebooks. Here he can be seen sketching out some of the diagrams which appeared in his letter, including a large table of ratios, and performing the calculations and tabulation of musical intervals and ratios which underlay them.12 The letter set out several elements which were to recur in all his subsequent discussions of the subject, as well as what amounted to Wallis’s programme for the study of music. Most prominent was what has more recently been called the ‘coincidence’ theory of consonance.13 On the assumption that musical sound consisted of or was somehow associated with a series of distinct pulses or vibrations, the idea of the coincidence theory was that reasonably frequent coincidence between those vibrations was a requirement for two sounds to be perceived as harmonious. That was only possible if the frequencies of the two sets of vibrations formed a ratio of small whole numbers. Thus Wallis suggested that two strings that are Unisons, be therefore Harmonious, because (supposing them to haue one common beginning) the Vibrations of the one doe exactly answer

4

John Wallis: Writings on Music to those of the other: And next unto these, Octaves; because that, the Vibrations of the more Acute being twice as many in the same time, every vibration of the more Graue or slower string, is coincident with every second Vibration of the Quicker or more Acute[.]14

The argument continued for other consonances; the perfect fifth was associated with a ratio of 3 : 2 and the perfect fourth with a ratio of 4 : 3. By this means the traditional association of mathematical ratios with musical intervals was transformed from an observed fact about the harmoniousness of combinations of strings of different lengths into a mechanical explanation founded on the properties of musical sound and capable in principle of being investigated experimentally. Wallis did his best to avoid committing himself, but made it clear that he considered this account adequate as at least a working hypothesis concerning the phenomena of consonance. His subsequent discussion depended upon it quite heavily, since he relied at more than one point on an assessment of the relative degree of consonance of musical intervals which derived from the coincidence theory: more coincidences meant greater harmoniousness. Further, his discussion of musical ratios in this letter consistently worked with the relative frequencies of sounds, rather than (as was more traditional) with the lengths of the strings that would produce them. A second important element, and one that was to remain characteristic of Wallis’s musical thought, was what is known variously as the just intonation, the just scale, or the syntonic diatonic scale. In his hands this was a mathematical description specifying the position of the diatonic notes within the octave, in terms of their relative frequencies of vibration.15 The octave (with ratio 2 : 1) was divided into fifth (3 : 2) and fourth (4 : 3) and the fifth into major (5 : 4) and minor (6 : 5) thirds. The major third was further divided into major (9 : 8) and minor (10 : 9) tones. In each case the division was by the same procedure: double the terms of the original ratio and form a middle term as their mean; this middle term would produce two new ratios with the original terms. Thus, for example, 2 : 1 is equal to 4 : 2; the mean is 3, and the new ratios into which 4 : 2 may be divided are 4 : 3 and 3 : 2. Finally, the diatonic semitone was defined as the difference between a fourth and a major third. With this set of intervals, a complete scale could be built up.16 Characteristic of this scale was the distinction between two different sizes of tone; by contrast, in both the ‘Pythagorean’ scale of many medieval and early modern theoretical discussions (including that of John Birchensha), and in the mean-tone scale favoured by some performers, all whole tones were equal.17 The fact that this scale did indeed occur as one of the many which Ptolemy had listed in his Harmonics added an ancient veneer to a set of musical ratios whose serious discussion by theorists was less than two centuries old when Wallis was writing.18 In Wallis’s account, certain questions followed from this construction of the scale. The octave, the fifth, and the major third could each be divided mathematically into two parts, producing musically useful results: why not the fourth? Wallis gave

Introduction

5

some attention to the intervals that would result from a mathematical division of the fourth, with ratios 7 : 6 and 8 : 7.19 Another feature of Wallis’s account of music was his scepticism about ‘modern’ musical theorists (though precisely whom he meant is a difficult question: see below). Some spoke of the tone and semitone as though one was unproblematically half of the other, something which would introduce into music surd numbers which Wallis considered ‘absolutely Unmusicall’.20 Some wrote as though six tones with ratios 9 : 8 equalled an octave with ratio 2 : 1, which had been shown to be false as long ago as Euclid’s Sectio canonis. Some described a ‘Pythagorean’ tuning all of whose tones had ratio 9 : 8, and whose major thirds, with the ratio 81 : 64 as a result, would according to the coincidence theory sound dreadful. This last was indeed a commonly cited reason for the adoption of the just intonation by writers of the sixteenth and seventeenth centuries. For Wallis, too, it showed that the ear and hand were doing something different from and better than what he took to be prescribed by modern theory. Thus, the sounds wee sing, are not the same wee prick (according to the scale as it is now supposed to be divided) … the ear, in these niceties, guiding the Voice, better than the Scale (if a little erroneous) can do.21

The aim of Wallis’s mathematical ratiocinations was therefore to produce a correct theory, one which would better describe what ear and hand were in fact doing. Wallis did not remark on the possibility that his description was, in turn, more ‘nice’ than anything that ear and hand could really achieve. In these two letters Wallis acknowledged by name only the Euclidean Sectio canonis and the De institutione musica of Boethius. It is not clear that he had any other sources for his remarks about ancient music: the terminology, the ratios, and the ascription of certain ideas to Pythagoras which we find in his letters could all have come from these two texts. (The Savilian statutes left the choice of music textbooks to the professor himself, although the wording – ‘canonics or music’, ‘canonicam, sive musicam’ – suggested the Euclidean Sectio, while Boethius had long been a standard reference on music for university curricula.)22 Vague references to ‘those Ancient or Modern Musicians that I have read’, ‘our modern Composers’, and ‘the moderns’ seem intended to give the impression of some acquaintance with material more recent than Boethius, but here the question of Wallis’s sources is more complex. He acknowledged no source for the just intonation and the coincidence theory; neither did he specifically claim originality for those ideas. It is not plausible that Wallis had arrived at those two pieces of modern musical mathematics independently of those who had written about them over the previous century, although there is no particular reason to suppose that Oldenburg knew that. Oldenburg, however, ‘produced’ Wallis’s letters at the meeting of the Royal Society on 18 May, apparently telling Wallis he had done so in another letter – now lost – written shortly afterwards. Wallis had not specifically consented to this,

6

John Wallis: Writings on Music

although it was common for letters to the secretary of the Royal Society to be read to the society. ‘It was ordered, that these letters should be referred to the president to peruse and consider them, and to make report thereof to the society’.23 The president, William Brouncker, was, with another fellow, Walter Charleton, one of the only two men in England who had published on the mathematical theory of music and he might have been expected to raise questions about the sources of Wallis’s ideas.24 In fact he does not seem to have made the requested report. We have no record of just what Oldenburg conveyed to Wallis about the reading of his letters. But Wallis responded in a rather unexpected way. Instead of acknowledging his dependence on earlier writers, he wrote a third letter to Oldenburg specifically claiming independence from them.25 To two of the copies of his second letter now extant – made apparently for Robert Boyle and Narcissus Marsh – he also added a postscript to much the same effect. ‘I find’, Wallis wrote in the postscript, that our modern Writers of the Theory of Musick (in this & part of the last Century) haue, upon like Principles, divided the Monochord much after the same proportion as I haue done.26

These ‘modern Writers’ included Johannes Kepler; the postscript also mentioned ‘our practicall Musicians’, without naming them, and Ptolemy’s Harmonics, which Wallis insisted he had not seen when the bulk of his long letter was written. In his new letter to Oldenburg he added ‘Mersennus’ to the list of his sources ‘& some others’. He repeated his insistence that he had spent ‘very little time, & very few thoughts’ on music. This is a difficult action to explain, and efforts to understand it are hindered by the loss of Oldenburg’s letters to Wallis. A charitable interpretation is that notes Wallis had made at the time of earlier reading in Kepler and Mersenne formed the basis of his second letter without him recollecting how far those notes recorded other men’s ideas rather than his own. A different possibility is that Wallis relied not on his own notes but on unpublished text from, or discussion with, a third party, not realising how widespread were the ideas involved. It is perhaps more likely that Wallis was to some degree deliberately uncandid, but what he could have hoped to achieve by this, or whom he could have imagined he would deceive, is not clear. As far as we know no contemporary challenged him about the matter. Thus the details of Wallis’s debts to earlier musical writers remain in part unclear. The coincidence theory seems likely to have come from Mersenne – discussions of coincidence and frequency were to be found in his main Latin work, the Harmonicorum libri of 1636 – or perhaps from Galileo (in the Discorsi e dimostrazioni of 1638). It had also been discussed by Giovanni Battista Benedetti in his rather more obscure Diversarum speculationum … liber of 1585; although the Bodleian Library owned a copy of the book, we have no direct evidence that Wallis was acquainted with it.27 Wallis’s admission in his third letter that he had studied Mersenne makes it almost impossible to believe that he had not done so before writing the second letter.

Introduction

7

A source Wallis did not acknowledge in 1664, although he would do so in his later writings on music, was Descartes’ Compendium musicæ, published in 1650. Some of Wallis’s terminology, notably the word ‘schism’ for the discrepancy with ratio 81 : 80 between different forms of some pitches, seems unmistakably to derive from here.28 Where Wallis had found the just intonation is open to doubt. It had been quite widely discussed for over a century, since Zarlino’s presentation of it in the Istitutioni harmoniche in 1558.29 In 1682 Wallis would name Zarlino himself as a source; at this earlier stage either Mersenne or Descartes could have given Wallis the information he put in his second letter. Also unclear as to their source are Wallis’s claims about the views and practices of ‘Modern Musicians’. Most, perhaps all, of what Wallis said on this subject could in fact have been his response to Boethius, but it would be pleasant to think he had consulted some writer more modern and more practical. The language of tones and semitones – ‘notes’ and ‘half notes’ – to which he referred, with the implication that one was half of the other and thus that six tones made an octave, could have been found in many introductions to music from early modern England: the @ cliefe which is comon to euery part, is made thus @ or thus # the one signifying the halfe note and flatt singing: the other signifying the whole note or sharpe singing.30

Thus Thomas Morley in 1597; in Wallis’s time John Playford put it like this: the B fa or B flat doth alter both the name and property of the Notes before which he is placed, and they are called Fa, making them halfe a tone or sound, lower then they were before.31

Wallis’s assertion that practitioners described the tone as always and only bearing the ratio 9 : 8 is more mysterious; few practitioners had much to say about mathematical ratios in respect of musical pitch. Wallis may simply have been thinking of Boethius. One of the few seventeenth-century theorists who would fit the bill was John Birchensha, and it is curious therefore that Wallis’s letters were prompted by Oldenburg’s report of his appearance before the Royal Society. Wallis’s letters, though, seem to indicate that details like this were absent from what Oldenburg had reported to him. Certainty about these questions is regrettably elusive, and thus the beginnings of Wallis’s thinking about music and the exact nature of his reliance on the writings of others are to some degree irrecoverable. What is certain is that by the end of May 1664 he had read some of the key texts: Ptolemy and Euclid among the ancients; Boethius, Kepler, Mersenne, and very probably Descartes among the moderns. He was known to Oldenburg and now to the Royal Society to be interested in music, and he had committed himself to positions concerning musical methodology and the correct description of the diatonic scale which he would never substantially modify.

8

John Wallis: Writings on Music

In 1665–6 the Royal Society suspended its London meetings as a result of the outbreak of plague. Several fellows moved to Oxford, and during the autumn a series of informal meetings took place there. On 28 September Robert Moray wrote to Oldenburg from Oxford that yesterday 7. or 8. of our Society met at Mr Boiles … there was, S[ir]. W. Petty: Dr Cox. Dr Wallis; C. Graunt, & I. Sr P. Neile, & Mr Williamson come … Wee talked much of the Monochord … Wee intend to prosecute Musick, & meet at least once a week.32

Boyle left Oxford for Stanton some time between 18 November and 9 December: if the meetings continued weekly until he did so there would have been at least eight, perhaps up to eleven of them. We can be confident that musical subjects continued to be discussed, since the following March the meetings were mentioned to the Royal Society: DR WALLIS being asked, what experiments had been made during the last summer at the committee of the society, which had frequently met at Oxford, related, that, among others, there had been tried divers musical experiments; whereof he mentioned some, but referred for more particulars to Mr. BOYLE, who had caused them to be put into writing. Mr. OLDENBURG was desired to write to Mr BOYLE to communicate them.33

Oldenburg duly relayed the request to Boyle,34 but Boyle never complied. At a meeting in April at which he was present, Mr. BOYLE was called upon for the experiments of sounds, made at Oxford the preceding summer by some of the members of the society, and said by Dr. WALLIS to have been put in writing by him: to which he answered, that they were not perfect.35

The matter does not appear again either in the minutes of the Royal Society or in Boyle’s correspondence. It is possible that the discussion of various vibrational effects eventually published in Boyle’s 1685 Essay of the Great Effects of Even Languid and Unheeded Local Motion owed something to the meetings of 1665, but the notes taken at them do not seem to have survived.36 Wallis’s involvement shows that his interest in music continued and that it could take an experimental turn; sadly it tells us nothing more specific. ‘A new Musical Discovery’: the letters of 1677 The music historian Sir John Hawkins (1719–89) reports that in 1675 Wallis contributed £1 to the refurnishing of the Oxford Music School with instruments

Introduction

9

and books.37 But our next substantial evidence for Wallis’s interest in music dates from slightly later and concerns a discovery connected with nodes of vibration, which was made at Oxford during the 1670s. As Wallis put it: it hath been long since observed, that if a Viol string, or Lute-string, be touched with the Bow or Hand; another string on the same or another Instrument not far from it, (if an Unisone to it or an Octave, or the like,) will at the same time tremble of its own accord. … But adde this to the former observation; That, not the whole of that other string doth thus tremble, but the several parts severally, according as they are unisones to the whole or the parts of that string which is so struck.38

In other words, sympathetic resonance could cause a string to vibrate not just as a whole, but as a set of two, three, four, or more parts. It could be demonstrated, as Wallis explained, that the points which separated those parts remained stationary. The antiquary Anthony à Wood (1632–95) is informative about the affair. In his account William Noble, MA, of Merton College, made the discovery first, during 1673. He passed it on only to ‘one or more friends’. Some time later Thomas Pigot, BA, of Wadham College, a younger and ‘a more forward and mercurial man’, came to the same discovery apparently independently, despite his lesser skill in practical music. Wood judged that Pigot, unjustly, ‘got the glory of it among most Scholars’. Having taken his MA in 1678 Pigot went on to be vicar of Yarnton near Oxford, a fellow of Wadham, ‘and afterwards Chaplain to James Earl of Ossory’, in whose service he died in 1686. He became a fellow of the Royal Society in 1681 and published a paper in the Philosophical Transactions in his own right in 1683, dealing with an earthquake at Oxford. Noble, for his part, became a chaplain at Christ Church and died there in 1681.39 From Pigot the discovery passed to Robert Plot (1640–96), antiquary, fellow of the Royal Society, and keeper of the Ashmolean Museum. Plot received (and may well have commissioned) a full discussion of the phenomenon and its implications from the scholar and musical enthusiast Narcissus Marsh, who was by this time principal of St Alban Hall, Oxford. (Marsh hosted a weekly music meeting at his lodgings in Oxford during the 1660s and 1670s.40) This Plot printed in his Natural History of Oxfordshire in 1677, giving credit to both Pigot and Noble. The brief treatise covered 11 pages and went into some considerable detail about the nature and behaviour of sound and the different phenomena that could be observed in relation to sympathetic strings of unequal length.41 Information on the subject also passed to Wallis, who wrote it up in a letter to Henry Oldenburg dated 14 March 1676/7. Wallis credited Pigot and Noble, mentioned Plot’s Natural History, but failed to mention Narcissus Marsh. The letter was read to the Royal Society, and Oldenburg reported that the society had received it enthusiastically and proposed to print it in the Philosophical Transactions, ‘which I believe you will not oppose, especially since you adde something of your owne observations, which perhaps Dr Plot takes no notice of in his book’.42 Wallis added

10

John Wallis: Writings on Music

a postscript, dated 27 March, remarking that ‘it will be convenient to do it quickly, that it may be abroad as soon as the other Book; & not be like stale news’. The letters were published in the Philosophical Transactions later in 1677 (Chapter 2 below).43 They were printed under a title which emphasised the supposed novelty of the matter by describing it as it as ‘a New Musical Discovery’, and the printed text omitted the aside in which Wallis mentioned Plot’s book. The same matter was republished in Latin as chapter 107 of Wallis’s De algebra in 1693; the introductory and concluding remarks were somewhat rearranged, but the main exposition of the new musical observations was simply a translation of what had appeared in the Transactions.44 One change was a few extra remarks on the phenomenon of sympathetic resonance itself, including that the faint motion of the sympathetic string could be displayed by placing a small piece of paper on it. Another was an acknowledgement that it had been Marsh who first showed the phenomenon to Wallis in 1676. Indeed, Marsh had been a fellow of Exeter College – the college through which Wallis was incorporated MA in 1649 – until he became principal of St Alban Hall in 1673,45 and Marsh at some stage acquired a copy of Wallis’s second, long letter to Oldenburg on music of 1664;46 since both wrote on the subject, it would be strange if they had not discussed Noble and Pigot’s discovery. Wallis was certainly aware of the existence of Marsh’s much more extensive discussion of the phenomenon when he wrote his letters to Oldenburg; it is not quite clear whether he had seen it. Thus, once again, Wallis produced a discussion about music whose relationship to the ideas of other men is not completely clear. His keenness to pre-empt its appearance in Plot’s Natural History looks a little churlish; perhaps Wallis was aware of this, since he attempted to excuse himself in the letter of 27 March: ‘I think that [sc. the Transactions] a proper place for communicating new discoveries: which perhaps would be less universal if onely in the other book [sc. Plot’s Natural History].’ It is also not clear how much there was of genuine novelty about the discovery, which evidently drew some attention at least among the natural philosophers of Oxford. The fact that a bowed or plucked string can be made to sound at a range of pitches (the fundamental and its harmonics) by touching it at certain places (the vibrational nodes) was essential to the functioning of the trumpet marine, a bowed stringed instrument whose pitches were the harmonics of a single string and which enjoyed some popularity during the late seventeenth and early eighteenth centuries.47 (The use of harmonics in playing instruments of the violin family seems to date from the 1730s.) The trumpet marine was described by Glarean in the sixteenth century and by Mersenne earlier in the seventeenth; Pepys heard a performance at Charing Cross in 1667. Particularly relevant to Noble and Pigot’s discovery is that fact that some trumpets marine possessed sympathetic strings; Praetorius provided an illustration in 1620.48 All of this may have provided some motivation for Noble a­ nd Pigot to set up the apparatus they did: a string forced, not by touching at a node but by sympathetic resonance, to vibrate at a pitch higher than its fundamental, and

Introduction

11

whose behaviour – visibly vibrating not as a whole but as separate parts, with stationary points in between those parts – could be observed. The novelty lay less in the string’s behaviour, which was substantially that of the string(s) in a trumpet marine, but in their deliberately isolating and studying it and characterising it as a natural philosophical phenomenon. The whole affair, indeed, amounted to a re-description of a phenomenon which had previously held musical rather than natural philosophical interest. Wallis, in his paper, also remarked on a related novel observation that he ‘took notice of upon occasion of making trial of the other’. If a string was struck at a vibrational node – its midpoint, or a point a third or a quarter of the way along – it would ‘give no clear Sound at all; but very confused’. Wallis accounted for this by reference to ‘the contemporary vibrations of the several unisone parts’,49 an obscure phrase which may have been intended to indicate that when struck or bowed in the normal way a string vibrated at several frequencies simultaneously. The thought of Francis North (1637–85) seems to have been running along very similar lines during the same year, 1677. In his Philosophical Essay of Musick North commented at some length on the ways that the sound of a string or pipe could vary or ‘break’. ‘In a false string two or more sorts of vibrations are blended’, while if a musical string be so struck, that the whole string is not removed out of its place to cause the greatest vibration in the middle, there must be a crossing of vibrations: for before the motion communicated to the farther end, the part of the string that was struck may have restored it self to its first place. This crossing of vibrations, if the string be true, will be upon equall parts, and produce an Octave fifth, or some other Chord. The Trumpet marine that sounds wholly upon such breaks, is a large and long monochord play’d on by a Bow near the end, which causes the string to break into shrill Notes. The removing the thumb that stops upon the string gives measure to these breaks, and consequently directs the Tone to be produced.50

The fact that these ideas – the vibration of strings at more than one frequency at once, the ‘breaking’ of such patterns of vibration and the trumpet marine as an example – were evidently circulating among natural philosophers may suggest that Wallis did no more in his Transactions paper than recite what he had heard from others. But in fact he claimed – and there seems no good reason to doubt him – that he had seen and tested the phenomena for himself. Indeed, he stated that the phenomenon of sympathetic resonance on which the observation depended had been tested in a number of different configurations: lutes and viols would ‘answer to’ one another; a viol would answer to a chamber organ but rather less well to a harpsichord. (In the version of 1693 Wallis said slightly more about the transmission of vibrations from one string to another: a string vibrated at a rate determined by its tension, striking the surrounding air; the

12

John Wallis: Writings on Music

air struck the other string, which having the same tension was apt to receive the vibrations, unlike others of a different tension.51) These observations occasioned one of Wallis’s rather rare remarks about the mechanical nature of sound: the Metal-strings, though they give to the Air as smart a stroak, yet [one] not so diffusive as the other … . But Wind-Instruments give to the Air as communicative a concussion, if not more, than that of Gut-strings[.]52

Evidently there was in his mind some understanding of the nature of sound in which ‘smart’, ‘diffusive’ and ‘communicative’ meant specific, and different, things. Unfortunately Wallis left no evidence here or elsewhere to enable us to say just what they meant. ‘The Harmonics of the Ancients compared with Today’s’: Wallis’s appendix to Ptolemy’s Harmonics, 1682 In 1682 Wallis published an edition of Ptolemy’s Harmonics based on his study of manuscripts in university and private collections in Oxford: materials in the Bodleian Library, St John’s College, Magdalen College, and the collections of Isaac Vossius and Thomas Gale.53 In all, he collated 11 manuscript sources to produce a critical text of one of the most substantial and important of Greek musical treatises. He also provided a Latin translation, notes, and an appendix discussing the ancient science of harmonics and its relationship to modern musical theory: ‘The Harmonics of the Ancients compared with Today’s’ (Chapter 3 below). Ptolemy’s treatise, written in Egypt in the second century, was a complex, balanced account of musical theory and contained, as well as a wealth of information about different forms of the scale and the approaches taken by different groups of theorists, his description of an astonishingly refined apparatus for the experimental investigation of musical pitch.54 In view of what we have already seen, it is no surprise that Wallis found the text of interest. The motivation for such a demanding project as the edition and translation seems to have included a desire to see the whole Greek musical corpus in print: the 1652 Antiquae musicae auctores septem of Marcus Meibom included all but three of the known surviving Greek musical treatises, and by the end of his life Wallis had produced editions and translations of all of those three. The Greek text of Ptolemy had not been printed at all before Wallis’s edition, and the Latin translation of Antonio Gogava issued in 1562 was judged unsatisfactory by commentators including Marcus Meibom, an anonymous writer in the Philosophical Transactions, and François-Joseph Fétis (1784–1871), as well as Wallis himself.55 Added to this, perhaps, was Wallis’s sense of his responsibilities as Savilian Professor of Geometry: in particular his responsibility to the materials on mathematical sciences to be found in the Bodleian. Henry Savile’s 1609 donation of manuscripts to the library, indeed, included copies of writings by Ptolemy and

Introduction

13

Aristides Quintilianus on music. Wallis was not the first Savilian Professor to devote his attention to these and the Bodleian’s other ancient musical manuscripts. In 1627 the scholar John Selden employed Peter Turner, Wallis’s predecessor as Professor of Geometry, to ‘translate and collate’ Greek musical manuscripts including those of Gaudentius and Alypius.56 Edmund Chilmead (1610–54), scholar and musician, also worked on Oxford’s Greek musical manuscripts and began an edition of the Harmonic introduction of Gaudentius which, in the event, was forestalled when Meibom’s Auctores septem got under way.57 Meibom’s edition of Aristides’ Liber de arte musica was based on the collation of copies in Oxford libraries by Gerard Langbaine, provost of Queen’s College, and others.58 Furthermore, about 1673 Edward Bernard, Savilian Professor of Astronomy, and John Fell, dean of Christ Church and vice-chancellor of Oxford University, had announced a project for an ambitious, multi-volume edition of ancient mathematical texts, which was to take in among other subjects architecture, astronomy, geography, and music. The project failed to find a patron and lapsed; it is possible to see Wallis’s work as an editor of ancient mathematical texts as, in part, a resurrection of it.59 As well as the 1682 edition of Ptolemy on music, that work included editions of two treatises of Archimedes in 1676, and in 1688 of a treatise by Aristarchus and a (new) fragment of the writings of Pappus. In 1930 Ingemar Düring, the editor of Ptolemy’s Harmonics whose work at last rendered Wallis’s edition obsolete, paid tribute to his predecessor: ‘Wallis’s Harmonics is one of the very first editions to include an apparatus criticus. The edition must be described as exemplary for its time’.60 Wallis’s careful identification and description of his manuscript sources has come in for particular praise; it rested on his knowledge of the Oxford libraries and the Savilian manuscript collection in particular.61 Wallis’s ‘The Harmonics of the Ancients’ was by far his longest and most considered discussion of musical matters. It drew on a much wider range of sources than his previous musical work and covered a wider range of topics. It dealt with four main subjects: terminology, pitch and its study, Greek musical theory, and modern musical theory. Perhaps not surprisingly, Wallis struggled to control his material, and the text seems imperfectly organised. Concerning terminology,62 Wallis set out the opinions of a range of ancient authors on such terms as ‘music’ and ‘harmonics’, as well as on ‘pitch’ and ‘tone’. The names of the individual pitches were surveyed with similar rigour, and the modern system of notation and solmisation discussed. Terminological and philological matters recurred throughout ‘The Harmonics of the Ancients’, with Wallis commenting in detail on possible faults in the texts he had studied, inconsistencies in the use of terms and the appearance and possible interpretation of rare or confusing words. He was the first to remark, for instance, on the displacement of some matter within the text of Aristoxenus.63 All of this was in a sense the raw material for some of what would follow. Next Wallis addressed the study of pitch and the nature of what might be called ‘pitch space’:64 how was it to be conceived? What did ‘movement’ through it mean?

14

John Wallis: Writings on Music

How finely could it be subdivided? He gave some attention to the ‘Aristoxenian’ view of pitch and the proper way to study it: the octave or double octave as a continuous space to be divided up much as one might divide up a strip of land. Thus ‘there is no absolutely smallest interval (just as there is no absolutely shortest line)’, and any space of notes can hold innumerable sounds (as a line does points). And like a point so a whole line can be moved through a greater or smaller distance ad infinitum; so too sounds and groups of sounds.65

Wallis, in common with many other mathematicians of his period, was interested in vanishing and infinitesimal quantities;66 these remarks suggest that he may have connected his ideas about pitch with his ideas about number and the continuity of lines.67 But the main point of ‘The Harmonics of the Ancients’ was a detailed exposition of the content of Greek and modern musical theory in terms which were those of the so-called Pythagoreans: the octave or double octave not as a continuous space but as a ratio to be divided into a finite number of other ratios. In these terms Wallis worked through the construction of the Greek two-octave system of pitches called the greater perfect system; he showed how variation within this system was managed, by the system of genera and species; and how scales (‘modes’) were derived from it.68 Wallis’s interest in these matters was also historical; in one section he attempted to set out a history of Greek tuning theory based upon the different systems of pitch and the variations in terminology which he found in various Greek writers, together with his own speculations about the motivations for the changes between one system and another.69 To some degree this account, and Wallis’s general sense of the shape of Greek musical history, was distorted by his accepting the thencommon view that the treatise of Cleonides – a late Aristoxenian work of possibly the second century a.d. – was a work of Euclid from around 300 b.c. This historical dimension extended into the final section, and perhaps the main point, of ‘The Harmonics of the Ancients’: the comparison of Greek harmonic theory with its more recent counterpart.70 Wallis believed that the two were in essence the same; as a notice of Wallis’s edition of Ptolemy in the Philosophical Transactions put it, ‘the several particulars [of Greek music], are, for the most part, retained in ours, but very differently expressed’.71 Thus the Greek modes (for the diatonic genus) could be correlated with the medieval ‘church’ modes, producing at least the impression of chronological development from one to the other. Wallis cited a number of medieval and renaissance writers: Boethius, Sigebert, Henricus Stephanus, Guido Aretinus, Faber Stapulensis, Glarean; but it is not clear how much familiarity he had really acquired with medieval modal and hexachord theory. His grasp of these matters does not seem to have been as secure as his understanding of Greek theory, and modal theory was a subject on which later music historians would criticise ‘The Harmonics of the Ancients’.72

Introduction

15

This approach meant that for Wallis the details of Greek theory could ultimately provide insights into early modern tuning; the choice between one tuning or another for the diatonic scale could be construed as the choice between one species or another of the diatonic genus. Wallis’s discussion of the Pythagorean and syntonic diatonic scales in ‘The Harmonics of the Ancients’ was closely related to that in his letter to Oldenburg of 14 May 1664.73 He introduced his ideas in substantially the same order; the mathematics he deployed was essentially the same, and some of the supporting diagrams remained the same. Naturally, however, certain things had changed in the 18 years since the letter was written. Most notably, Wallis seems to have become less convinced of the usefulness of the coincidence theory and its account of consonance in constructing and justifying the syntonic diatonic scale. Although the theory was described in ‘The Harmonics of the Ancients’ and diagrams illustrating it were displayed, in the text Wallis never explicitly displayed a correlation between numbers and pitches. For most of the discussion he thus left it unclear whether the musical mathematics he was developing dealt with frequencies, as did the coincidence theory, or lengths of musical strings, as did his Greek sources. A few passages explicitly discussed string lengths, but nowhere was the construction of the scale explicitly based on considerations about frequencies.74 There were other developments in the details of Wallis’s ideas. He was now, for example, prepared to consider intervals larger than an octave as consonances; and he explored a little further than in 1664 the implications for the scale of dividing the perfect fourth into two by mathematical means.75 He also pursued the underlying thought by considering not just divisions of the fourth and the minor third into two parts but also the division of some intervals into three parts, a procedure which would further increase the number of possible intervals and pitches. Perhaps the corollary of this further theoretical elaboration was what seems to have been an increased degree of pessimism about the usefulness of the ear in the detailed study of musical tuning, particularly in distinguishing one system of tuning from another.76 (Another source for this new pessimism seems to have been Ptolemy’s critique of Aristoxenian methods.77) Thus the harmonic canon could reckon differences of sounds ‘more precisely than the sense can perceive’, and Wallis judged that ‘more precise discussion concerns speculation more than practice’. Thus the just scale was now, in Wallis’s view, a product more of reason than of sense.78 This development of his ideas allowed Wallis to characterise the views he ascribed to practical musicians as substantially those of Aristoxenus as far as methods were concerned: although ‘those who speak more precisely sometimes dispute’ concerning the just intonation, ‘later musicians, particularly practical musicians, right down to our own time mostly side with Aristoxenus, measuring intervals by tones and semitones’. This was still, though, and somewhat paradoxically, in the context of an ascription to practitioners of a belief in a Pythagorean scale of equal tones, each with ratio 9 : 8.79 Despite all this, Wallis still believed that the voice ‘without instruction’ could provide – for instance – a distinction amounting to a ratio of 81 : 80 between

16

John Wallis: Writings on Music

different forms of a pitch when it was needed.80 This unresolved tension between pessimism about the capabilities of the ear and optimism about the practice of (vocal) performers would remain a feature of Wallis’s musical writings. Wallis’s debts to modern writers were a good deal clearer in ‘The Harmonics of the Ancients’ than they had been in 1664 (or would be in his later musical writings), yet they still raise questions. It is not always clear that Wallis had fully assimilated what he had consulted. The list of moderns cited is not long, and in several cases it is not clear whether Wallis had seen their works at first hand. Mersenne was given some credit: on the ordering of intervals according to their degree of consonance, for instance, Wallis deferred to him by name.81 To Zarlino, Wallis now ascribed the reintroduction of the just intonation;82 and Kepler was mentioned as a proponent, Galilei as an opponent.83 Other names received only the briefest of acknowledgements. Descartes, for instance, received a single passing mention. So did the polymath Johann Heinrich Alsted (1588–1638); Wallis had quite possibly been led to the musical section of his 1620 Encyclopedia by the 1664 translation of John Birchensha.84 So, too, did John Gerard Voss (1577–1649), for what his remarks on music in De quatuor artibus popularibus (1650) were worth; once again a personal connection may have directed Wallis’s attention, since Voss’s son Isaac was now resident in England. Another name mentioned was that of Andreas Papius of Ghent (1542–81): evidently Wallis had seen his 1581 De consonantiis. Conversely, there were some odd omissions. The absence of references to Athanasius Kircher or Edmund Chilmead is particularly strange, since both had discussed Greek musical notation in print.85 Chilmead’s brief essay would be mentioned in Wallis’s later correspondence with Thomas Salmon, and it is possible he was simply unaware of it at this stage.86 Kircher’s Musurgia universalis cannot have been unknown to Wallis at least by reputation (Oxford libraries now hold 11 copies of the Musurgia, one of which is part of the Savilian collection), and we can only suppose that his failure to cite it here or anywhere else in his musical writings was a deliberate policy. In places, too, the ideas of the idiosyncratic Italian theorist Pietro Mengoli (1626–86) seem clearly discernible, yet Wallis did not name him.87 John Hawkins would detect a dependence on Francisco de Salinas (1513–90), another individual not acknowledged by Wallis;88 he would also suggest that Wallis depended more closely on Antonius Hermannus Gogava’s 1562 translation of Ptolemy than he acknowledged: [Bottrigaro] corrected Gogavinus’s Latin version of Ptolemy in numberless instances, and that to so good a purpose, that Dr. Wallis has in general conformed to it in that translation of the same author, which he gave to the world many years after.89

In a similar vein, Fétis would claim that Wallis had used the work of the French librarian and astronomer Ismael Bouillaud (1605–94).90 All of this suggests that

Introduction

17

Wallis was still keen to give an impression of independence from his sixteenthand seventeenth-century predecessors that was not quite candid. A notice of Wallis’s edition of Ptolemy’s Harmonics was printed in the Philosophical Transactions for January 1683 (Chapter 4 below). This brief text indicated that Wallis: ‘our learned Professor [of Geometry]’, ‘the Doctor [of Divinity]’, yet believed the edition to be ‘a Task well agreeing with his Province’. It is not clear who wrote the notice, but the secretaries of the Royal Society at the time were Francis Aston and Robert Plot, and the latter, whom we met above, seems the obvious candidate.91 ‘Your Most Obliged & Faithfull Servant’: Thomas Salmon The period of Wallis’s work on his edition of Ptolemy’s Harmonics was also the period of preliminary work and writing for another text about the mathematics of musical tuning: the Proposal to Perform Musick, in Perfect and Mathematical Proportions (1688) of Thomas Salmon (1647–1706).92 Salmon was a country clergyman and musical amateur who published three books on musical subjects. He is best known today for the celebrated pamphlet war with Matthew Locke, principally concerned with a proposed reform of musical notation, in which he engaged during 1672–3; he is also known to have had some contact with John Birchensha, John Rogers, the Steffkin family, and James Paisible, among other musicians.93 The Salmon and Wallis families may have been connected or acquainted for some time, and it is possible that Salmon and Wallis became acquainted during Salmon’s time as a student at Oxford in the 1660s.94 Wallis was the addressee of Salmon’s reply to Locke’s criticisms in 1672, which was presented as an open letter to the Savilian Professor: further evidence that well before the appearance of the edition of Ptolemy’s Harmonics or even the publication of the letter to Oldenburg in Philosophical Transactions in 1677 Wallis was known to some to be particularly interested in musical matters. In that reply, A Vindication of an Essay to the Advancement of Musick, Salmon stated that Wallis had been responsible for his first induction into matters of musical mathematics: it was you that spoil’d me, by letting me know, That the satisfaction an Octave created, did proceed from an exact duple-proportion[.]95

How Wallis had let him know this did not appear, though it could have been in the course of lectures delivered as part of Wallis’s Savilian duties. Salmon also made the following intriguing remark about the English solmisation system: our own Countrey-men have much out-done all the Musicians in the world … in so ordering the Monosyllables, that the same should always signifie an interval of the same proportion.

18

John Wallis: Writings on Music Which, Sir, was first taken notice of by your self, in your most ingenious Letter of March 20 1664/5, to the R. Society, representing Musical Harmony to the eye, in one of Mr Henry Laws his airs, rendred in Parallelograms.96

If the Royal Society received such a letter it left no record of doing so, and the letter itself is not to be found in the society’s archive.97 If Salmon was nevertheless correct, Wallis had interested himself in some novel but irrecoverable form of musical notation about a year after his surviving musical letters of 1664. But although we have no evidence that Wallis challenged Salmon’s claim, it is perhaps likely that Salmon was mistaken. Salmon’s later writings on music focussed quite squarely on the mathematics of pitch and tuning. We have a draft treatise possibly dating from the 1680s, a letter to Wallis from 1684 and Wallis’s reply, and the 1688 Proposal. Salmon solicited approval from both of the Savilian professors for his Proposal after it had been set up in type. Edward Bernard, Professor of Astronomy, provided the briefest of commendatory letters, but Wallis contributed ‘remarks’ which approached the length of the original book. Those remarks clarified technical points and added practical suggestions, which Salmon did not take up, about how to realise the just intonation on a viol; Salmon favoured interchangeable fingerboards with fixed frets, Wallis individually moveable frets. These ‘remarks’ Salmon duly had printed with his book; they are edited in the second Salmon volume of the present Ashgate series.98 After its publication Salmon developed the ideas of the Proposal in two further manuscript treatises, and after Wallis’s death he achieved in 1705 the unique distinction of being allowed to stage a demonstration musical performance before the Royal Society to illustrate his musical ideas. A subsequent paper in the Transactions, written by Salmon, and brief correspondence with Sir Hans Sloane, the secretary, are our final evidence for his musical activities before his own death in 1706. These quite voluminous writings consistently shared certain interests with those of Wallis, although they also consistently diverged from Wallis in their approach to the study of music. In particular, Salmon’s experimental bent – most of his writings aimed in one way or another to bring about practical change and to facilitate the practical demonstration of his musical ideas – was not shared by Wallis. If Salmon was able to make contact with London practitioners and instrument makers in a way that Wallis never did, to ‘set the Mechanicks to work’99 on his ideas, Wallis, for his part, never showed any interest in so doing. Indeed, for all he had written about the theory of music, Wallis never seems to have acquired a knowledge of practice: according to the antiquary Thomas Hearne (1678–1735) Dr. Wallis understood nothing of the practice of musick. A certain gentleman having read his Ptolemy, and believing that the doctor was well skilled in the practice, as well as the theory, of musick, went one day and intreated the doctor to assist him in obtaining the practice. The doctor ingenuously confessed, he knew nothing of it.100

Introduction

19

Neither did Salmon’s earlier, controversial work on the proposed reform of musical notation find any echo in Wallis’s writings. Conversely, Salmon showed no interest at all in the coincidence theory of consonance. The main point of contact between the two men concerned the just intonation, which Salmon set out in all of his writings on musical pitch. He constructed the scale in a similar way to Wallis, by repeated division of a ratio into smaller parts; like Wallis, too, he addressed on one occasion the questions of why such division should be applied to some intervals and not others, and of what would happen if it was applied to the fourth.101 Moreover, when Salmon extended the principles of the just intonation to provide a complete chromatic scale, he did so in the same way as Wallis. Wallis’s version of the chromatic scale was published in 1698; Salmon’s in 1705.102 The tricky question of whether, within that scale, the major or the minor tone should stand lower in the interval C–E, on which Wallis vacillated in both 1664 and 1682, was resolved by him in his 1688 remarks on Salmon’s Proposal in favour of placing the major tone lower, a solution which thereafter became Salmon’s settled view of the matter.103 Thus it is certainly the case that Salmon’s writings on music during the period 1672–1705 and his career as a musical amateur which culminated in his appearance before the Royal Society in 1705 drew something both of general inspiration and of mathematical detail from the work of John Wallis. Despite the complementarity of the two men’s approaches in some respects, however, influence seems to have flowed only one way; Wallis’s musical writings do not seem to owe anything to Salmon. The corpus of musical writing we have from Thomas Salmon tells us something of Wallis’s reputation and his willingness to contribute ideas to the work of others; it does not add anything of substance to our knowledge of the content of Wallis’s musical ideas. ‘A thing wherein I am so little acquainted’: musical publications in 1698 Fifteen years after his translation of Ptolemy and three decades after his first foray into musical mathematics, Wallis returned once again in print to musical subjects. His musical contributions in 1698 included four surviving letters, of which three were printed in the Philosophical Transactions during that year. The first two concerned the mathematics of tuning and should be read together.104 The first, which appeared in the March issue of the Philosophical Transactions (Chapter 5 below), replied to a question from an unknown source and concerned the ‘section of the musical canon’, a transparent reference to the Euclidean Sectio canonis and an opportunity, possibly concocted by Wallis himself, to re-present in public his thinking about the mathematics of musical tuning. Ostensibly he was prompted by a specific question: how to divide the musical string into 12 semitones, and whether the solution given in Christopher Simpson’s 1667 Compendium of Practical Musick was correct. There is nothing to indicate who had proposed the question (if anyone really had); possible candidates include

20

John Wallis: Writings on Music

Thomas Salmon and Samuel Pepys, who would be the explicit addressee of Wallis’s next musical paper in the Transactions. Simpson’s just intonation and construction of semitones in fact adopted an approach similar to that of Wallis; they could possibly have been one of his sources in 1664. In fact Wallis dismissed Simpson at once: ‘What Method is used by Simson (in the Book mentioned) to divide a String or Chord, I know not: Nor have I the Book at Hand to consult.’105 Instead he gave an account broadly similar to those of 1664 and 1682, including the mathematical construction of the first few consonances and the fact that since six tones do not make an octave, some solution other than the Pythagorean scale was needed. Once again Wallis was pessimistic about the ear; here he doubted whether it could distinguish between the semitone of equal temperament and his mathematically true semitones.106 Compression rendered all of this less lucid and Wallis’s conclusion in favour of the just intonation less convincing than in his earlier discussions of the same material. There were changes, though, compared with Wallis’s earlier writings on musical pitch. This text contained no mechanical account of consonance or even of sound and hence none of the diagrams which had supported Wallis’s earlier discussions of the coincidence theory of consonance. Perhaps as a result, but perhaps also because of the context of a discussion explicitly about the division of strings, there was no mention of the frequencies of sounds. By far the most significant change in this text was that Wallis now took the division of intervals into near-equal parts a stage further than he had before. ‘More precise discussion concerns speculation more than practice’, he had written in ‘The Harmonics of the Ancients’ in 1682, but here he overturned that judgement, introducing a precise specification for the positions of the semitones. Both the letters of 1664 and ‘The Harmonics of the Ancients compared with today’s’ of 1682 divided the octave (2 : 1) into fifth (3 : 2) and fourth (4 : 3), the fifth into major (5 : 4) and minor (6 : 5) thirds, and the major third into major (9 : 8) and minor (10 : 9) tones. This text also divided the major tone into semitones of 18 : 17 and 17 : 16, and the minor tone into semitones of 20 : 19 and 19 : 18. Together with the diatonic semitone of 16 : 15 (defined, as ever, as the difference between a fourth and a major third), this made five different sizes of semitone, and provided a complete specification for the chromatic notes of the scale.107 In the context of Wallis’s discussion of a mathematically defined scheme for the division of intervals, this was fairly natural; the division of the tones was performed by the same mathematical means as the divisions of octave, fifth, and major third. But as a proposal for placing, say, the frets on a viol it had serious – and obvious – disadvantages. Wallis would mention in his next paper the fact that his system would permit a musician to play in only one key, re-tuning being needed should the key change. Worse – and not mentioned by Wallis – passing uses of chromatic notes would never in this scheme produce pure harmonies with the other notes of the scale. This was by contrast with Wallis’s previous discussions of the tuning of chromatic notes, in 1664 and in his remarks on

Introduction

21

Salmon’s 1688 book, when he had suggested a solution that would ensure they made pure harmonies with at least some diatonic notes.108 In this connection another personal relationship may be mentioned: that with William Holder. Holder has been mentioned above as one of the several objects of Wallis’s printed attacks; the two men clashed initially over the question of teaching the deaf to speak. (It is interesting that on the London comic stage the idea of teaching the deaf to speak had by 1670 become attached to the name of John Birchensha, the apparent stimulus for Wallis’s first foray into musical writing.)109 Both men also published on the calendar: Holder’s 1694 Discourse concerning time appeared while Wallis was contending with others on the same subject in a pamphlet exchange. Although it did not mention Wallis, the Discourse did include ‘a brief Account of the Author’s New Hypothesis, concerning the Natural Production and Differences of the Letters of the Alphabet’, in fact a re-statement of Holder’s position on phonetics and the teaching of the deaf, manifestly aimed at Wallis and his supporters.110 Thus, it seems likely that Holder’s Treatise of the natural grounds and principles of harmony, also published in 1694, was intended to be read in relation to Wallis’s work on music. The Treatise is known to historians for the informative collection of surviving correspondence related to its printing.111 It was reprinted in 1701 and 1731 and would be admired by both Burney and Hawkins for its erudition and clarity.112 Although there was no direct indication in either man’s work that their publications on music were an extension of their rivalry, there were some marked similarities between what they had to say. Holder set out in his book the just intonation, using similar methods to Wallis: the repeated division of intervals into smaller intervals by mathematical means. But his decision about the placement of the major and minor tones within the scale was different, as was his construction of semitones. Indeed, Holder mentioned and rejected the very chromatic scale that would be adopted by Wallis in 1698: Music would have seem’d much Easier, if the Progression of Dividing had reached the Hemitones: I mean, If, … the dupled Terms of Tone Major 18 to 16, thus divided, had given Usefull and Proper Hemitones 18 to 17, and 17 to 16. But there are no such Hemitones found in Harmony[.]113

Instead Holder constructed semitones as the differences between other intervals, in such a way that so far as possible they made pure harmonies with other notes of the scale. Wallis made no mention of Holder in his paper. His choice of chromatic scale thus allows us to place Wallis in an – albeit small – network of discussion about such matters. As mentioned above, this same division of tones into semitones was in fact described by Christopher Simpson, though without mathematical details.114 And Wallis’s use of this form of chromatic scale is also the clearest evidence we have for the relationship between his and Salmon’s musical ideas at this stage in their careers. Salmon set out the same scale, with the

22

John Wallis: Writings on Music

same mathematical rationale, both in a manuscript written probably in the 1680s and in further re-workings of his musical ideas written around 1704.115 Indeed, the original impulse for such a division of tones into mathematically determined semitones may well have come from Salmon, although it seems quite likely that mathematical details had been supplied to him by Wallis. By 1698 Wallis was prepared to commit himself in print to the chromatic scale which resulted, but the surviving correspondence does not allow us to say more. Wallis’s next musical paper in the Transactions, printed in the May issue (Chapter 6 below), contained his final statement of his views on the tuning of musical instruments. It is remarkable for the evidence it provides of direct contact between the author and a musical-instrument builder, a representative of a world with which Wallis seems otherwise to have had little contact. René Harris (c. 1652–1724), a native of Brittany, was indeed a somewhat unusual individual. A member of an organ-building family, a recusant Catholic, and an organ builder in his own right since 1681, he was a rival of the wellknown ‘Father Smith’, and is known to have worked on more than 60 organs in one capacity or another.116 Earlier in 1698 Harris had publicly claimed to be able to divide a semitone into 50 distinguishable parts – a claim which engaged both his pre-eminence as an instrument builder and the acuity of his musical ear – and invited interested parties to a demonstration at his house: Whereas the Division of half a Note (Upon an Organ) into 50 Gradual and distinguishable parts has been declar’d by Mr. Smith, as also by the Generality of Masters, to be impracticable: All Organists, Masters, and Artists of the Faculty are, together with the said Mr. Smith, invited to Mr. Harris’s house in Wine-Office Court, Fleetstreet on Easter Monday next, at Two of the Clock in the Afternoon, to hear and see the same demonstrated.117

Later the same month Harris felt able to assert that the demonstration had been a success, and to enlarge both his claims and his invitation. Whereas the Division of half a Note (Upon an Organ) into 50 Gradual and Distinguishable parts, was performed by Mr. Harris, on Easter Munday, to the full satisfaction of the Persons of Quality and Masters that were present; And whereas the said Mr Harris intends a further Division of half a Note, viz. into One Hundred parts (and this, as before, not Mathematically, but purely by the Ear) all Masters and others of curious and Nice Ears, are invited to the said Mr. Harris’s House, in Wyne Office Court, Fleetstreet, on the 10th of May, at Three of the Clock in the Afternoon, to hear and see the Performance, and to be inform’d (if any doubt) of its Usefulness.118

Though Harris provided few details, we can deduce something about his demonstration. Harris implied that there was a qualitative difference between his divisions of the semitone into 50 and into 100 parts, and the reader’s first

Introduction

23

thought is of an instrument with, on the one hand, 600, and on the other, 1,200 pitches per octave. (This seems to have been how Wallis, who gave no indication he had seen the instrument in question, understood Harris’s claims.)119 But an organ – even an experimental instrument with just one stop and an extremely limited range – which possessed 100 separate pipes for each semitone, would have been unthinkably massive in a private house, and would have required either an enormously complex keyboard or a normal keyboard linked to the pipes in an enormously complex way. The expense, for an instrument destined only for private demonstration, would have been insupportable. Harris’s statement that the division was performed ‘by the Ear’ was ambiguous; it could have meant that the 1,200 pipes of the octave had been individually tuned by ear, but it may suggest instead that the organ involved pipes whose pitch could be continuously adjusted. We perhaps do not go far wrong if we imagine a (broadly trombonelike) mechanism for lengthening and shortening each pipe, perhaps controlled by a set of levers from near the keyboard.120 Just what satisfaction such a device can have produced for ‘Persons of Quality and Masters’, or how edifying the laborious enumeration of 50 or 100 different positions for one lever can have been, it is difficult to imagine. But Harris’s assertion of the division’s ‘usefulness’ surely points to a belief on his part that an organ which could be flexibly tuned, just as a harpsichord could, would possess real advantages for musical performance over an instrument that was locked to one particular tuning. Harris’s experimental organ seems to make no further appearances in the sources, and it is not completely clear how it is to be related to the text presented in this volume. As Wallis recounted, Harris approached Pepys, who directed him to Wallis. There followed some ‘little discourse’ with Wallis, who wrote back to Pepys in a letter dated 27 June, later printed in the Transactions. Harris apparently initiated this contact with Pepys and Wallis himself; it is not clear why. Wallis wrote of his asking ‘my [sc. Wallis’s] opinion about perfecting an Organ’. Perhaps Harris had become dissatisfied with the tuning of an organ he was building or repairing; possibly he was trying to impugn the quality of a rival’s organ; or possibly the reference was to his experimental instrument. On the other hand, publicity for his experiments and ideas may have been more in Harris’s mind than advice about organ-building itself. Wallis’s text in fact said nothing about any specific instrument, but set out the just intonation in the same form as his previous paper, with its mathematical rationale. Wallis was aware of Harris’s interest in the subdivision of the semitone: he took a moment to sketch the mathematical proof that an equal such subdivision (Harris’s advertisements had made no such stipulation) could produce only irrational ratios at least as unmusical, from a Pythagorean point of view, as those of equal temperament. This was an extension of Wallis’s proof in earlier texts that dividing the tone into two equal parts produced an ‘Unmusicall’ result; here the division could be into any number of equal parts and the conclusion would be the same.121 Rejecting the equal subdivision of the semitone therefore, and pointing out that his own just-intonation scheme was locked to a single key, Wallis

24

John Wallis: Writings on Music

concluded that the most practicable tuning for an instrument which could not be re-tuned was equal temperament. Wallis seems to have missed, in this text and the contact with René Harris to which it bears oblique witness, the opportunity to engage with practitioners’ knowledge and their concerns. It does not seem that he had modified his position, expressed in 1682, that instrument makers ‘necessarily take the nearly true for true’: that the compromises involved in building real instruments were inimical to the mathematically precise descriptions of musical sound in which he was interested. Although he adverted twice to the correction of the voice by the ear, this was for him a way to suggest that the musical instincts represented by the singing voice came closer to mathematical perfection than modern theory allowed for. Perhaps as a result, Wallis’s survey of different strategies for tuning an organ was far from complete, failing to consider either the temperaments which were in practical use, such as mean-tone temperament and Werckmeister’s irregular systems, or the possibility of an equal subdivision of the octave into a number of pitches which was not a multiple of 12, such as had been repeatedly suggested in print by this date as a means of producing better approximations of the pure intervals than equal temperament would achieve.122 Wallis’s third paper in the 1698 Transactions was also his final word in print concerning music (Chapter 7 below). It appeared in the August issue, and addressed not the technical questions of the theory of tuning seen in his earlier forays into the subject, but the puzzling fact that the widely attested effects of music in the ancient world were not to be found in the seventeenth century. In fact, this letter was only a fragment from a somewhat larger correspondence with a ‘Mr. Andrew Fletcher’. The individual most probably intended was Andrew Fletcher of Saltoun (1653?–1716), the Scottish political theorist, who spent some time in London during the 1690s. The letter printed in the Transactions survives in a copy Wallis sent to Hans Sloane, the secretary of the Royal Society.123 Dated 18 August 1698, this is in Wallis’s hand; there are a few scribal hesitations, but no clear evidence that this was other than an accurate copy of what had been sent to Fletcher. The title – ‘A Letter of Dr John Wallis to Mr Andrew Fletcher, Concerning the Strange Effects Reported of Musick in Former Times, beyond what is to be found in Later Ages’ – appears in the manuscript, presumably indicating that, as in other similar cases, Wallis expected the letter to be printed in the Philosophical Transactions. An endorsement records that it was indeed read to the society on 31 August and printed in the Transactions. Meanwhile Fletcher replied in a letter which is now lost, apparently clarifying what he had intended to ask Wallis. Wallis wrote to him again more briefly on 27 August. On 5 September he sent a copy of this second letter to Sloane, with the suggestion that it be printed as a postscript to the first.124 Finding that his previous letter had in fact already been printed, Wallis mentioned the matter again in his next letter to Sloane on 21 September: I find (by the Transactions which you were so kind as to send me) that my Postscript came too late to accompany my letter of Aug. 18. (which I did scarce

Introduction

25

expect to have seen it abroad so soon.) You may somewhere mention these Errata.125

Wallis listed three ‘errata’ and moved on to other subjects. The correspondence was apparently initiated by a question proposed to Wallis through ‘a Friend’. The only hint concerning the friend’s identity comes at the beginning of Wallis’s second letter: ‘If I did mis-take your Directions by D.G. and have over-done what was desired; pray excuse me that fault.’ The initials ‘D.G.’ could perhaps be those of David Gregory (1659–1708), Savilian Professor of Astronomy at Oxford and one of Wallis’s friends and correspondents. No evidence has been found which would clarify this. In his first letter Wallis took the question at issue to be ‘whence it is that these great effects which are reported of Musick in Former Times … are not as well found to follow upon the Musick of Later Ages’. In his reply Fletcher apparently intimated that he had intended a rather different question: ‘How, what the Ancients called Musick … might be of such good use in Education … influencing men to Virtue, and directing their Morals & Conversation.’ Not where the effects of music had gone, then, but how they might yet be harnessed in education. As Wallis indicated, his first letter had ranged quite widely enough to address this subject too. The general tenor of his remarks was that the increased complexity of modern music acted against its effectiveness: a thought developed from briefer remarks on the subject in ‘The Harmonics of the Ancients’ and one which he developed, rather unexpectedly, using a culinary metaphor. His conclusion was that modern music aimed at pleasure rather than the production of pure affections (emotions) and that were modern musicians to turn their hands to simple unaccompanied melodies suitably presented, the emotional effectiveness of ancient music could be recovered. Wallis’s discussion of music and the emotions, and in particular the ways in which the accoutrements of music – accompaniment, gesture, and so on – might impact the emotions, nevertheless seems superficial. Concerning Wallis’s beliefs about the mathematics of music, we may note that in these letters he did not consider the possibility that mathematical imperfections in modern music were to blame for its ineffectiveness, a possibility which had been explored by other English music theorists including John Birchensha and Thomas Salmon.126 This was of a piece with Wallis’s position in his other writings: that the true musical proportions were in fact realised in at least some musical performances of his day, although the language and theory of performers tended to obscure the fact, and the use, as he believed, of equal temperament for keyboard instruments mitigated it in practice. In the 1690s a Greek manuscript containing Byzantine hymns and anthems came into the possession of the scholar and librarian Humphrey Wanley (1672– 1726), having apparently been ‘taken from the Turks at Buda about the year 1686’.127 Wallis’s opinion of the text was apparently solicited, and he gave it in a letter whose date and address have been lost and which was printed by Hawkins in 1776.128 References to his editions of Porphyry and Bryennius (see below) as

26

John Wallis: Writings on Music

ready for publication ‘soon’ probably place the letter in the late 1690s. (Wanley wrote in June 1698 that he had acquired the manuscript ‘lately’, but it is not certain that Wallis was writing after he did so.129) In the letter Wallis described the manuscript as to one who either had not seen it or could not read the Greek script in which it was written: possibly Arthur Charlett, Master of University College, or another representative of the Oxford libraries, to whom Wanley had offered the manuscript for sale. It was a substantial volume of over 400 leaves, and Wallis stated that he had only ‘cursorily perused’ it. Of interest is that the addressee of the letter apparently considered Wallis an expert in musical matters. As Wallis described it, the manuscript contained ‘an account of the musical notes then in use, their figures, names, and significations’, as well as ‘near a thousand’ pieces of music. Characteristically, he looked on it as being of interest for the evidence it provided for musical history and remarked that ‘it will require some sagacity and study to find out the full import of it, and to be able to compare it with our modern music’.130 The notation he judged to differ both from that of the ancient Greeks and from that of Guido; the text contained as far as he could tell no trace of multi-voice composition. He expressed his satisfaction that this new evidence for ‘musical compositions of the ancient Greeks’ had come to light, although of course this was Byzantine rather than earlier material. In 1693–99 Wallis’s Opera mathematica were printed, in three volumes, of which the third (1699) contained his edition of Ptolemy and the appendix.131 The text and diagrams were unchanged compared with the first edition of 1682, except for a handful of small amendments. Also contained in this volume were editions, previously unpublished, of Porphyry’s commentary on Ptolemy and of Bryennius’s Harmonics, with Latin translations and notes.132 The edition of Porphyry was in fact printed from a text written out in 1695 by Humfrey Wanley, owner of the Byzantine musical manuscript described by Wallis.133 As Wallis remarked, ‘Adeoque jam habemus editos, Veteres (qui extant) Græcos Musicæ Scriptores omnes’: between them he and Meibom had now produced (new) editions and Latin translations of the entire surviving Greek musical corpus.134 Evidently Wallis still believed that such work was ‘part of his province’. Wallis also revisited musical matters at least once more in his correspondence (and there is the possibility that further discoveries in Wallis’s correspondence remain to be made). In 1701 a letter to Edward Tyson (1651–1708), physician, anatomist, fellow of the Royal Society and associate of Robert Plot, contained a discussion of the structure of the ear.135 Wallis rehearsed matter that was, as he acknowledged, well known by this date to anatomists; he was at the same time not afraid to speculate about the functions of some of the structures to be found within the ear. Much of what he said was concerned with strictly anatomical matters and some with the apprehension and judgement of speech rather than musical sound. But some of his remarks bore on music. The tympanum, and a thinner membrane which he believed covered it, he suggested might

Introduction

27

answer one another, much like the Base and Treble strings in a Lute or viol; or like two Drums, braced in Consert one to the other. The Base-String (with a slow Vibration) gives a strong Lowd Sound, which may be heard further-off; the Treble-string (with a Quick Vibration) gives a brisk, nimble, acute sound, (and in the same proportion, more Acute, as it is quicker;) which, duly tuned, give a Consert in Musick.136

Despite this, Wallis considered that the mere sensation of pulses upon the tympanum did not suffice to account for the variety of sensations of sounds: I do not deny, but that the Pulse of the Air on the Tympanum, may (by compressing the Inner Air) assert the Auditory Nerve; as in the Beating of a Drum. But this, being granted, goes but a little way, on to the whole busyness of Hearing. … The Variety of Musick in a Drum, lies in the Diversification, & due Mixture, of Longer & shorter successive Intervalls between Stroke and Stroke. It gives no account of what may arise from the successive Variegations of Tones (As doth the Lute or Viol;) much lesse of the Articulation of different Letters in Speech: Which are the two great things in Hearing.137

This hints at exposure to, and perhaps sympathy for, theories of sound and hearing based on a Cartesian understanding of the air as possessing both gross and subtle components;138 it is not obvious whether it was, or was intended to be, compatible with Newton’s account of sound published in 1687.139 Interestingly from the point of view of musical theory it does not really suggest a great deal of confidence in the coincidence theory to account for the sensations of musical sound. As in 1698, Wallis did not repudiate the theory, but simply ignored it. This letter is the last known discussion of musical topics by Wallis before his death in 1703. It indicates that by the end of his career Wallis could still be interested in the mechanical phenomena of sound and its apprehension but at the same time that he had no high degree of confidence in any specific mechanical explanation for those phenomena. Perhaps he still believed, as in 1664, that examination by experiment was needed in order to settle the matter. ‘Very elaborate and judicious’: Wallis’s legacy in musical scholarship Wallis’s edition of Ptolemy’s Harmonics was not superseded until the twentieth century (and the only printed version which had preceded it was Gogava’s Latin translation of 1562). There was therefore a long period during which it was a virtually unavoidable point of reference for those interested in the text. But early responses to the edition of Ptolemy are unfortunately almost unknown. At least 60 copies of the first edition survive today.140 Early owners included Charles Boyle, fourth Earl of Orrery,141 Narcissus Marsh,142 Isaac Hawkins Browne (the poet [1706–60] more probably than his son),143 Isaac Newton,144 Robert Hooke,

28

John Wallis: Writings on Music

and Francis North.145 A possibly later owner was John Rutherford of Edgerston (there were several individuals with that name).146 In the nineteenth century a copy was owned by the mathematician Augustus de Morgan.147 The colonial officer and mathematically minded scholar James Logan (1674–1751) added manuscript notes on two inserted leaves and the last flyleaf; we are not aware of any other copy with substantial annotations.148 Printed catalogues show that several copies were offered for sale during the eighteenth century; prices ranged from 1s 6d to 5s. Unfortunately only one of these can be assigned to the library of a specific individual: that of ‘R. Morgan, D.D. Late Canon of Hereford, and Rector of Ross’, sold in Ross in 1747.149 Nevertheless, responses to Wallis’s ideas can be found. His discussion of various aspects of music, notably its effect on the passions, was cited by various theologians in the early eighteenth century seeking to legitimise the role of music in church liturgy. In the sermons which accompanied the annual Three Choirs Festival, inaugurated by Thomas Bisse around 1616, a case was regularly made that the liturgical use of music had the function of preparing the soul for prayer and worship, given the assumption that terrestrial music anticipated the celestial music which was to be the language of the afterlife.150 This argument was developed by Bisse himself in the annual sermon for 1726 with reference to a comprehensive range of musical issues currently under philosophical investigation; after listing a number of contemporary theoreticians in support of his discussion, he singled out Wallis for special mention, and more specifically his ‘Appendix to Ptolemy’s Harmonicks’, which he said was ‘esteemed by all, particularly Dr. Aldrich, a masterpiece of Criticism’.151 It is implausible that Wallis did not himself have well-informed views on these liturgical and theological issues, which he would have debated with Oxford contemporaries such as Henry Aldrich, who was himself both a noted theologian and musical expert.152 In fact, there is no explicit discussion of liturgical issues relating to music in any of Wallis’s own sermons, even when he touched on eschatological matters. The music historians of the eighteenth and nineteenth centuries also took an interest in Wallis’s musical scholarship; on the whole they admired it. John Hawkins (1719–89) was particularly enthusiastic. In his General History of the Science and Practice of Music (1776) he relied on Wallis’s edition for his citations of Ptolemy and the appendix for some of his information about other Greek theorists including Aristoxenus and Aristides Quintilianus.153 He reproduced a number of Wallis’s diagrams illustrating the Aristoxenean account of the genera and the modes.154 Hawkins judged ‘The Harmonics of the Ancients’ ‘very elaborate and judicious’ and quoted from it at length on the modern forms of the diatonic scale, and on the modes.155 His opinion is perhaps summed up in his remark that ‘Had [Pepusch] considered how little Salinas, Mersennus, Kircher, and Dr. Wallis have left unsaid on this part of musical science [sc. ‘arithmetical calculations of ratios’], he might possibly have turned his thoughts another way.’156

Introduction

29

Charles Burney (1726–1814) was on the whole less unequivocally enthusiastic, although he quoted from ‘The Harmonics of the Ancients’ on Greek terminology and cited Wallis on the ancient modes in his 1776–89 General History of Music.157 He perhaps relied less on Wallis than did Hawkins, and treated him more critically. On the ancient modes, for instance, he considered Wallis’s account to have been superseded by that of Francis Eyles Stiles in 1760,158 while on the question of whether ancient music knew counterpoint or part-writing he acknowledged at some length that opinion had varied and that several more recent writers had responded specifically to Wallis. Thus, what Wallis had to say about the matter was a starting point rather than the last word.159 Burney was also slightly guarded in the praise he bestowed on Wallis’s edition of Ptolemy: ‘Some parts of [Ptolemy’s] disputes and doctrines are now become unintelligible, notwithstanding all the pains that our learned countryman Dr. Wallis bestowed on him’.160 More minor writers also made use of Wallis’s work. François-Joseph Fétis (1784–1871) reported, for example, that Jean-Antoine Ducerceau (1670–1730) employed Wallis’s notes on Ptolemy to support his interpretation of certain musical matters in Horace.161 But by the time Fétis was compiling his Biographie universelle (1834) Wallis had become a figure of whose musical work less was known for certain. Fétis seems to have relied in part on Hawkins for his biographical information about Wallis and gave a somewhat garbled list of his musical publications which included three non-existent papers.162 He also remarked, revealing something of the relative obscurity into which Wallis had now fallen among writers on music, that several recent authors had mistakenly given his name as Walker.163 Nonetheless, Fétis knew Wallis’s edition of Ptolemy, and had considerable praise for it. He judged that Wallis was badly mistaken only ‘dans la traduction qu’il donne des modes grecs au second livre de Ptolémée’, in effect concurring with Burney that Wallis’s account of the modes had now been superseded by newer scholarship.164 On the editions of Porphyry and Bryennius, it is the same picture of guarded praise: Malgré le mérite incontestable de ces éditions, il est à regretter que Wallis n’ait pu consulter d’autres manuscrits que ceux dont il s’est servi, car tous ceuxci sont du seizième siècle et sortent de la même source. Les manuscrits de la bibliothèque impériale de Paris … lui auraient fourni en plusieurs endroits un texte plus correct, et des éclaircissements sur des passages qu’il n’a pas bien entendus.165

*** Wallis’s reflections on music resulted in editions of three ancient texts, the substantial appendix to Ptolemy’s Harmonics, and four papers in the Philosophical Transactions, as well as various items in his correspondence. Despite his disclaiming musical expertise in 1664, it is clear that he had by that date already given some attention to musical matters and acquired some reputation for being

30

John Wallis: Writings on Music

interested in the subject. That reputation grew over the following decades, and Wallis was consulted as a musical expert by Thomas Salmon, Samuel Pepys, Andrew Fletcher, and René Harris. He was also handed musical information by colleagues in Oxford, possibly including Robert Plot and Narcissus Marsh. As well as Henry Oldenburg, manuscript copies of his first long discussion of music reached Robert Boyle and Marsh. His edition of Ptolemy circulated widely, and his musical work may have been deliberately rivalled by William Holder. Wallis positioned his musical expertise somewhat differently from those who might most naturally be compared with him in Restoration England: Thomas Salmon, Francis and Roger North, and William Holder. Unlike them he wrote his main work on the subject in Latin; unlike them he reserved his vernacular expositions for private correspondence or for the pages of the Philosophical Transactions. His interest in music engaged philology, and indeed his attention to the texts of Ptolemy, Porphyry, and Bryennius may have been the most important part of his legacy to musical scholarship. On the other hand, he was comparatively uninterested in practitioners’ knowledge, and over time even the coincidence theory of consonance seems to have come to hold less appeal for him. His silence on the relationship of musical sound to speech is remarkable, given that Wallis was one of the main contributors to the study of language in his time. So is the absence of any mention of music from his sermons. He seems, on the whole, to have preferred to keep musical scholarship firmly within the bounds implied by its status as part of the quadrivium. Thus at a time when others were building accounts of the mathematics of music on the foundations of mechanical explanation and practical experiment, he erected the edifice of the just intonation on mathematical grounds alone. His was not, on the whole, a path that found followers in the eighteenth century, and his textual scholarship fell eventually out of date. In a world in which the judgement of the ear was increasingly accustomed to trump theoretical speculation, what he had to say about the mathematics of the scale did not always seem relevant to later generations. His work nevertheless illustrates uniquely the nexus of classical scholarship, mathematics, and curiosity about the mechanical world which he brought to bear on the theory of music. Editorial Policy The texts presented retain original spelling, punctuation, capitalisation, italics, paragraph breaks, indentation, and blank lines. Editorial matter, including reports of pagination or foliation and supplied headings, is placed within square brackets. Catchwords are ignored, except where they are incorrect. In Chapters 1 and 7: manuscript underlining is rendered as italics; insertions are placed in angle brackets; abbreviations involving the thorn or the notation ~ (for omitted m, n, or ti) are silently expanded, as are the following common abbreviations: wch, wth, 2d, 3d, Sr;

Introduction

31

footnotes record deletions and endorsements in later hands, as well as variants between the different manuscripts; illegible small deletions are disregarded. In the endnotes to each text we have endeavoured to elucidate those passages which might otherwise be obscure, to draw attention to related passages in the other texts, and to shed some light on Wallis’s sources of information. We have not attempted to comment in detail on the content or the correctness of Wallis’s assertions about ancient musical theory or about the mechanics of musical sound. A Note on Mathematical Notation Throughout the sources, fractions are written or printed vertically, with the numerator directly above the denominator. In this edition diagonal fractions (3/4, for example) are substituted. Wallis regularly used one piece of mathematical notation which is now obsolete. The sentence /2 ) 2/1 ( 4/3

3

for example, will be found in his letter to Oldenburg of 14 May 1664.166 It expresses a division, set out somewhat like a modern ‘long division’. The central term is divided by the left-hand term, and the result is placed at the right. We would now write /1 ÷ 3/2 = 4/3.

2

Notes 1 Necessary background to Wallis’s work on Greek musical theory can be found in Martin L. West, Ancient Greek Music (Oxford, 1992); on the earlier reception of that tradition, see Thomas J. Mathiesen, Apollo’s Lyre: Greek music and music theory in antiquity and the Middle Ages (Lincoln, 1999). 2 Philip Beeley and Christoph J. Scriba (eds), The Correspondence of John Wallis (Oxford, 2003–). 3 Modern discussions of Wallis’s life include Domenico Bertoloni Meli, ‘Wallis, John (1616–1703), Mathematician and Cryptographer’, in ODNB; and the introductions to the volumes of Beeley and Scriba (eds), The Correspondence of John Wallis. Important information is contained in Wallis’s own sketch of his life in a letter published in Thomas Hearne (ed.), Peter Langtoft’s Chronicle (Oxford, 1925), vol. 1, pp. cxl–clxx; and in Christoph J. Scriba, ‘The Autobiography of John Wallis, F.R.S.’, Notes and Records of the Royal Society 25 (1970): 17–46, but the value of this as an ‘autobiography’ is doubtful. 4 Michael Hunter, The Royal Society and its Fellows 1660–1700: the morphology of an early scientific institution (Chalfont St Giles, 1982), pp. 146–7; Hunter describes Wallis’s activity thus: ‘Frequent corresp. and slightly active’. See also Gail Ewald Scala, ‘An Index of Proper Names in Thomas Birch, “The History of the Royal

32

John Wallis: Writings on Music

Society” (London, 1756–1757)’, Notes and Records of the Royal Society 28 (1974): 263–329. 5 Published sermons include The resurrection asserted in a sermon preached to the University of Oxford on Easter-day, 1679 (Oxford, 1679) and Three sermons concerning the sacred Trinity (London, 1691). 6 Information from ESTC. 7 There were also quarrels with Pierre de Fermat, Christiaan Huygens, Bernard Frénicle de Bessy, and Gilles Personne de Roberval: see J.F. Scott, The Mathematical Work of John Wallis, D.D., F.R.S. (1616–1703) (London, 1938), pp. 80–82; Philip Beeley and Siegmund Probst, ‘John Wallis (1616–1703): Mathematician and Divine’, in Luc Bergmans and Teun Koetsier (eds), Mathematics and the Divine: a historical study (Amsterdam, 2005), pp. 443–4; Jacqueline A. Stedall, ‘John Wallis and the French: his quarrels with Fermat, Pascal, Dulaurens, and Descartes’, Historia Mathematica 39 (2012): 265–79. 8 G.R.M. Ward (ed.), Oxford University Statutes. Volume I. Containing the Caroline Code, or Laudian Statutes (London, 1845), pp. 273–4 (274). 9 Oxford, Bodleian Library, MS Don d. 45, fols 7–93. 10 See Christopher D.S. Field and Benjamin Wardhaugh (eds), John Birchensha: writings on music (Farnham, 2010), pp. 15–17. 11 Thomas Birch, A History of the Royal Society of London (London, 1756–7), vol. 1, p. 418. 12 Oxford, Bodleian Library, MS Don. d. 45, a fat volume of mathematical drafts in Wallis’s hand ranging from 1651 to 1685; the musical material covers only fols 306v–307r and is undated; material apparently written after it on fol. 301v is dated 1664, but no more precise information about the date of these drafts can be gleaned. 13 H. Floris Cohen, Quantifying Music: the science of music at the first stage of the scientific revolution, 1580–1650 (Dordrecht, 1984), pp. 75–8, 90–97. 14 Wallis to Oldenburg, 14 May 1664, fol. 2v (p. 48–9 below). 15 Cohen, Quantifying Music, pp. 3–6; J. Murray Barbour, Tuning and Temperament: a historical survey (East Lansing, 1951), pp. 89–105; Mark Lindley, Lutes, Viols and Temperaments (Cambridge, 1984), passim. 16 See Wallis’s diagram, p. 53 below. 17 Barbour, Tuning and Temperament, pp. 1­–4, 25–44. 18 Ptolemy, Harmonics, 2.14–15; see GMW, pp. 346–55. 19 Compare Wallis to Oldenburg, 7 May 1664, fol. 1r : ‘That there bee many Distances, or Diastemata, of which no notice is taken; is very evident’ (p. 45 below). Others, including Mersenne, had investigated these ratios, and would continue to do so, and, although it seems the thought was Wallis’s own, it is a strange coincidence that in the same year, 1664, Christiaan Huygens also wrote a manuscript treatise in which he speculated about the musical use of this pair of intervals: see Cohen, Quantifying Music, pp. 225–8. Robert Moray, another fellow of the Royal Society, was corresponding with Huygens at this time, but there is no evidence that his ideas could have come to Wallis’s attention. 20 Wallis to Oldenburg, 14 May 1664, fol. 5v (p. 53 below). 21 Wallis to Oldenburg, 7 May 1664, fol. 1r (p. 45 below). 22 Ward, Oxford University Statutes. Volume I., p. 273; Statuta Universitatis Oxoniensis (Oxford, 1857), p. 241. 23 Birch, History, vol. 1, pp. 425–6; information from EMLO. The Royal Society’s interest in musical matters is documented in Leta Miller and Albert Cohen, Music in the Royal Society of London 1660–1806 (Detroit, 1987) and Penelope M. Gouk,

Introduction

33

‘Music in the Natural Philosophy of the Early Royal Society’ (Ph.D. thesis, Imperial College London, 1982). 24 Walter Charleton, Physiologia Epicuro–Gassendo–Charletoniana: Or a Fabrick of Science Natural, Upon the Hypothesis of Atoms (London, 1654), esp. pp. 222–3; René Descartes, Renatus Des-Cartes Excellent Compendium of Musick and Animadversions of the Author, ed. and trans. anon. [trans. Walter Charleton, ed. William Brouncker] (London, 1653). 25 There had been an intervening letter on other subjects, written on 16 May: see Birch, History, vol. 1, p. 426; EMLO. 26 Wallis to Oldenburg, 14 May 1664, fol. 10r (p. 62 below). 27 Marin Mersenne, Harmonicorum libri (Paris, 1636); Galileo Galilei, Discorsi e dimostrazioni matematiche intorno è due nuove scienze (Leiden, 1638); Giovanni Battista Benedetti, Diversarum speculationum … liber (Turin, 1585); Catalogus librorum bibliothecæ publicæ quam vir ornatissimus Thomas Bodleius Eques Auratus in Academia Oxoniensi nuper instituit (Oxford, 1605), p. 290; see Cohen, Quantifying Music, pp. 75–8, 90–97, 103–11. 28 René Descartes, Musicae compendium (Utrecht, 1650), pp. 31–3. 29 Gioseffo Zarlino, Istitutioni harmoniche (Venice, 1558). 30 Thomas Morley, A Plaine and Easie Introduction to Practicall Musicke (London, 1597), p. 3. 31 John Playford, A breefe introduction to the skill of musick for song & violl (London, 1654), p. 8. 32 A. Rupert Hall and Marie Boas Hall (eds), The Correspondence of Henry Oldenburg (Madison, Milwaukee and London, 1965–86), vol. 2, p. 530 (Moray to Oldenburg, 28 Sept. 1665); see also pp. 537 (Boyle to Oldenburg, 30 Sept. 1665), 555 (Oldenburg to Boyle, 10 Oct. 1665). 33 Birch, History, vol. 2, p. 68. 34 The Correspondence of Henry Oldenburg, vol. 3, pp. 61–2 (Oldenburg to Boyle, 17 Mar, 1665/6); see also p. 66 (Boyle to Oldenburg, 19 Mar, 1665/6). 35 Birch, History, vol. 2, p. 83. 36 Michael Hunter (ed.), The Boyle Papers: understanding the manuscripts of Robert Boyle (Aldershot and Burlington, VT, 2007); Miller and Cohen, Music in the Royal Society of London. 37 John Hawkins, A General History of the Science and Practice of Music (London, 1776), vol. 4, p. 376. 38 Wallis to Oldenburg, 14 Mar. 1677 (p. 70 below). 39 Anthony à Wood, Athenae oxonienses. An exact history of all the vvriters and bishops who have had their education in the most ancient and famous University of Oxford … (London, 1691–2), vol. 2, pp. 870, 881; on Pigot, see Hunter, The Royal Society and its Fellows, pp. 200–201. 40 Penelope M. Gouk, ‘Speculative and Practical Music in Seventeenth-Century England: Oxford University as a case study’, International Musicological Society Congress Report XIV (Bologna, 1990), pp. 199–205, at p. 201. 41 Robert Plot, The Natural History of Oxford-Shire (Oxford, 1677), pp. 288–99. 42 Oxford, Bodleian Library, MS Add D. 105, fol. 48; The Correspondence of Henry Oldenburg, vol. 12, letter 3081, p. 233. 43 Wallis’s paper is discussed in Ll. S. Lloyd, ‘Music Theory in the Early “Philosophical Transactions”’, Notes and Records of the Royal Society of London 3 (1941): 149–57, at pp. 149–50.

34

John Wallis: Writings on Music

44 John Wallis, Johannis Wallis … De algebra tractatus historicus & practicus (Oxford, 1693), pp. 466–8. 45 Muriel McCarthy, ‘Marsh, Narcissus (1638–1713), Church of Ireland archbishop of Armagh’, in ODNB. 46 See Introduction to Chapter 1 below. 47 See C. Adkins and A. Dickinson, A Trumpet by any Other Name: a history of the trumpet marine (Buren, 1991); C. Adkins, ‘Trumpet marine’, in GMO. 48 Guy Oldham et al., ‘Harmonics’, in GMO, citing Jean-Joseph Cassanea de Mondonville, Les sons harmoniques: sonates à violon seul avec la basse continue (Paris, 1738). 49 Wallis to Oldenburg, 14 Mar. 1677 (p. 72 below). 50 Francis North, A philosophical essay of musick directed to a friend (London, 1677), p. 18. On North’s work, see in particular Jamie C. Kassler, The Beginnings of the Modern Philosophy of Music in England: Francis North’s A Philosophical Essay of Musick (1677) with comments of Isaac Newton, Roger North and in the Philosophical Transactions (Aldershot, 2004). 51 Wallis, De algebra, p. 466. 52 Wallis to Oldenburg, 27 Mar. 1677 (p. 73 below). 53 Claudii Ptolemæi Harmonicorum libri tres. Ex. Codd. MSS. Undecim, nunc primum Græce editus (Oxford, 1682), sig. b1r–b2r. 54 See Andrew Barker, Scientific Method in Ptolemy’s ‘Harmonics’ (Cambridge, 2000). 55 Marcus Meibom, Antiquae musicae auctores septem (Amsterdam, 1652), introduction to Aristoxenus, Elementa harmonica (separately paginated), sig. *2r; ‘Κλαυδιου Πτολεμαιου Ἁρμονικων Βιβλια Γ’ [Claudius Ptolemy’s Three Books on Harmonics], Philosophical Transactions 13 (1683): 20–21 (Ch. 4 below); François-Joseph Fétis, Biographie universelle des musiciens et bibliographie générale de la musique (2nd edn; Paris, 1866–8), vol. 4, p. 48; vol. 7, p. 154; cf. vol. 8, p. 409. 56 Mordechai Feingold, The Mathematician’s Apprenticeship (Cambridge, 1984), p. 156. 57 See Mordechai Feingold and Penelope M. Gouk, ‘An Early Critique of Bacon’s Sylva Sylvarum: Edmund Chilmead’s “Treatise on Sound”’, Annals of Science 40 (1983): 139–57; Mattias Lundberg (ed.), Studies on Marcus Meibom (Copenhagen, forthcoming). 58 Feingold, The Mathematician’s Apprenticeship, p. 156. 59 M. Reeve, ‘John Wallis, Editor of Greek Mathematical Texts’, in Aporemata: Kritische Studien zur Philologiegeschichte, vol. 2: Editing Texts / Texte edieren, ed. G.W. Most (Göttingen, 1998), pp. 77–93, at pp 79–80, citing T. Smith, Vita clarissimi et doctissimi viri Edwardi Bernardi (London, 1704), pp. 23–6. 60 I. Düring, Die Harmonielehre des Klaudios Ptolemaios (Gothenburg, 1930), pp. xciii–xciv; cited in Reeve, ‘John Wallis’, p. 89. 61 Reeve, ‘John Wallis’, pp. 86–7, 93. 62 John Wallis, ‘The Harmonics of the Ancients compared with Today’s’, in Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium (Oxford, 1699), pp. 153– 82, at pp. 153–6 (pp. 77–84 below). 63 Fétis, Biographie universelle, vol. 7, p. 137. 64 Wallis, ‘The Harmonics of the Ancients’, pp. 153–5 (pp. 77–82 below). 65 Ibid., p. 172 (p. 123 below). 66 John Wallis, Johannis Wallisii … operum mathematicorum pars altera qua continentur de angulo contactus & semicirculi, disquisitio geometrica. De sectionibus conicis

Introduction

35

tractatus. Arithmetica infinitorum: sive de curvilineorum quadraturâ, &c (Oxford, 1656); Katherine Neale, From Discrete to Continuous: the broadening of number concepts in early modern England (Dordrecht, 2002). 67 Jacqueline Stedall, A Discourse Concerning Algebra: English algebra to 1685 (Oxford, 2002), pp. 155–82. 68 Wallis, ‘The Harmonics of the Ancients’, pp. 163–6 (pp. 101–­8 below). 69 Ibid., pp. 159–63 (pp. 91–101 below). 70 Ibid., pp.173–82 (pp. 124–43­below). 71 ‘Κλαυδιου Πτολεμαιου Ἁρμονικων Βιβλια Γ’, p. 21 (p. 206 below). 72 See p. 29 below. 73 Wallis, ‘The Harmonics of the Ancients’, pp. 176–81 (pp. 130–41­below). 74 The tacit use of string lengths appears for instance in the division of the fifth into major and minor thirds on p. 179 (p. 136–7 below). 75 Wallis, ‘The Harmonics of the Ancients’, p. 180 (pp. 139–40 below). 76 Ibid., p. 167 (p. 111­below). 77 Ibid., p. 169 (p. 115­below). 78 Ibid., p. 176 (pp. 131–2 below). 79 Ibid., p. 170 (pp. 118–19­below). 80 Ibid., p. 179 (p. 138­below). 81 Ibid., p. 182 (p. 143 below). 82 Ibid., pp. 173, 176 (pp. 125, 131­below). 83 Ibid., p. 176 (p. 131­below). 84 Johann Heinrich Alsted, Templum Musicum; or the Musical Synopsis of the Learned and Famous Johannes–Heinricus–Alstedius, being a Compendium of the Rudiments both of the Mathematical and Practical part of Musick, of which Subject not any Book is extant in our English Tongue, trans. John Birchensha (London, 1664). 85 Edmund Chilmead, ‘De musica antiquâ Græcâ’, in Αρατου Σολεως Φαινόμηνα καὶ Διοσημεῖα. Θεωνος Σχόλια. … Accesserunt Annotationes in Eratosthenem et Hymnos Dionysii. (Oxford, 1672), pp. 47–69, at pp. 49–51 et passim. 86 Benjamin Wardhaugh (ed.), Thomas Salmon: writings on music (Farnham, 2013), vol. 2, pp. 38, 50, 90. 87 See p. 140, n. 71 below. 88 Hawkins, History, vol. 1, p. 74. 89 Ibid., vol. 3, p. 208. 90 Fétis, Biographie universelle, vol. 7, p. 136: ‘Bouillaud a rapporté quelques passages de ce traité [sc. Ptolemy’s Harmonics] dans des notes sur Théon de Smyrne, et les a éclaircis. Son travail n’a pas été inutile à Wallis.’ 91 Hunter, The Royal Society, pp. 79, 82. 92 Thomas Salmon, A Proposal to Perform Musick, in Perfect and Mathematical Proportions (London, 1688). 93 Wardhaugh, Thomas Salmon, vol. 1, pp. 6–9, 14; vol. 2, pp. 23–6. 94 Ibid., vol. 1, p. 18. 95 Thomas Salmon, A Vindication of an Essay to the Advancement of Musick (London, 1672), p. 10. 96 Ibid., pp. 14–15; see Wardhaugh, Thomas Salmon, vol. 1, p. 151, n. 50. 97 See Miller and Cohen, Music in the Royal Society of London. 98 Wardhaugh, Thomas Salmon, vol. 2, pp. 108–16. 99 London, British Library, MS Sloane 4040, fol. 104r; Wardhaugh, Thomas Salmon, vol. 2, p. 175.

36

John Wallis: Writings on Music

100 Thomas Hearne, Reliquiae Hearnianae: The Remains of Thomas Hearne, M.A. of Edmund Hall. Being Extracts from his MS Diaries, Collected, with a few notes, by Philip Bliss (London, 1869), vol. 2, p. 79; see also John Wallis’s letter of 1700 printed in Collectanea, First Series (Oxford: Clarendon Press, 1885), pt VI, p. 317. Wallis apparently considered that ‘the business of dancing, singing, playing on musick and the like’ was ‘in an university … rather an hindrance, than a promotion, of other studies’. 101 Wallis, ‘The Harmonics of the Ancients’, p. 180 (p. 139–40 below); John Wallis, ‘Remarks on the Proposal to perform musick, in perfect and mathematical proportions’, in Salmon, A Proposal to Perform Musick, in Perfect and Mathematical Proportions, pp. 29–41, at p. 40; Cambridge University Library, Add. MS 3970, fols 1–11: Thomas Salmon, ‘The Division of a Monochord’, fols. 5–6; Wardhaugh (ed.), Thomas Salmon: writings on music, pp. 151–3. 102 Wallis, ‘A Question in Musick’ (Ch. 5 below); Wallis to Pepys, 27 June 1698 (Ch. 6 below); Thomas Salmon, ‘The Theory of Musick reduced to Arithmetical and Geometrical Proportions’, Philosophical Transactions 24 (1705): 2072–7, 2069. 103 Wallis to Oldenburg, 14 May 1664, §13 (pp. 55–6 below); Wallis, ‘The Harmonics of the Ancients’, p. 179 (pp. 137–8 below); id., ‘Remarks on the Proposal to perform musick, in perfect and mathematical proportions’, pp. 37–8. 104 See Lloyd, ‘Musical Theory’, pp. 154–6. 105 Wallis, ‘A Question in Musick’, p. 81 (p. 208 below). 106 Ibid., pp. 81–2 (p. 208–9 below). 107 See Wallis’s diagram on p. 217 below. 108 Wallis to Oldenburg, 14 May 1664, §17 (p. 59 below); Wallis, ‘Remarks on the Proposal to perform musick, in perfect and mathematical proportions’, 39–40. 109 Field and Wardhaugh (eds), John Birchensha, p. 25; Montague Summers (ed.), The Complete Works of Thomas Shadwell (London, 1927), vol. 1, pp. 191, 217–18, 222. 110 William Holder, A Discourse concerning Time (London, 1694), pp. 106–13. 111 H. Edmund Poole, ‘The Printing of William Holder’s “Principles of Harmony”’, Proceedings of the Royal Musical Association 101 (1974): 31–43. 112 Charles Burney, A General History of Music, from the earliest ages to the present period (London, 1776–89), vol. 3, pp. 598–9; Hawkins, History, vol. 4, pp. 541–4; see also vol. 1, p. 309. 113 Holder, Treatise, pp. 162–3. 114 Christopher Simpson, A Compendium of Practical Musick (London, 1667), pp. 102–9. 115 Wardhaugh, Thomas Salmon, vol. 2, pp. 3, 9–10. 116 David S. Knight, ‘Harris, Renatus [René] (c.1652–1724), Organ Builder’, in ODNB; Michael Gillingham et al., ‘Harris (i): (2) Renatus Harris’, in GMO; see Susi Jeans, ‘Renatus Harris, Organ-Maker, his Challenge to Mr Bernard Smith, Organ-Maker’, The Organ 61 (1982): 129–30. In her Newtonian papers, Jeans cited Andrew Freeman: ‘Dr. Blow, master on the Harpsicord to her late Majesty Queen Anne, prov’d it on an Organ then standing in his Workhouse in Wine Office Court in Fleet Street, now in St Andrew’s Church in Holborn. Mr. Harris did this to let the World know that he was Master of his Art, because Mr. Smith, who made St. Paul’s Organ, “had reported that he had no Ear to Musick”’ (Andrew Freeman, The Organ 26 [1947]: 178–9). 117 The Post Boy, 9–12 April 1698; see Michael Tilmouth, ‘A Calendar of References to Music in Newspapers Published in London and the Provinces (1660–1719)’, Royal Musical Association Research Chronicle 1 (1961), p. 23. Knight suggests that Harris may have been referring to 50 gradations of volume rather than a subdivision of

Introduction

37

pitch, but this seems at odds with both the plain sense of the words and the subject of Wallis’s response to Harris, unless he had misunderstood him very badly (‘Harris, Renatus [René] (c.1652–1724), Organ Builder’). 118 The Post Boy, 28–30 April 1698; see Tilmouth, ‘A Calendar’, p. 23. 119 See Wallis to Pepys, 27 June 1698, p. 256: ‘multiplying intermediate Pipes’ (p. 219 below). 120 Compare Galileo’s remarkable description of an organ pipe ‘in two pieces, one inserted in the other’, which ‘may be lengthened and shortened at will’ in The Assayer (1623): Stillman Drake (ed.), Discoveries and Opinions of Galileo (New York, 1957), p. 251. 121 Wallis to Pepys, 27 June 1698, p. 255 (p. 218 below); cf. Wallis to Oldenburg, 14 May 1664, fol. 1v (p. 47–8 below). 122 See Barbour, Tuning and Temperament, pp. 107–11, 133–44. 123 London, Royal Society, Early Letters W2 no. 77. 124 London, Royal Society, Early Letters W2 no. 78. 125 London, Royal Society, Early Letters W2 no. 79. 126 See Thomas Salmon, An Essay to the Advancement of Musick, by casting away the perplexities of different cliffs. and uniting all sorts of musick … in one universal character (London 1672), p. 3. 127 London, British Library, MS Lansdowne 763, fol. 124; cf. fol. 125, Wanley’s account of the manuscript; see also London, Society of Antiquaries, MS 264, fols 96r–98v, 128r–130v. There has been some more recent uncertainty as to the identity of the manuscript Wallis described: the British Library’s online catalogue entry for Lansdowne 763 states that it is certainly not the article afterwards described by [Wanley] in the Harl. Catal. No 1613, as was stated in the former Catalogue of the Lansd. MSS. See a letter from him on the same subject to Dr. Char[l]ett, printed in Hawkins’s Hist. of Musick, vol. i. p. 392. If this is correct, it would seem that the facsimile printed by Hawkins (History, vol. 1, pp. 394–5) is not of the manuscript Wanley and Wallis described. 128 Hawkins, History, vol. 1, pp. 391–2. 129 Ibid., p. 392. 130 Ibid., p. 391. 131 John Wallis, Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium, pp. 1–182. 132 Ibid., pp. 183–355, 357–508. 133 Reeve, ‘John Wallis’, p. 82. According to Reeve, the printer’s copy for the two editions is at Oxford, Bodleian Library, MSS Savile 53, 54. 134 Wallis, Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium, sig. a3r; cf. Hawkins, History, vol. 4, p. 485. 135 Oxford, Bodleian Library, MS Add. D105, fols 124–5 (a draft of the letter sent), 126– 7 (a copy of the draft). 136 Ibid., fol. 126v. 137 Ibid., fol. 126v. 138 See Benjamin Wardhaugh, Music, Experiment and Mathematics in England, 1653– 1705 (Farnham, 2008), pp. 80–82; Penelope M. Gouk, ‘Some English Theories of Hearing in the Seventeenth Century: before and after Descartes’, in Charles F. Burnett, Michael Fend, and Penelope M. Gouk (eds), The Second Sense: studies in

38

John Wallis: Writings on Music

hearing and musical judgement from antiquity to the seventeenth century (London, 1991), pp. 95–113. Tyson was aware of the anatomical work of Claude Perrault (see Anita Guerrini, ‘Tyson, Edward (1651–1708), physician and anatomist’, in ODNB) and may have known his work on sound (Claude Perrault, Essais de physique, ou Receuil de plusieurs traitez touchant les choses naturelles, vol. 2: De la bruit [Paris, 1688]); it is possible that Wallis had become aware of such ideas by this route. 139 Isaac Newton, Philosophiæ naturalis principia mathematica (London, 1687), bk 2, §8. 140 British Isles (38 copies) – Aberdeen: University Library (2 copies); Birmingham: Central Libraries; University Library; Bristol: University of Bristol Arts and Social Sciences Library, Special Collections; Cambridge: Cambridge University Library (2 copies); Cambridge University, St. John’s College; Cambridge University, Trinity College (2 copies); Cambridge University, King’s College (2 copies); Dublin: Marsh’s Library; Durham: Cathedral Library; Edinburgh: National Library of Scotland; Edinburgh University Library; Glasgow: University Library (2 copies); Liverpool: University Library; London: British Library; Royal Society Library; Senate House Library, University of London (2 copies); Manchester: John Rylands Library, University of Manchester; Oxford: Bodleian Library (3 copies); Oxford University, Brasenose College Library; Oxford University, Christ Church (3 copies); Oxford University, Jesus College; Oxford University, Museum of the History of Science; Oxford University, New College; Oxford University Queen’s College (2 copies); Peterborough: Cathedral Library; Sheffield: University Library; Warminster: Longleat House Old Library; Windsor: Eton College Library. France (2 copies) – Paris: Bibliothèque nationale de France (2 copies). Germany (2 copies) – Göttingen: Niedersachsische Staats- und Universitatsbibliothek (2 copies). USA (14 copies) – Cambridge, MA: Harvard University Library; Chicago: Newberry Library; Los Angeles: University of California, Los Angeles, William Andrews Clark Memorial Library; New Haven, CT: Yale University, Sterling Memorial Library (2 copies); New York: Columbia University Library; New York Public Library (2 copies); Philadelphia: Library Company of Philadelphia; Providence: Brown University, John Carter Brown Library; San Marino, CA: Henry E. Huntington Library and Art Gallery (2 copies); Urbana: University of Illinois at Urbana-Champaign; Washington DC: Folger Shakespeare Library; Canada (1 copy) – Toronto: University Library. Australia (1 copy) – Melbourne: State Library of Victoria. This list is based on that in ESTC, supplemented using COPAC and checked with the individual libraries’ catalogues. ESTC also reports copies in (1) Ely cathedral library, which seems to have passed to the Folger Shakespeare Library in 1961; (2) Cashel Cathedral Library and Gloucester Cathedral Library, neither of which appears in COPAC, which has a possibly more up-to-date holdings report for the cathedral libraries; (3) Trinity College Library (Dublin), where the copy is listed in the printed catalogue of 1872 but not the modern online catalogue; (4) Oxford University, Worcester College Library, a copy which does not appear in the library’s online catalogue. 141 Copy in Christ Church, Oxford, shelfmark: OO.3.7. 142 Copy now in Marsh’s Library, Dublin, with Marsh’s Greek motto on the title page. 143 Copy in University of Bristol Arts and Social Sciences Library, system number: 000646802.

Introduction

39

144 Copy in Trinity College, Cambridge. 145 Information from COPAC. British Library, MS Sloane 1039, fols. 118r–119r contain notes by Robert Hooke on Wallis’ edition of Ptolemy. 146 Copy now in Henry E. Huntington Library and Art Gallery, call no.: 432049. 147 Now in Senate House Library, London. 148 Library Company of Philadelphia, shelfmark: Log 197.Q. 149 A catalogue of part of the books of R. Morgan, D.D. Late Canon of Hereford, and Rector of Ross (Gloucester, 1747), p. 8, lot 71; John Whiston and Benjamin White, Bibliotheca utilissima & elegans. A catalogue of many thousand volumes ([London], [1759]), p. 57, lot 1730, price 2s; John Whiston and Benjamin White, [A] catalogue [of the] entire l[ibrar]ies of the following gentle[men], lately deceased, Edward Smith, Esq. of Edmonthorpe, ... Henry Bromfield, ... ([London], [1765]), p. 59, lot 1759, price 2s; Benjamin White, A catalogue of a large collection of curious and useful books in most sciences and languages; including the Libraries of The late Rev. Mr. Botham, ... ([London ], [1774]), p. 64, lot 1760, price 2s; Thomas Payne and Son, A catalogue of near forty thousand volumes of curious and rare books; containing the libraries of an eminent prelate, Sir James Porter, W. Negus, ... ([London], [1778]), p. 81, lot 2608, price 3s; Thomas Payne and Son, A catalogue of a large and curious collection of books; Containing several libraries lately purchased, In all Languages and Faculties ... ([London], [1779]), p. 69, lot 2089, price 2s; Lockyer Davis, L. Davis’s catalogue of a very large and valuable collection of books. In the Greek, Latin, French, Italian, Spanish, and English Languages ([London], [1790]), p. 89, lot 3170, price 5s; Joshua Cooke, A catalogue of useful and valuable books in all branches of literature; in which are contained the library of the Rev. John Noel, deceased: ... . ([Oxford ], [1791]), p. 25, lot 951, price 1s 6d; John Hayes, Food for book-worms. Price one shilling, (to be allowed in purchase, or on returning it,) a catalogue of books, for 1791 ... ([London], [1791]), p. 140, lot 4626, price 3s 6d; Joshua Cooke, A catalogue of useful and valuable books in all branches of literature; including the libraries of the Right Hon. and Rev. Lord Tracy ... ([Oxford], [1793]), p. 20, lot 646, price 2s 6d. 150 On the early history of this festival, and its seventeenth-century precursors, see Daniel Lysons, History of the Origin and Progress of the Meeting of the Three Choirs (London, 1812). On the issue of music as the language of the afterlife, see David Cram, ‘The Changing Relations between Grammar, Rhetoric and Music in the Early Modern Period’, in Rens Bod et al. (eds), The Making of the Humanities (Amsterdam, 2010), pp. 263–82, at pp. 272–6. 151 Thomas Bisse, Music the delight of the sons of men. A sermon preached at the Cathedral Church of Hereford, at the Anniversary Meeting of the Three Choirs, Glocester, Worcester, Hereford, September 7, 1726 (London, 1726), p. 37. 152 Henry Aldrich (1647–1710) was an active promoter of musical activities in Oxford University circles. Thomas Hearne made a similar note on Aldrich’s esteem for Wallis in his diary for 17 June 1718, ‘Dr. Aldrich used to say, Claudius Ptolemy’s Musica, published by Dr. Wallis, was Dr. Wallis’s masterpiece’: see Hearne, Reliquiae Hearnianae, vol. 2, p. 79. 153 Hawkins, History, vol. 1, pp. 70, 90. 154 Ibid., pp. 87 (see also pp. 132–3), 155. 155 Ibid., pp. 104–7, 149–52, 174–5. 156 Hawkins, History, vol. 5, p. 403. 157 Burney, History, vol. 1, pp. 21, 52.

40

John Wallis: Writings on Music

158 Francis Eyles Stiles, ‘An Explanation of the Modes or Tones in the antient Græcian Music’, Philosophical Transactions 51 (1760): 695–773. 159 Burney, History, vol. 1, pp. 123–31, citing Stillingfleet’s Principles and Power of Harmony and Fraguier’s Memoire (but see vol. 1, pp. 131–2). 160 Burney, History, vol. 1, p. 460. 161 Fétis, Biographie universelle, vol. 3, p. 66 (and vol. 2, p. 236), citing Ducerceau, ‘Dissertation adressée au père Sanudon, où l’on examine la traduction et les remarques de M. Dacier sur un endroit d’Horace, et où l’on explique par occasion ce qui regarde le tétracorde des Grecs’, Mémoires de Trévoux LII, pp. 100–141, 284–310. 162 Fétis, Biographie universelle, vol. 8, p. 410. 163 Ibid., p. 407. 164 Ibid., p. 409; see also vol. 7, p. 136. 165 Ibid., vol. 7, p. 135. 166 Fol. 4v (p. 51 below).

Chapter 1

Letters to Henry Oldenburg, May 1664 Editorial Note The three letters presented here were addressed to Henry Oldenburg, the secretary of the Royal Society, during May 1664. They were prompted by a lost letter from Oldenburg of 4 May, reporting on the letter and personal appearance of John Birchensha received by the Royal Society during April, and by a second lost letter from Oldenburg informing Wallis that the first two letters presented here had been read to the society on 18 May.1 The content of Wallis’s letters, and in particular his evasiveness about his dependence on earlier writings, are discussed in the Introduction (see pp. 2–8 above). The second letter, one of Wallis’s most substantial discussions of the theory of music, should be read in part as his response to the writings of his modern predecessors, acquaintance with which he would acknowledge in a postscript and in the third letter. At the same time, these three letters set out an agenda for the mathematical study of music that was distinctly Wallis’s own. As Wallis’s earliest surviving discussion of music theory and an important source for ‘The Harmonics of the Ancients compared with Today’s’ (Chapter 3 below), they therefore possess considerable interest. For each letter the version that was sent is now preserved in the archive of the Royal Society: MS Early Letters W1, nos 7–9. Each was copied into the society’s ‘Letter Books’ and subsequently into the ‘Letter Book Copy’; these copies have no independent authority and are disregarded here.2 For the second letter there are two additional copies. We denote the three copies A, B and C.3 A: London, Royal Society, MS Early Letters W1, no. 8. The letter as sent, in Wallis’s hand and dated 14 May 1664. B: London, Royal Society, Boyle Papers, vol. 41,4 fols 23r–30r. In Wallis’s hand. This copy is juxtaposed in Robert Boyle’s papers with the unique manuscript of John Birchensha’s ‘Compendious Discourse … of Musick’.5 The simplest explanation is that both items were produced for Boyle’s use and at his request, probably shortly after Birchensha’s appearance before the Royal Society on 27 April 1664. Unfortunately there is no evidence, either in Boyle’s own correspondence or published works or elsewhere, to confirm this, and it is possible that one or both of these items became associated with Boyle’s papers in other circumstances or even after his death.

42

John Wallis: Writings on Music

C: Dublin, Marsh’s Library: Z3.4.24. Not in Wallis’s hand.6 As with B, it seems most probable that this copy was solicited and produced soon after 27 April 1664. The most natural assumption is that it was made for Narcissus Marsh, who was at this date associated with the same Oxford college as Wallis and was certainly interested in musical topics. All three versions have the appearance of fair copies. But, setting aside matters of spelling and punctuation (and the placement of paragraph breaks and the choice of parentheses or parenthetical commas), there are about 240 places where the three texts differ. Most of these are very minor matters of re-drafting – the substitution of octave or eight for octave, the choice of 5 to 4 rather than 5/4, the transposition of so do to do so – while a few are accidental omissions of words from one of the copies. Nearly all of these minor differences show agreement between two of the texts against the third. No pair of texts agrees significantly more frequently than any other. Moreover, we have not located any clearly incorrect reading which occurs in more than one of the texts. These facts are difficult to reconcile with any suggestion that one of the texts is a copy of another; instead they suggest quite strongly that all three are in fact copies of a single lost original.7 There are also about 20 places where the three texts show larger differences, with passages of up to a whole paragraph absent or much shorter in one or two of the versions. C – which is not in Wallis’s hand – is nearly always the longest and contains at least nine significant passages which are absent from both A and B, six which are absent from A only (in one case the sense in A is impaired as a result) and one which is absent from B only. While some omissions are thus common to A and B, where those two texts have compressed versions of a passage they are invariably different. Other explanations cannot be ruled out, and there are some strange features of this set of texts. But the most natural explanation seems that the lost original text resembled C and was shortened by Wallis for transmission to Oldenburg (A) and Boyle (B). Those two copies were made independently, but Wallis retained from one to the other a record – or a memory – of most of the major excisions he had made. A copy (C) was also made for Narcissus Marsh by another scribe, from Wallis’s original, longer text. To copies B and C were added the date and address from A, giving them the somewhat misleading appearance of straightforward copies of that letter to Oldenburg. To both was also added a postscript acknowledging, as Wallis acknowledged to Oldenburg on 25 May that his ideas were similar to those of other modern writers: it is shorter in B than in C, probably due to a lack of space on the page. The lost original could have been drafted at any time before copy A was made on 14 May 1664; indeed, the length and coherence of the text suggest that it was not composed from scratch in the week which separated it from Wallis’s previous letter to Oldenburg. But the references to Birchensha and to

Letters to Henry Oldenburg, May 1664

43

Oldenburg’s letter must have been introduced after Wallis received Oldenburg’s account of Birchensha’s ideas dated 4 May. Copy A can be securely dated to 14 May. If copy B was solicited by Boyle this was presumably soon after he had heard from Oldenburg of the reading of Wallis’s letter to the Royal Society on 18 May. Copy C bears the date 14 May and must have been made after (or on) that date, but no secure terminus ad quem can be given for it; it may have been made during the same period in the second half of May 1664. For the first and third letters we use Wallis’s single autograph as copy-text. For the second we give here an edition of C, the longest version of the text and on our hypothesis the closest to the original. Differences from A and B which exceed two words in length or substantially affect the sense are reported in the footnotes. The original is unfoliated; here we impose a foliation so that fol. 1r contains the start of the letter’s text. Throughout our edition we have silently expanded common contractions: yor, wch, ye, yt, sr and disregarded both scribal hesitations where a word is deleted and then restored, and illegible deletions. Insertions are signalled using angle brackets. These letters have been published in both the correspondence of Oldenburg and that of Wallis.8

44

John Wallis: Writings on Music

Figure 1.1 The opening of John Wallis’s letter to Henry Oldenburg of 14 May 1664, in the copy (C) made for Narcissus Marsh. Dublin, Marsh’s Library, Z3.4.24, fol. 2r. By kind permission of the Governors and Guardians of Marsh’s Library in whom copyright remains vested.

Letters to Henry Oldenburg, May 1664

45

Text [London, Royal Society, MS Early Letters W1, no. 7]a [fol. 1r] Oxon May. 7. 1664. Sir

I thank you for the large account you gave mee in your last, of what the Society hath been of late intertained with, as to the business of Musick. I do not profess myself to be a Master in it; & therefore what I have been formerly unsatisfyed in, I have been willing to impute to my imperfect knowledge; not having applyed myself so particularly to that, as to some other parts of Mathematicks. But that there is a great defect in the Theory thereof (at lest as to thoseb Ancient or Modern Musicians that I have read) I think is without question. Arising mostly from hence; that many of those who have applyed themselves to the Theory of it, have either been but Transcribers, or else but imperfect Geometers. (Out of which I must yet except Euclides Sectio Canonis; which is perfectly Geometricall. And what Ptolemy’s Harmonicks are, I know not; as not having seen them)9 The Distances of Diapason, Diapente, Diatessaron, & Tonus; or, as they are now called Eights, Fifts, Fourths, & a Note; are well stated by Pythagoras his notion of them;10 as answering the proportions, 2/1, 3/2, 4/3, 9/8. And consonant hereunto, the compounds of them; as Diapason & Diapente, Diapason & diatessaron, disdiapason &c; that is 3/1, 8/3, 4/1, &c. But their Hemitonici, & Dieses, &c, are very obscurely at best, if not imperfectly deliver’d. wanting much of Illustration, & (I doubt) somewhat of Correction. That there bee many Distances, or Diastemata, of which no notice is taken; is very evident: (&, that some of them are not lesse musical, than divers of those allready observed:) which it will not be hard for one that is both a good Musician (in the Theory) & a good Geometer (as to the nature of Proportions)11 to supply. And the person you mention (though I know him not) hath, I suppose, well considered it. I am apt to think allso, that the sounds wee sing, are not the same wee prick (according to the scale as it is now supposed to be divided) in divers cases; but somewhat different from them: the Ear, in these niceties, guiding the Voice, better than the Scale (if a little erroneous) can do.12 But I leave these considerations to those who have more particularly considered the matter, than I have yet done. And I suppose it very reasonable that the Gentleman be incouraged in his design; especially if such a person as your letter represents him. His censure of the Condition wherein c‹Musick› stands at present; I concur with: His Propositions are good.d, 13  Later endorsements: Dr Wallis’s Letter to Mr Oldenburg concerning musick, and the Translation of Uleg-beg. In another hand: (read May 18.64. Enterd LB.1.149) b  Illegible word. c  Deleted: it d   Fol. 1r ends here; the remainder of the letter is concerned with other matters. a

John Wallis: Writings on Music

46

[fol. 2r] Sir Your very humble servant.   John Wallis. [fol. 2v] For my very worthy friend Mr Henry Oldenburgh, at the Lady Ranalagh’s house in the Palmal, in St James’s fields London.a [Dublin, Marsh’s Library: Z3.4.24] [fol. 1r]b

Oxon May. 14. 1664.

For Mr. Henry Oldenburg, Secretary to the Royall Society, Londonc Sir, Though (as I said in my last)d I doe not look upon my self as so much a master in Musick, but that much of the obscurity or dis-satisfaction which I haue mett with, may be well imputed to my own imperfect Skill, & the few thoughts I haue imployd in that Theory; Nor is my opinion so Authentick, as that upon my own Authority I shall presume to invocatee any thingf which hath been, for so long a time, so generally receiu’d: Yet since you seem to invite me to it,14 I shall venture to giue you a brief account of my present un polished thoughts: which whether they will be worth pursuing, or not; I shall leaue to better Judges To this purpose; I shall select an Octaue out of the scale or Gamm-ut: suppose, from E la mi, to e la mi15

 Later endorsement: Dr Wallis’s letter to Mr Oldenburg concerning Musick and the translation of Uleg-beg / May 7th. 1664 b  Editorial foliation; original is unfoliated. c  AB place this address elsewhere. d   B lacks parenthesis. e  AB: innovate f   B adds: in that a

Letters to Henry Oldenburg, May 1664

47

Which is wont to be sung by these notes. La Fa Sol La Mi Fa Sol La. Whereof all but Fa, are supposed [fol. 1v] to rise a full Tone, or (as they now call it) a whole Note. & Fa, an Hemitone, or Half-Note. So that between the eight Sounds, are seven Intervalls, containing Six Tones; or rather, Fiue Tones, & two Hemitones But forasmuch as it is confessed, (& is by Euclide demonstrated in his Harmonicks)a, 16 that six whole Tones, are more then an Octaue or Dia-pason: Because that 9/8 (the proportion of a Tone) being six times compounded, is more then 2/1 (the proportion of a Dia-pason) /8 × 9/8 × 9/8 × 9/8 × 9/8 × 9/8 = 531441/262144 >

9

/262144 = 2/1

524288

Therefore, allowing to all the Tones their full measure; they doe profess, by the Hemitone, to mean, not precisely Half a Tone; but somewhat less then half a Tone: (bviz: so much as the Diatessaron is more then two Tones; or Dia-pente then three Tones: which is 256/243:) Yet giue it the name of Hemitonicum,c because it is very near it. Three Tones Dia-pente Hemitone 9 /8 × 9/8 × 9/8 = 729/512:) 3/2 (1536/1458 = 256/243 Two Tones Dia-tessaron Hemitone /8 × 9/8 = 81/64) 4/3 (256/243 That is 3.000000/2.847654x proxime

9

 AB lack in his Harmonicks  C lacks ( c  AB: Hemitonium a

b

John Wallis: Writings on Music

48

Which is less then half a Tone, or √9/8 a = 3/√8. That is, 3.000000/2.828427x proxime[.] Anb this I thought necessary expressely to mention; that I might not seem to wrong the Ancients. For, though possibly some of our modern Composers may not heed it: the Ancients did clearly discern, that a Dia-tessaron was somewhat less then two Tones & an half; & a Dia-pente, then three and an half. [fol. 2r] This being the constitution of the Ancients, in their Genus Diatonicum; (according to which Euclide proceeds in sectione Canonis; & which the moderns haue followed:)c without saying any thing of their Genus Chromaticum, or Enarmonium; I shall shew, wherein I think, it may be amended to good advantage; or rather, some mistakes rectifyed; (for indeed my opinion is, that the voice doth make truer Musick, then what the notes, according to this Hypothesis, doe direct.)17 And here (without laying a new foundation) I shall proceed upon the Hypothesis of Pythagoras; that the Diastemata, or Intervalls of Sounds are conveniently expressed by Rations or Proportions:18 to wit, in what Proportion one sound is more Acute, or more Graue, then another compared with it. As if the Acuteness of the one, be sayd to be double, Treble, Quadruple, Sesquialter, Sesquitertian, &c. to that of the other. By what Methods this may be determined most conveniently; I will not take upon me to define: as whether a Hammer, half so heavy; will giue a sound, twice so Graue:d or, a string stretched with a double weight, will giue a sound doubly Acute: Or a string, half so long; will sound, twice as high: And the like in other proportions. I shall not I say here define; whether all or any of these ways will determine it. Because I conceiue, it may be proper enough for thee society to examine the truth of these Hypotheses, by good severe Experiments, rather then to take them, meerly upon the Authority [fol. 2v] of those that relate them.19 Nor shall I strickly inquire into the Physicall cause of Harmony, or sweetness between some sounds; & of discord, in others. As, whether two strings that are Unisons, be therefore Harmonious, because (supposing them to haue one common beginning) the Vibrations of the one doe exactly answer to those of the other:20 Unisons

And next unto these, Octaves;   B: √ 3/8  AB: and c   moderns haue followed A: moderns do allso follow B: Modern[deleted: s] ‹Composers› do generally follow d  AB: Acute e   B adds Royal a

b

Letters to Henry Oldenburg, May 1664

49

Octaves

because that, the Vibrations of the more Acute being twice as many in the same time, every vibration of the more Graue or slower string, is coincident with every second Vibration of the Quicker or more Acute: & in fifths; Fifths

because the Quickness of Vibrations being in the one sesquialter to that in the other, every second Vibration of the one is coincident with every third of the other: which may be therefore less harmonious than octaues (because of a less frequent coincidence;) yet more then fo‹u›rths; where, the celerity of Vibrations being in Sesquitertian proportion, every fourth of the one will be coincident with every third of the other: Yet will this be more Harmonious than a Tone; Tone

whose proportion being sesqui-octave, the coincidence will not be more frequent then for every ninth of the one to Answer to every Eighta of the other: Whether this, I say, be the true Physicall cause of Harmony & Discords in Sounds (though it be a very specious & promising account of it:) I will not positively determine: because I would not [fol. 3r] Even here exclude a strict examination by Experiment.21 But that this, or somewhatb hereunto, is the true cause & Measure of Harmony & Discord in Sounds, wee must grant; or else, that we are yet to seek for a foundation to build upon. I shall therefore lay this for a foundation to proceed upon (at least till some better be discovered) that upon the proportion of celerity in the Vibrations, or at lest (which since Pythagoras, hath been generally receiud) upon the proportion of Acuteness & Gravity (whence soever this proportion doth arise) dependeth the Harmony & discord of Sounds. Next; that those proportions, according to which such coincidence will be more frequent (& the fewestc vibrations not coincident) are the more Harmonious; & the contrary, either more discordant, or at least less Harmonious. Upon these Principles I thus proceed. a

  B: Eighth   B adds equivalent c  B: Fewer b

John Wallis: Writings on Music

50

Unisones 1/1 1. Thata it is manifest, that, of all sounds compard, the Unisones are most Harmonious. For, the proportion of the Celerity of Vibrations (or of what els shall be found to be substituted in the room thereof) being as 1 to 1,b that is equal: the Vibrations of the one & of the other will be all respectiuely coincident[.] My meaningc is (which I interpose by way of Caution) that it will so be, upon supposition that in both strings they haue one common beginning; or which is Equivalent, in case any two vibrations are coincident, all the rest [fol. 3v] respectiuely will be so too. And with this caution I would be understood all along; which I once mention for all.d Octaue, or Dia-pason. 2/1. 2 It is all-so manifest, that next unto Unisons, those are most Harmonious, where the proportion is double; or as 2 to 1. For then euery Vibration of the slower will be coincident with every second Vibration of the swifter. That is, of every three Vibrations (Two in the one string, & one in the other) there will be Two coincident. And to this it is that the Ancients do accommodate their Octaue, or dia-pason. So tuning the two Extreams of the Octachord, that the Acuteness of the one should be double to that of the other. And those sounds that so differ, are calld Octaues or eights, not as if the number 8 had any thing peculiar in the proportion: But because this proportion took in the whole compass of the Octachord; which they esteemed to contain a perfect Systeme;22 & the sounds or Places considered therein, as it was then devided, being in number 8. To which eight sounds, were fitted, in the Harp, eight strings.e & conformable hereunto, are the names of Dia-pente & Dia-tessaron (Fifths & Fourths) as extending to so many of those strings.f which would haue been otherwise nam’d, had the sounds or Places considerd within this compass, been more or fewer. And this of Octaues, is of all Intervalls the most Harmonious. Such is the Intervall La La; from E la mi, to e la mi [fol. 4r] A fift, or Dia-pente. 3/2 3. The next Harmonious Intervall, (within the compass of an Octaue; for of such I am now speaking;)23 is the sesquialter Proportion, or as 3 to 2. Wherein every second vibration of one string is coincident with every third of the other.  AB: First  Altered from: one to one c  A: reasoning d  A lacks which I once mention for all e  A lacks this sentence f  B: Strings or Sounds a

b

Letters to Henry Oldenburg, May 1664

51

So that of every fiue vibrations (three in one string & two in the other) two are coincident.24 This they call a Dia-pente, or Fifth: because it extended to fiue of those eight sounds or strings in their perfect Systeme. Such is the intervall La Mi, (from E to b:) or La La (from a to e.) And is supposed to contain three Tones & an Hemitone. A Fourth or dia-tessaron. 4/3. 4. The next to this (I mean, within the compass of an Octaue; & so of the rest;) is the sesquitertian proportion; or as 4 to 3. Wherein every fourth Vibration of the one is coincident with every Third of the other: So that of every Seven Vibrations, (that is four in the one string & Three in the Other) there be two coincident. This they call a Dia-tessaron, or Fourth; as extending to Four of those eight Musicall Sounds or Strings, of which they took notice in the Octachord. Such is the Intervall La La. (from E to a.) or Mi La (from b to c.)a & is supposed to contain Two Tones & an Hemitone. 5. It is also manifest, that these two Intervalls, of dia-pente & dia-tessaron, as they are both Harmonious; so being put together, they make up a Dia-pason. for if the two proportions, of 3 to 2, & of 4 to 3, be compounded; the result is that of 2 to 1b [fol. 4v] or if the proportion of 2 to 1,c be abated by that of 3 to 2; the result is, that of 4 to 3. or if abated by that of 4 to 3. the result is that of 3 to 2. As is manifest in the definition & Doctrine of Compounding proportions. / 2 × 4/ 3 = 2/ 1. d 3 / 2 ) 2/ 1 ( 4/ 3. 4 /3 ) 2/1 (3/2.e, 25 3

So that the Dia-pente & Dia-tessaron, are each the others Complement to a Dia-pason: & that of a Dia-pason equal to both these put together. If, from E we rise a Dia-pente to b, (La Mi,) the remainder from b to e, (Mi La,) is a dia-tessaron. If from E we rise a Dia-tessaron to a, (La La,) the remainder from thence to e, (La La,) is a Dia-pente. If from e to b, we rise a Dia-pente (La Mi,) & again from b to e, a Dia-tessaron, (Mi La) the whole from E to e, (La La,) will be a Dia-pason.

 AB: b to e  Altered from: one c  Altered from: one d  AB = 4/2 = 2/1 e  B: (6/4 = 3/2 a

b

John Wallis: Writings on Music

52

Thus in numbers; if we express the more graue sound of an Octaue,a by 6; & consequently, the higher by 12 (in double proportion to it,) the number 9, which is to 6, as 3 to 2, or a dia-pente; will be to 12, as 3 to 4, which are therefore a dia-tessaron. & the Number 8, which is to 6 as 4 to 3. (a dia-tessaron) will be to 12, as 2 to 3; which are a diapente. which is the intendment of that so frequent proposition in Musick, that a dia-pente & a dia-tessaron make a dia-pason. A Tone 9/8. 6. The difference or distance between a dia­-pente & a dia-tessaron; that is between the [fol. 5r] proportions of 3 to 2, & of 4 to 3; is that of 9 to 8; or the sesqui-Octave. So that of every 17 Vibrations (to vilb 9 on the one string, & 8 on the other) Two are coincident.c For if that of 3 to 2, be abated by that of 4 to 3; the result is that of 9 to 8. /3 ) 3/2 ( 9/8d

4

And this distance they call a Tone. As La Mi, from a to b. For that from E to b, being a Dia-pente; & that from E to a, a Dia-tessaron, that from a to b, is the Intervall or difference between them. Thus far, I think, & no farther, Pythagoras did proceed: viz, to determine the Intervalls of Dia-pason, dia-pente,e Diatessaron, & Tonus; by the proportions of 2/1, 3/2, 4/3 & 9/8. And, consequently to determine Four of those eight sounds, whereof their Octachord consists.f  AB adds: or Diapason  Read: to wit c  AB lack this sentence d  A lacks this calculation e  Deleted: & f  AB add: in this manner a

b

Letters to Henry Oldenburg, May 1664

53

The four sounds, La La Mi La, being thus determined by their proportions to one another; the other four Fa Sol Fa Sol, are neither by Pythagoras in like manner determined, for ought appears to us.26 Nor haue yeta any of his fellows extant,b as low as Boetias’s time, [fol. 5v] & a good while after, persuedc his steps in designing of them: But haue very inconveniently devided the two Tetrachords, La Fa Sol La, & Mi Fa Sol La; making La Fa, & Mi Fa, Hemitones; & Fa Sol, Sol La,d perfect Tones equall to that of La Mi, from a to b.27 I shall therefore first shew the inconvenience of such a Division; from hence; That whereas it is manifest that between the proportions of 4/3, & 9/8, there be many more Harmonious then 9/8 (which is that of a Tone) though less Harmonious than 4 /3 (that of a dia-tessaron) such as 5/4, 6/5, 7/6, 8/7, &c:28 for these they haue made no proportion,e nor taken any notice of them: Which yet are undeniably Musicall, & (as we shall after shew) of frequent use, at least some of them; whereas those they haue put in stead of them, are much more dissonant then that of a Tone. Anf then I shall after shew, how they may with much more advantage be divided. Half a Tone √9/8 = 3/√8. 7. I Say therefore, that if La Fa, & Mi La be Hemitones; & if by an Hemitone, they mean, precisely half a Tone: then is this proportion absolutely Unmusicall. For the Tone being 9/8, the Hemitone must be √9/8, or 3/√8; (For this twice compounded, makes √9/8 × √9/8 = 9/8.) Now the number 3 being incommensurable to √8, (the sureg root of 8:) it is not possible, that Vibrations in such proportions, if they haue one   B lacks: yet   fellows extant AB: Followers c   as low ... persued A: to this day, (whom I haue yet met with,) followed d  A adds: (in both places,) e  AB: provision f  AB: And g  AB: surd a

b

John Wallis: Writings on Music

54

common beginning, can ever after be coincident. And the like is to be understood of all proportions between terms incommensurable.29 The Ancients Hemitone 256/243 Or if, by Hemitone, they mean that proportion, of 256/243 before described: the terms are then indeed commensurable; but there must pass (after their common beginning) 499 Vibrations, (that is 256 [fol. 6r] uppon one string, & 243 on the other) before any two will be again coincident; which is very far from Harmonious.30 Perfect Thirds, or Two Tones. 9/8 × 9/8 = 81/64. 8. The Interval Fa La, from F to a, or c to e,) if it be a perfect ditone, as they Suppose it to be; that is two intire of 9/8;a the proportion will be 81/64: & there must (after the common beginning) pass 145 Vibrations before any two be ‹again› coincident: which therefore must be a much harsher Intervall then that of a Tone. Whereas yet, in experience, thirds are found to be a good concord. Imperfect Thirds or Tone & Hemitone. 9/8 × 256/243 = 32/17. 9. The same is to be said of the Intervalls La Sol & Mi sol, (from E to G, & b to d:) which are imperfect thirds; & are supposed to Consist of a Tone & Hemitone: & consequently, (supposing the Hemitone as before) that proportion will be 32/27 (being compounded of 9/8 & 256/243:) & therefore 59 vibrations before there be a new coincidence: Yet experience tells us, That Imperfect Thirds, make no ill musick. Adde also, that by this reconing, Imperfect Thirds should be better musick then perfect Thirds, (because 59 are fewer then 145) which the ear doth contradict.b, 31 Thisc division therefore of the Tetrachord being so incongruous, &d answering to experience, (for certainly Thirds as they are wont to be Sung, make better Musick then this Hypothesis will afford:) I shall next shew, how I conceiuee the Tetrachord may be divided with more advantage, & more complying with experience; prosecuting therein the same Steps as in the Division of the Octachord into Diapente & Dia-tessaron. Diatrion32 major or True Thirds. 5/4 = 80/64 10. I say therefore, that (in persuance of the former principles) next to 4/3 or Sesquitertian (the proportion of a [fol. 6v] Dia-tessaron) succeeds 5/4, the Sesquiquartan. Where every fifth Vibration of that one string, will be coincident with every fourth of the other: So that in nine Vibrations (5 in the one String, & 4 in the other)f two will be coincident. Which though it be less Harmonious then that of a Dia-tessaron, or 4/3: yet it is more Musicall then a Tone or 9/8.

 AB: intire Tones  AB lack this sentence c  B: The d  AB add: not e  A lacks I conceiue f  A lacks the parenthesis a

b

Letters to Henry Oldenburg, May 1664

55

And this I would make the Intervall Fa La, (from F to a, or c to e,) which is not perfectly a Ditone or two entire Tones of 9/8,a (that is 81/64 = 9/8 × 9/8:) but is very near it, (5/4 being equall to 80/64,)b & is much more Harmonious: For here the coincidence returns at 9 Vibrations on the two Strings; & there, not till 145.c Imperfect Hemitone 16/15. 11. And consequently the Hemitone Mi Fa, or La Fa from E to F, or b to c (which is confessed not to be precisely Half a Tone) will be, not 256/243 (as it hath beend thought to be;) but (which is neare to a Harmony) 16/15: So that in 31 Vibrations, (16 on one String, & 15 on the other) two will be coincident.f For if 4 /3, the proportion of a Dia-tessaron, be abated by 5/4, the proportion of a Diatrion; the result is 16/15 for the residue of that Intervall, Mi Fa, or La Fa.g, 33 5

/4 ) 4/3 ( 16/15.h

Which we are content, at present, to call an Hemitone, as being near it. Dia-trion minor imperfect Thirds 6/5 12. The Intervali La Fa (from a to c) consisting of La Mi (a perfect Tone) & Mi Fa (that imperfect Hemitone) is 6/5, which results from the composition of 9 /8 & 16/15. 9

/8 × 16/15 = 144/120 = 6/5j

So that in 11 Vibrations (6 on the one String & 5 on the other) two will be coincident.k Which is, though less Harmonious than 5/4, yet more than 9/8. [fol. 7r] And the same will be the proportion of La Sol or Mi Sol, (from E to G, or b to d,) in case we make Fa Sol a perfect Tone of 9/8.l And Sol Mi (from G to b) will then be 5/4; equall to that of Fa La, (from F to a, or c to e) aboue mentioned.m, 34

 AB lack or two ... 9/8  AB lack the parenthesis c  AB lack For here ... till 145 d   hath been A: is e  AB: nearer f  AB lack So that ... be coincident g  AB lack Mi Fa, or La Fa h  A lacks this calculation i  Deleted: Fa j  A lacks this calculation k  AB lack this sentence l  AB lack of 9/8 m  A lacks will then ... aboue mentioned a

b

John Wallis: Writings on Music

56

13. Then for dividing Fa La (from F to a, or c to e) if we make Fa Sol a perfect Tone of 9/8 a (which is not inconvenient;) then is 10/9 the proportion:b (So that of Nineteen Vibrations, 10 in one string & 9 in the other, Two will be coincident)c for if we abate 5/4 by 9/8, the result is 10/9.35 /8 ) 5/4 ( 40/36 = 10/9.d

9

So that the simple intervalls, or Diastemata, with the principall of the compounds, or Systemata, within the compass of one Octaue, will stand in this order.

14. There be yet two other Superparticular Proportions, 7/6 & 8/7, (the SesquiSexton & the Sesqui-Septiman; or as we use to express them in English, Once with a Sixt part, & once with a seventh part;) fore which, being more Harmonious then a Tone or 9/8, it may perhapsf be [fol. 7v]g expected we should allow a place.h, 36 Which may be done, if need be,i by interposing a sound between Fa and Sol (which may be called Fa sharp) which may so divide the Upper diatessaron, Mi La, (from b to e,) or the lower, La La, (from E to a,) in such manner, as that, with one of its terms, it may make the proportion of 7/6; with the other, 8 /7: which being compounded make 8/6, or 4/3.  AB lack of 9/8  AB add of Sol La c  AB lack this parenthesis d  A lacks this calculation e  AB lack (the Sesqui-Sexton ... part;) for f  A lacks perhaps g  Deleted: expressed h  AB: expected to have a place allotted [B adds: for] them i  AB lack if need be a

b

Letters to Henry Oldenburg, May 1664

57

/7 × 7/6 = 8/6 = 4/3.

8

And it may be done in either of these forms, according as you please to make the greater of the two, in the Lower or higher place.

Both together, stand thus,a

But because there will be a necessity to rest somewhere, (since proportions may be thus broken in infinitum;) & because experience teacheth us that the voice cannot well express less Intervalls, in a continued sequence, then those of the eight Sounds in a Dia-pason, (However, by leaps, it may fall upon intermediate Sounds, denoted usually by Flats & sharps; amongst which possibly, these may be found, though not attended:) Therefore, I think, we may well rest in the number of Places for[fol. 8r]merly receiu’d, without introducing these new proportions. a

  Bracketed letters and dotted braces are editorial in the following diagram

John Wallis: Writings on Music

58

And that the rather, because these cannot be designd without taking in another Prime number, (viz: 7.) besides those of 2. 3.a & 5, which serue to Design all those Antecedent proportions. for though 7/6 & 8/7, be expressed in smaller numbers, than, 9/8, 10/9, or 16/15, yet these being not Prime, but composite numbers, may be resolued into such as are designable, by 2. 3. 5, without taking in any other prime number. For 8 is but 2 × 2 × 2, & 9 = 3 × 3, & 10 = 2 × 5, & 15 = 3 × 5, & 16 = 2 × 2 × 2 × 2. Whereas 7 being a Prime number, cannot be so resolued. I am content therefore at present, to leaue out the proportions of 7/6, & 8/7; as less Harmonious.b 15. There be also some super-partientc, 37 proportionsd as 5/2 5/3, &c: which I take to be very Harmonious, & doe not want their use in Musick; at least some of them.e (though I know that Euclide, in the beginning of his Sectiof Canonis, excludes all as dissonous, which are not either Multiple, or super-particular, but upon a reason which, to me, doth not seem conclusiue)38 But they are such as either doe not occurre within the compass of a single Octaue; As 5/2,g 8/3, 9/2, 9/4, 2/5,h &c. which are greater than 2/1. Or could not be expressed without inserting such a sound asi we mentioned between Fa & Sol, As 7/4, from E to such a sound between c & d, as shall be, to j b, in the proportion of 7/6, (for 3/2 × 7/6 = 7/4:) & 7/2 & 7/3 (which are also more than an Octaue) from EE, & from B, to the same sound:k And 9/7, from b to such a sound interposed between F & G,l whose intervall to a shall be 8/7, (for 8/7 × 9/8 = 9/7:) & 12 /7, ‹from› the same interposed sound to e, (for 3/2 × 8/7 = 12/7:) & 7/5, from a Sound between F & G (whose intervall from ‹a› should be 7/6) to c (for 7/6 × 6/5 = 7/5:) & such others.m Or, at lest could not be well expressed,n withouto [fol. 8v] makingp too much confusion in the scheme; as 5/3, from G to e, & 8/5, from E to c; & 9/5, from E to d, &c: On which therefore I shall here insist no further.  Altered from: 5  AB lack And it may be ... less Harmonious c  A: superparticat d   B adds: ‹and Multiple-superparticular› e  AB lack at least some of them f  B: Treatise De Sectione g   B adds 7/2, 7/3, h  B: 12/5 i   such a sound as AB: the sound j  AB add that mi k  A lacks & 7/2 ... same sound: B has ‹Or 7/2, (from EE to the same sound:)› l  Altered from: g m  A lacks & 7/5 ... such others. B lacks & such others n   B lacks well o   Or ... making: A: Or, not without p  AB lack making a

b

Letters to Henry Oldenburg, May 1664

59

16. Having thus determined the singlea Intervalls, within the compass of a Diapason; it will be easy to determine (by compounding two or more proportions) all compounded Intervalls, whether within a single, or a double, or Treble dia-pason; or further yet if there be occasion. As for example; in a dis-diapason, from E to ee: The severall proportions will be these

17 To these we are to add further, such others as shall arise, upon altering the Key of b Flats, & sharps; Whereby divers of the Intervalls which here seem Harsh are mollified & sweetned. as for Instance (A Flat in b, d. bb, dd, which instead of a full Tone of 9/8, (as now) shall be supposed to make them rise but a Hemitone of 16 /15, will make their intervalls from F, which here are 45/32, 27/16 45/16, 27/8, to become, 4 /3, 8/5, 8/3, 16/5, & b, f, which now is 64/45 will be 3/2. A sharp in b, d, bb, dd, which shall [fol. 9r] be supposed to raise them ‹an› Hemitone of 16/15, will make their Intervalls from F,c which now are 45/32, 27/16, 45/16, 27/8, to become 3/2, 9/5, 3/1, 18/5, And b, f which now is 64/45 will become 4/3.39 And the like of many Others.d With  B: simple  AB: by c  Deleted: to d  AB lack whereby divers ... many Others a

b

John Wallis: Writings on Music

60

which I shall not trouble you at present,a having thought it sufficient to giue youb this Specimen. 18.c Nor will it be necessary (because tis obvious) to intimate that what is here fitted to the Octaue from E to e, upon supposition that b is not Flat:d doth (upon altering the Key) agree to that from a to aa, in case a flat be in b: or to that from D, to d, in case a flat be also in e. la mi. And the like otherwhere according as the place of mi varies.e

19.f I might add also, that in dividing the Dia-trion, Fa Sol La; whereas I make Fa Sol, 9/8; & Sol La, 10/9: itg had been free to doe the contrary: But, of the two that seemd to me the more Eligible. Indeed it had been proper enough to haue made three Intervalls, (for so many must be understood)h to wit, twice 10/9, & a middle one between them, which would be 81/80, which midle intervall might indifferently be added, sometime to the one, sometime to the other, as occasion should serue:40 In like manner as an Octaue; or dia-pason is divided, into two Dia-tessarons, & a Tone between them; which Tone being added to this or that of the Dia-tessarons, makes it a Dia-pente. But, as there, when we rise on eight at Twice, it is usuall first to rise a Fifth, & thence a Fourth (putting the bigger Intervall in the lower place) so I choose here to make Fa Sol 9/8; & Sol La 10/9. [fol. 9v]

 A: at present trouble you B: at present trouble the Scheme  AB lack you c  AB lacks 18 d  B: that there is no b Flat in b e  AB lack: And the like ... varies f  AB lacks 19 g  C repeats it h  AB lacks this parenthesis a

b

Letters to Henry Oldenburg, May 1664

61

So, from b to e, the Intervalls may be thus disposed, (with or without that sound between Fa, & Sol, for the proportions of 7/6 & 8/7;)a, 41

Not that the voice can in a continued sequence well distinguish all these sounds; (as neither can it, those commonly designd by flats & sharps:) but that it may (as in flats & sharps is usuall) by leaps fall on each of these, from some further Intervall.b 20.c But being willing to innovate as little as possible, I haue therefore onely abated the Tone Sol La by 81/80, as much increased in lew thereof, the Imperfect Hemitone, Mi Fa, or La Fa. But, how much this one Amendment hath made the Intervalls more Harmonious, will be very evident to any one, who shall but take a

  In the first diagram the position of the 6/5 bracket has been adjusted editorially.  AB lacks But, as there, ... further Intervall c  AB lacks 20 b

John Wallis: Writings on Music

62

the pains to make a like scheme of all the Intervalls in a Dis-dia-pason according to the ancientlya receiud Hypothesis. (as I haue done according to thisb) & compare it with mine. [fol. 10r] I doe not descend to the particularities of Smaller Intervalls; nor to dispute the Names of these: because I doubt I may haue sufficiently tired you already. And for the same reason, many other things must be omitted. If what hath been said, be thought reasonable; it will be easy to ad more: If not, there is too much already. How far this may agree to Mr. Birchinsha’s thoughts or Notions (of which your Letter speaks)c concerning the same subject; I cannot tell. For your letter intimating only the Generalls, both as to Defects in Musick, & as to the proposals he makes of supplying those defects: it doth not at all appear from thence, what his particular thoughts are, & by what methodes he intends a Restitutiond To Conclude; I shall the rather hope you will pardon me the present trouble I giue you in so prolix a letter; because you haue in part occasioned it, by that of yours, to Sir    Your very humble Servant,    John Wallis. Oxon. May 14. 1664.e Post-script. Since I had written this; I find that our modern Writers of the Theory of Musick (in this & part of the last Century) haue, upon like Principles, divided the Monochord much after the same proportion as I haue done: (Though our practicall Musicians generally take littlef notice of it) And Ptolemy, before them, in his Harmonicks, which when this letter was written I had not seen) Tis therefore a wonder, how it come to be lost in succeeding ages; or that he should be quite deserted till now of late: For Keppler is the first whome I find to haue recourse to that notion.g, 42

 A lacks anciently  AB add restitution c  A lacks this parenthesis d   B lacks How far this ... a Restitution e   B lacks these two lines and places here: For my very worthy / Friend, Henry Oldenburgh / Esquire; Secretary to the / Royall Society f   B: no g  A lacks Oxon. May 14. ... that notion. B lacks And Ptolemy ... that notion. a

b

Letters to Henry Oldenburg, May 1664

[London, Royal Society, MS Early Letters W1, no. 9] [fol. 1r]a Sir,

63

Oxon. May. 25. 1664.

Since my last to you (about 10 days since) I haue taken occasion to consult some modern writers of the Theory of Musick. Which had I done a little sooner, I should have omitted all or much of what I then wrote. Not that I see cause to retract what was then sayd, concerning the Proportion of Sounds, & the Distribution thereof. But, because I find some of them have been before mee. For though from Pythagoras downwards, till a good while after Boethius, (how long after, I am not certain;) no other Intervalls (for aught I find) were taken in to the Diatonick Systeme, but Tones of 9/8; & Hemitones of 256/243; (& such as were compounded of these:) And, that even in our days, the Scale, & Instruments fitted thereunto, (at lest, as used by ordinary Practicioners,) seeme to take notice of no other but Equal Tones, & Equall Hemitones:43 Yet I, since, find, (by what I meet with in Mersennus & some others,) that our Modern Theorists (at lest since Keppler) admit a double Tone, (a Greater & a Lesser) viz. one of 9/8, another of 10/9, (whose difference, 81/80, they call a Comma; & their Aggregate, 5/4, a Ditone;) &, consequently, an Hemitone, or Limma, of 16/15; much different from the Pythagorean, of 256/243.44 Which are the very same Intervalls with those of mine; & are in the same Methods disposed: and very possibly may have been first discovered, upon a like inquiry with that of mine. Which though it do confirm me in the notion: yet I should not have thought it necessary to propose, had I been aware, it had been so far allready prosecuted, by those whom I had either not before consulted, or not lately; &, but very superficially if at all: as having imployed but very little time, & very few thoughts, (as I did at first promise,) in this Theory. However, it isb (at most) but so much labor lost, in doing a second time, what had been done before: & perhaps you will not be displeased to see that drawn together in a clear & brief Synopsis, which might have cost as much time to recollect out of others: & the Methode so clear, as that it may, with the greatest ease, be pursued, (according to Pythagoras his first notion,)45 to as many subdivisions of proportion as you please. But, how far it will be fit to prosecute; I shall leave to those who have spent more thoughts upon it than I have done. And rest, Your very humble servant   John Wallis. [fol. 2v] These For my very worthy friend, Mr Henry Oldenburgh, at the Lady Ranalagh’s house in the old Palmal, in St James’s fields. London.   Latest endorsement: Another letter of Dr Wallis to M. Oldernburg, concerning [deleted: the same subject] Musick. In another hand: enter’d L.B.1.173. b  Deleted: but a

64

John Wallis: Writings on Music

Notes 1 Beeley and Scriba (eds), The Correspondence of John Wallis, vol. 2, pp. 114 (Oldenburg to Wallis, 4 May 1664 [lost]), 114–17 (Wallis to Oldenburg, 7 May 1664); 118–37 (Wallis to Oldenburg, 14 May 1664); 137–9 (Wallis to Oldenburg, 25 May 1664); see also Hall and Hall (eds), The Correspondence of Henry Oldenburg, vol. 2, pp. 179–81, 190–201, 202–3. 2 London, Royal Society, Letter Book Original 1, pp. 149–51, 152–64, 173–4; Letter Book Copy 1, pp. 173–6, 177–92, 200–202. 3 Beeley and Scriba denote these w1, w2, w3: The Correspondence of John Wallis, p. 118. 4 Beeley and Scriba incorrectly give the volume as no. 40: ibid. 5 For an edition of the ‘Compendious Discourse’ and discussion of its association with Boyle, see Field and Wardhaugh (eds), John Birchensha, pp. 93–178. 6 Though there are some similarities with Wallis’s hand, there are consistent substantial differences, most notably in the forms of r and e. 7 We therefore differ from Beeley and Scriba’s suggestion that B is a copy of A and that C is based on B: The Correspondence of John Wallis, p. 118. 8 See n. 1 above. For Wallis to Oldenburg, 14 May 1664, Hall and Hall print text A only; Beeley and Scriba print A with notes indicating the main variants in C, plus the whole of the conclusion of C. 9 Euclid’s Sectio canonis had been edited by Ioannes Pena (Paris, 1557) and Conrad Dasypodius (Strasbourg, 1571), but Wallis is perhaps more likely to have seen it in the Antiquae musicae auctores septem of Marcus Meibom (Amsterdam, 1652). Ptolemy’s Harmonics had not yet been printed in Greek, although Antonio Gogava had printed a Latin translation in Venice in 1562. It may well have been true that Wallis had not seen the text at this stage. 10 No musical writing attributed to Pythagoras survives. The ‘notion’ which Wallis here ascribes to him could in fact derive from Boethius: see De institutione musica 1.10. 11 The insistence that a theorist of music should be ‘A good Geometer’ amounted to a claim that Wallis himself was suited to such work. ‘The nature of Proportions’ might more naturally have been called the province of the arithmetician (see e.g. Aristotle, Posterior Analytics 7–9, 12). The view that mathematical music was a geometrical study recalls in particular the work of Kepler and Descartes: see Johannes Kepler, Harmonices mundi libri quinque (Linz, 1619); René Descartes, Musicæ compendium (Utrecht, 1650); cf. H. Floris Cohen, Quantifying Music, p. 18, fig. 4. 12 This sentence is crucial to an understanding of what Wallis intended to achieve in his musical writings; for him it was and remained axiomatic that the description of music provided by musical notation and by the writings of practitioners (Morley, Simpson, Playford, and others) was not a true description of the sounds that were heard in seventeenth-century performances. Thus Wallis’s mathematical theory would be an attempt to provide a true description: an accurate account of what the ear and the voice apparently achieved by innate instinct. See Introduction, p. 5 above. 13 Evidently Oldenburg had provided a description of Birchensha and his ideas which reflected well on the man (he was probably around 60 and would be described by others as a figure worthy of respect) and reported his claims that modern musical theory was defective, while leaving Wallis in the dark as to the details: cf. Wallis to Oldenburg, 14 May 1664, fol. 10r (p. 62 below). Ironically, what Birchensha believed

Letters to Henry Oldenburg, May 1664

65

was that the just intonation was an error and the Pythagorean the only true scale: see e.g. Field and Wardhaugh, John Birchensha, pp. 116–17, 132. 14 Oldenburg’s letters to Wallis being lost, we cannot know just what form this seeming invitation took, but he was on occasion keen to encourage the study of music. In 1665 he would reassure Robert Boyle that ‘the Inquiry about Sounds is worthy of Philosophers’: Hall and Hall, The Correspondence of Henry Oldenburg, vol. 2, p. 555. 15 Wallis’s diagram and music example set out the modern notation, the letter names, and the (English) solmisation names for a scale from E to e; also shown are the sequence of tones and semitones between consecutive pitches, and one way of dividing this octave into fourth, tone and fourth. This last was to have enduring importance for Wallis’s discussion of the relationship between modern and ancient theory: cf. fol. 8r (p. 58 below); Wallis, ‘The harmonics of the ancients’, pp. 160–62 (pp. 93–9 below). 16 Euclid, Sectio canonis, proposition 9; GMW, p. 199. 17 Cf. Wallis to Oldenburg, 7 May 1664, fol. 1r: ‘the sounds wee sing, are not the same wee prick’; that is, modern notation is a misleading guide to practice (p. 45 above). 18 Wallis uses ‘proportion’ as a synonym for ‘ration’, modern ‘ratio’. This was a matter of some variation in seventeenth-century English usage; the word ‘ratio(n)’ as a translation of Latin ratio and Greek logos in their sense of ‘a relationship of quantities’ was new to the language in the 1650s and the correct rendering of Greek analogia (in either English or Latin) a matter of some doubt. 19 The relationships to which Wallis alluded were those involved in the legendary discovery by Pythagoras of the numerical nature of harmony (Boethius, De institutione musica 1.10): the story of the ‘harmonious blacksmith’. Vincenzo Galilei was apparently the first to demonstrate their falsity in the case of the hammer and the weight. Wallis’s reading in Mersenne – acknowledged in Wallis to Oldenburg, 25 May 1664 (p. 63 below) – could in principle have informed him of that falsity, and it is surely significant that the different versions of this letter show a hesitation about the effect of the hammer’s weight on the pitch produced. Wallis surely had not performed the experiment for himself. In the event, the Royal Society did perform experiments during the summer of 1664 which determined the relationship between the length and tension of a string and its pitch: see Cohen, Quantifying Music, pp. 78–85; Wardhaugh, Music, Experiment and Mathematics in England, 1653–1705, pp. 99–106. 20 For this, the ‘coincidence’ theory of consonance, see Introduction, pp. 3–4 above. As discussed there, Wallis’s account of the theory is likely to have relied on Mersenne. Wallis’s insistence on ‘one common beginning’ or the importance that the series of ‘vibrations’ envisaged be in phase with one another – considerations to which he would return in §§7, 8 (p. 53–4 below) – was completely novel at this date; it would be considered by both Francis North and Isaac Newton in the next decade: North, A Philosophical essay of musick directed to a friend, pp. 8–11; H.W. Turnbull (ed.), The Correspondence of Isaac Newton (Cambridge, 1959–77), vol. 2, pp. 205–8; Kassler, The Beginnings of the Modern Philosophy of Music in England. 21 Wallis would continue throughout the remainder of this letter to use ratios which represented frequencies, higher numbers representing higher pitches. This put him at odds with his Greek sources, where ratios were almost invariably ratios of the lengths of sounding strings, and perhaps indicates a stronger commitment to the coincidence theory of consonance than he seems to acknowledge here. His suggestion that the matter receive ‘a strict examination by Experiment’ was not immediately taken up by

66

John Wallis: Writings on Music

the Royal Society, but Robert Hooke’s demonstrations with toothed wheels from 1676 onwards would add a good deal of evidence to the theory: see Wardhaugh, Music, Experiment and Mathematics, pp. 107–10. 22 See the much more extensive discussion of the history of Greek tuning systems in Wallis, ‘The Harmonics of the Ancients’, pp. 159–62 (pp. 91–9 below). 23 See ibid., pp. 177–8 (p. 134 below) for a justification of the restriction to intervals smaller than an octave. 24 Concerning the status of the fourth, see Introduction, pp. 4, 15, 19 above. 25 The line breaks between these three mathematical statements are editorial. In modern notation the second and third items would read: 2/1 ÷ 3/2 = 4/3 2/1 ÷ 4/3 = 3/2. The three lines thus set out the justifications in terms of ratios for the facts that a fifth plus a fourth equals an octave, an octave minus a fifth equals a fourth, and an octave minus a fourth equals a fifth. 26 What Wallis seems to have meant was that the remaining four pitches of the scale had not, as far as he knew, been placed according to mathematical principles similar to those governing the placement of the four discussed so far: ‘prosecuting therein the same Steps as in the Division of the Octachord into Diapente & dia-tessaron’, as he would put it in §6 (p. 54 below). 27 The Pythagorean scale described here and in what follows was set out in ancient sources including Ptolemy (Harmonics 73: the ‘ditonic diatonic’ and the ‘diatonic of Eratosthenes’; GMW, pp. 349–50). It was also favoured by John Birchensha – see n. 13 above – but it is not clear on what evidence Wallis believed it was common among modern theorists: see Introduction, p. 15. 28 Wallis, here, tacitly assumed that ratios of musical interest must be superparticular, that is with the first term exceeding the second by 1. See his remarks on the subject in §15 (p. 58 below). 29 Incommensurable means that the ratio between two numbers cannot be expressed as a ratio of two whole numbers. If such is the case for a ratio of frequencies, it follows that no whole number of one set of vibrations can ever be equal to a whole number of the others. 30 This section of the text considers harmoniousness in a somewhat different sense from the rest of Wallis’s discussion. 31 Wallis’s appeals to experience in §§8, 9 (p. 54 below) – and in particular the judgement that major thirds are ‘better musick’ than minor thirds – need not necessarily be taken as literally the result of his own observations of music. These sections may be dependent on a source in which the motivation for the syntonic diatonic scale was ascribed in part to the demands of musical experience. 32 ‘Diatrion’ seems to be Wallis’s own neologism, on the pattern of ‘diapente’ and ‘diatessaron’ (H.G. Liddell and R. Scott, Greek–English Lexicon [revd edn, Oxford, 1996] cite διάτρῖτος). 33 The diatonic semitone is still, as before, defined as the difference between a fourth and a major third, but with the new major third of 5/4 it takes a new value: 16/15 instead of 256/243. 34 The introduction of the ratio 6/5 here relies wholly on its being the next item in the sequence 2/1, 3/2, 4/3, 5/4 … of ratios previously discussed, as though its musical consequences were not yet clear. Yet the fluent discussion of the sizes of particular

Letters to Henry Oldenburg, May 1664

67

thirds within the scale makes it clear that Wallis already had a complete model of the scale in mind, in which this new ratio had an important part: cf. Wallis, ‘The Harmonics of the Ancients’, p. 179 (p. 136–8 below). 35 The decision whether the major third should consist of major tone plus minor tone or minor tone plus major tone would exercise Wallis in his later expositions. Nevertheless, the choice he made here, to place the major tone below the minor, would become his settled preference hereafter. See Wallis, ‘The Harmonics of the Ancients’, p. 179 (p. 137 below), id., ‘A Question in Musick’, p. 83 (p. 210 below); Introduction, p. 19 above. 36 See Introduction, p. 5 above; Wallis, ‘The Harmonics of the Ancients’, p. 178–81 (pp. 136–42 below). 37 See n. 28 above. A ‘superpartient’ ratio was one in which the first term exceeded the second by more than 1. The term was sometimes restricted to ratios smaller than 2 : 1, in which case 5/2 would not be superpartient but multiplex superparticular (the first term exceeds some multiple of the second by 1). The different versions of the text show Wallis hesitating between this and a looser sense. 38 See Euclid Sectio canonis, 149; GMW, pp. 192–3, nn. 6, 8. Euclid’s reason seems to depend on the fact that in Greek a ratio is ‘spoken of under a single name’ only if it is multiple or superparticular. 39 Wallis would omit detailed discussion of the placement of the ‘Flats and Sharps’, the black notes of the keyboard, from ‘The harmonics of the ancients’, but in 1698 approached the subject with both care and idiosyncrasy: see Introduction, pp. 20–21 above; Wallis, ‘A Question in Musick’, pp. 83–4 (p. 210 below); Wallis to Pepys, 27 June 1698, pp. 252–3 (pp. 216–17 below). At this stage his thinking was rather different. Conceiving of key signatures as transposing his whole scale to a new pitch level (see §18 [p. 60 below]), he understood the effects of an individual flat in a key signature as placing the pitches it affected in a new relationship with their surroundings. In his example, pitches such as b and d (and g) which stand a major tone above their lower neighbour, are moved by a flat so as to be a diatonic semitone above their lower neighbours: therefore the effect of a flat is to move them by an amount equal to 135/128, the difference between a major tone (9/8) and a diatonic semitone (16/15). This was why he used b and d as examples: that it led him to describe an impossible key signature (b@ d@) need not indicate that his grasp of the use of key signatures in practice was weak. Turning to the effect of sharps, Wallis retained b and d as his examples but envisaged a sharp in the key signature as moving them by 16/15, a diatonic semitone. 40 The division of the major third C–E into a major tone and a minor tone could be carried out in two different ways, placing those intervals in two different orders and resulting in two different positions for D differing by 81/80. The mathematical division of the fourth B–E into intervals 7/6 and 8/7 could likewise be carried out in two different ways, resulting in two different pitches both of which Wallis denoted C# (compare the diagrams on fol. 7v [p. 57 above]). In the final diagram in §19 he set out all seven of the resulting pitches in a single diagram. This was not a scale of pitches intended ever to be heard consecutively, but a catalogue of possibilities which might be used in different contexts. 41 Wallis never published his table of ‘all the Intervalls … according to the anciently receiud Hypothesis’, which would presumably have resembled the large table on fol. 8v (p. 59 above) and would indeed have illustrated the rather higher proportion

68

John Wallis: Writings on Music

of intervals which involved large numbers – and therefore were less harmonious by Wallis’s mathematical principles – which resulted. 42 See Introduction, pp. 6–7; cf. Wallis to Oldenburg, 25 May 1664 (p. 63 below). In his letter to Oldenburg of 14 May 1664, Wallis had acknowledged an acquaintance with Euclid, Sectio canonis and Boethius, De institutione musica (pp. 45, 47, 48, 53, 58 above); here he adds unspecified ‘modern Writers of the Theory of Musick (in this & part of the last Century)’ including Kepler; ‘our practicall Musicians’, and Ptolemy’s Harmonics, which he insisted he had not seen when the bulk of the letter was written. 43 See Introduction, p. 7. 44 See Introduction, pp. 6–7; compare the postscript to Wallis to Oldenburg, 14 May 1664 (p. 62 above). Here he added ‘Mersennus’ to the list of his sources, ‘& some others’; these ‘others’ would most naturally have included Kircher and Descartes. 45 Compare p. 45 n. 10 above; by ‘Pythagoras his first notion’ Wallis apparently means the procedure of dividing ratios arithmetically.

Chapter 2

Letters to Henry Oldenburg, March 1677 Editorial Note Wallis wrote to Henry Oldenburg on musical subjects again in 1677, dating his letters from Oxford on 14 and 27 March. They contained a discussion of the phenomenon of vibrational nodes (‘a new Musical Discovery’).1 The letters as sent survive as London, Royal Society, MS Early Letters W2, Nos. 36 and 37. These are the source for this edition. The first letter was read to the Royal Society on 22 March; the substance of both was printed in the Philosophical Transactions just a few weeks later: issue 12/134 (23 April 1677): 839–42. The printer of the Transactions introduced numerous changes to punctuation and spelling, but made no larger alterations. This version of the text is used to supply a few words and letters now lost in the manuscript due to damage to the paper. Readings from the Philosophical Transactions are denoted by the siglum PT; W denotes the manuscript reading. A Latin version of the text was included in Wallis’s De algebra in 1693, with some alterations and rearrangements.2 The first letter was also printed in Thomas Birch’s 1772 edition of the works of Robert Boyle, where it was implicitly identified as being addressed to Boyle.3 Birch did not identify his source, but his text is the same as that in Wallis’s manuscript, save for changes of spelling and punctuation. Possibly Birch simply made a mistake about the addressee of the manuscript letter now in the Royal Society’s archive (which has lost its cover and hence its address), but it is also possible that Boyle had received a separate copy of the letter which is now lost. Hall and Hall printed the two letters (from the manuscripts) in their edition of Oldenburg’s correspondence; Michael Hunter, Antonio Clericuzio, and Lawrence Principe reprinted Birch’s text of the first letter in their edition of Boyle’s correspondence.4 Wallis’s lucid discussion of his subject was rendered somewhat opaque by the diagrams which accompanied it in his manuscript, which the printers of the Transactions faithfully followed. Each of the first four diagrams illustrates a pair of musical strings sounding different pitches: in the first case AC sounds an octave above αγ; in the second, AD is sounds a twelfth above αδ; in the third, AE sounds two octaves above αε; finally in the fourth diagram AG sounds a fifth above αη. Yet each of the four printed diagrams shows two strings of the same length. The implication is that the differences in pitch were to be effected by changing the tension of the strings (or conceivably their thickness), but Wallis did not say so,

John Wallis: Writings on Music

70

and the reader accustomed to think of pitch as related to the lengths of strings is naturally confused. Text [London, Royal Society, MS Early Letters W2, no. 36] [recto]a Oxford, March. 14. 1676/7. Sir, I have thought fit to give you notice of a discovery that hath been made here, (about 3 years since, or more,) which I suppose may not be unacceptable to those of the Royal Society, who are Musical, & Mathematical.5 ’Tis this. Whereas it hath been long since observed, that if a Viol string, or Lute-string, be touched with the Bow or Hand; another string on the same or another Instrument not far from it, (if an Unisone ‹to it or› an Octave, orb the like,) will at the same time tremble of its own accord. The cause of it, (having been formerly discussed by divers,) I do not now inquire into.6 But adde this to the former observation; That, not the whole of that other string doth thus tremble, but the several parts severally, according as they are unisones to the whole or the parts of that string which is so struck. For instance.c ‹Supposing› AC to be an upper Octave to αγ; & therefore an Unisone to each half of it, stopped at β:

Now if, while αβ is open, AC be struck; the two halves of ‹this› other, that is αβ, & βγ, will both tremble; but not the middle point at β. Whichd will easyly be observed if a little belte of paper be lightly wrapped about the string αγ, & ‹re›moved successively from one end of the string to the other. In like manner, if AD be an upper Twelfth to αδ:

 Endorsement: Read Mar: 22: 76. / Trans: 134.  Deleted: other concord c  Deleted: If d  Deleted: which e  Unclear in W; PT has bit a

b

Letters to Henry Oldenburg, March 1677

71

& consequently an Unisone to its three parts equally divided in β γ. Now if, αδ being open, AD be struck: its three parts αβ, βγ, γδ, will ‹severally› tremble; but not the points β γ. Which may be observed in like manner asa ‹the former.› In like manner, if AE be a double Octave to αε:

the four quarters of this will tremble, when that is struck; but not the points β γ δ. So if AG be a Fifth to αη;

& consequently eachb ‹half› of that stopped ‹in› D, an Unisone to each third part of this stopped in γ ε: While that is struck, each part of this will tremble severally, but ‹not the points› γ ε: & while this is struck, each part of that will tremble, but not the point D. The like will holdc in lesser concords;7 but lesse remarkably as the number of divisions increases. This was first of all (that I know of) discovered by Mr William Noble, a Mr of Artsd of Merton College; & by him shewed to some of our Musicians about 3 Years since; & after him by Mr Thomas Pigot, a Batchelour of Arts & Fellow of Wadham-College;8 who, giving notice of it to some others, found that (unbeknown to him) the same had been formerly taken notice ofe by Mr Noble, & (upon notice from him) by others: & ‹it› is now commonly known to our musicians here. ‹Of this you will find a particular ‹notice› is taken in Dr Plotsf ‹Natural› [Historyg] of Oxfordshire, now in the Presse, & will ‹be› soon abroad.›h, 9 I adde this further, (which I took notice of [verso] upon occasion of making triall of the other;) that the same string, asi  Deleted: before  Deleted: part c  Deleted: hold? d  Deleted: & Fellow e  Deleted: it f  Deleted: Description g  Text lost in W. h  This insertion is placed at the side of the page, cued with a hash. It is omitted in PT. i  Deleted: AC a

b

John Wallis: Writings on Music

72

‹αγ›, being struck in the midst ata ‹β›,b (each part being Unisone to the other,) will give no clear sound at all; but very confused. And not onely so (which others allso have observed, [thatc] a string doth not sound clear if struck in the midst:) but allso if αδ be struck at β or γ; where one part is an Octave to the other. And in like manner, if αε be struck at β or δ, the one part being a double Octave to the other. And so if αζ be struck in γ or δ;

the [one partd] being a fifth to the other. And so in o[there] like consonant divisions: But still the ‹lesse› remarkable as the number of divisions increaseth. This & the former, I judge to depend upon one & the same cause; viz. the contemporary vibrations of the severallf unisone parts; which make the ‹one› tremble at the mo[tiong] of the other: But, when struck at the respective points ofh di[visions,i] the sound is incongruous, by reason that the point is distu[rbedj] which should be at rest. I adde no more, but that I am Yours &c John Wall[isk]l [London, Royal Society, MS Early Letters W2, no. 37] [recto] March 27. 1677. Oxford. Sir, I received yours, & thank you for it, & the Transactions in it. Those I want for this last year, are March, July, October, November, December. The Gentlemen you recommended (those of them that came to me) I indeavoured to make sensible that your recommendation was not insignificant. What you mention concerning inserting my last in the Transactions; I like well inough; because I think that a proper place for communicating new discoveries: which perhaps would be less  Deleted: B  Deleted: wil c  Text lost in W. d  Text lost in W. e  Text lost in W. f  Deleted: y parts g  Text lost in W. h  W: at i  Text lost in W. j  Text lost in W. k  Text lost in W. l  Endorsement: Rec. March 16.77. a

b

Letters to Henry Oldenburg, March 1677

73

universal if onely in the other book. You may adde allso, as a further remark to the same purpose; (which I forgot to insert in the last;) That: “A ‹Lute-string or› Viol-string will thus answere, not onely to aa consonant string on the same or a neighbouring Lute or Violl; but to a consonant Note in windInstruments. Which ‹was› particularly tryed, on a Violl, answering to consonant Notes on a Chamber-Organ, very remarkably. But not so remarkably, to the wirestrings of an Harpsichord. Which whether it were because of the different texture in Metal-Strings from that of Gut-strings: or (which I rather think) because the Mettal-Strings,b though they give to the Air as Smart a Stroke, yet not so diffusive, as the other: I list not to dispute. But Wind-Instruments give ‹to the Air› as communicative a concussion, if not more, than that of Gut-strings.10 And wee feel thec wind-scotd seats ‹on which› wee sit, or lean, to Tremble ‹constantly› at certain Notes ‹on›e the Organ or other wind-Instruments; as well as at the same Notes on a Bass Violl. I have heard allso (but cannot aver it) of a ‹f Thin, Fine,› VeniceGlass cracked with the strong & lastingg sound of a Trumpet or Cornet, ‹(near it)› sounding an Unisone or ‹a› Consonant Note to that of theh Tone or Ting of the Glass. And I do not judge the thing very unlikely; though I have not had the opportunity of making the Triall.”11 This you may (if you think fit) adde as a Post-script to the former letter; or insert it in a convenient place.i And, if you do it; it will be convenient to do it quickly, that it may be abroad as soon as the other Book;12 & not be like stale news. I am Yours &c John Wallis. [verso]j These For Mr Henry Oldenburg in the Palmal neer St James’s London.

 Deleted: like?  W: Mettal-String c  Deleted: ‹would› seats we sit. d   PT: Wainscot e  Deleted: of f  Deleted: fine g  Deleted: tooting? h  Deleted: ?Ting which ? i  Deleted: I am j  Endorsement: Rec. March. 31. 77. a

b

74

John Wallis: Writings on Music

Notes 1 See Introduction, pp. 8–12 above. 2 John Wallis, De algebra tractatus historicus & practicus, pp. 466–8; see Introduction, pp. 10 above. 3 Thomas Birch (ed.), The Works of … Robert Boyle (London, 1744), vol. 5, pp. 515–6; id. (ed.), The Works of … Robert Boyle (new edn; London, 1772), vol. 6, pp. 460–61. 4 Hall and Hall (eds), The Correspondence of Henry Oldenburg, vol. 13, pp. 224–6; Michael Hunter, Antonio Clericuzio and Lawrence Principe (eds), The Correspondence of Robert Boyle (London, 2001), vol. 4, pp. 438–40. 5 Fellows of the Royal Society who were ‘Musical and Mathematical’ would have included the members of the committees set up in 1662 and 1664 to examine the musical writings of John Birchensha: notably William Brouncker, Thomas Baines (Gresham Professor of Music), William Petty, Robert Moray, Robert Boyle, and John Pell: see Penelope M. Gouk, ‘Music in the Natural Philosophy of the Early Royal Society’ (Ph.D. thesis, London, 1982). 6 The phenomenon of sympathetic resonance which Wallis describes had been discussed for instance by Francis Bacon in Sylva sylvarum (London, 1627), p. 72. See also Penelope M. Gouk, Music, Science and Natural Magic in Seventeenth-Century England (New Haven and London, 1999), pp. 169–70 and Christopher Marsh, Music and Society in Early Modern England (Cambridge, 2010), pp. 14–15. 7 With the phrase ‘lesser concords’ Wallis avoided discussion of which intervals were to be considered concordant: cf. Wallis to Oldenburg, 14 May 1664, §§14, 15 (pp. 56–8 above). 8 Concerning William Noble and Thomas Pigot, see Introduction, p. 9 above. 9 On Plot’s Natural History, and the piece in it by Narcissus Marsh (whom Wallis does not mention in these letters) on this subject, see Introduction, p. 9, above. 10 A theory of the nature of sound must have lain behind these assertions, which distinguish between the ‘smartness’ and the ‘diffusiveness’ or ‘communicativeness’ of a ‘stroak’ or ‘concussion’ given to the air by a musical instrument. ‘Smartness’ presumably referred to the speed at which an impulse was delivered, while its ‘communicativeness’ was in this context apparently a measure of the distance over which it was transmitted. We have no more information about Wallis’s views on the nature of sound at this time, and the possibilities were quite numerous: see Gouk, ‘Some English Theories of Hearing in the Seventeenth Century’, pp. 95–113; Wardhaugh, Music, Experiment and Mathematics, pp. 71–82. In particular, Wallis need not have supposed that any necessary relationship existed between the speed and the magnitude of an impulse. 11 Compare the reports of vibrational effects in Robert Boyle, A Treatise of … Languid and Unheeded Motions (London, 1685); see Introduction, p. 8 above. The passage which Wallis placed in quotation marks was included in the printed version of his letter in the Philosophical Transactions, as a ‘Postscript’. 12 The ‘other book’ was Robert Plot’s Natural History of Oxfordshire: see p. 9.

Chapter 3

‘The Harmonics of the Ancients compared with Today’s’: Appendix to Ptolemy’s Harmonics, 1682 Editorial Note The appendix which Wallis provided for his 1682 edition of Ptolemy’s Harmonics was his longest account of the mathematical theory of music. As discussed in the Introduction,1 it seems to have been intended to perform more than one function, illustrating not just the relationship of ancient musical theory to modern but also John Wallis’s technical mastery of the contents of the Harmonics and providing a forum for Wallis to publish his own ideas about modern musical theory. In the final sections he in fact re-worked, sometimes quite closely, material from his letters to Oldenburg of 18 years previously. The basis for this edition is the text which appears in volume 3 of Wallis’s 1699 Opera mathematica (pp. 153–82): this was printed under his supervision and may be taken to represent his intentions where it differs from the 1682 text. In footnotes to the Latin text we report differences from the 1682 text, as well as the errors noted in the printed lists of errata in the 1699 edition, and Wallis’s own manuscript corrections to his copy of that edition.2 No further editorial emendations have been introduced; occasional apparent faults in Greek words and phrases have been allowed to stand. This is a difficult text for the translator, both because of the complexities of Wallis’s intentions and because he evidently had difficulty finding satisfactory Latin terminology with which to render technical Greek terms in his translation of the Harmonics. In some cases terms bore both literal and technical meanings in the original Greek; in other cases the Latin musical terminology inherited from the Middle Ages complicated matters. Wallis frequently resorted to Greek terms, sometimes transliterated into the Latin alphabet, in the attempt to make his meaning clear; occasionally he even used an English term within the Latin text. The English version presented here is intended as far as possible to be comprehensible to readers with neither Greek nor Latin, and it takes readability as its priority. An attempt has been made to render Wallis’s terminology consistently, but for readers who are concerned with the details of that terminology crossreference to the Latin text, which follows, will of course be vital. One particular issue concerns the names for musical intervals. In Latin, Wallis used both the Latin terms quarta, quinta, octava and the transliterated Greek terms diatessaron,

76

John Wallis: Writings on Music

diapente, diapason. It is possible that he intended a subtle difference of meaning; nevertheless, rather than introduce Hellenisms into the English translation, we have opted to translate both sets of terms using the standard English terms fourth, fifth, octave. In the interest of providing a readable text in English, Wallis’s in-text citations have been moved to footnotes in the translation and put in modern style. Wallis’s references to ancient musical works are to the 1652 edition of Meibom for Aristides Quintilianus, On Music; Aristoxenus, Harmonic Elements; Alypius, Introduction to Music; Bacchius, Introduction to the Art of Music; Euclid, Division of the Canon; ‘Euclid’ (i.e. Cleonides), Introduction to Harmonics; Gaudentius, Harmonic Introduction; Martianus Capella, The Marriage of Philology and Mercury, book 9. They are to his own edition of 1682 for Claudius Ptolemy, Harmonics. Quotations from the Greek are as far as possible also placed in the footnotes, as are Wallis’s briefer asides on strictly philological matters. Greek technical terms in the Latin text, where Wallis uses idiosyncratic seventeenthcentury accentuation and orthography, have where appropriate been given in the translation in the form standardly found in modern editions of ancient Greek texts. Editorial section headings and page numbers are placed in square brackets. The present editors’ commentary appears in the endnotes to the English text.

The Harmonics of the Ancients compared with Today’s

77

Text [p. 153]

The Harmonics of the Ancients compared with Today’s

[The Definition of Music] Music, μουσική – or as Aristoxenus puts it, ‘the science of song’,a or Aristides Quintilianus ‘the science of song and those things which touch on song’b – was understood by the ancient Greeks in a broader sense than many of us now understand the word. We learn this from Aristoxenusc (and likewise from other musical works edited by Meibom) and others; also from Meibom’s notes both here and on Euclid’s Introductio musica:3 Meibom, following Porphyry, lists music’s parts as harmonics, rhythmics, metrics, organology, poetics, [and] acting.d We also learn it from John Gerard Voss, who divides music into harmonics, rhythmics, and metrics;e Alypius does similarly,f and to these Aristoxenus adds organology.g Likewise Voss adds to the three forms of practical music – harmonics, organology, and hydraulics – a fourth, orchestria or dance.h And most of all we learn from Aristides Quintilianus, who expands other people’s definition – ‘the art of what is fitting in sounds and motions’ – to ‘knowledge of what is fitting in bodies and motions’, and lists and describes its various parts.i What we mainly call ‘music’ nowadays, they instead called ‘harmonics’,j which is principally concerned with ‘systems’ and ‘tones’.k It deals with the voicel insofar as it is considered with respect to high and low pitch.m Ptolemy gives the term ψόφοςn a wider sense, using it of any sound; and he defines harmonics as the perceptive capacity for differentiating sounds as high or low.o a

  περὶ μέλους ἐπιστήμη: Aristoxenus, p. 1.   ἐπιστήμη μέλους καὶ τῶν περὶ μέλους συμβαινόντων: Aristides Quintilianus, p. 5. c  Aristoxenus, p. 1. d   ‘Euclid’, p. 41. e   J.G. Voss, De scientiis mathematicis [Amsterdam, 1650/1660], chapter 22. f  Alypius, p. 1. g  Aristoxenus, p. 32. h   J.G. Voss, De quatuor artibus popularibus [Amsterdam, 1650/1660], chapter 4. i   τέχνη πρέποντος ἐν φωναῖς καὶ κινήσεσιν; γνῶσις τοῦ πρέποντος ἐν σώμασι καὶ κινήσεσι: Aristides Quintilianus, pp. 6, 8. j   ἁρμονική. k   Cf. Aristoxenus, p. 1: συστήματά τε καὶ τόνοι. l   φωνή. m   ὀξύτης καὶ βαρύτης. n   Ptolemy, p. 1. o   αὶ ἐν τοῖς ψόφοις, περὶ τὸ ὀξὺ καὶ τὸ βαρὺ, διαφοραί. b

John Wallis: Writings on Music

78

[Pitch] A sound set at a fixed height or depth they called a pitch;a Nicomachus defines this as ‘a tensing of the voice, appropriate for singing but lacking duration’.b Aristides calls it ‘melodic tension’,c and Ptolemy says ‘a pitch is a sound having a single tone’.d Aristoxenus says, ‘a sound is a voice falling on one tension’;e and likewise Gaudentius.f Sometimes it is also called a ‘tone’,g or a certain tenor of the voice ‘taken as one kind of tension’.h But this word is polysemous: it means different things in different places. And the former, ‘pitch’,i is sometimes used specifically of those notes or strings, limited to a certain number, which are counted among the Perfect System, like the proslambanomenos and those which follow it. A voice which goes from higher to lower, or from lower to higher, is said to ‘move’j and specifically to ‘move in respect of place’.k What lies in between a high and a low pitch they call a ‘place’,l or space through which the voice moves.m They divide this movement in place,n into two kinds: the one continuouso in which the voice moves constantly through a space, resting nowhere along the way; the other ‘discrete’p or in distinct intervals, in which the voice, jumping from one stopping-point to another, rests here or there for a little while. They use the former of those speaking or engaged in dialogue,q and call this movement ‘discursive’.r They use the latter of singers,s and they call it ‘melodic movement’t or [movement] appropriate for singing. Indeed, those who do this are not said

a

  φθόγγος.   φωνῆς ἐμμελοῦς ἀπλατῆ τάσις: Nicomachus, p. 7. c   τάσις μελῳδική: Aristides Quintilianus, p. 9. d   φθόγγος ἐστὶν … ψόφος, ἕνα καὶ τὸν αὐτὸν ἐπέχων τόνον: Ptolemy, p. 17. e   φωνῆς πτῶσις ἐπὶ μίαν τάσιν, ὁ φθόγγος: Aristoxenus, p. 15. f  Gaudentius, p. 3. g   τόνος. h   παρ᾽ ἓν εἶδος τὸ τῆς τάσεως εἰλημμένος: Ptolemy, p. 16. i   φθόγγος. j   κινεῖσθαι. k   κατὰ τόπον κινεῖσθαι. l   τόπος. m  Aristoxenus, pp. 3, 8, 9, 10; Aristides Quintilianus, pp. 8, 9; Nicomachus, pp. 3, 4; Gaudentius, pp. 2, 3; ‘Euclid’, p. 2. n   ἡ κατὰ τόπον κίνησις. o   συνεχής. p   διαστηματική. q   οἱ διαλεγόμενοι. r   λογική: Meibom interprets this as ‘rational’, but I prefer the term ‘conversational’. s   οἱ μελῳδοῦντες. t   κίνησις μελῳδική. b

The Harmonics of the Ancients compared with Today’s

79

to speak,a but to sing.b And they teach that the latter, not the former, should be considered here. Likewise Ptolemy distinguishes sounds of varying pitchc into continuousd and discrete;e he says the former are irrelevant to harmonic theory and the latter proper to it.f Aristides Quintilianus inserts a third, ‘intermediate motion’,g in between these two movements, which is that of those reciting a poem.h (I understand him to be cited by Boethius as ‘Albinus’, as if that Albinus [p. 154] were the same as this Aristides or he translated Aristides into Latin.i At any rate, Gerard Voss cites Albinus (from Cassiodorus) as a Latin writer, as does Boethius.j) A voice passing from a lower to a higher pitch is said to be ‘tautening’;k passing from a higher to a lower pitch it is ‘relaxing’.l The former movement or passing is called ‘tautening’;m the latter ‘relaxation’.n What is reached by tautening is called high-pitched;o what is reached by relaxation is called low-pitched.p But that ‘ongoing position’ of the voice,q whether reached by tautening or relaxation, is called a tensionr or ‘holding’ of the voice.s This range of high and low pitches may seem limitless in its nature, in terms of both extendibility and divisibility. But in practice it needs to be delimited and defined, whether with respect to what makes the sound or to what judges it; that is, the voice or the hearing. Indeed, the human voice, or even that of an instrument, cannot produce either vastly huge or excessively small vocal distinctions, nor can the hearing judge them. (Thus Aristoxenus and Nicomachus,t and not very

a

  λέγειν.   ᾄδειν. c   ψόφοι ἀνισότονοι. d   συνεχεῖς. e   διωρισμένοι. f   ἁρμονικῆς ἀλλότριοι; οἰκεῖοι: Ptolemy, pp. 16, 17. g   κίνησις μέση. h  Aristides Quintilianus, p. 6 i   Ptolemy, 1.12. j  Voss, De scientiis mathematicis, chapter 60; Boethius, De musica, 1.26. k   ἐπιτείνεσθαι. l   ἀνίεσθαι. m   ἐπίτασις. n   ἄνεσις. o   ὀξύτης. p   βαρύτης. q   μονὴ καὶ στάσις φωνῆς. r   τάσις. s  Aristoxenus, pp. 10, 11, 12, 13; Aristides Quintilianus, pp. 8, 9; Gaudentius, pp. 2, 3; ‘Euclid’, p. 2. t  Aristoxenus, pp. 13, 14; Nicomachus, pp. 4, 5. b

John Wallis: Writings on Music

80

differently Ptolemy, except that he makes the bounds of hearing wider than those of the voice.a) And so Aristoxenus accepts no smaller distinction in use than the smallest diesis,b which he makes a quarter of a tone (and similarly Aristides),c and scarcely any larger than a twenty-note or twenty-one-note span (but not three octaves) if one talks about the human voice, or three octaves if one talks about other instruments: in particular if one is talking about the voice of a particular person or particular instrument such as a reed or pipe. He does not denyd that the compass of the voice can be extended to three or four octaves or even further if several are being compared: for instance if the very high voices of boys and women are compared with the very low ones of men, or the very high range of higher-pitched pipes with the very low one of lower-pitched pipes. [Intervals] What is delimited by two pitchese of differing tensionf with respect to high and low pitch, Aristoxenus calls an interval.g This has almost the same meaning as what he earlier called place,h but in a much stricter sense, as ‘tension’i is habitually used for the stricter sense of the term ‘pitch’. And it is, he says, ‘like a difference of tensions’ or ‘a space placed between two sounds, that is able to hold sounds which are lower than the higher bound of that interval but higher than the lower bound’.j That which is composed of many intervals he calls a system.k He sets out different kinds of intervals.l Indeed, some are larger in size than others, some smaller. Again, some are consonant, others dissonant.m And some are compounded, others uncompounded.n Some belong to one genus,o others to another; namely the diatonic, chromatic, or enharmonic genus. Finally, some are rational, others irrational.p a

  Ptolemy, pp. 15, 16.  Aristoxenus, pp. 14, 20. c  Aristides Quintilianus, p. 13. d  Aristoxenus, p. 21. e   φθόγγοι. f   τάσις. g   διάστημα. h   τόπος. i   τάσις. j   διαφορά τις τάσεων; καὶ τόπος δεκτικὸς φθόγγων, ὀξυτέρων μὲν τῆς βαρυτέρας τῶν ὁριζουσῶν τὸ τάσεων διάστημα, βαρυτέρων δὲ τῆς ὀξυτέρας: Aristoxenus. k   σύστημα. l  Aristoxenus, p. 16. m   σύμφωνα, διάφωνα. n   σύνθετος, ἀσύνθετος. o   γένος. p   ῥητός, ἄλογος. b

The Harmonics of the Ancients compared with Today’s

81

He makes the same classification of systems (except that there is no uncompounded system), but he adds other distinctions. So, as far as position is concerned, some are disjunct, others conjunct, others mixed.a Again, there are continuous systems and overlapping ones.b Again, simple, duple and multiple systemsc and other things of that kind.d (And Aristides Quintilianus, Gaudentius, and Euclid do not very differently.e) [Melody: The Consonances] From sounds and intervals comes melody:f not only that ‘conversational’g melody which is heard in conversation (for speech also has its melody and accent), but ‘harmonic melody’,h or modulated song. It does not, however, come from any things combined anyhow – what is unharmonised,i and not harmonic, can still consist of sounds and intervals – but from those which have suitable ‘synthesis’j (putting together) and the necessary composition for the organisation of each song.k So, compared with each other according to the ratio of the intervals, sounds are either concinnous,l suitable for song, or inconcinnous,m unsuitable for song and to be rejected from it.n Among concinnous sounds, they call some consonant,o others dissonant:p the former are more welcome to the ears; the latter, though not altogether unsuitable, less welcome. They listed the types of consonant intervals, called consonances.q First the two simpler and less complete ones, the fourth and the fifth:4 the Aristoxenians reckon both of these with the judgement of their ears, but the Pythagoreans lay down the ratio of the former as sesquitertia, or as 4 to 3, and of the latter as sesquialtera, or as 3 to 2. Composed of these two is the ‘perfect’ octave, to which the Pythagoreans accordingly assign the duple ratio, because 4/3 × 3/2 = 2. And those further composed, a

  διεζευγμένος, συνημμένος, μικτός.   συνεχής, ὑπερβατός. c   ἁπλοῦς, διπλοῦς, πολαπλοῦς. d  Aristoxenus, p. 17. e  Aristides Quintilianus, pp. 12–16; Gaudentius, pp. 4, 5; ‘Euclid’, pp. 8, 18. f   μέλος. g   λογῶδες. h   τὸ ἡρμοσμένον μέλος. i   ἀνάρμοστος. j   σύνθεσις. k  Aristoxenus, p. 18. l   ἐμμελεῖς. m   ἐκμελεῖς. n   Ptolemy, p. 18. o   σύμφωνοι. p   διάφωνοι. q   συμφωνίαι. b

John Wallis: Writings on Music

82

the octave-plus-a-fourth (whose ratio 8/3 is equal to 4/3 × 2), the octave-plus-a-fifth (whose ratio is triple, since 3/2 × 2 = 3), and the double octave (whose ratio is quadruple, since 2 × 2 = 4): and any others one might want to add, such as a fourth or fifth combined with two or more octaves.a, 5 The Pythagoreans, however, do not accept one of these as a consonance, namely the octave-plus-a-fourth: because its ratio is neither multiple nor superparticular, and these are the only kinds they would admit as consonances.6 But Ptolemy would have it admitted, [p. 155] despite that exception. He argues in particular that an octave added to any consonance makes a consonance.b (Aristoxenus too had stated this earlier.c) Further, Ptolemy places the octave and the double octave ahead of the other consonances, as well as those made up of more octaves, giving them the particular name homophonies,d or unisons.e Others call them ‘correspondents’,f reserving the term ‘unison’ for sounds of the same pitch.g And the same people call the fifth a paraphone,h and likewise the octave-plus-a-fifth (and even a fifth added to more than one octave) as one might expect, in addition to the fourth or the fourth added to one or more octaves; and these they call by the general term ‘consonances’.i They call the fourth and the fourth-plus-an-octave paraphones, just like the fifth and the fifth-plus-an-octave; but the octave and double octave they call correspondents. But Gaudentius understands by paraphones those which come between consonances and dissonances, like the ditone, tritone and their like.j [Dissonances] The difference between the first two consonances (the fourth and fifth) they call a ‘tone’, which the followers of Aristoxenus reckon by the judgement of the ear, but the Pythagoreans define as the sesquioctave ratio (since 3/2 = 4/3 × 9/8).7 But it is admitted to the list of consonances by neither; nor are its parts, or even the ditone, the tritone or other intervals allowed in song (except those previously mentioned). Content to call these ‘concinnous sounds’k or even ‘concinnities’,l they did not call them consonances, neither those smaller than the fourth like the diesis, hemitone, tone, trihemitone, or ditone, nor those which are interspersed a

  Ptolemy, 1.5.   Ptolemy, 1.7. c  Aristoxenus, pp. 20, 45. d   ὁμοφώνιαι. e   ὁμόφωνα. f   ἀντίφωνα. g  Aristides Quintilianus, p. 12; Gaudentius, p. 11. h   παράφωνον. i   σύμφωνα: see Gaudentius (notes), pp. 35 (on Bryennius), 36 (on Psellus and Theo of Smyrna). j  Gaudentius, p. 77. k   ἐμμελεῖς. l   ἐμμελίαι. b

The Harmonics of the Ancients compared with Today’s

83

between the consonances like the tritone, tetratone, pentatone, and their like. Thus Euclid and all the others.a But taken individually there are innumerable pitchesb in any space,c if not infinitely many, just as in every continuum there are points; which Aristoxenus asserts clearly, and Aristides and Euclid, and [indeed] the matter speaks for itself.d And so there are also intervalse interspersed between them that are likewise innumerable. [The Pitches in Use] But of all these, musicians choose those they consider concinnousf or suitable for song, to call pitches par excellence (and in this sense Aristides distinguishes pitches and tensions).g And likewise, intervals are especially to be understood as those which occur between sounds in this sense. The ancients listed eighteen such sounds in each genus (diatonic, chromatic, and enharmonic). They are set out by Euclid in the following order:h proslambanomenos, hypate hypaton, parhypate hypaton, lichanos hypaton (enharmonic, chromatic, or diatonic), hypate meson, parhypate meson, lichanos meson (enharmonic, chromatic, or diatonic), mese, trite synemmenon, paranete synemmenon (enharmonic, chromatic, or diatonic), nete synemmenon, paramese, trite diezeugmenon, paranete diezeugmenon (enharmonic, chromatic, or diatonic), nete diezeugmenon, trite hyperbolaeon, paranete hyperbolaeon (enharmonic, chromatic, or diatonic), nete hyperbolaeon. And the same names are very often retained in Latin. And they are set out with little difference in Nicomachus and Gaudentius:i proslambanomenos, hypate hypaton, parhypate hypaton (enharmonic, chromatic, or diatonic), lichanos hypaton (enharmonic, chromatic, or diatonic), hypate meson, parhypate meson (enharmonic, chromatic, or diatonic), lichanos meson (enharmonic, chromatic, or diatonic), mese, trite synemmenon (enharmonic, chromatic, or diatonic), paranete synemmenon (enharmonic, chromatic, or diatonic), nete synemmenon; paramese; trite diezeugmenon (enharmonic, chromatic, or diatonic), paranete diezeugmenon (enharmonic, chromatic, or diatonic), nete diezeugmenon; trite hyperbolaeon (enharmonic, chromatic, or diatonic), paranete hyperbolaeon (enharmonic, chromatic, or diatonic), nete hyperbolaeon. a

  ‘Euclid’, pp. 8, 13.   φθόγγοι. c   τόπος. d  Aristoxenus, p. 26; Aristides, p. 9; Euclid, p. 2. e   διαστήματα. f   φθόγγοι ἐμμελεῖς. g   φθόγγοι and τάσεις: Aristides, p. 9. h  ‘Euclid’, pp 3, 4, 5, 6, 7. i  Nicomachus, p. 27; Gaudentius, pp. 11, 18. b

John Wallis: Writings on Music

84

There we see not only the lichanos and paranetai but also the parhypatai and tritai placed in three different ways for the three genera (enharmonic, chromatic, or diatonic): which is all one, when the sounds of the latter are ‘mobile’, just as those of the former.8 But I do not see why Euclid and the others omitted this about parhypatai and tritai. [Two Philological Matters] I give warning in passing of that passage in Gaudentius, p. 11 v. 10, which I think is missing: if I am not mistaken, I believe it should be restored thus: enharmonic parhypate, [chromatic parhypate and diatonic parhypate, and enharmonic lichanos,] chromatic lichanos, and diatonic lichanosa – which agrees with what he has on p. 18 v. 21. Indeed, whenever he says this – parhypatai, lichanoi, tritai and paranetai are mutable sounds; and thus what is proper to each genus is to be understood by the common name. For example, enharmonic lichanos meson, chromatic lichanos meson, diatonic lichanos mesonb – he adds ‘the same is to be said about paranetai, parhypatai and tritai’.c Next, it should be noted that the term lichanos is often omitted and paranete likewise: as when for enharmonic lichanos meson appears simply enharmonic meson, but the word lichanos is understood. And likewise for the others. And in the word proslambanomenos, agreement (of gender) with φθόγγος (sound) is seen; in the others, hypate, mese, and so on, agreement with χορδή (string). And ἡ λιχανὸς (feminine) must be written, and likewise [p. 156] ἐναρμόνιος, διάτονος whenever these occur (it is ὁ καὶ ἡ ἐναρμόνιος and ὁ καὶ ἡ διάτονος). But in Bryennius (and only him, if I am not mistaken) is found προσλαμβανομένη. Likewise ὑπάτη, παρυπάτη, νήτη, παρανήτη are taken as substantival nouns (in apposition to the noun χορδή), as is clear from the genitive plurals with circumflexes, ὑπατῶν, παρυπατῶν, νητῶν, παρανητῶν. But μέσων, τρίτων, συνημμένων, διεζευγμένων, ὑπερβολαίων are taken as adjectives. We pointed this out in our note on Ptolemy 2.5.

a

  οἷον παρυπάτη ἐναρμόνιος, [καὶ παρυπάτη χρωματικὴ, καὶ παρυπάτη διάτονος, καὶ λιχανὸς ἐναρμόνιος,] καὶ λιχανὸς χρωματικὴ, καὶ λιχανὸς διάτονος. b   λιχανὸς μέσων ἐναρμόνιος, λιχανὸς μέσων χρωματικὴ, λιχανὸς μέσων διάτονος. c   ὁ δὲ αὐτὸς λόγος καὶ περὶ τῶν παρανητῶν τε, καὶ παρυπατῶν καὶ τρίτων.

The Harmonics of the Ancients compared with Today’s

85

[The Pitches in Use] The same names as above are listed (for the diatonic genus) by Martianus Capella,a adding in turn Latin names he coined (how accurately for all of them, I would not like to say); thus. proslambanomenos hypate hypaton parhypate hypaton diatonic hypaton hypate meson parhypate meson diatonic meson mese trite synemmenon diatonic synemmenon nete synemmenon paramese trite diezeugmenon diatonic diezeugmenon nete diezeugmenon trite hyperbolaeon diatonic hyperbolaeon nete hyperbolaeon

Adquisitus Principalis principalium Subprincipalis principalium Principalium extenta Principalis mediarum Subprincipalis mediarum Mediarum extenta Media Tertia conjunctarum Conjunctarum extenta Ultima conjunctarum Prope-media Tertia divisarum Divisarum extenta Ultima divisarum Tertia excellentium Excellentium extenta Ultima excellentium

[Notation and Solmisation] For each of these strings or sounds, the Greeks had (in each genus: enharmonic, chromatic, or diatonic) a particular symbol or note,b with which it could be designated. This is clear from Bacchius, Aristides, Boethius, and others, but above all from Alypius and Gaudentius. Voss too (and others he cites) seems to remark on the same matter.c Meibom devotes great effort to the reconstruction of these notes, especially in his notes on Alypius, where they are to be seen. And to be versed in these notes was a large part of the practice of music: how to write down a melody in notes after

a

  Martianus Capella, p. 179.   σημεῖον. c  Vossius, De quatuor artibus popularibus, chapter 4, section 10; Vossius, De scientiis mathematicis, chapter 22, section 14. b

John Wallis: Writings on Music

86

hearing it and then to sing it from the written form. This Aristoxenus calls ‘the art of noting’,a and Meibom in his note on Alypius ‘semiotics’.b For these notes, we have in today’s music the so-called Guidonian scale, at least for the diatonic genus, as invented by Abbot Guido Aretinus around 1070 (according to John Gerard Voss) or 1028 (according to Sigebert).c Some call this the hand, because they sometimes used to represent it in the form of a hand for ease of remembering, as can be seen in the work of Mersenne and others; some call it the Gamut, taking its name from its first or lowest pitch. Using some lines (ascending by steps, like stairs, scalae) and the spaces in between, they represent the places of single pitches,d arranged so that they rise from low to high. No other sign is needed for this than the position itself. In fact, in our music the variety of signs which is seen (such as larges, longs, breves, semibreves, etc.) refers not to pitch but to time, according to which a pitch is to be sung either longer or shorter. It pertains to the rhythmic part of music, which is concerned with rhythm or modulation according to time – short and long – not to the harmonic part, with which we are now concerned, that is with modulation according to pitch, high and low. But so that it can be better perceived how they relate to each other, we present both the modern scale and the ancients’ diagram (for the diatonic genus, which alone we now use) joined together [see Figure 3.1]. We have distinguished in the ancients’ diagram the sounds which they call mobile (shown by a dotted line) from those they call fixed or immobile (shown by a continuous straight line). Equally, in the modern scale we have distinguished those sounds which are customarily written in the space (as they say) from those written on the line. In both cases the intervals are observed, whether tones or semitones, as they are usually thought of in the diatonic genus. They print some of these lines, as many as necessary, leaving out the rest: say four or five (either from the bottom, or the top or the middle, according to the measure of each song). They designate one of them, say F fa ut or c sol fa ut or g sol re ut, with a particular symbol which they call a key, in order to mark which line of the scale it refers to; and the remainder are judged according to their positions in respect of this. But neither the diagram of the ancients nor our scale ends with these bounds in such a way that no further progress to low or high is possible. In our scale after the seven letters A, B, C, D, E, F, G, ascending, which designate the same number of strings or pitches, the same recur a second and a third time and, if necessary, more times, and these repetitions, so that they might be distinguished from each other, are usually distinguished by changing the form of the letters. Likewise, in descending to low pitch, the same is done in reverse order (and for a different G a

  παρασημαντική.   Aristoxenus, p. 39; σημειωτικήν: Alypius, p. 66 (note). c  Voss, De scientiis mathematicis, chapter 22; Voss, De quatuor artibus popularibus, chapter 4. d   φθόγγος. b

The Harmonics of the Ancients compared with Today’s

87

they write the Greek Γ, from which the whole scale’s name came to be Γ ut or Gam-ut). And similar letters (say A a, or G g, etc.) mark out octaves.9 In the same way in the diagram of the ancients, after reaching from proslambanomenos to nete hyperbolaeon in ascending to high pitch – a string which is held to be the same as the proslambanomenos of the diagram recurring above – there recur as before hypate hypaton, parhypate hypaton, etc. Similarly, descending to low pitch, below proslambanomenos – which is the same as the nete hyperbolaeon of the diagram placed above – there recur if necessary, in reverse order, paranete hyperbolaeon, trite hyperbolaeon, etc.a And similar names of strings (say mese, mese or paramese, paramese or, since they are held to be the same, proslambanomenos, nete hyperbolaeon, etc.) encompass a double octave. [p. 157] The words (in the Guidonian scale) are derived from letters. They are taken from a certain hymn (by one Paul, a Roman deacon) in honour of St John the Baptist, which Voss gives in chapter 4 of his Four Popular Arts, as do others. UT queant laxis REsonare fibris MIra gestorum FAmuli tuorum SOLve pollutis LAbiis reatum (Mersenne and others add to these Sapphics Adonius’s Sancte Johannes, and Alsted O pater alme: but these are beyond our present aim.) He took the initial syllables from these six half-lines – ut, re, me, fa, sol, la – for all the strings in their sequence, namely Γ A B C D E. They recur after an octave as G a b c d e, and again as g aa bb cc dd ee. But the seventh note is always absent: the place of F or f lacks a note (for to have fa and ut in the same place belongs to a different conjunction);10 and likewise anywhere, whether before ut or after la. And it is a wonder that Guido did not see and make provision for this. Some more recent scholars, among them Mersenne, supply the note Si or something similar. They could, from the start of Adonius’s verse, have supplied the note sa. [The Modes] The use of these notes is this: re and mi make (as the Greeks call it) a ‘diezeuctic tone’ (by which two disjunct tetrachords are separated, and which, added to either of them, makes a pentachord or fifth). To them correspond, in their natural place (which is Ptolemy’s Dorian mode and our ‘hard’ scale),11 the mese and paramese; and likewise, an octave away from them, the proslambanomenos and hypate hypaton. That is, in our scale a b and A B, assuming of course that mi lies in B, b, bb. And corresponding to this is the hexachord ut re mi fa sol la (which occurs in the scale in seven positions) in its first, fourth and seventh positions [p. 158]. These three positions are considered as one, since the same notes recur at the distance of an octave.12 a

  Ptolemy, 2.11.

88

Figure 3.1

John Wallis: Writings on Music

John Wallis’s diagrammatic representation of the ancient and modern scales. Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium (Oxford: E Theatro Sheldoniano, 1699), p. 157 (Oxford, Bodleian Library, shelfmark: Savile B 12). By permission of the Bodleian Libraries, University of Oxford.

The Harmonics of the Ancients compared with Today’s

89

But where the song changes by mutation ‘according to modes’ (as the Greeks call it) or mi is moved to another place, re and mi are ‘by force’ (as the Greeks say), mese and paramese, whatever position they occupy. And likewise they are proslambanomenos and hypate hypaton ‘by force’. Say a so-called flat sign is placed on B, b, bb. Mi is moved to E, e; and the hexachord is sung, as it occurs, in its second and fifth positions. This is Ptolemy’s Mixolydian tone (or mode), in which ‘by force’ mese and paramese are, in terms of position, paranete and nete diezeugmenon: that is, d and e in our scale. If another flat sign is placed on E, e, by which mi is transferred to a, aa, the hexachord is sung as in its third and sixth positions: this is Ptolemy’s Hypolydian mode, in which ‘by force’ mese and paramese are ‘by position’ lichanos meson and mese; that is G and a in our scale.13 These three cases, and no more, are recognised in the Guidonian scale. But since his time more recent musicians have added other key signatures, by which they indicate that mi is to be sung on any string, as required, so as to correspond to Ptolemy’s other modes or tones. (As we showed in our note on Ptolemy, book 2, chapter 11.)14 Indeed, by placing a third flat sign on a, mi is thus moved to d; this is Ptolemy’s Lydian mode. Again, adding a sharp sign to F fa ut, with which the note fa naturally agrees (as they say), this changes to mi and the place of mi becomes F, f; this is Ptolemy’s Hypodorian mode. But it is moved from there to C, c, cc, if a similar sharp sign is repeated there, and this is Ptolemy’s Phrygian mode. And from there, finally, mi is moved to Γ, G, g, if there too is found the same sharp sign, and this is Ptolemy’s Hypophrygian mode. All of these we have set out in the place mentioned. [Hard b, Soft b, and Modern Solmisation] The origin of these signs for flat and sharp is the following. In Greek music, on reaching mese (that is, our a la mi re) from proslambanomenos by rising, it was unclear whether to continue to the conjunct tetrachord (synemmenon) or the disjunct tetrachord (diezeugmenon), and so, whether the next string (which is b fa b mi) should be higher by a semitone (for trite synemmenon) or a complete tone (for paramese). In the first case, the string would be lower (by a semitone) than in the latter case, that is (as they liked to say) ‘softer’: and the latter would be correspondingly higher than the former, that is more taut, or harder. And so in the first case, they designated that string with a rounded b which they called ‘soft b’, from the case in which b, as the note fa, rose a semitone from a la mi re. This is usually called a ‘soft scale’. In the latter case, they designated that string with a square b (which has clearly been corrupted into the symbol #), which they called ‘hard b’. And this is usually called a ‘hard scale’ (or even, as happens often enough now, they left out that signature, since this scale is said to be ‘natural’ and so to need no signature). In this case, it goes up by a tone to the note mi. This happens because this string has two names – b fa, b mi – and ‘soft b’ is sung for the note fa, which is reached by rising a semitone, ‘hard b’ for the note mi, which is reached by rising a complete tone. But the ‘soft’ and ‘hard’ signs, which had at first been

John Wallis: Writings on Music

90

intended to distinguish only the string b fa b mi, now mean as much when applied to any string. That is, if a string is higher than the next by a tone, by the addition of a ‘soft’ (that is, lower-pitch) [flat] sign it is lowered by a semitone, and so it becomes only a semitone sharper than the next. On the other hand, one which is only a semitone higher than the next, is made higher by a semitone when the ‘hard’ (that is, sharp) sign is added, and so becomes a complete tone sharper than the next.15 But our musicians nowadays leave out the terms ut re and use only mi fa sol la.16 And they place mi as the key signature indicates and sing upwards fa sol la, fa sol la (after which mi comes again), and downwards la sol fa, la sol fa (and then mi returns again). And of these fa is said to be a semitone higher than the next below, the rest higher by a complete tone. Now those who speak more precisely assign to the note sol the major tone (with the sesquioctave ratio [9 : 8]) and to the note la the minor tone (with the sesquinona ratio [10 : 9]); and so to fa, for the semitone, they assign the sesquidecimaquinta ratio [16 : 15], so as to complete the sesquitertia ratio [4 : 3] of the tetrachord; but to mi they all assign the sesquioctave ratio [9 : 8].17 But those who (ignoring the difference between major and minor tones) indiscriminately assign the sesquioctave ratio [9 : 8] to all tones, are the same ones who assign to the semitone the ratio of a lemmaa which is 256 to 243; or (speaking less precisely) half a tone (which is not very different from a lemma). And almost all practical musicians speak in this way, and do not usually think of intervals other than the tone and half-tone (and those made up of these).18 [The Division(s) of the Tetrachord] But having briefly set aside our scale, I return to the Greeks’ diagram. Where there are eighteen strings, as I said, in each genus, there are not always the same number of notes, but there are as many terms or possibilities.b Indeed, some of these notes are held sometimes to coincide with others, especially in the diatonic genus (paranete synemmenon with trite diezeugmenon, and nete synemmenon with paranete diezeugmenon, as already shown in the diagram), and eighteen are hardly distinguished elsewhere than in the enharmonic genus. But let us expound in order the reason for the names, going through the whole matter from the start. From these notes (or strings tuned to them) the ancients formed intervals and scales that are consonantc or suitable for song (as explained above): of these, some they called concords, others discords,d according to whether they were endede or bounded by concords or discords.f a

  λεῖμμα.   δυνάμεις. c   ἐμμελῆ. d   σύμφωνα, διάφωνα. e   ὡρισμένα. f  Aristoxenus, pp. 16, 17; Aristides, pp. 13, 14. b

The Harmonics of the Ancients compared with Today’s

91

[p. 159] However, the diastema or interval or distance which they used like a measure of the others (especially those called ‘Aristoxenians’, who measured intervals more by differences than by ratios) they called a tone,b not in the sense that it indicates a particular timbre of the voice but that it signifies a certain difference of [pitches]; and it was as much as the difference (in magnitude) between the first two concords, fourth and fifth. ‘Its half, third and quarter can be sung, but no smaller intervals than these.’c The size of this distance the Aristoxenians (who were not further concerned with this) allow to be judged by ear; but the Pythagoreans (who instead measured intervals by ratios or proportions) fixed it as the sesquioctave ratio [9 : 8].d In fact, since they had put the sesquialtera ratio, or 3 to 2, as the fifth, and the sesquitertia, or 4 to 3, as the fourth – as above – the difference between these, which should be a tone, they reckoned as the sesquioctava: that is, the rise from one sound to the other is 9 to 8, since 4/3 ) 3/2 ( 9/8.e, 19 The first consonant interval, or the first consonancef they made the ‘tetrachord’,g which they also called the fourth,h being the smallest of all (but which is the largest of all, they are unable to define).i This consonance, the fourth, they divided with four notes (whence its name) into three intervals, next to each other. They neither divided it equally nor in all genera similarly, but, informed by the judgement of the ear, into unequal intervals, and differently in that genus called enharmonic, in that called chromatic and in that called diatonic. And according to this different division of the tetrachord, they distinguished the three (as they called them) genera:j enharmonic, chromatic, and diatonic.k Of each tetrachord or fourth the two end notes (which delimit the whole) were called ‘standing’, ‘immobile’ or ‘resting’:l the two in between, ‘mobile’.m This was because both the distance between the former (which the Aristoxenians defined as two and a half tones, the Pythagoreans as the sesquitertia ratio [4 : a

  διάστημα.   τόνον. c  Aristoxenus, pp. 21, 46. d   λόγῳ ἐπογδόῳ. e  Euclid, p. 30; Ptolemy, 1.7 and elsewhere. f   τῶν συμφώνων διαστημάτων πρῶτον καὶ ἐλάχιστον. g   τετράχορδον. h   διὰ τεσσάρων. i   Thus Aristoxenus, pp. 20, 21, 22 and elsewhere (where for μέγεθος, p. 20, line 14 should appear μέγιστον, as Meibom rightly points out in his notes; and the matter speaks for itself). j   γενῆ. k   ἐναρμόνιος, χρωματικός, διατονικός: So Aristoxenus, pp. 21, 22, and elsewhere. l   ἑστῶτες, ἀκινητοί, ἠρεμοῦντες. m   κινητοὶ or κινούμενοι. b

John Wallis: Writings on Music

92

3]) and their place in the whole scale were stable and the same in all genera (and hence they were said to be at resta). But the distances of the latter – both the difference between them and their differences from the outer two – and their position in the whole scale, were different for the different genera (and hence they were said to moveb). And indeed it is according to the diverse place of the latter in the tetrachord that the various generac are named. And there are various species of these genera, which they used to call colours.d They called the most low-pitched note or string of the tetrachord the hypate,e the ‘uppermost’; and the most high-pitched the nete,f the ‘furthest’ or ‘bottommost’. Henricus Stephanus also realises this as regards the word nete,g and he thus says it refers to the ‘furthest or bottom-most’, in contrast to the hypate and the paranete, as being next to the bottom-most. For those who were the first to impose these names (contrary to what we do now) used ‘bottom-most’ for the most high-pitched and ‘uppermost’ for low-pitched.h, 20 And so on the seven-stringed lyre, he ascribes the hypate to Saturn and neate, or, contracted, nete, to the moon. And Boethius, in his Musica, always places low-pitched sounds at the top of the scheme and high-pitched sounds at the bottom. So the hyper-hypate and hyper-mese,i which occur afterwards, are [respectively] the next below the hypate and mese.j But so as to make the matter less clear, after ‘low’ began to be used for low-pitched and ‘high’ for high-pitched (as happens now), the Latin authors (following Martianus Capella and Boethius) translated hypate as principalis (rather than suprema) and the parhypate as subprincipalis.

a

  ἠρεμεῖν.   κινεῖσθαι. c   γένη. d   χρόας. And thus Euclid, p. 10. And Alypius, p. 2: τῶν φθόγγων οἱ μέν εἰσιν ἑστῶτες καὶ ἀκλινεῖς· οἱ δὲ κινούμενοι· ἑστῶτες μὲν οὖν λέγονται, ὅτι ἐν ταῖς τῶν γενῶν διαφοραῖς οὐ μεταπίπτουσιν· κινούμενοι δὲ, ὅτι ἐν ταῖς τῶν γενῶν διαφοραῖς μεταπίπτουσιν εἰς ἑτέρας τάσεις. [Some of the sounds are standing and not moving; others are moving: the standing ones are so called because they do not change in the different genera, the moving ones because they move to different pitches in the different genera.] e   ὑπάτην. f   νήτην or νεάτην. g   νήτη. h   And thus Nicomachus, p.  6: ὁ βαρύτατος φθόγγος ὑπάτη ἐκλήθη, ὕπατον μὲν τὸ ἀνώτατον· κατώτατος δὲ νεάτη, καὶ μὲν νέατον τὸ κατώτατον. [The deepest string was called the hypate, the hypate being the topmost and the bottom-most being nete: indeed nete was the bottom-most.] i   ὑπερυπάτη, ὑπερμέση. j   As if someone used ὕπατος [adjective] as if it were ὑπόατος [superlative of ὑπό, ‘below’]: although Homer, indeed, used it as if it were ὑπέρτατος [superlative of ὑπέρ, ‘above’], since he usually calls Jupiter ὕπατος κρειόντων [‘highest of kings’]. b

The Harmonics of the Ancients compared with Today’s

93

And Aristides Quintilianus uses the term prima rather than suprema.a Whence (betraying the meaning of ‘highest’ and ‘lowest’) hypatos and netosb are translated as Primus and Ultimus.21 The sounds in between (which are also movable) they called parhypate,c subsuprema or subprincipalis (the one next to hypate) and paraneted or penultima: the one lying next to the nete, the last. hypate   parhypate paranete nete

Suprema. la Subsuprema.   fa Penultima. sol Ultima. la

Thus the notes of the tetrachord were called when it was alone, as in the socalled lyre of Mercury, which Boethius says had four strings.e [p. 160] And this (except for using lichanos in the place of paranete in the same sense, although it may be a faulty reading) is almost how Aristoxenus sets out the order of the names (very well known and agreed) for any tetrachord.f I translate him thus. ‘Although there are many arrangements of the strings spanning the range of the aforesaid fourth’, namely, as he had just said, with two fixed outer sounds and two movable ones in between, ‘and each is specified by the names of one [of the strings]’, say, tetrachord of the suprema, of the media, of the conjuncta, of the disjuncta, etc., ‘there is one order (common to all) (namely nete, lichanos, parhypate, hypate) well known to all those who have engaged even casually in music.’ Meibom notes at this point that in order for the syntax to agree, the reading should either be ὅτι μία, or (as I take it) γνωριμώτατα rather than γνωριμώτατον as in some manuscripts; but he did not notice that (as I think, at least) μέσης is a mistake for νήτης, both here and again a little later. It would indeed be preferable a

  Aristides Quintilianus, p. 10: ὑπάτη ὑπατῶν, ὅτι τῶν πρώτου τετραχόρδου πρώτη τίθεται· τὸ μὲν πρῶτον, ὕπατον ἐκάλουν οἱ παλαιοί· παρυπάτη δὲ, ἡ παρ᾽ αὐτὴν κειμένη. [The hypate hypaton is what is placed first in the first tetrachord: the ancients called the first hypatos and the one lying next to it the parhypate.] And a little later: παρανῆται καλοῦνται, διὰ τὸ πρὸ τῆς νήτης κεῖσθαι· ἐπὶ δὲ ταύταις ἡ νήτη, τουτέστιν ἐσχάτη· νέατον δὲ ἐκάλουν τὸ ἔσχατον οἱ παλαιοί [paranetai are so called because they lie in front of the nete, and the nete is upon them, namely at the end; and the one at the end the ancients called neatos]. b   ὕπατος and νῆτος. c   παρυπάτη. d   παρανήτη. e   Boethius, 1.20, and elsewhere. f   Aristoxenus, p. 22: τῶν δὲ συγχορδιῶν, πλειόνων τε οὐσῶν, τὴν εἰρημένην τάξιν τῶν διὰ τεσσάρων κατεχουσῶν, καὶ ὀνόμασιν ἰδίοις ἑκάστης αὐτῶν ὡρισμένης· μία τίς ἐστιν ἡ μέσης [I read νήτης] καὶ λιχανοῦ καὶ παρυπάτης καὶ ὑπάτης, σχεδὸν γνωριμωτάτη τοῖς ἁπτομένοις μουσικῆς.

John Wallis: Writings on Music

94

here to have a discussion of one order for everything, rather than of one tetrachord placed ahead of the others. But, if someone tenacious should prefer to stick to the other reading instead of following my conjecture – as if Aristoxenus said these things about one tetrachord (that of mese) which is well known beyond the others and not about a single order of all – it is not a matter of such gravity that I should want to be anxious to dispute it. For even in that way this single one is proposed as the model of all, since the argument is the same in every respect. And what Aristoxenus says seems to favour this, where he expounds the tetrachord from mese to hypate (that is the meson hypaten tetrachord) as if it were an exemplar for the others,a just as he does in the place just mentioned. But after there ceased to be a single tetrachord and more tetrachords began to be created, in place of the parhypate (subsuprema) in some tetrachords there is the trite (that is, third from the neteb), ‘third from last’ or ‘antepenultimate’. In others, in place of the paranete (‘penultimate’) they say lichanos,c ‘index’; either because that string is struck with the index finger (as Aristides says)d or (as I should rather think) because the place of this note is the index to whether the genus is soft or tensed. For the further the lichanos is from the highest note of the tetrachord, the softer the genus is considered to be; the nearer it is, the more intense. This is what Ptolemy has,e and in fact speaking about these notes (the highest and its neighbour) which he calls ‘leading notes’f and their interval, he adds ‘which makes the difference (that is, of genus) between soft or tensed’.g That being so, it is evident that the trite in some tetrachords has the same value as the parhypate in others, and similarly for the lichanos in some and the paranete in others. Hence, tritai are called ‘parhypate-like’h and paranetai ‘lichanos-like’,i and likewise in others, so just as tritai and parhypatai have the same ratio, likewise lichanoi and paranetai. And similarly the lowest notes of any tetrachord are called either the hypatai or at least ‘hypaton-like’.j, 22 [Systems of Two Tetrachords] After the lyre of Mercury (or whosever it actually was), with its four strings (for the number of the four elements), there came the lyre of Orpheus, heptachordal,k a

 Aristoxenus, p. 46.   τρίτη ἀπὸ νήτης. c   λιχανός. d  Aristides, p. 10. e   Ptolemy, 2.3, p. 54. f   φθόγγους ἡγουμένους. g   οἵ τινες ποιοῦσι τὰς ἐπὶ τὸ μαλακώτερον ἢ τὸ συντονώτερον παραλλαγάς. h   παρυπατοειδεῖς. i   λιχανοειδεῖς. j   hypatoeides: Boethius, p. 11. k   ἑπτάχορδος. b

The Harmonics of the Ancients compared with Today’s

95

with seven strings (for the number of the planets). There are those, however, who say that Mercury himself made the lyre – which he had received from his forebears with four strings, the so-called lyre of Mercury – seven-stringed, and that Orpheus received this, handed on by Mercury.a But (we should not be excessively worried concerning the first inventor) there is agreement about this: this seven-stringed lyre, called Orphean, is made up of two conjunct tetrachords, so that the highest note of the lower tetrachord is the lowest of the higher one. The former they called the tetrachord hypaton,b ‘of the highest’; the latter the neton,c ‘of the last’. But so that the string common to both chords should not have two names (lowest of the highest, and highest of the lowest), they called it mese,d ‘middle’. The lowest note of the lower tetrachord they called simply hypate (not hypate hypaton) and the one next to it parhypate; and the highest note of the higher tetrachord they called simply nete (not nete neton) and the one next to it paranete. So that it should not remain nameless, the note of the lower tetrachord which had been paranete (which was now infelicitous as a name, because the name of nete had been changed to mese) was given the new name lichanos (or even hypermese).e And what had been the parhypate of the higher tetrachord (since its hypate had likewise become mese) was variously called either paramese (that is, next after mese) or trite (that is, third from nete). This we find from Boethius and Nicomachus.f The latter assigns the string hypate to Saturn (the highest of the planets), nete to the Moon (the lowest of the planets), and the others to the rest in order, except that he puts Mercury ahead of Venus, and this therefore he says should be emended as an error.g (But he equally claims, against others, that in the reverse order, the lowest should be assigned to the Moon, the highest to Saturn and similarly for those in between.)

a

 Nicomachus, p. 29.   ὑπάτων. c   νητῶν. d   μέση. e  Nicomachus, p. 7, and Meibom on him, p. 57. f   Boethius (loc. cit.); Nicomachus, pp. 6, 7. g  Nicomachus, p. 33. b

John Wallis: Writings on Music

96

[p. 161] And this is seen to be the case in the diagram above, in the part from hypate meson to nete synemmenon, or even from hypate hypaton to mese: or again from paramese to nete hyperbolaeon. After this lyre came (neglecting the number of planets) the lyre of Pythagoras with eight strings, octachordal.a He, keeping the names from the seven-stringed lyre almost as before, was the firstb to place a tone, the diazeuctic tone, between the two tetrachords which were previously conjunct. This, therefore, when added to each tetrachord, made a pentachord,c whose outer notes form a fifth. But for each tetrachord he in fact set up an octachord, not a heptachord, the end-strings of which were to constitute an octave (the chief consonance): this could not be heard in a heptachord, its outer notes not even being consonant. And thus, what had previously been a single string, called mese, now became two: the nete hypaton [‘bottom-most of the uppermost’], called now, as before, mese, since it had not moved much from the middle; and the hypate neton [‘uppermost of the bottom-most’], now called paramese (previously called trite). The latter differed from mese by a whole tone, and from trite by a semitone. And what had been the tetrachord neton synemmenon (‘of conjunct netai’) became the tetrachord neton diezeugmenon (‘of disjunct netai’). a

  ὀκτάχορδος.  According to Nicomachus, p. 7. c   πεντάχορδος. b

The Harmonics of the Ancients compared with Today’s

97

This is now seen to be the case in that part of the diagram from hypate meson to nete diezeugmenon. There, between two tetrachords (meson and diezeugmenon), a tone is interposed from mese to paramese, which makes a pentachord when joined to either tetrachord. And so, from hypate meson to mese is a fourth, to paramese a fifth; and from nete diezeugmenon to paramese is a fourth, to mese a fifth; and from hypate meson to nete diezeugmenon is an octave.

And in fact if the tetrachord meson were continued by a tetrachord synemmenon (‘of conjuncts’), we would have the ancients’ heptachord; if with the same tetrachord meson were taken a tetrachord diezeugmenon (‘of disjuncts’) after interposing a tone, we would have the (Pythagorean) octachord. From this it is clear why one of the two tetrachords is called synemmenon (‘of conjuncts’) and the other diezeugmenon (‘of separated [netai]’ or ‘of disjuncts’): since, of course, the former is conjoined with the tetrachord meson, the latter disjunct from it (by the interposed tone). Of the two tetrachords (whether in the heptachord of the ancients or in the octachord of Pythagoras) I have no doubt that they themselves called the one hypaton (because it was lower pitched) and the other neton (because it was higher pitched). And so in the heptachord we find neton synemmenon (the last of the conjunct [tetrachord]), in the octachord neton diezeugmenon (the last of the disjunct [tetrachord]). Gaudentiusa retains these names,b and I should not have thought, as Meibom does, that this was wrong. Others very often, and Gaudentius sometimes, say simply synemmenon and diezeugmenon, leaving out the word neton (understood but safely omitted), for the same reason that, speaking briefly, they a

 Gaudentius, pp. 8, 9, 10.   Gaudentius, pp. 8, 9, 10: συνημμένων νητῶν and διεζευγμένων νητῶν.

b

John Wallis: Writings on Music

98

similarly say hypaton diatonos, meson diatonos, etc., for what should properly be (and in many places are) termed lichanos hypaton diatonos, lichanos meson diatonos, etc. Meanwhile, it seems, an eighth string was added to the seven handed down, either by Terpander or Timotheus of Miletus or someone else altogether: not, however, in between the two tetrachords (as in the lyre of Pythagoras) but lowerpitched than both, and thus was called either hyperhypate,a being lower than hypate, or proslambanomenos or proslambanomene,b, 23 as if it were ‘taken outside’, beyond the two tetrachords, so as to complete the consonance of an octave (as did not occur with the seven-stringed lyre of conjunct tetrachords). The [other] names from the seven-stringed lyre were retained as before, in this way:

And Bryennius has this in places,c and he places his seven tones or modes little differently from how Ptolemy distributes his seven. And this is called the Pythagorean lyre with eight strings, just as the former, without the added string, is called the old-fashioned lyre of Mercury with seven strings.d And what they call the Perfect System, with fifteen strings, he means is made up of two lyres of this kind put together, of which one (from proslambanomenos to mese) is called the lower-pitched, the other the higher-pitched,e and the nete of the lower and the proslambanomenos of the higher (being one and the same string) are known by the shared name mese. And similarly the heptachord [p. 162] bounded by hypate hypaton and mese he calls ‘the lower ancient seven-stringed lyre of Mercury’, and

a

  ὑπερυπάτη, as in Boethius, 1.20.  Bryennius, passim. c   Bryennius, 2.4, 3.1, and elsewhere. d   Πυθαγόρου ὀκτάχορδος λύρα; ἀρχαιότροπος ἑπτάχορδος λύρα ἑρμοῦ. [Bryennius] b

1.1.

e

  Πυθαγόρου ὀκτάχορδον λύραν βαρυτέραν/ὀξυτέραν.

The Harmonics of the Ancients compared with Today’s

99

the one bounded by paramese and nete hyperbolaeon he calls the higher sevenstringed lyre of Mercury.a But I come back to the earlier distribution of the octachord (which seems more valid and which Ptolemy follows in arranging his tones or modes),b namely the one which has a diazeuctic tone placed between the two tetrachords. This, for its time, was considered a ‘perfect system’, and hence this union of eight strings was not called ‘dia-octon’,c after the number of strings (as the unions of four and five strings are called fourth and fifth), but dia-pason, as if it was the union of all strings.d [Scales of Three or Four Tetrachords] Since then, more recent scholars have placed between the two tetrachords (hypaton and neton) another tetrachord (meson) in the middle. So now it is not just a mese string but a tetrachord of mesai, middle strings, that has been placed in the middle position. And as they had called the middle string of the heptachord mese, for the same reason they called the middle tetrachord in a system of three tetrachords meson (‘of the middle strings’), keeping the names of the top and bottom (or ‘deepest’) tetrachords as before. (By ‘top’ understanding the lowpitched, and by ‘bottom’ or ‘deepest’ the high-pitched, contrary to what we do today.) And so from the heptachord and octachord (depending on whether the tetrachord neton was conjunct or disjunct, synemmenon or diezeugmenon), they inserted another tetrachord to make the decachord and hendecachord. And indeed this tetrachord meson (of middle strings) particularly belongs between the two parts of the conjoined system, hypaton (of the highest strings) and synemmenon (of the lowest of the conjunct strings), as Aristides clearly explains.e But it can be reasonably taken to refer to the tetrachord of disjunct strings,f although this is not exactly in the middle of the two.g

a

  Ἑρμοῦ τρισμεγίστου ἀρχαιότροπον βαρυτέραν λύραν ἑπτάχορδον/ὀξυτέραν.   Ptolemy, 2.8, 2.9, 2.10, 2.11. c   δι᾽ ὀκτώ. d  See Ptolemy, 2.4, 3.1, and Aristoxenus, p. 16. e   Aristides, p. 10: τῶν μέσων τετράχορδον· τοῦτο γαρ μόνον μεταξὺ θεωρεῖται τῶν τε ὑπατῶν καὶ τῶν συνημμένων. f   νητῶν διεζευγμένων. g   Boethius, 1.20. b

John Wallis: Writings on Music

100

Here hypate, parhypate and lichanos appear twice, with the addition first of hypate, then of mese, and together with mese they make two conjunct tetrachords, to which hypate meson (which is the same as hypaton nete) is common. After mese the tetrachord neton is either conjunct (as in the heptachord) and makes up a lyre of ten strings, decachordala (from hypate hypaton to nete synemmenon in the above diagram); or it is disjunct (as in the octachord) and makes up a lyre of eleven strings, hendecachordalb (from hypate hypaton to nete diezeugmenon in the above diagram). a

  δεκάχορδον.   ὲνδεκάχορδον.

b

The Harmonics of the Ancients compared with Today’s

101

Finally, to extend the range of song further (keeping what had been the names of the tetrachords, hypaton, meson, and neton), after the ‘final’ tetrachord neton, whether conjunct or disjunct, they would add yet another high-pitched tetrachord, called the hyperbolaeona as if wandering, circulating, or going out; for you would rightly call those who are beyond the final points extravagant, ‘wanderers-out’. Or, in less querulous terms, which many people also use, outstanding. This tetrachord hyperbolaeon is disjunct from the tetrachord neton synemmenon but conjunct with the tetrachord neton diezeugmenon. These names, neton synemmenon, neton diezeugmenon, and neton hyper-[p. 163]bolaeon are kept in their entirety, as was said above, by Gaudentius.b And in Euclid we read neton synemmenon, neton diezeugmenon, neton hyperbolaion, before Meibom expunged the word neton.c The same is seen in Porphyry’s commentary on Ptolemy. But for the most part, among others neton is omitted (but understood), and the usual terminology is simply tetrachord synemmenon, diezeugmenon, and hyperbolaeon. Last of all, since there is still one note missing from the whole double octave, the proslambanomenos (a sound ‘adjoined’ or acquired) was added at the bottom, one tone lower than hypate hypaton. The final result is that mese (which had been so called in the first place because it was the middle string of the heptachord) returns home to its middle position, exactly an octave both from proslambanomenos and from nete hyperbolaeon (separating the two octaves as it had previously separated two tetrachords). And that completes Ptolemy’s Perfectd System of fifteen strings, as in the earlier diagram. [The Perfect System] Who was first to introduce this or that string into the whole collection I cannot say, since it is neither certain enough nor do the authors agree on the matter. We have the names of some of them in Nicomachus and Boethius:e but we should not be unduly concerned about what each did. So, the whole Perfect System is made up of four tetrachords and two tones (and these at the same time make up two octaves, namely one from proslambanomenos to mese, the other from mese to nete hyperbolaeon). Four tetrachords, I say (each of which we sing as, from the bottom, the note la or mi, then, rising, fa sol la): either two conjoined pairs (hypaton and meson, and diezeugmenon and hyperbolaeon) as in the system called disjunct; or three conjoined (hypaton, meson, and synemmenon) and one separate (hyperbolaeon), as in the system called conjunct (the third tetrachord is conjoined with first two a

  ὑπερβολαίων, or even, as Gaudentius, pp. 9, 10, νητῶν ὑπερβολαίων.  Gaudentius, pp. 8, 9, 10, 18. c  As Meibom himself says in his notes on that passage, p. 6. d   τέλειον. e  Nicomachus, pp. 29, 35; Boethius, 1.20. b

John Wallis: Writings on Music

102

through mese). And two tones (which, from a lower note la, always rise through a whole sesquioctave tone [i.e. with ratio 9 : 8] to mi), which they call diezeuctic or ‘diazeuxes’: one from proslambanomenos to hypate hypaton (which they call the lower-pitched diazeuxis), the other, in the disjunct system, from mese to paramese (which they call the higher-pitched diazeuxis). Or in the conjunct system (at least if it is continued to a fourth tetrachord), instead of the latter there is a tone from nete synemmenon to nete diezeugmenon. And this Boethius states, and Bacchius.a In fact, overall there are five tetrachords and three tones, although not used at the same time. For when the tetrachord diezeugmenon is used, and so the diazeuxis from mese to paramese, the tetrachord synemmenon is omitted, as is the tone from nete synemmenon to nete diezeugmenon. And vice versa: when the latter are used, the former are omitted. Again, in the conjunct system they scarcely remember the tetrachord hyperbolaeon or the tone appended to it, but they end the system at nete synemmenon as if they did not exist, making a system not of fifteen but of eleven strings. Now it will not be enough to say hypate, parhypate, and lichanos to designate the strings (as before), but it will rather be necessary to say in which tetrachord: whether hypaton or meson. Nor is it enough to say trite, paranete, and nete, but, again, it will be necessary to say of which tetrachord: synemmenon, diezeugmenon, or hyperbolaeon. It is from this (and for the reasons already given) that the names and distribution of strings arise in the Greek system. The terms lichanos and trite could comfortably have been abandoned, after the need for two terms to designate strings had arisen, since they had been introduced to avoid such duplication; and the terms paranete and parhypate restored to their rightful place so that (instead of lichanos) the terms hypaton paranete and meson paranete would be used; and (instead of trite) synemmenon parhypate, diezeugmenon parhypate, hyperbolaeon hypate. For it is no less convenient to say paranete hypaton than lichanos hypaton, and likewise for the rest. But the custom has been that the previously introduced terms were retained, even after the reason for which they had been introduced has ceased. [The Genera and their Species] But to return to the fourth. According to its various division, different so-called genera (as it is said) arise. Thus ‘genus’ is how the tetrachord or its four sounds is divided and distributed.b As any one tetrachord is divided, so too are all the other tetrachords in the same genus.   Boethius, p. 10, though on p. 18 he extends the conjunct system only to a diapason plus a diatessaron. Bacchius, p. 10. b   ποὺ τεττάρων φθόγγων διαίρεσις, Euclid, p. 1; or ποιὰ τετραχόρδου διαίρεσις καὶ διάθεσις, Gaudentius, p. 5. a

The Harmonics of the Ancients compared with Today’s

103

By common agreement, there are three genera: enharmonic, chromatic, and diatonic.a But [writers] do not agree about the species of these genera which are to be enumerated, which they call ‘colours’.b Nor do they agree on the division of the tetrachord according to its genera, which different people divide in different ways in each of the genera. Aristoxenus arranges the whole fourth (which he considers to be made up of two tones and a half) as follows.c, 24

a   γένη ἐναρμονίος, χρωματικός, διατονικός. See Aristoxenus, pp. 19, 44; Gaudentius, p. 5; Euclid, pp. 8, 9 10. b   χρόας: χρόα δέ ἐστι γένους εἰδικὴ διαίρεσις: Euclid, p. 10. c  Aristoxenus, p. 24, and elsewhere.

104

Figure 3.2

John Wallis: Writings on Music

John Wallis’s diagrammatic representation of the genera according to Aristoxenus. Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium (Oxford: E Theatro Sheldoniano, 1699), p. 164 (Oxford, Bodleian Library, shelfmark: Savile B 12). By permission of the Bodleian Libraries, University of Oxford.

The Harmonics of the Ancients compared with Today’s

105

In the enharmonic genusa he makes the interval from hypate to parhypate, and that from parhypate to lichanos (or paranete), a diesis: in fact a diesis quadrantalis, which contains one quarter or three twelfths of a tone. And he calls it an enharmonic diesis and the smallest of all, no smaller interval being sung.b But from there [lichanos] to nete he makes a ditone; indeed, an uncomposed ditone,c [p. 164] an undivided interval of two tones. And so if a tone has twelve parts, each diesis will have three parts, a ditone twenty-four and the whole [tetrachord] thirty parts: 3 + 3 + 24 = 30. In the soft chromatic genusd he has a diesis trientalise or four twelfthsf of one tone, both from hypate to parhypate and from this to lichanos. And from lichanos to nete he has twenty-two twelfths; that is, an undivided interval made up of a tone, a semitone and a diesis trientalis, or 12, 6 and 4 twelfths. For 4 + 4 + 22 = 30. In the hemiolic chromatic genusg he has a diesis hemiolios (one-and-a-half harmonic dieses, and so four-and-a-half twelfths) from hypate to parhypate and from there to lichanos. And from there to nete he has twenty-one twelfths; that is an undivided interval made up of seven dieses quadrantales, or of a tone, a semitone and one diesis quadrantalis, that is from 12, 6 and 3 twelfths: 41/2 + 41/2 + 21 = 30. In the tonic or toniaeanh chromatic he has a semitone (of six twelfths) from hypate to parhypate, and from this to lichanos. And from the latter to nete he has a trihemitonei or sesquitone [tone and a half]; that is an undivided interval made up of a tone and a semitone, or 12 and 6 twelfths, that is 18: for 6 + 6 + 18 = 30. In the soft diatonicj he has a semitone, or six twelfths, from hypate to parhypate. From this to lichanos he has three quarters of a tone, or a semitone with a diesis quadrantalis, or an undivided interval made up of three dieses quadrantales, that is, 9 twelfths. And so from lichanos to nete he has 15 twelfths, or an undivided interval consisting of five dieses [p. 165] quadrantales, or of a tone and one diesis quadrantalis: for 6 + 9 + 15 = 30. In the tensed [or syntonic]k diatonic he has a semitone from hypate to parhypate; from parhypate to lichanos and likewise from this to nete he has a tone: for 6 + 12 + 12 = 30. a

    c   d   e   f   g   h   i   j   k   b

γένος ἐναρμόνιον. διέσις τεταρτημόριος, διέσις ἐναρμόνιος: Aristoxenus, p. 46. δίτονος ἀσύνθετος. μαλακὸν χρωματικόν. τριτημόριος. δωδεκατημόρια. ἡμιόλιος. τοναίῳ. τριημιτόνιον. διατονικὸν μαλακόν. σύντονος.

John Wallis: Writings on Music

106

Euclid has the same genera, and divided in the same way.a, 25 And Gaudentius has it much the same.b And Ptolemyc recaps these same genera according to Aristoxenus’ opinion, but doubling the numbers (that is, dividing a tone into 24 parts, and so two-and-a-half tones into 60: namely so that there is no need of a fraction in the hemiolic chromatic). Namely: Aristoxenus’ Enharmonic, 6 + 6 + 48 = 60 Soft Chromatic, 8  + 8  +  44  =  60 Hemiolic chromatic, 9  +  9  +  42  =  60 Tonic chromatic, 12  +  12  +  36  =  60 Soft diatonic, 12  +  18  +  30  =  60 Tense diatonic, 12  +  24  +  24  =  60 So there are, to Aristoxenus’ thinking, three genera (as others think) but six colours,d as if there were that number of species of these genera: one in the enharmonic genus, three in the chromatic, two in the diatonic. In Aristoxenus’ time or before, it seems that the enharmonic genus was the most celebrated of these genera,e but performing it was considered difficult and so few people used it.f In Gaudentius’ time the diatonic, almost alone, was in use.g In Ptolemy’s time, all the diatonic genera and the tense chromatic but not the enharmonic or the soft chromatic were in use.h But in our day, scarcely anything is used other than the tense diatonic, or what is almost equivalent to it. [The Ranges of Various Sounds] The space in which parhypate moves (from the lowest to the highest parhypate) is an enharmonic diesis.i Namely, from the lowest parhypate (which is a diesis quadrantalis from hypate) to the highest (which is a semitone from hypate, that is two dieses of that kind) is a diesis quadrantalis, or an enharmonic diesis: it is through this space that parhypate wanders, in other words from the end of the first diesis quadrantalis to the end of the second, the semitone.

a

 Euclid, pp. 10, 11.  Gaudentius, pp. 5, 6. c   Ptolemy, 1.12. d   χρόαι. e  Aristoxenus, p. 2 and Meibom on him, pp. 76, 77. f  Aristoxenus, p. 23; Meibom, p. 92. g  Gaudentius, p. 6. h   Ptolemy, 1.16. i  Aristoxenus, pp. 23, 47; Euclid, p. 10. b

The Harmonics of the Ancients compared with Today’s

107

The space in which lichanos or paranete moves is a tone: for from mese or nete it is never less distant than a tone (as in the tense diatonic genus) nor further than a ditone (as in the enharmonic genus) and so it wanders through the space of a tone.a And these spaces do not overlap (with one coming within the boundaries of the other), nor are they disjunct (so that anything could come in between the two places), but they are continuous: where one ends the other begins. And the end of the first semitone from hypate is the shared boundary for the two spaces (the most intense parhypate and the least tense lichanos). And this is what Aristoxenus means: Their places (those of parhypate and lichanos) do not overlap but are joined by a common boundary. For where parhypate and lichanos come to the same pitch (the former by intension, the latter by remission), their spaces have a common boundary. It is both the place of parhypate when it is lower-pitched and the place of lichanos when it is higher-pitched. And in fact let it be so defined by these (the positions of the lichanos and parhypate). It is now necessary to address matters concerning the genera and colours.b A tetrachord in which the two lesser systems (from hypate to lichanos) taken together are less than the undivided third (from lichanos or paranete to nete) is called dense.c (This happens in four types, the enharmonic and the three chromatics). The opposite, where this does not happen, is non-densed (Martianus Capella called it ‘rare’), as in the diatonic genera. Here the two lesser systems are, together, either equal to the other (as in the soft diatonic) or larger (as in the tense diatonic). After all, in the former the three strings (hypate, paranete, and lichanos) are joined more densely or closely, in the latter more loosely or less densely.e Those sounds or strings which can enter the dense region (namely hypate, parhypate, lichanos, or what replaces them in any tetrachord) are called pycnif

a

  Aristoxenus, pp. 22, 46 (where for τρόπου, ‘mode’ we should read τόπου, ‘place’, p. 22, line 25); Euclid, p. 10. b   Aristoxenus, p. 23: οὐ γὰρ ἐπαλλάττουσιν οἱ τόποι, ἀλλ᾽ ἔστιν αὐτῶν πέρας ἡ συναφή, which I am not sure the editor has quite understood, for he carries on, ὅταν γὰρ ἐπὶ τὴν αὐτὴν τάσιν ἀφίκωνται ἥ τε παρυπάτη καὶ ἡ λιχανὸς, ἡ μὲν ἀνιεμένη, ἡ δὲ ἐπιτεινομένη, πέρας ἔχουσιν οἱ τόποι. καὶ ἔστιν, ὁ μὲν ἐπὶ τὸ βαρὺ, παρυπάτης· ὁ καὶ ἐπὶ τὸ ὀξὺ, λιχανοῦ (I delete τε καὶ παρυπάτης· but you may restore this if you care to) καὶ περὶ τούτων μὲν [λιχανοῦ τε καὶ παρυπάτης] οὕτως ὡρίσθω· περὶ τῶν δὲ κατὰ γένη τε καὶ τὰς χρόας, λεκτέον. c   πυκνός. d   ἄπυκνος. e  Aristoxenus, p. 24. f   πυκνός.

John Wallis: Writings on Music

108

(dense): namely, hypatai (and all hypate-likea strings) are called barypycni;b parhypatai (and parhypate-likec strings, i.e. tritai) are called mesopycni;d and lichanoi (and lichanos-likee strings, i.e. paranetai) are called oxypycnif – since the first are found in the lower-pitched part of the dense region, the second in the middle and the last in the higher-pitched part. Those which cannot enter the dense region are called apycni.g There are therefore (of the eighteen sounds) five barypycni: hypate hypaton, hypate meson, mese, paramese, nete hyperbolaeon (i.e. the ones which are hypaton-likeh in the tetrachords hypaton, meson, synemmenon, diezeugmenon, and hyperbolaeon). There are equally many mesopycni: parhypate hypaton, parhypate meson, trite synemmenon, trite diezeugmenon, trite hyperbolaeon (namely [p. 166] those which are parhypaton-like,i in the same tetrachords); and the same number of oxypycni: lichanos hypaton, lichanos meson, paranete synemmenon, paranete diezeugmenon, paranete hyperbolaeon (namely those which are lichanos-like,j in the same tetrachords). Of apycni there are three: proslambanomenos, nete synemmenon, nete hyperbolaeon (which are those which never enter the dense region). Apycni and barypycni are all fixed sounds; mesopycni and oxypycni are mobile ones.k These are the genera according to Aristoxenus and his followers, the so-called Aristoxeneans: they measure concords by intervals, and they consider the fourth to be comprised of two tones and a half. [The Pythagorean Approach] Pythagoras and his followers, the so-called Pythagoreans, considering Aristoxenus’ method of measuring concords by intervals to be of little account, designated them by the ratios of the sounds to one another.26 And in particular they reckoned the consonance of a fourth to be constituted not when the sounds differ by two-anda-half tones, but when they are in the sesquitertia ratio [4 to 3] to each other. And thus, in place of the three intervals added together to make two-and-a-half tones, they found three ratios which, composed, make up the sesquitertia or 4/3. And just as this may be done in one way or another, so they defined one and another genus.l a

  ὑπατοειδεῖς.   βαρύπυκνοι. c   παρυπατοειδεῖς. d   μεσόπυκνοι. e   λιχανοειδεῖς. f   ὀξύπυκνοι. g   ἄπυκνος. h   ὑπατοειδεῖς. i   παρυπατοειδεῖς. j   λιχανοειδεῖς. k  Aristoxenus, p.12; Alypius, p. 2. l   Ptolemy, 1.9. b

The Harmonics of the Ancients compared with Today’s

109

Ratios are not all indifferently admitted for this purpose, for designating sounds following one another continuously: but only multiple and superparticular ratios; not (unless forced) superpartient ratios, still less those which are inexpressible numerically.a, 27 Ptolemy shows divisions of the tetrachord of this kind, by many musicians, together:b

Ptolemy discusses in depth the judgement he reaches about all these divisions, and his reasons for creating his own and preferring it ahead of others.c [Mutation] But in the same genus all tetrachords or fourths are similarly divided; enharmonically in the enharmonic, chromatically in the chromatic, diatonically in the diatonic, and in each colourd of each genus according to its nature. If ever it happens differently – say, one tetrachord is divided enharmonically, another chromatically or diatonically; or one is divided as soft diatonic, another as tense diatonic (or similarly for other genera) – this is called mutation of genus, namely a shift from genus to genus.e This change or transition from one genus to another must be done on one of the fixed sounds, the bounds of the tetrachords, if it is to remain concinnous [melodic]: if, in the same tetrachord, there appeared both the parhypate of one genus and the lichanos of another, the transition would be a

    c   d   e   b

ἄλογος, ἄῤῥητος: Ptolemy, 1.5, 1.15, 1.16. Ptolemy, 2.14, and in particular those of Archytas, at 1.13 and his own at 1.15, 1.16. Ptolemy, 1.12, 1.13, 1.14, 1.15, 1.16. χρόα. Euclid, p. 20: μεταβολὴ κατὰ γένος.

John Wallis: Writings on Music

110

inconcinnous [unmelodic].a The same should also be observed in other transitions, namely of tone, of system, or of melopoeisis (the composition of melody), which will be discussed in turn.b For, he says, the word mutationc is used in four ways. These are mutation in genus,d already discussed, mutation in mode,e which will be discussed later, mutation in system,f as when there is transition from a disjunctg to a conjuncth system or the latter to the former (which will later be shown to be no different from moving from the Dorian mode to the Mixolydian or vice versa). And change in melopoeisis:i as, for instance, when a greater ‘dilatation’ (a diastaltic characterj) signifying magnificence, manly strength of spirit, heroic deeds, and the emotions corresponding to them, as in tragedies and their accompaniments, changes – through the customs or the ingenuity of musicians – to become more contracted (a systaltic characterk), indicating a humbler state of mind and a greater effeminacy, as in the emotion of lovers, of laments, pity, and the like, or more calmed (a hesychastic characterl), indicating a calm, free, peaceful state of mind, as in hymns, encomia, advice, and the like. Or vice versa.m Change in genusn is explained at greater length by Ptolemy.o He says which genera appropriately change with each other and that the movement from one genus to another must be made in that tone, the diazeuctic, which separates the two tetrachords (the higher-pitched in one genus, the lower-pitched in another) and is common to both genera. But now all this teaching about the different genera is almost out of use. This is because for many ages [p. 167] we have had only one musical genus in use. Whether this is Aristoxenus’ intense diatonic of tones and semitones, at least roughly speaking; or Ptolemy’s ditonic diatonic of tones and lemmasp (which are very little different from semitones), which Euclid adopted in his Sectio canonis, which many musurgi or practical musicians down to Boethius and almost to our a

 Euclid, ibid.  Euclid, ibid. c   μεταβολή. d   μεταβολὴ κατὰ γένος. e   κατὰ τόνον. f   κατὰ σύστημα. g   διεζευγμένος. h   συνημμένος. i   κατὰ μελοποιίαν. j   ἤθος διασταλτικόν. k   συσταλτικός. l   ἡσυχαστικός. m  Euclid, p. 21. n   κατὰ γένος μεταβολή. o   Ptolemy, 1.16, 2.15. p   λείμματα. b

The Harmonics of the Ancients compared with Today’s

111

own day have followed; or Ptolemy’s tense diatonic, by major and minor tones and quasi-semitones (concerning which those who speak more precisely sometimes dispute), this is not the place to discuss in greater detail (as we will do later).28 In fact these things differ so little from each other that the judgement of the ear can hardly, if at all, distinguish them; and roughly speaking they are generally considered the same: more precise discussion concerns speculation more than practice.29 [The Principal Consonances and their Compounds] Now, thus far we have talked about the tetrachord or fourth. They call this the first consonance, and it is according to its various divisions that the various genera we speak of arise. The Aristoxeneans make it two-and-a-half tones; the Pythagoreans define it as the sesquitertia ratio [4 : 3]. But if one whole tone were added (above or below) to a tetrachord thus divided, there arises a system consisting of five strings,a or a fifth.b They call this the second consonance. And the Aristoxeneans make it three-and-a-half tones (since 21/2 + 1 = 31/2), but the Pythagoreans define it as the sesquialtera ratio [3 : 2] (since 4/3 × 9/8 = 3/2). They make the octachord,c composed of these two (the tetrachord and pentachord): a system of eight strings. (One string is common to the tetrachord and pentachord; so the total number of strings is eight, not nine.) And this is the third consonance. The Aristoxeneans make this six tones (since 21/2 + 31/2 = 6), or rather five tones and two semitones (since two are in fact semitones), but the Pythagoreans define it as the duple ratio (since 4/3 × 3/2 = 2/1). But they call it not a ‘dia-octon’d (after the number of strings, as in fourth and fifth) but a diapasone (‘through-all’), what earlier writers considered a Perfect System.f But Ptolemy gives this name only to a two-octave system.g This consonance is almost considered a unison, the sounds which delimit it being effectively like a single sound. When added to any consonance – fourth, fifth, or octave itself – either once or more than once, they understand the result to be, in turn, a consonance; and they attribute this property only to this consonance.h Ptolemyi famously disagrees with the Pythagoreans about this, since they wish to exclude the octave-plus-a-fourth from the consonances, because it contains the ratio of 8 to 3 (since 2/1 × 4/3 = 8/3), which is neither multiple nor superparticular a

  πεντάχορδος.   διὰ πέντε. c   ὀκτάχορδος. d   δι’ ὀκτώ. e   διὰ πασῶν. f   σύστημα τέλειον: Aristides, p. 16; Ptolemy, 2.4. g   Ptolemy, ibid. h  Aristoxenus, pp. 20, 45; Euclid, p. 13; Ptolemy, 1.6. i   Ptolemy, ibid. b

John Wallis: Writings on Music

112

(those being the only ones they agree to be consonances) but superpartient, namely it is (as they say) the dupla superbipartiens tertias [8 : 3 or 22/3], since 8/3 = 22/3.a, 30 They therefore make the octave-plus-a-fourth the fourth consonance, a system of eleven strings which the Aristoxeneans make eight-and-a-half tones (since 6 + 21/2 = 81/2), or, if you prefer, seven tones and three semitones; the Pythagoreans define it by the ratio of 8 to 3 (since 4/3 × 2/1 = 8/3). They make the octave-plus-a-fifth the fifth consonance: twelve strings; nineand-a-half tones according to the Aristoxeneans (since 6 + 31/2 = 91/2) or eight tones and three semitones; a triple ratio according to the Pythagoreans (since 2/1 × 3/2 = 3/1). They make the double octave the sixth consonance: fifteen strings; twelve tones according to the Aristoxeneans (since 6 + 6 = 12) or ten tones with four semitones; according to the Pythagoreans, a quadruple ratio (since 2/1 × 2/1 = 4/1). And Ptolemy goes this far and no further in setting out the limits of the double octave system, which is called the Perfect System.b Not because there are no more (for the nature of things allows this process to go on ad infinitum) but because these seem to him to be enough for the present, and the rest, beyond these, are in effect only repetitions of the same. Others add a seventh consonance, the double-octave-plus-a-fourth: eighteen strings; fourteen-and-a-half tones according to the Aristoxeneans (since 6 + 6 + 21/2 = 141/2) or twelve tones and five semitones; according to the Pythagoreans, having the ratio 16 to 3 (since 4/1 × 4/3 = 16/3). Or even an eighth concord, namely double-octave-and-a-fifth: nineteen strings; fifteen-and-a-half tones according to the Aristoxeneans (since 6 + 6 + 31/2 = 151/2) or thirteen tones and five semitones; of the sextuple ratio according to the Pythagoreans (since 2/1 × 2/1 × 3/2 = 6/1). Beyond this they do not extend the range of the voice,c but limit it within the triple octave, whether we are considering a human voice or an instrumental one:d if we are considering one particular instrument or the voice of a single person. Aristoxenuse does not deny that in different pipes or reeds, or comparing a boy’s or woman’s voice with a man’s, the rangef may be extended, perhaps to three or four octaves, or even more.31 And the range of the so-called Guidonian scale is the same.g Thus, from A re to ee la is a double-octave and a fifth, and from Γ ut to ee la (just one tone more) is still within the triple octave. (Thus I think it is reasonable to agree with Meibomh [p. 168] when he affirms boldly but incautiously that ‘the extent of the Guidonian scale is the same’, namely ‘fifth plus octave’; again, ‘likewise the greatest system a

  Ptolemy, 1.5, 1.6.   Ptolemy, 1.4. c   φωνῆς τόπος. d  Aristoxenus, pp. 20, 45; Euclid, p. 13. e  Aristoxenus, p. 21. f   διάστασις. g  As Meibom remarks on Aristoxenus, p. 90, and on Euclid, p. 45. h   Meibom, loc. cit. b

The Harmonics of the Ancients compared with Today’s

113

for Guidonians, which they call the great perfect scale, is twenty-stringed or of twenty notes, that is, double-octave plus fifth’, when in fact it is one tone greater.) It is impossible to define which is the largest of the consonances: by the nature of song itself (as has been said) a concord can be enlarged ad infinitum.a And a dissonance may be both enlarged and diminished ad infinitum, since there is no absolutely smallest interval (just as there is no absolutely shortest line);b but they take the smallest that can be sung to be the enharmonic diesis (as described above), which they make a quarter of a tone. Our musicians of today normally refer to these intervals of which we have spoken by other names.32 So, the diatessaron they call a fourth (which should be understood thus: of the strings bounding the interval one is, counting inclusively, the fourth from the other). The diapente they call a fifth (for a similar reason), the diapason, an octave; the diapason-and-diatessaron, an eleventh; the diapason-anddiapente, a twelfth; the disdiapason, a fifteenth; the disdiapason-and-diatessaron, an eighteenth; the disdiapason-and-diapente, a nineteenth; the trisdiapason (which is the ninth consonance) a twenty-second, and similarly for the rest. [Pythagorean and Aristoxenean Views of Interval] Now above it has often been mentioned that consonances (and indeed dissonances) were differently reckoned by the Aristoxeneans and the Pythagoreans, because the former designate the consonances by intervals (reckoning by subtraction), but the latter designate the same intervals by ratios (reckoning by division). For instance: a fourth is of two tones and a half, said Aristoxenus; but Pythagoras said it was in a sesquitertia ratio [4 : 3]. Likewise for the rest. It will be opportune here to discuss the opinion of each more fully. Pythagoras, then, and those called Pythagoreans after him. Considering sound to be made by the movement of air that has been struck, they consider the various levels of pitch to arise from the various speeds of the trembling motion. Thus with double the speed the level of pitch is doubled, and they make this the octave consonance.33 And likewise for the other consonances. So they assigned the duple ratio to the octave consonance; to the fifth the sesquialtera [3 : 2]; to the fourth the sesquitertia [4 : 3], and they derived the rest by calculation from this, using multiplication and division, or the composition of ratios.c But Aristoxenus and his followers considered this to be excessively precise and had recourse to the judgement of the ears and wished to derive from that source the first principles from where the rest were to be deduced. Crying that those who pursued the details of ratios acted in vain, they asserted that the judgement of the ears showed they were often contrary to the phenomena. (And perhaps this was because some of the Pythagoreans had sometimes assigned ratios not as carefully as was right, as Ptolemy sometimes admits.) a

 Aristoxenus, p. 20.  Aristoxenus, pp. 20, 46. c  Euclid, p. 23; Boethius, 1.30. b

114

John Wallis: Writings on Music

And so Aristoxenus places sensory judgement in the first place and thence deduces his demonstrations. This is different from the geometers, who start from the intellect. He puts it thus: A geometer has no need of the power of sense. – But the musician puts the precision of sense in almost the first place. – For we judge the sizes of intervals by hearing, but we perceive their force in the end by the intellect.a So he deduces his demonstrations from sensory phenomena, and he takes others (namely the Pythagoreans) to task for being excessively precise, who determine intervals by the proportions of numbers. This, too, is cited by the very learned John Gerard Voss in chapter 19 of his On Mathematical Sciences, as having been said by Aristoxenus against the Pythagoreans. He says: Of intervals or systems, we will proceed to give demonstrations deduced from the phenomena (of sense). Not as those who came before us (the Pythagoreans) who, preferring something else before this method, rejected the sense as insufficiently accurate and treated the causes as a matter of ratiocination. And they said that some proportions are numeric and are the ratios of speeds to each other: in which high and low pitch consist. Thus they prefer ratios, which are most alien to the subject, and most contrary to the phenomena themselves.b And much the same wherec he says that from sensory phenomena should arise principles and postulates,d from which the rest is then to be demonstrated; they are not to be sought, going beyond the limits of the matter concerned, from the nature of the voice or the motion of the air. And so they used to call the first consonance, found pleasing to the ears, a fourth; the one next to this, equally pleasing to the judgement of the ears, they called a fifth. They used to take these as givens, as though approved by the a   Aristoxenus, p. 33: Ὁ μὲν γὰρ γεωμέτρης οὐδὲν χρῆται τῇ τῆς αἰσθήσεως δυνάμει. – τῷ δὲ μουσικῷ σχεδόν ἐστιν ἀρχῆς ἔχουσα τάξιν ἡ τῆς αἰσθήσεως ἀκρίβεια. – τῇ μὲν ἀκοῇ κρίνομεν τὰ τῶν διαστημάτων μεγέθη· τῇ δὲ διανοίᾳ θεωροῦμεν τὰς τούτων δυνάμεις. b   Aristoxenus, p. 32: Καὶ τούτων [διαστημάτων] ἀποδείξεις πειρώμεθα λέγειν ὁμολογουμένας τοῖς φαινομένοις. Οὐ καθάπερ οἱ ἔμπροσθεν· οἱ, ἀλλοτριολογοῦντες, καὶ τὴν μὲν αἴσθησιν ἐκκλίνοντες, ὡς οὖσαν οὐκ ἀκριβῆ· νοητὰς δὲ κατασκευάσαντες αἰτίας· καὶ φάσκοντες λόγους τέ τινας ἀριθμῶν εἶναι, καὶ τάχη πρὸς ἄλληλα, ἐν οἷς τό τε ὀξὺ καὶ βαρὺ γίνεται· πάντων ἀλλοτριωτάτους λόγους λέγοντες, καὶ ἐναντιωτάτους τοῖς φαινομένοις. c  Aristoxenus, pp. 43, 44. d   ἀρχαί and ὁμολογούμενα.

The Harmonics of the Ancients compared with Today’s

115

common agreement of ears (and by the fact that very many, not to say all, judge the same intervals, by the hearing, to be consonances and grateful to the ear). And they used to call the difference between these given intervals, namely how much one is higher- or lower-pitched than the other, a tone, and they took this also to be a given and to be discerned by the judgement of the ears. Therefore they took this (the tone, I mean) as a known measure, validated by the agreement of the ears, in order to reckon the remaining musical intervals from it: semitones, dieses, other smaller intervals such as half-, third-, and quartertones, etc., and larger intervals; as though each was made up of a number of tones, [p. 169] or certain parts of a tone. And in particular, a fourth is made up of two tones and a half; and from this they gathered the other intervals by calculations (using addition and subtraction). Thus a fifth is made up of three tones and a half (since, by its construction, it is a tone larger than a fourth); an octave of six tones (since, in the judgement of the ears, it is made up of both [a fifth and a fourth]). And the same for the rest. Sometimes Aristoxenus seems to speak uncertainly concerning whether a fourth (on which the rest depend) is really precisely two tones and a half, or whether it comes close to that, differing by only the smallest amount.a Likewise, whether a fourth is to be measured from one of the smaller intervals (the semitone or the diesis), or is in fact incommensurable with them all: in other words whether what is called a semitone is precisely half a tone, and thus a fifth part of a fourth, whether an enharmonic diesis is precisely a quarter of a tone, and whether a chromatic diesis is a third of it, etc. And whether a ditone is precisely eight times the smallest diesis or a little less.b And similarly wherec he means that taking a third or a quarter of a tone is one thing, but singing a tone divided into three or four equal parts is another; as if he did not exactly mean what is understood by third or quarter, but something close to it. (Unless, instead, he means that a third or quarter may be reckoned exactly but that three thirds or four quarters consecutively cannot, which is not an impossible interpretation). But perhaps he did not say this not because he was unsure about it, but because he had not yet expounded the matter (as he will go on to do, p. 56), and so he says only this much about these obscure matters, amounting to the same points or differing only very slightly. [An Aristoxenian ‘Test’] This is how he sets out his proof. After he has determined by definition that ‘the magnitude of a tone is the same as the excess of a fifth over a fourth’, he sets out to prove ‘that the magnitude of a fourth itself is two tones and a half’,d taking the a

  For instance Aristoxenus, p. 24.   Aristoxenus, p. 28: μικρῷ τινι παντελῶς καὶ ἀμελῳδήτῳ ἔλαττον. c  Aristoxenus, p. 46. d  Aristoxenus, p. 56. b

John Wallis: Writings on Music

116

judgement of the hearing for a postulate, in almost these words (except that an error obscures the passage, as Meibom points out in his notes). So, let us take a fourth (judged by the hearing), say AB. Let a ditone be removed from each of its ends, say AC and BD (he had previously, on p. 55, said how this was to be done with the aid of hearing).

And so the remainders AD, BC will be equal, since equals have been taken away from equals. Then from these ends D, C (using the judgement of the ears) let two fourths DE, CF be taken, in the higher-pitched direction from the lower end and vice versa. And so the extra parts of these, beyond the limits of the original fourth, will be equal to one another for the same reason as before (as were AD, BC; and thus the fourths extend beyond the ditones). These things being prepared, let the furthest points thus established, F, E, be examined by sensory judgement. For if these are perceived as dissonant, a fourth is not made up of two tones and a half; but if they are perceived as consonant (which, on p. 24, he takes to be a phenomenon proved by experiment), namely a fifth, it is clear that it consists of two tones and a half.

He proves this conclusion as follows. Since FC is a fourth and FE a fifth, the excess of the latter beyond the former, CE, is a tone. This is bisected at B. BC, the excess of a fourth above a ditone, is therefore half a tone, and so a fourth is two tones and a half.34

And though he had taken as a lemma that ‘the sounds furthest apart F, E, although certainly consonant, make no consonance but the fifth’, he proves it thus: ‘For this interval is greater than a fourth (by the excesses which were added to the original fourth), and less than an octave’. Which he proves thus: For what is made up of these excesses is, at least, less than a ditone – since a fourth exceeds a ditone by less than a tone: for all agree that a fourth is greater than two tones, but less than three – and so it is much less than a fifth (which, together with a fourth, makes a fifth). So FE is certainly not an octave. And thus if the interval FE is consonant, since it is larger than a fourth and smaller than an octave, it is necessarily a fifth, as the only consonance (in the judgement of the sense) intermediate in magnitude between fourth and octave.

This very argument of Aristoxenus is summarised by Ptolemy.a And he shows how weak it is. In fact, in a variation as tiny as that in question, between a semitone a

  Ptolemy, 1.10.

The Harmonics of the Ancients compared with Today’s

117

and a lemma (namely about 1/128 of a lemma) it is impossible (since even those intervals themselves hardly differ) for the ear to distinguish so precisely. And indeed if the ear were able on one occasion to be wrong to that extent, it would be much more significant in the whole course of the argument, where fourths are measured three times and ditones twice, in opposite directions, by the judgement of the ears. The result is that the judgement of the ears should not be accepted in such a subtle matter. And Euclid himself, elsewhere an Aristoxenian who repeatedly treads in his tracks in the Introductio harmonica,35 demonstrates in his Sectio canonis that the lemma, by which the fourth exceeds the ditone, is less than a semitone: but on the Pythagorean principle. And likewise Aristides Quintilianus (also an Aristoxenean).a Perhaps even Aristoxenus himself (who speaks hesitantly in places, as we said, and concludes his demonstration hypothetically, as we have seen: ‘if they are found consonant, etc.’) means something other than that the interval by which a fourth exceeds a ditone is (insofar as we can judge it by the hearing) a semitone. [The Pythagorean Alternative] The Pythagoreans by contrast reckoned the [p. 170] relations of sounds to each other by ratios, as Ptolemy reports,b together with the starting points they used. And although he may note that some people had done this less cautiously in some respects, nonetheless he denies it should be rejected as a method; he demonstrates it himself, and he contents himself with correcting what had carelessly been done wrongly by others, and confirms what had been rightly set up. And he does not dismiss the judgment of his ears, but confirms his case thereby. The most important part of the matter returns here. The Pythagoreans considered sounds compared with each other, more or less consonant or dissonant, that is pleasing or displeasing to the ears, according to the accuracy of the ratios between them.36 And so they connected the more significant ratios to the more significant consonances. In particular they assigned the duple ratio, 2 to 1, the first after the ratio of equality, to the octave, the most pleasing of all, at least after the unison; the sesquialtera, 3 to 2, to the fifth; the sesquitertia, 4 to 3, to the fourth. And just as the octave is composed of the fifth and fourth, of which the fifth is the larger, so the duple ratio is made up of the sesquialtera and sesquitertia ratios, of which the sesquialtera is the larger: since 3/2 × 4/3 = 2/1. And so they assigned to a tone, by which a fifth exceeds a fourth, the sesquioctave ratio, since 4/3 × 9/8 = 3 c /2. To explain all this, four numbers were normally used (the minimum for this purpose): 12, 9, 8, 6. For 12 to 6 is the octave, 9 to 6 (and 12 to 8) the fifth, 8 to 6 (and 12 to 9) the fourth, and 9 to 8 the tone. And the rest were determined from these few things that had been established. a

 At the start of book 3.   Ptolemy, 1.5 and what follows. c  He goes into greater depth on this in 1.5 and 1.7. b

John Wallis: Writings on Music

118

Ptolemy, however, confirms this mainly because if lengths of strings (other things being equal) were to be in the duple ratio, they would display to the ears an octave; and if sesquialtera, a fifth; if in sesquitertia, a fourth; if in sesquioctave, a tone; and the others similarly. Thus depth of pitch of sounds is always considered proportional to the lengths of strings, while height of pitch is the opposite, reciprocala to the same lengths.b, 37 From this basis it is easy to overturn the doctrine of those who make a fourth to be two tones and a half, a fifth three-and-a-half, an octave six tones, and what follows from this.c Instead of the semitone what is called the ‘lemma’d is to be substituted, that is the ‘remainder’ (namely what remains after taking two tones from a fourth), with the ratio 256/243: which is a little less than a semitone. But a tone cannot be divided into two or more equal parts (consonant ones, that is) of this kind.38 I believe this method – judging the differences between sounds by ratios, not intervals – is what Ptolemy callse the ‘harmonic canon’,f with which differences of sounds can be reckoned more precisely than the sense can perceive. And it is from this that those who spoke thus are called canonists;g the others, musicians.h How far this method is more powerful than the Aristoxenean (measuring sounds by intervals) Ptolemy shows by many things.i And what Aristoxenus and those who speak on his side call ‘diastems’j (‘distances’ or ‘intervals’) Ptolemy calls by other names, always avoiding that word (in his own explanations, not theirs). He calls them differences,k excesses,l magnitudes,m species,n and ratioso or the like: never (as far as I remember) ‘diastems’p (at least if the last three chapters were not written by Ptolemy himself, as the Scholium at the start of book 3, chapter 14 suggests).39 But later musicians, particularly practical musicians, right down to our own time mostly side with Aristoxenus, measuring intervals by tones and semitones. a

  ἀντιπεπονθότα.  This he goes into in depth in chapters 8 and 11, adding the monochord canon. c  This he shows in chapters 10 and 11. d   λεῖμμα. e   Ptolemy, 1.2. f   κανὼν ἁρμονικός. g   κανονικοί. h   μουσικοί. i   Ptolemy, 1.9. j   διαστήματα. k   διαφοραί. l   ὑπεροχαί. m   μεγέθη. n   εἴδη. o   λόγους. p   διαστήματα. b

The Harmonics of the Ancients compared with Today’s

119

Yet by ‘semitone’ they mean ‘lemma’, which is not very different from a semitone, and when necessary they say so. They call the difference between a lemma and a whole tone an apotome, with a ratio of 2187/2048: this is usually called a major semitone and the former a minor semitone.a And they call the difference between them a ‘comma’.b But those who deal with ‘speculative’ music more precisely, designate the differences between sounds by ratios, in the Pythagoreans’ way. But enough of these things. [The ‘Species’ of Consonances] Next we are to consider what they call species – which they also call figures – of each consonance,c namely fourth, fifth, and octave.d In order to understand these better, let us take two conjunct tetrachords divided alike according to same genus, whatever it may be: say (effectively) from hypate hypaton to mese, which we usually sing to the names mi fa sol la fa sol la. In other words, from the bottom or lowest-pitched note mi we sing (ascending to higher pitches) three intervals twice, which we signify by the names fa sol la, fa sol la. It is certain that, for tetrachords similarly divided to the one we have here, it is the same magnitude of consonance from hypate to hypate as from parhypate to parhypate, and the same from lichanos to lichanos and from hypate meson to mese. [For,] since the magnitude is equal (for equal tetrachords similarly divided) from hypaton hypate to [hypaton] parhypate and from meson hypate to [meson] parhypate, something intermediate added to both will have the same property: for instance, from hypate hypaton to hypate meson and from parhypate hypaton to parhypatee meson [will also be the same magnitude]; and for the same reason from lichanos hypaton to lichanos meson, and even from hypate meson to mese.40 But their constituent [intervals] are, of course, distributed differently, since the smallest one of all, which [p. 171] falls in the lowest position of one (say, the one from hypate to parhypate) is in the highest position of the other, and similarly for the rest. And this diverse order or distribution of parts is what is called species or figure. Ptolemy writes: ‘[A species is] how (i.e. in what order) are distributed within their specific boundaries the particular (i.e. the characteristic) ratios of each genus’.f And what he calls the characteristic ratiosg of each genus are, for the octave and fifth, a diazeuctic tone (which we designate with the term mi); for the

a

 So Boethius, 2.27, 2.29.   Boethius, 3.6. c   εἴδη καὶ σχήματα. d  Aristoxenus, pp. 6, 74; Euclid, pp. 13, 14, 15, 16; Gaudentius, pp. 18, 19, 20; Ptolemy, 2.3. e  Original has paranete. f   Ptolemy, 2.3: εἶδός ἐστι ποιὰ θέσις τῶν καθ᾽ ἕκαστον γένος ἰδιαζόντων, ἐν τοῖς οἰκείοις ὅροις, λόγων. g   λόγους ἰδιάζοντας. b

John Wallis: Writings on Music

120

fourth, the two leading sounds,a that is, the highest-pitched (which we designate with the term la).41 Here ‘leading’b is said of high [pitches] and ‘following’c of low ones;d although elsewhere, even in the same writers, low pitch is held to precede and high to follow. As in Ptolemy book 2, chapter 5, where ‘before’e is said of the lower pitch and ‘after’f of the higher. And Aristidesg says ‘lowest-pitched’h for ‘first’,i and ‘highest-pitched’j for ‘last’k. And I advise readers to take care which sense should be taken in which place. But there are as many species in each of these consonances as there are intervals or ratios from which they are held to be made up. And so in a fourth (since three intervals are made by four strings) there are three species; in a fifth (for a similar reason) four species; in an octave, seven. In each genus the species which has its characteristicl ratio or note (that is, la in a fourth and mi in a fifth or octave) in first place from the top, is called the first species; if it is in second place it is called the second; if in third place the third, and so on. And so, in the fourth, the first species is the one contained by sounds that are, as they say, barypycni:m, 42 from hypate to hypate, which we sing (from the lowest note, going upwards) to the notes fa sol la. The second is contained by mesopycni:n from parhypate to parhypate, sung to sol la fa. The third is contained by oxypycni:o (from lichanos to lichanos), sung to la fa sol. The species from hypate meson to mese (sung as fa sol la) is not different from the first, since all [the intervals] recur in the same order from hypate hypaton to hypate meson and from there to mese, and so it does not form a fourth species but repeats the first. We do not include the lowest note from which [the system] ascends in our scheme, or at least we write it in a different character, because here we describe not so much sounds as ratios or intervals. a

  ἡγούμενοι.   ἡγούμενος. c   ἑπόμενος. d   Ptolemy, ibid. and 1.12, 1.13, 1.14, 1.15, 1.16, et passim. e   πρό. f   κατά. g  Aristides, pp. 10, 11. h   ὕπατος. i   πρῶτος. j   νῆτος or νέατος. k   ἔσχατος. l   ἰδιάζων. m   βαρύπυκνος. n   μεσόπυκνοι. o   ὀξύπυκνοι. b

The Harmonics of the Ancients compared with Today’s

121

So let us take two disjunct tetrachords (with a diezeuctic tone interposed, which we designate as the note mi), so that we have a fifth.43 For example, from hypate meson to nete diezeugmenon, where after first singing fa sol la we sing mi and then fa sol la again. And so in the consonance of a fifth there are (as it is easy to gather from what has already been said about the fourth) four species or figures, according to whether its characteristic note mi is in the first, second, third, or fourth position from the highest pitch. They are sung, after the lowest pitch, to the notes: 1. fa sol la mi; 2. sol la mi fa; 3. la mi fa sol; 4. mi fa sol la.

The next species we would expect [in this diagram] – c d e f, to be sung, after the lowest note mi, to the notes fa sol la fa – would not be a species of the fifth, but what is called a false fifth: because the diazeuctic tone (designated mi) is not part of it, except if by mutating the key (as it is called) and putting a sharp sign on f, the note which would otherwise become fa turns into mi. (For putting the note mi at the bottom and rising from there leaves the diazeuctic tone, placed below it, outside the pentachord.)

John Wallis: Writings on Music

122

Finally let us take two conjunct octaves, with which to construct the species of the octave. Namely from proslambanomenos to nete hyperbolaeon, sung to the notes la, mi, fa sol la, fa sol la, mi, fa sol la, fa sol la. In these we will find seven species of octave, by the same method as before. If we want to go beyond this, the first species repeats itself at the octave, as above for the fourth. And if we go backwards [to the octave] which is bounded by the strings A and a, this will be nothing but the seventh species repeated. For from proslambanomenos to mese the same [intervals] recur in the same order as from mese to nete hyperbolaeon. [p. 172]

And it is from this (the fact that two octaves are needed in order to have every species of that consonance) that Ptolemy does not have the octave as a perfect system, as earlier writers had thought, but the double octave.a Although after the single octave, effectively the same sounds return (which he deals with himself when he considers tones or modes and their number),b nonetheless its species are not all held within the range of one octave, which he argues should happen in the perfect system. a

  Ptolemy, 2.4.   Ptolemy, 2. 8, 2.9.

b

The Harmonics of the Ancients compared with Today’s

123

And these seven octave species are each given a name. The first is called Mixolydian, the second Lydian, the third Phrygian, the fourth Dorian, the fifth Hypolydian, the sixth Hypophrygian, the seventh either Locrian or Hypodorian and common.a [The Modes] Finally, it remains to talk about modes or tones.b ‘Tone’c occurs in at least four senses:d 1) for a sound, as in ‘heptatonic cithara’,e the name given to one with seven sounds or strings; 2) for an interval, as when the fifth is said to differ from the fourth by a tone (i.e. the sesquioctave ratio [9 : 8]) or mese to stand a tone away from paramese; 3) for a position of the notes, like the tones called Dorian, Phrygian, Lydian, etc., in the sense just considered (some call them ‘modes’, with the same sense); 4) for a tension [pitch register], as when someone is said to use the high-, low- or middle-pitched tone (or tenor or tension) of the voice.f And so ‘mode’ or ‘tone’, as they are used here, refer to the position[s] of the notes, when not a single sound but a whole series or system of notes is sung, higher- or lower-pitched. Just as, for us, mi is sung now in b fa b mi, now in e la mi, now in a la mi re, etc., so for them it effectively appeared, to coin a phrase, ‘in the position of paramese’ (which amounts to the same thing as our mi), now in paramese position, now in nete diezeugmenon position, now in mese position, etc. In other words, the sound which by reason of the song takes the place of paramese, is taken at that pitch level which on another occasion might be either paramese itself, or nete diezeugmenon, or mese, etc. And the remaining sounds of the whole song are allotted their pitch according to their ratio to this one.44 Thus the modes, or ‘tones’ in this sense, are potentially infinite in number, like sounds.g Indeed, like sounds, so too groups of sounds can move in height or depth of pitch, more or less, with unlimited variety; since, as said before, any space of notes can hold innumerable sounds (as a line does points).45 And like a point so a whole line can be moved through a greater or smaller distance ad infinitum; so too sounds and groups of sounds, so that for this reason it is impossible to specify the number of tones. In reality and in practice, a host of modes is usually defined, differently by different people, either considering the distance between the tones furthest apart or the number of tones between those bounds or by the ratio or interval by which those tones are separated from each other. a

 Euclid, pp. 15, 16; Gaudentius, p. 28; Bacchius, pp. 18, 19.   Thus Alypius, p. 2: τοὺς λεγομένους τρόπους τε καὶ τόνους. c   τόνος. d  As Euclid notes, p. 19. e   ἑπτάτονος φόρμιγξ. f   ὀξυτονεῖν, βαρυτονεῖν, ἤ μέσῳ τῳ τῆς φωνῆς τόνῳ κεχρῆσθαι. g  As Ptolemy says, 2.7. b

John Wallis: Writings on Music

124

The ancients set up at least three modes from the start:a Dorian, Phrygian, and Lydian, each of which was one tone away from the next: that is, the sesquioctave ratio [9 : 8]; and so the outer two were two tones apart. The Phrygian mode was one tone higher than the Dorian, and equally the Lydian was one tone higher than the Phrygian; so the Lydian was two tones higher than the Dorian. Aristoxenusb set up thirteen modes, which were as follows: Hypermixolydian (also called Hyperphrygian); two Myxolydians, higher- and lower-pitched (of which the higher is second in order and is sometimes called Hyperiastian, and the lower is sometimes called Hyperdorian); two Lydians (also called Aeolians), higher- and lower-pitched; two Phrygians, one low-pitched (also called Iastian), the other high-pitched; one Dorian; two Hypolydians, higher- and lower-pitched (the latter also called Hypoaeolian); two Hypophrygians (the lower-pitched of which is also called Hypoiastian); Hypodorian. The lowest of these [p. 173] is the Hypodorian; each of those which follow, from lowest- to highest-pitched, is higher than its neighbour by a semitone. Others set up fifteen modes, separated each from another by a semitone.c In fact, above the thirteen already mentioned two higher-pitched ones are added, the Hyperlydian and Hyperaeolian. This is what they mean. The proslambanomenos of the Hypodorian mode, the lowest-pitched of all, they consider the lowest-pitched possible sound that either the human voice can produce or the ear judge distinctly: and so all lower things are not sounds but bangs or indistinct murmurs. They make the proslambanomenos of the Hypoiastian (or the lower Hypophrygian) a semitone higher than that of the Hypodorian (and so the hypate of the latter is a semitone higher than that of the former, and so on for the other [notes]): thus the Hypoiastian proslambanomenos is in a middle position between the Hypodorian proslambanomenos and hypate hypaton. They make the proslambanomenos of the higher Hypophrygian one semitone higher still, and thus a whole tone higher than that of the Hypodorian with whose hypate hypaton it will coincide. And so on for the rest, as is seen in the table at the foot of Ptolemy, book 2, chapter 11. Ptolemy argues against those who make the tones or modes rise by semitones in this way.d And he shows that the use of the various modes was not only introduced so that the tenor of an entire melody could be made higher- or lower-pitched: indeed, a higher- or lower-pitched voice for the singer, or the accommodation of the musical instrument to such ranges, would suffice for that. Indeed, if attention were paid to this alone, one could add to the fifteen modes already set out as many more as were desired, as far as the stretche of the human or instrumental voice permitted, and insert in between as many more as were desired. a

 As Ptolemy says, 2.6 and elsewhere.  Referred to by Euclid, p. 19. c  Alypius, p. 2 and Meibom on this passage, which we repeat in what was set out on Ptolemy, 2.11. d   Ptolemy, 2.7, 2.8, 2.9, 2.10, 2.11. e   διάτασις. b

The Harmonics of the Ancients compared with Today’s

125

But for this reason there was introduced a way to make a transition from mode to mode in the course of one song, which they call mutation of mode:a we do it by changing the key signature with the addition of sharp or flat signs. So it is necessary to take care not so much that the modes be separated by constant intervals (tones or semitones) but that the transition be made through consonances (fourth, fifth, or even octave). Not that the transition is always into the nearest mode, either a tone or semitone away, but more often and more pleasingly to one which differs from the first by a consonance (fourth, fifth, or octave). But a mode which differs from another by a octave should not be considered so much another mode as the same, or almost the same, continued; for the same reason that sounds differing by a octave were said above to be considered the same or almost the same.46 (This is exactly the same as saying, in today’s music, that mi is placed on B mi or b fa b mi or bb fa bb mi: it does not change the mode; for if mi became another of them the others would still be treated the same if the melody reached them.) Also, for the same reason, modes which differ by more than an octave (namely by a octave-plus-fourth or octave-plus-fifth) should be treated like ones which differ by the the same amount minus the octave: that is, by a fourth or fifth. And so the most distant modes are limited to an octave, since those there or beyond are to be considered no different from those within. And because of this, [Ptolemy] says there are altogether seven distinct modes: Mixolydian, Lydian, Phrygian, Dorian, Hypolydian, Hypophrygian and Hypodorian.47 Today’s music admits the same number, with different key signatures.b (Glarean, though, in a good volume called Dodecachordon, argues for twelve modes, as you will see. In his time the Ptolemaic tense diatonic genus, which Zarlino later recalled, had not yet been reintroduced, so everyone spoke in the Aristoxeneans’ manner, dividing intervals into tone, tone and semitone (or lemma) and moving up through the modes by semitones.) [The Modes and the Modern Key Signatures] Ptolemy, however, uses this method in designating his seven modes (which ours also follow by changing key signatures). To understand these better at this point, the schema which appears in the note on Ptolemy, book 2 chapter 11 should be consulted.48 First of all he treats the Dorian mode (which is the middle one), because it has the effective mese in its natural place; and so too paramese, which is our mi: that is, as he says, ‘the effective mesec in the position of mesed’, and so the effective paramese in the position of paramese (that is, as we say, mi in b fa b mi), and a

  μεταβολὴ κατὰ τόνον.  As we show in the note on Ptolemy, 2.11. c   μέση δυνάμει. d   μέση θέσει. b

John Wallis: Writings on Music

126

similarly for the rest. With us this corresponds to an empty key, without flat or sharp signs. Second he takes the mode a fourth higher than this, because it therefore has its effective mese, a fourth higher, in the position of the Dorian paranete diezeugmenon, and so the [effective] paramese, which is our mi, is nete diezeugmenon. That is, as we say, mi is in e la mi; and he calls this mode Mixolydian. For this we put a mollis sign, a ‘flat’,49 on b fa b mi, excluding mi from there, and place mi in e la mi. From this he concludes elsewherea that there is no need for the system they call conjunct, from proslambanomenos to nete synemmenon, since it is enough to make the transition, in mese, from the Dorian mode to the Mixolydian. Indeed, after two conjunct tetrachords in the Dorian mode, from hypate hypaton to mese, that is from B mi to a la mi re, there follows [p. 174] a third in the Mixolydian from its hypate meson (the Dorian mese) to its mese, that is from a la mi re to d la sol re: so there are now three conjunct tetrachords, namely from B mi (the Dorian hypate hypaton) to d la sol re (the Mixolydian mese). Third, since he cannot take another higher fourth from there without exceeding the octave in the middle of which he had set the Dorian mese, he takes instead a lower fifth (equivalent to taking a higher fourth, because the sounds taken in this way, differing by an octave, can be treated as the same). So the mese of this mode (a fifth lower than that of the Mixolydian) is the Mixolydian lichanos hypaton, that is, Dorian lichanos meson; and the paramese is the Dorian mese: that is, in our terms, mi is in a la mi re. And he calls this mode Hypolydian. For this, after the first flat sign already placed on b fa b mi, we place a second flat sign on e la mi, excluding mi also from there, and we move mi to a la mi re. Fourth, since from here he could take neither a lower fifth nor a lower fourth without exceeding the octave previously mentioned, he takes a mode higher by a fourth than the Hypolydian, and calls it the Lydian. So its mese is the Hypolydian paranete diezeugmenon, and its paramese is the Hypolydian nete diezeugmenon, that is the Dorian paranete diezeugmenon; in our terms, mi is in d la sol re. For this, after the two already placed on b and e, we place a third flat sign on a la mi re, so that mi, excluded from there, moves to d la sol re. Fifth, just as the Mixolydian was taken a fourth higher than the Dorian, so from the same Dorian he takes the (low-pitched) Hypodorian a fourth lower than the Dorian. So its mese is the Dorian hypate meson, and its paramese (which is our mi) is the Dorian parhypate meson, that is, in our terms mi is in F fa ut. To signify this, we put a sharp sign (a ‘sharp’)50 on F fa ut (and omit flat signs), so that the string which would otherwise have to rise a semitone and be treated as fa, is raised a whole tone above the neighbouring string and treated as mi. Thus the next higher string will be a semitone away and be called fa; mi eventually returns, after an octave, in the higher-pitched f fa ut. Sixth, from the Hypodorian placed thus, since another lower fourth cannot be taken without exceeding the said octave, the Phrygian mode is taken a fifth higher a

  Ptolemy, 2.6.

The Harmonics of the Ancients compared with Today’s

127

(which amounts to the same thing, a octave distant). So its mese is the Hypodorian nete diezeugmenon, that is the Dorian paramese; and its paramese is the Dorian trite diezeugmenon: that is, in our terms mi is in c fa ut. For this, as well as the sharp sign previously placed on F fa ut, another sharp sign is placed on c fa ut. So the string which used to rise a semitone from its neighbour and was treated as fa now rises a whole tone and is called mi. It is therefore only a semitone below d sol re ut, and the latter is, for this reason, treated as fa, with mi returning again either as cc sol fa above or C fa ut below. Seventh, from the Phrygian placed thus the Hypophrygian is taken a fourth lower. So its mese is the Phrygian hypate meson, that is the Dorian parhypate meson; and its paramese, which is our mi, is the Dorian lichanos meson: so, in our terms, mi is in G sol re ut. For this a third sharp sign is needed for G sol re ut, leaving the others untouched. So the string which (by previously making F fa ut sharp), had to rise a semitone and be treated as fa now rises a whole tone and is called mi. And a la mi re, a semitone higher than G sol re ut, is called fa, with mi returning on g sol re ut above and Γ ut below. After establishing these, it is founda that the distance from the Mixolydian mode to the Lydian is a lemma, or loosely speaking a semitone. From here to the Phrygian is a tone; to the Dorian is another tone; from Dorian to Hypolydian is a lemma; from Hypolydian to Hypophrygian a tone; and from there to the Hypodorian is also a tone. Finally he concludes that these seven tones are enough and there is no room for more, since all the strings of the aforementioned octave are now occupied. Since each string from hypate meson to paranete diezeugmenon (inclusive) is the mese of some mode, the question does not arise of what the mese would be of some mode falling between the seven. In other words, since the effective mese of the Hypodorian is in the position of hypate meson, and that of the Hypophrygian in that of parhypate meson, and no string lies between these, there is no question of a mese of an intermediate mode called (as Aristoxenus supposes) the Hypoiastian or lower Hypophrygian. And what can be said of mese can similarly be said of paramese, our mi. In other words, when we use our terms, with the modes just established the note mi appeared on every string. For in the Hypodorian mi is in F (and so also in the f an octave away from it); in the Hypophrygian it is in G (and so also in Γ and g an octave away); in the Hypolydian it is in a (and so also in A and aa); in the Dorian it is in b (and so also in B and bb); in the Phrygian mi is in c (and so also in C and cc); in the Lydian it is in d (and so also in D and dd); and finally in the Mixolydian it is in e (and so also in E and ee). And so the question does not arise of another string for mi, in some other mode which coincides with none of these. Some people add to these seven an eighth, the Hypermixolydian, but Ptolemyb says this is nothing other than the Hypodorian, from which it differs by an octave. And thus enough about tones or modes. a

 As in chapter 10.  ibid.

b

John Wallis: Writings on Music

128

[Melopoetics] Now we have analysed almost the whole harmonics of the ancients. Euclid sets out the seven parts of it: concerning sounds, intervals, genera, systems, tones, modulations, and melopoetics.a [p. 175] All of these we have now dealt with except melopoetics, which today we call the composition of song.51 Concerning this they look at what was said above about consonances, since in sequences of notes, the more consonant should be preferred, other things being equal, to the less consonant. Just as in putting letters together to make words, not every one is suitable to follow every other, but a choice is made; equally, in the composition of a song, not every interval can properly follow every other, but only as the nature of song demands.b There is no consideration here of what is appropriate in the construction of the diagram or, as we now say, scale. For instance, that after two dieses or two semitones (say in the chromatic or enharmonic genera) a third cannot be placed immediately above, but the least that can follow above is the remainder of a fourth; nor anything smaller than a tone, below. Rather, they consider the intervals which can follow in turn in the course of a melody, whether high- or low-pitched: either sounds adjacent in the scale or ones separated to some extent. As for instance that after a dense or a non-dense interval (such as between hypate and lichanos in any genus) no smaller interval can be sung next than – upwards – what remains of a fourth or – downwards – a tone.c There are many things of this kind throughout almost the whole of the third book of Aristoxenus: for instance, a dense interval is not to be sung next to another dense interval; neither the whole nor any part of it.d Two ditones should not be placed in succession,e nor, in the [en]harmonic or chromatic genus, two tones,f and not in the soft diatonic either. But it is different for the intense diatonic, where three tones may follow in succession, though no more.g And many other things of this kind. He deduces almost all of them from this assumption: that (in the whole diagram) either the fourth [string] from a given sound must make a fourth, or the fifth a fifth, whether or not, indeed, a diazeuctic tone comes between.h This being established, he draws the rest of his conclusions. Here, too, they consider the various names for the ‘flows’ or movements of the voice, whether high or low in pitch: such as ‘flowing’, ‘twining’, ‘repetition’,52 a

  Euclid, p. 1: περὶ φθόγγων, περὶ διαστημάτων, περὶ γενῶν, περὶ συστημάτων, περὶ τόνων, περὶ μεταβολῆς, περὶ μελοποιίας. And similarly, Alypius, p. 1; Aristoxenus, pp 34, 35, 36, 37, 38. b  Aristoxenus, p. 27. c  Aristoxenus, pp. 27, 28, 29 and again pp. 52, 53, 54. d  Aristoxenus, p. 62. e  Aristoxenus, p. 63. f  Aristoxenus, p. 64. g  Aristoxenus, p. 65. h  Aristoxenus, pp. 58, 59.

The Harmonics of the Ancients compared with Today’s

129

‘extension’, etc.a Likewise ‘dissolution’, ‘projection’;b again, ‘remission’, ‘intension’, ‘staying’, ‘standing still’;c again, synaphe, diazeuxis, hypodiazeuxis, episynaphe, hypsynaphe, paradiazeuxis, hyperdiazeuxis; and again prolepsis, eklepsis, prolemmatismos, exlemmatismos, melismos, prokrousis, ekkrousis, prokrousmos, ekkrousmos, komismos, teretismos, diastole,d and other names of this kind which are found in Bacchius and Bryennius.e But I do not feel it worth dwelling on these. It remains to say that the melopoetics of the ancients was simple and (at least as far as I can see) used only one voice, as we would say. So its ‘harmony’f pertained to the sequence of sounds, namely how a preceding sound related to the following one. According to Aristoxenus: the understanding of music arises from these two things: sense and memory. For it is necessary to sense what is happening, and to retain in the memory what has happened. Otherwise it is impossible to grasp what happens in music.g Hence, from the collation of an earlier sound with the one being heard now, we start to hear ‘harmony’. But what is found in music nowadays, the consent of two, three, four, or more so-called parts or voices (sounds concinnous among themselves which are heard at the same time), was unknown to the ancients as far as I can see. Some words which appear in Ptolemy may seem to hint at such a thing, which he says is needed for the division of the monochord more than for other instruments, since there is only one hand plucking and it cannot reach places that are far apart at the same time.h This makes me think that several strings were sometimes struck at the same time; nonetheless I think it happened rather rarely, for perhaps one or two sounds: not in, as they say, continuous parts agreeing with one another (our bass, tenor, countertenor, descant, etc.), or the so-called divisions or twitterings which harmonise in a

  ἀγωγή, πλοκή, πεττεία, τονή. The meaning of these names is to be found in Euclid, p. 22; Aristoxenus, p. 29; Bryennius, 3.10. b   ἔκλυσις, ἐκβολή. c   ἄνεσις, ἐπίτασις, μονή, στάσις. d   συναφή, διάζευξις, ὑποδιάζευξις, ἐπισυναφή, ὑποσυναφή, παραδιάζευξις, ὑπερδιάζευξις: πρόληψις, ἔκληψις, προλημματισμός, ἐκλημματισμός, μελισμός, πρόκρουσις, ἔκκρουσις, προκρουσμός, ἐκκρουσμός, κομπισμός, τερετισμός, διαστολή. e   Bacchius, pp. 9, 11, 12, 13, 14, 19, 20, 21, 22; Bryennius, 3.3. f   concentus. g   Aristoxenus, pp. 38. 39: ἐκ δύο γὰρ τούτων ἡ τῆς μουσικῆς σύνεσίς ἐστιν, αἰσθήσεώς [sic] τε καὶ μνήμης· αἰσθάνεσθαι μὲν γὰρ δεῖ τὸ γενόμενον· μνημονεύειν δὲ τὸ γεγονός· κατ᾽ ἄλλον δὲ τρόπον οὐκ ἔστι τοῖς ἐν τῇ μουσικῇ παρακολουθεῖν. h   Ptolemy, 2.12: ἐπιψαλμός, σύγκρουσις, ἀναπλοκή, καταπλοκή, σύρμα, καὶ ὅλως ἡ διὰ τῶν ὑπερβατῶν φθόγγων συμπλοκή.

130

John Wallis: Writings on Music

later song. Of these I find scarcely any trace, and not one for certain, in the music of the ancients. And so I am completely convinced that neither was the ancients’ music more accurate than ours, nor did those prodigious effects which they often recall (by Orpheus, Amphion, Timotheas, etc.) ever touch the minds of men: except, you might say, through the exaggeration of historians, or perhaps they touched an inexperienced people because of the great rarity of music, rather than its excellence.53 For the moment I readily admit this, since today’s musicians play almost for themselves alone, to please the soul, rather than, as the ancients seem to have affected, to move the emotions to and fro. It is altogether possible that they were more skilled than us in moving the emotions. Furthermore, their simpler music, with a single voice, did not obscure the text set as our more compendious music does; hence, for instance, tragic texts being produced with tragic acting, tragic verses and tragic sound, all of which they put together in their music, [p. 176] it is little wonder that they aroused tragic emotions, no differently from us when performing tragedies: and likewise for the other emotions. But our composers give up these precepts for composition, whether for simple songs or for several voices sounding together, and concerning the consent of sounds either in sequence or simultaneously (on which we have received very little from the ancients); and it is not now necessary to pursue these things further, nor does the brevity demanded by the present work allow it. So, enough concerning these matters. [The Modern Scale] Finally it remains to discuss a question to which we return here. Concerning the genera which were mentioned above and their colours, we said above that only one out of all of them has, for many centuries, been accepted in use; and everyone agrees this is the diatonic (leaving out the enharmonic, all the chromatic and the other diatonic species). But there is doubt whether this is what is called the tense diatonic of Aristoxenus, or the ditonic diatonic of Ptolemy, or the tense diatonic, also of Ptolemy. So, when, from a low-pitched mi or la we sing a higher fourth, with the notes fa sol la, remarking the same number of intervals, how large is each interval? The first view is that of Aristoxenus: since he reckons a fourth to consist of two tones and a half, he wants the interval of fa, the lowest interval, to be a semitone, and the other two, sol and la, whole tones. This is Aristoxenus’ tense diatonic, and almost all musicians speak in this way, down to this day, at least where they speak loosely. But those who treat the matter with more precision understand by ‘semitone’ not exactly half a tone but something smaller. Euclid, though an Aristoxenean, demonstrated this: though I do not know if he was the first. Having accepted the Pythagoreans’ principles about measuring the intervals of sounds by ratios,

The Harmonics of the Ancients compared with Today’s

131

and their tone with a sesquioctave ratio [9 : 8] (though he mentions tones and semitones like the Aristoxeneans in his Introductio harmonica),54 he shows in his Sectio canonis that what remains of a fourth after two tones is less than a semitone. He then began to call this interval a lemma, with the ratio 256/243.55 Now, if a fourth contained two tones and a half, an octave, which is two fourths and a tone, would contain six tones. But an octave, whose ratio is duple, is less than six tones, because the sesquioctave ratio six times composed is more than the duple ratio. An octave, therefore, is less than six tones and a fourth less than two-and-a-half. 9 /8 × 9/8 ×9/8 ×9/8 ×9/8 ×9/8 = 531441/262144 > 524288/262144 = 2/1. The next opinion is that of those who, in place of tone and semitone, substitute tone and lemma; so if ever they say semitone they mean lemma, which differs a little from a semitone. Thus they understand the note fa to signify a lemma, sol and la two tones. That is, 256/243 × 9/8 × 9/8 = 4/3. This is Ptolemy’s ditonic diatonic, though it was described by Euclid already, before Ptolemy. (And there was even Eratosthenes’ diatonic, as mentioned above.) Musicians almost down to our own time have seen things this way. And Ptolemy himself, although he gives the other diatonic genera, does not reject this, reporting all the others who talk in this way about the matter rather than keeping only to his own pleasure, as he himself says.a And Boethius divided the tetrachord in this way, and after him Guido Aretinus, Faber Stapulensis, Glarean, and others.56 Until, at the end of the last century, Zarlino (perhaps the first, around a hundred years ago), Kepler and others decided that Ptolemy’s tense diatonic should be taken up again: although Vincenzo Galilei argues against Zarlino in his dialogue on music. Thus there is a third opinion: that of those who, following Ptolemy, substitute the sesquidecimaquinta ratio (16/15) for the semitone or lemma; and call this, too, a semitone. And in place of the second tone (the ratio 9/8 was previously given to both tones) they have a so-called minor tone, with the ratio 10/9. Thus they make up the fourth from the ratios 16/15 × 9/8 × 10/9 = 4/3, the note fa indicating the ratio 16/15, sol 9/8 and la 10/9. This is Ptolemy’s tense diatonic, and it is the same as Didymus’ diatonic except that the latter changes the order and has 16/15 × 10/9 × 9/8 = 4/3.57 And since they called 16/15 a major semitone, they gave the minor semitone the ratio 25/24, so that taken together with 16/15 it makes up a minor tone 10/9 = 16/15 × 25/24: and it is the difference between a so-called major third and minor third. (Mersenne adds two other semitones: namely one with the ratio 135/128, which with 16/15 makes up 9/8, a major tone; and one with the ratio 27/25, which with 25/24 again makes up 9 /8, a major tone.58) But since these genera, as we have called them, differ from each other so little that the ears’ precision can scarcely distinguish them, if at all (since the ratio 16/15 differs from that of the lemma, 256/243, and that of the major tone, 9/8, from the ratio 10 /9, by the ratio 81/80; which is so small that the ear can hardly distinguish it from the unison) we must judge by reason rather than sense which opinion should be a

  Ptolemy, 1.16.

John Wallis: Writings on Music

132

preferred (since sense will not be excessively troubled in permitting either). And reason favours the latter opinion quite considerably. [Pythagorean Theory Placed on a Physical Basis] So, to go through the matter again from the beginning. The view of the Pythagoreans is not displeasing: they hold that sound arises from the vibration of air that has been struck and deduce the physical cause of degree of pitch from the speed of this vibrating motion, so that the faster or denser are the comings and goings (of the vibrating string) the higher in pitch is the sound, and the slower and less frequent they are, the lower-pitched it is: likewise, following the Pythagoreans’ view, Porphyry (in his commentary on Ptolemy’s Harmonics) and others.59 And thus it is that one string trembles to the motion of the other when equally tense strings are placed next to each other: [p. 177] for the string is apt to make simultaneous movements, as though spontaneously, and so when it is moved by the very soft touch of the moving air, it undergoes a similar vibration.60 And so it seems that either this or something like it must be considered the physical cause of sounds’ pitch. Following this (or whatever is eventually put in its place, a question which it is inappropriate to discuss in detail here), it will be necessary to find the physical cause of consonance and dissonance by the same method. So, the more often the comings and goings of two strings, struck next to each other, coincide, the more pleasing those strings sound together to the ears; the more rarely (and so the more frequently they clash), the less pleasing they sound.61 Hence, you might say, two unison strings strike the ears most pleasantly of all; for, assuming that they either start at the same time or at least coincide at some point, the coincidence of their vibrations will last as long as they continue [to sound]. But if, through starting at different times or some other reason, there is no coincidence, a similar alternation will endure: or even, not improbably, each, struck with the other’s strokes (carried by the common air), will be impeded in its own motion by the other’s until, little by little, after a few strokes they will come together in motion or simultaneous vibration.62 Unisons. 1/1.

Next after equality comes the duple ratio, just as, after unisons come those sounds which sound a diapason, which we call octaves: since their speed or rather pitch is in the duple ratio. Every second vibration of the faster [string] coincides with each vibration of the slower, so out of every three vibrations – two from one string, the third from the other – two coincide.

The Harmonics of the Ancients compared with Today’s

133

Diapason. Octaves. 2/1.

After this, fifths, which sound a diapente; their pitch, speed, or frequency of return is in the sesquialtera ratio or as 3 to 2. Since every third [vibration] of the faster vibrating [string] coincides with every second of the slower, out of every five vibrations – three of one string, two of the other – two coincide. Diapente. Fifths. 3/2.

Finally fourths, sounding a fourth, with the sesquitertia ratio, or as 4 to 3: with coincidence in every seven vibrations of the two strings. Diatessaron. Fourths. 4/3.

These are all generally called consonances, and, by the judgement of the ears, they are more consonant in order as the numbers denoting their ratios are smaller, whence the coincidences of their vibrations are more frequent. After all these the tone follows after a long interval, whose ratio is the sesquioctave, or as 9 to 8: so after one coincidence another does not come until after seventeen vibrations of the two strings (namely nine of one and eight of the other). Tone. 9/8.

Whether this is the true physical cause of the pitch of sounds, which the Pythagoreans assume, not without reason, and others after them; and, further, whether the degree of pitch is proportional to the frequency of vibrations in a given time: I do not know if this has yet been adequately proved by experiment, and therefore I would not want to affirm it too boldly. But Ptolemy has showna that depth of sounds is certainly proportional, other things being equal, to the length of the strings (and so pitch is inversely proportional to length) and the phenomena prove it. Also that the ratios which are more pleasing to the ears, other things being equal, are those signified by smaller numbers, viz. 1/1 [is more pleasing] than a

  Ptolemy, 1.8.

John Wallis: Writings on Music

134

/1, this than 3/2, and this than 4/3; and equally for the rest. This is enough for our present purpose. According to this hypothesis, they consider the octave to be the simplest and most complete consonance: its sounds are in the duple ratio, as 2 to 1 (though some, on occasion, disagree about the name, whether it should be termed consonant or ‘idem-sonant’, same-sounding). And they want this to be so near a consonance that sounds which differ by an octave are treated as the same or almost the same: homophonesa in Ptolemy’s term, unisons in ours. (This is because a multiple ratio, twice composed, makes a multiple one, but no other ratio, whether superparticular or superpartient, twice composed, makes either a multiple or a superparticular ratio, which alone are considered consonances or at least simple ones.)63 And so an octave-plus-fourth they hold to be the same consonance as a simple fourth; likewise an octave-plus-fifth as a simple fifth; and similarly for the rest. This both Ptolemy and, before him, Aristoxenus wrote, in the places cited above. Thus there is no reason for us here to be concerned with intervals that are greater than an octave, since their ratios always turn out to be duple to those which they exceed by an octave. In other words, since the ratio of the fourth is 4/3, that of an octave-plus-fourth will be [p. 178] the duple to this, 8/3, and likewise for the rest. And so it is enough to consider the octave, and those intervals which are smaller than it and into which the octave is divided. Take the octave, for example, which lies in our scale from E to e, which we sing to the notes la, fa sol la, mi, fa sol la: so its end sounds are in the duple ratio, 2 to 1. 2

But this ratio of 2 to 1, since they were unable to divide it into two equal parts expressible in numbers (since no ratio expressible in numbers can result in the duple ratio when twice composed) they divided into two near-equal parts, representable by the smallest numbers possible.64 So instead of 2 to 1 they took the doubles, with the same ratio, 4 to 2, so that a 3 might come in the middle, dividing a

  ὁμόφωνοι.

The Harmonics of the Ancients compared with Today’s

135

the first ratio in two: 4. 3. 2 or 4/3 × 3/2 = 4/2 = 2/1. And so the octave la la, from E to e, they divided into two intervals, fifth and fourth, with the ratios 3/2 and 4/3: namely either into la mi (from E to b) and mi la (from b to e), with the fifth in the lower part and the fourth in the higher; or into la la (from E to a) and la la (from a to e), with the fourth in the lower part and the fifth in the higher, singing la mi la or la la la, according to whether the central tone la mi (with the ratio 9/8), was taken with the lower or higher fourth (with the ratio 4/3), since 4/3 × 9/8 = 3/2. And so, of the eight sounds of the whole octave, four, at least (designated by the notes la la mi la) are defined by their ratios to one another: that is, as the numbers 12, 9, 8, 6; dividing the octave into two fourths with a tone interposed.

Thus far went Pythagoras, unless I am mistaken, and, I believe, no further; determining these four intervals – octave, fifth, fourth, and tone – by the ratios 2/1, 3 /2, 4/3, and 9/8. And Boethius records that the tetrachord of Mercury was like this, and after him Zarlino.a But Pythagoras did not define the four remaining sounds (designated by the notes fa sol fa sol), nor is there agreement among others concerning their division, that is, the division of each tetrachord into three intervals. Very many, following Euclid, divide a tetrachord this way in the diatonic genus (omitting the others mentioned earlier): into tone, tone and lemma, in fact in such a way that the interval designated by the note fa (whether from E to F or from b to c) is always a lemma, with the ratio 256/243, all the rest being whole tones with the ratio 9/8, as we have often said. But this is found to be somewhat inconvenient. For despite the fact that 256/243 is neither a multiple nor a superparticular but a superpartient ratio (namely 113/243, one-and-thirteen-two-hundred-and-forty  Boethius, 1.20; [Gioseffo] Zarlino [Le istitutioni harmoniche (Venice, 1562)], vol. 1, part 2, chapter 1 [pp. 58–9]. a

John Wallis: Writings on Music

136

thirds), they rely on sounds which are close, at least as much as possible.a This, further, turns out to be inconvenient because it is reckoned by such large numbers that after one coincidence of vibrations, they do not coincide again until a further 499 vibrations of the two strings have passed. This seems unharmoniousb and very discordant, yet when the sounds mi fa or la fa are heard next to one another we hear nothing horrible or unwelcome to the ears. But after the sesquitertia ratio 4/3 = 11/3 there follow many superparticular ratios before the sesquioctave, which is a tone: namely the sesquiquarta, sesquiquinta, sesquisexta and sesquiseptima (5/4, 6/5, 7/6, 8/7), all of which this division of the tetrachord overlooks as discordant even though it retains the 256/243 of the lemma. Again, the interval mi sol or la sol, the so-called tri-hemitone which they call a minor third, from b to d or E to G, will by this division have the ratio 32/27 = 256/243 × 9 /8. But the ear testifies that this interval is more consonant than a tone, whose ratio is 9/8: and so its ratio should be expressed in smaller numbers. Likewise the interval fa la, called a ditone, which they call a major third, from F to a or C to e, will have the ratio 81/64 = 9/8 × 9/8. This [ratio] is less concordant not only than a tone but also than the minor thirds just mentioned; but this interval, albeit by the judgement of the ears, is more consonant than either the tone or the tri-hemitone. So, since this division of the tetrachord, which is Euclid’s and Eratosthenes’ diatonic genus, and Ptolemy’s ditonic diatonic, brings such great inconveniences, the one we named before is more valuable: Ptolemy’s tense diatonic genus, which more recent scholars also embrace: Zarlino, Kepler, Mersenne, Descartes, and others.65 [The Modern Scale Constructed by Ratios] They, then, have the interval mi (meaning the interval from the next lower sound), which is the diazeuctic tone, as a major tone with the sesquioctava ratio, 9/8, as in all the others. The interval fa, which [p. 179] is taken as a semitone, is in the sesquidecimaquinta ratio, 16/15; sol, in turn, a major tone in the sesquioctave ratio 9 /8; but la, called a minor tone, in the sesquinona ratio 10/9. This division of the tetrachord is formed in the same way as that of the whole octave into a fifth and a fourth, or into two fourths with an interposed tone. A fifth la la with mi interposed (e.g. from a to e) is, as we have often said, in the sesquialtera ratio 3/2 and, like others which are expressed66 by numbers other than squares, this [ratio] cannot be divided into two equal numerically expressible parts. So let it be divided, like the duple ratio before, into near-equal partsc designated by the smallest possible numbers. Namely, instead of 3, 2, take their doubles, with the same ratio: 6, 4, so that a 5 may appear in between. The three numbers 6, 5, 4 divide the ratio in question (3 to 2 or 6 to 4) into two parts, 6/5 × 5/4 = 3/2. Similarly a

 Euclid, p. 24; Ptolemy, p. 34.   ἐκμελέστατον. c   λόγοι πάρισοι. b

The Harmonics of the Ancients compared with Today’s

137

a fifth la la (from a to e) will be divided into two near-equal intervals: la fa, from a to c, with the ratio 6/5, which they call a minor third; and fa la, from c to e, with the ratio 5/4, which they call a major third.67 And since la mi, the diezeuctic tone, is in the ratio 9/8, mi fa, what is left, will be in the ratio 16/15: since 9/8 × 16/15 = 6/5 or 9/8) 6/5 (16/15. Or in fact, since mi la, a fourth, is in the ratio 4/3 and fa sol in the ratio 5/4, if we take the one from the other mi fa will remain, in, as before, the ratio 16/15, since 5/4) 4/3 (16/15 or 5/4 × 16/15 = 4/3. Similarly, as we just said, fa la is in the ratio 5/4 which, for the reason mentioned several times, cannot be divided into two numerically expressible equal parts. So by doubling, as before, the numbers 5, 4 and interposing a middle one, 10, 9, 8, two near-equals are made: 10/9 × 9/8 = 10/8 = 5/4. One corresponds to the interval fa sol, the other to sol la. In fact the larger of the two, 9/8, is assigned to the interval sol la according to Didymus, as mentioned above, and the smaller, 10/9, to fa sol. On the contrary, however, Ptolemy and more recent scholars assign the ratio 9/8 to the interval fa sol, calling it a major tone, and the ratio 10/9 to the interval sol la, calling it a minor tone.

But, if necessary, you may do either. For as an octave is divided into two fourths with an interposed tone which, added to either, makes a fifth, so the ditone fa la may be divided into two minor tones (with the ratio 10/9) with an interposed comma (with the ratio 81/80) which, added to either, makes it into a major tone. But as to why they assigned the minor tone to the interval sol la rather than fa sol, I think this is the main reason. When, as happens in a fifth, three tones follow consecutively using the notes fa sol la mi, of which one is a minor tone with the ratio 10/9 and the other two major tones with the ratio 9/8, if the minor tone is placed in middle position, sol la, this will make a concordant ditone with each of the end [notes]: with the ratio 5/4 in both cases, since 9/8 × 10/9 = 5/4. But if a major tone is placed in the middle, this will make the ratio of 5/4, as before, with one of the ends, but a discordant ditone with the other, with the ratio 81/64, since 9/8 × 9/8 = 81/64.68

138

John Wallis: Writings on Music

But this is not so inviolable that, if necessary, a major tone cannot be substituted for a minor to avoid a schism, as they call it, or vice versa.69 Just as in the course of a song, if a tone is required in a place where there happens to be a semitone or vice versa, the substitution of one for another is indicated by sharp or flat signs: equally, if instead of a minor tone a major tone is needed or vice versa to make the song more concordant, one will have to be substituted for the other. For instance, when sol fa sol la follow each other consecutively, the end sounds are customarily considered to be a true fifth; but for this to happen a major tone must be substituted for one of the two minor tones, otherwise it will fall short of a just fifth by a comma, 81/80. And likewise in other places, as necessary.

However, this is not usually indicated by adding some sign as in the other case, with the tone and semitone; rather the voice provides it without instruction, for the Guidonian notation which we use does not work with such accuracy. Indeed in Guido’s time the major tone was not distinguished from the minor; so there was not a precise part of a tone to be called a semitone, but they were content [just] to distinguish tones from semitones. They were unconcerned whether the tones were major or minor, or precisely what portion of a tone a semitone was, except at least that it should be about half. And if anything more subtle were sought, it would have to be achieved with the aid of the ear.

The Harmonics of the Ancients compared with Today’s

139

What has been said about the users of this notation in general will be understood to apply to instrument [p. 180] makers in particular; for they certainly cannot attend accurately to these details, but instead necessarily take the nearly true for true. But this is a digression. An octave so divided, therefore – Ptolemy’s tense diatonic genus – looks like this in modern music.

[Extensions of the Modern System] A division of the fourth mi la (from b to e) could be achieved similar to the kind we have already devised for the fifth. For by doubling the numbers 4, 3 and interposing one in the middle to give 8, 7, 6, the ratios 8/7 × 7/6 = 8/6 = 4/3 would be produced, and so a new note would be added between fa and sol (to be called sharp fa or flat sol), which would make the ratio of 8/7 with one of the two notes mi la, and 7/6 with the other. Again, the tri-hemitone mi sol or la fa (from b to d or a to c) could be similarly divided into the ratios 12/11 × 11/10 = 12/10 = 6/5: putting a new note between fa and sol, which would have the ratio 12/11 to one of the two notes mi sol, and 11/10 with the other; and between la and mi a note which would have the ratio 12/11 to one of the two notes la fa and 11/10 to the other. In fact these and many other divisions, made not only by doubling the numbers but also by tripling them so as to interpose two numbers and divide the ratio into three parts, might be applied, and many notes interposed, if the ancients’ enharmonic, chromatic and the various diatonic genera were adopted again, which have now been out of use for many centuries.70

John Wallis: Writings on Music

140

But today’s music does not attempt such subtlety; nor can either voice or ear easily distinguish such small things, at least as we judge today. I do not deny that, with the aid of the ears, the tongue might perhaps make some sound of this kind – tense c or slack d, shown with a sharp or flat sign – more subtly than the written notes indicate. Yet our modern masters of musical practice, or at least our instrument makers, who generally treat tense c and slack d as the same, intend that what is indicated by those sharp or flat signs should be realised as a sound which is as if in the middle between c and d. Nor do they enquire more closely exactly what ratio it has to the adjacent sounds, being content to count in tones and semitones loosely understood. Again, as it is necessary to stop at some point, because the matter does not allow divisions of this kind to proceed to infinity, today’s musicians are content with the divisions handed down, and go no further in dividing the octave. For although the ratios 7/6 and 8/7, which they overlook, seem more significant than 9/8 and 10/9, which they accept, inasmuch as they are designated by smaller numbers, we must nonetheless consider that the number 7 which appears in the designations of the former ratios is a prime or incomposed number, but the numbers 8, 9, 10 used in the latter are composed from smaller ones (i.e. 8 = 2 × 2 × 2; 9 = 3 × 3; 10 = 2 × 5), each of which is less than 7, and so the mind perceives these more quickly when reckoning ratios. For this is more efficient, and it is what the mind picks up more quickly – for example, dividing something into two equal parts and again into two so as to make a quarter [is easier] than dividing it once into three so as to make a third part, even though 3 is a smaller number than 4. Or, again, dividing first into two, and then again into two and finally a third time into two, to make an eighth part, [is easier] than dividing once into seven to make a seventh part. And similarly doubling and doubling again to make a quadruple [is easier] than tripling once; and equally doubling, doubling again and doubling a third time to make an octuple [is easier] than septupling once to make a septuple. And Ptolemy noted this, in this context.a, 71 And for this reason, while today’s musicians overlook the ratios 7 /6, 8/7 (not to mention 11/10, 12/11, etc.), as ones that demand prime numbers like 7, 11 or greater to express them, which the ancients accepted into music, they still, not unwisely, allow the ratios 9/8, 10/9, 16/15, etc., designated by larger numbers which are composed from smaller ones. They now accept no ratio except those designated by the prime numbers 1, 2, 3, 5 or by those composed from these. This in turn is why a double octave, whose ratio is 4/1, is considered more consonant than an octave-plus-fifth, whose ratio is 3/1, even though the number 3 is less than 4; since 4 (= 2 × 2) is composed of smaller primes. And often the judgement is the same for others, although in ranking consonances not everyone always has the same opinion. From these ratios, as given, of adjacent sounds, we form by composition the ratios for all sounds, however remote. We provide an example of these for the individual sounds in a double octave, which is Ptolemy’s perfect system from a

  Ptolemy, p. 3.

The Harmonics of the Ancients compared with Today’s

141

proslambanomenos to nete hyperbolaeon; that is, in our scale, from A re to aa la mi re above. It would easily be understood how much more consonant are the ratios thus given than if the fourth were divided (following others) into tone, tone and lemma, 9 /8 × 9/8 × 256/243 = 4/3, if all the ratios of sounds to each other up to the double octave were gathered together according to this plan, just as we have gathered them for [this] division.72 [p. 181] The ratios between the sounds in a double octave.

John Wallis: Writings on Music

142

Out of these the ratios within the double octave (i.e. from mi to mi) are generally considered to be the main ones. 1 to 16 10 9 6

1 15 9 8 5

5 4 3 8 5 16 9 15 2

4 3 2 5 3 9 5 8 1

Equitone Semitone called Minor second; mi fa Minor tone, Medial second; sol la. Major tone, Major second; la mi, fa sol. Tri-hemitone, (called by some Semi-ditone,) Minor third; mi sol, la sol. Ditone, Major third; fa la. Diatessaron, Fourth; mi la. Diapente, Fifth, la mi. Lesser hexachord, Minor sixth; mi sol. Greater hexachord, Major sixth; fa la. Lesser heptachord, Minor seventh; mi la. Medial heptachord, Medial seventh; la sol Greater heptachord, Major seventh; fa mi. Diapason, Octave; mi mi.

Of these the second and seventh are generally considered dissonances; the rest consonances. The unison, octave, fifth, and fourth (with their repetitions after an octave) are considered perfect; the major and minor thirds and sixths, with their repetitions, imperfect consonances. Perfect consonances73 1

4 11 18

5 12 19

8 15 22

3 10 17

6' 13' 20'

6 13 20

Imperfect 3' 10' 17'

Beyond these, the following appear by accident, all of them dissonances.

The Harmonics of the Ancients compared with Today’s

143

These arise either when a semitone takes the place of a tone or vice versa (which can be fixed by adding a sharp or flat sign), or when a minor tone take the place of a major one or vice versa. Although the difference is only a comma, 81/80, this case is usually emended or concealed in performance by the human voice, even without an instruction; with instruments [p. 182] it is generally so tempered that such a small difference is scarcely perceived.74 Just as a true fourth and fifth make an octave or diapason, so too false fourths with false fifths; and likewise thirds with sixths and seconds with sevenths respectively: for example a major third with a minor sixth and likewise for the others. To be reckoned with all these, if we go beyond an octave, are the same ratios respectively composed with the duple ratio once, twice or more times: that is, the same intervals compounded once, twice or more with an octave, generally called their repetitions. I think treating thirds and sixths as consonances was done with good reason, albeit without the authority of the ancients. Some recent scholars, against whom Andreas Papius of Ghent disputed a hundred years ago in a special treatise, have had doubts about the fourth; contrary to the opinion of all the ancients. The ancients were not unanimous about the fourth added to an octave, since of course an octaveplus-fourth has a ratio of 8/3 = 4/3 × 2, which is neither a multiple nor a superparticular ratio. Ptolemy, however, rejects this objection, as mentioned above, and our scholars overlook it in the cases of the major and minor sixth. But all the ancients, unanimously agreeing, counted the simple fourth among the consonances.75 But it is true that the fourth is rarely accepted on its own in the composition of melodies, and this is not because it is not in itself a consonance but because the power of the fifth on the other side, which sounds an octave to it and is almost heard within it, almost overshadows the fourth and renders it less suitable and thus less often used.76 The method and order by which the consonances which we have enumerated should be used in melopoetics and mixed with dissonances, on which the ancients said rather little, but recent scholars rather more, is not for the present work to discuss in depth. Anyone wanting more should go to those who teach music more broadly as a profession. But they may like to know in what order the consonances are to be reckoned according to their dignity: for instance whether an octave-plus-fifth, whose ratio is 3/1, is greater than a simple fifth, whose ratio is 3/2, or even than a double octave, whose ratio is 4/1; since it is made with smaller numbers and so coming-together [coincidence] returns in fewer vibrations – or the opposite, because the fifth seems simpler than an octave-plus-fifth, and although 4 is a larger number it is still composed of smaller primes, namely 2 and 2 of which each is smaller than 3. And likewise, whether the ratios 7/6, 8/7, made up of smaller numbers, or 9/8, 10/9, 16/15, whose numbers are composed of smaller primes, are greater. And similar things to these. On this, see Mersenne.a   Mersenne [Harmonicorum libri (Paris, 1636/7)], book 4, propositions 18, 22, etc.

a

144

John Wallis: Writings on Music

At any rate, it is agreed that, other things being equal, a ratio explicable in smaller numbers is greater than one with larger numbers; that is, one in which the coincidence of vibrations happens more frequently is greater than one in which it is rarer; or, again, one in which the number is composed of smaller primes is greater than one [where it is composed] of larger ones. I even add (a criterion from nature rather than from our minds) [that a ratio is greater] when, one of the two strings being struck, it brings a larger vibration to the other.77 But when one of these indices contradicts another, not every one has the same opinion which to prefer, and I do not put myself forward as judge. Equally I say nothing about other things: in a brief little treatise a longer investigation of them is not to be expected. THE END Notes 1 See pp. 12–17 above. 2 Oxford, Bodleian Library, shelfmark: Savile Gg 3. 3 i.e. Cleonides, Isagoge harmonike, which had been circulated and, at least since 1566, published as a work of Euclid. The attribution of this late work (second-century a.d.) to Euclid (fl. 300 b.c.) caused some difficulties for Wallis’s account of the history of Greek theory and terminology: see esp. Wallis, ‘The Harmonics of the Ancients’, p. 163 (pp. 101–2 below). 4 Concerning the translation of both diatessaron, diapente, diapason and quarta, quinta, octava as ‘fourth’, ‘fifth’, ‘octave’, see the Editorial Note (p. 76 above). 5 Cf. Wallis to Oldenburg, 14 May 1664, §15 (p. 58 above). Wallis now admitted intervals larger than an octave among the consonances. 6 Cf. ibid. §15 (p. 58, n. 37 above). There this issue was avoided by considering only intervals up to an octave. 7 Cf. ibid. §6 (pp. 51–2 above); Wallis, ‘The Harmonics of the Ancients’, p. 159 (pp. 90–91 below). 8 See pp. 163–4 (pp. 102–6 below). 9 This presentation of the letter names of notes and their repetition at the octave, which precedes the Introduction of the solmisation syllables in this text, may be compared with Thomas Salmon’s 1672 proposal to abandon solmisation in favour of what he considered the more natural and readily comprehensible letter names: Salmon, An Essay to the Advancement of Musick. 10 This rather opaque aside seems to refer to the possibility of providing the seventh note of the scale with a syllable – mi – by means of the ‘mutation’ by which old fa becomes new ut. 11 Wallis’s scala duralis or ‘hard scale’ is equivalent to the modern major scale. Mindful of the dangers of anachronism in the matter of modal and tonal theory, we have avoided the modern term in our translation. 12 See the diagram of the gamut in Figure 3.1 (p. 88 below), where the seven positions

The Harmonics of the Ancients compared with Today’s

145

for the hexachord are illustrated and Wallis’s meaning becomes clear. 13 For Wallis’s discussion of the Greek system of modes or ‘tones’, see pp. 172–4 (pp. 123–7 below), where Ptolemy’s modes are listed. 14 In his edition, Wallis added to the tables of modes in Harmonics 2.11 scales in modern notation using up to three sharps or flats. 15 Cf. Wallis to Oldenburg, 14 May 1664, §17 (p. 59 above), where a similar account of the effects of key signature sharps and flats was given. 16 ‘Our musicians’ were here English musicians: cf. p. 157 (p. 87 above). 17 Cf. Wallis to Oldenburg, 14 May 1664, §19 (p. 60 above); Wallis, ‘The Harmonics of the Ancients’, p. 179 (p. 137 below), where the decisions about where to place the major and minor tones in the diatonic scale were discussed further. 18 Cf. Wallis to Oldenburg, 25 May 1664 (p. 63 above), where the use of the Pythagorean tuning, with all tones equal, was ascribed to unnamed ‘ordinary Practicioners’. 19 Cf. Wallis to Oldenburg, 14 May 1664, §6 (p. 53 above); Wallis, ‘The Harmonics of the Ancients’, p. 155 (p. 82 above). The final clause would be written thus in modern mathematical notation: 3/2 ÷ 4/3 = 9/8. 20 See Andrew Barker, ‘Words for Sounds’, in C.J. Tuplin and T.E. Rihll (eds), Science and Mathematics in Ancient Greek Culture (Oxford, 2002), pp. 22–35; id., The Science of Harmonics in Classical Greece (Cambridge, 2007), p. 21. 21 Compare the list of Latin terms for Greek pitches given on p. 155 (p. 83 above). 22 We have not succeeded in identifying with certainty the edition of Boethius’s De musica to which Wallis referred in this and a subsequent page reference. 23 Compare Wallis’s remark on p. 156 (p. 87 above) about the rare term proslambanomene. 24 In the diagram which follows, the figures, unusually, do not refer to ratios of string lengths or frequencies. Instead they denote an equal division of the fourth into 30 parts. Similarly in the table on p. 165 (p. 106 above), the numbers refer to a division of the fourth into 60 equal parts, not to the lengths of strings. 25 Once again this is a reference to the late text of Cleonides: see p. 144, n. 3 above. Euclid in the Sectio canonis made no mention of such Aristoxenian formulations. 26 Compare the discussion in Wallis to Oldenburg, 14 May 1664, fol. 2 (p. 48 above), of the sense in which pairs of sounds might have ratios. 27 Multiple ratios were those in which the larger term was an exact multiple of the smaller. In a superparticular ratio the difference between the two terms was exactly 1; in a superpartient ratio it was more than 1: cf. Wallis to Oldenburg, 14 May 1664, fol. 8r (p. 58 above), where he considered Euclid’s reason for rejecting other ratios from music inadequate. On ratios inexpressible numerically in this context, see Peter Pesic, ‘Hearing the Irrational: music and the development of the modern concept of number’, Isis 101 (2010): 501–30. 28 See pp. 176–8 (pp. 130–36 below); cf. Wallis to Oldenburg, 14 May 1664 (p. 53 n. 27 above); on Wallis’s belief that practical musicians adopted the Pythagorean scale in some form, a belief which seems to have been slightly modified here, see Introduction, p. 15 above. 29 See further p. 176 (p. 130 below); cf. Wallis to Oldenburg, 14 May 1664, §§7–9 (pp. 53–4 above), where he claims that the ear could by its assessment of degrees of consonance falsify the Pythagorean description of the scale. 30 Cf. Wallis to Oldenburg, 14 May 1664 §15 (p. 58 above, with n. 37): here Wallis adopted a broader meaning of ‘superpartient’. ‘Dupla superbipartiens tertias’ (such terms were rendered into English rarely and not always consistently) signified that

146

John Wallis: Writings on Music

the first term of the ratio was equal to twice the smaller term plus two of its three parts. Wallis, perhaps because of his study of Ptolemy, was now prepared to consider intervals larger than an octave as consonances: cf. Wallis, ‘The Harmonics of the Ancients’, p. 154–5 (p. 81–2 above). 31 Compare the slightly longer discussion of this subject on p. 154 (p. 79–80 above). 32 See Editorial Note (p. 76 above); the terminological distinction Wallis discussed in this paragraph does not appear in our English translation. 33 For most Greek writers the pitch of a sound was associated with its velocity; occasionally, as in Euclid, Sectio canonis 148–9, it was associated instead with the frequency of oscillation of a string or the frequency of other kinds of impacts on the air: see GMW, pp. 7, 9–10. 34 The strategy described here in effect checked the size of the interval which remained when two tones were subtracted from a fourth. That interval was added, twice, to a fourth, producing a result which Aristoxenus claimed was a pure fifth. Since the difference between a fourth and a fifth was by definition a tone, the interval in question must be exactly half a tone.   The interval which was claimed to be a fifth was equal, by the construction given, to three fourths minus four tones, and would have, if the fourths and ditones were tuned pure, a ratio of 262144/177147, differing from a pure fifth by more than 23 cents. It therefore seems likely that if Aristoxenus in fact carried out this test he was tuning the intervals somewhat impure from the beginning. See Barker’s discussion in GMW, pp. 169–70, n. 114. 35 Once again (see nn. 3, 25 above) the attribution of Cleonides’ Introductio to Euclid led Wallis into interpretative difficulties. 36 The notion that consonance and dissonance were matters of degree dependent on the accuracy (acuminium) with which ratios are realised is hard to locate (to say the least) in the ancient writings: it was prominent, however, in the musical thinking of Thomas Salmon, on whom, see Introduction, pp. 17–19 above. 37 Cf. Wallis to Oldenburg, 14 May 1664, fol. 2 (pp. 48–9 above); Wallis, ‘The Harmonics of the Ancients’, p. 177 (p. 132–3 below), where slightly different accounts were given of how pitch could be quantified so as to justify musical ratio theory. 38 See Wallis to Oldenburg, 14 May 1664, §7 (p. 53–4 above), where Wallis showed why the tone could not be divided into two equal harmonious parts. 39 Part of Bk 3, ch. 14 had been lost and had been reconstructed by a late Byzantine scholar. Wallis seems to have understood the scholium concerning the matter (it was ambiguous) to refer to the rest of Bk 3, not just the rest of ch. 14. 40 In this paragraph Wallis showed that the interval from parhypate hypaton to parhypate meson, that from lichanos hypaton to lichanos meson, and that from hypate meson to mese, were all equal to the interval from hypate hypaton to hypate meson: namely, a fourth. In what follows he noted that the internal division of each of these fourths was different; the distribution of intervals within such a fourth was called its species or figure. 41 In a given genus, the different species of an interval could be characterised by the position at which some particular smaller interval appeared within it: for example, the species of fifth may be distinguished by which of the four possible positions the ‘diezeuctic’ tone held. See GMW, pp. 322–3, nn 27, 28. 42 On the meaning of these Latinised Greek terms – barypycni, mesopycni, oxypycni – see p. 165 (p. 108 above). They specify the position of the pyknos, the group of three pitches lying (relatively) close together within the tetrachord.

The Harmonics of the Ancients compared with Today’s

147

43 That is, so that we have fifths of every species. 44 ‘Mode’ or ‘tone’ thus denoted in this sense the internal constitution of the octave, unlimited in principle but in practice related, in part, to the seven species identified in the previous section. 45 See p. 155 (p. 83 above). 46 See p. 167 (p. 111–12 above). 47 That is, the seven modes corresponded to the seven species of octave listed on p. 172 (p. 122 above). 48 The diagram to which Wallis seems to refer appeared on p. 73 of his edition, and was as follows (slightly simplified). It showed roughly the position which mese occupied in each of the seven ‘tones’, in relation to the whole two-octave system. nete hyperbolæon paranete hyperbolæon trite hyperbolæon nete deizeugmenon paranete diezeugmenon trite diezeugmenon paramese mese lichanos meson parhypate meson hypate meson lichanos hypaton

mixolydian mese lydian mese phygian mese dorian mese hypolydian mese hypophrygian mese hypodorian mese

parhypate hypaton hypate hypaton proslambanomenos In Wallis’s notes on this chapter each mode was conceived as a two-octave scale (here a one-octave scale) whose disposition of intervals within the octave was determined by the means which had been discussed in the preceding paragraphs. For each mode, the pitches were named from proslambanomenos to nete hyperbolaeon: these were their positiones in Wallis’s Latin translation of Ptolemy 2.11; here he called them ‘effective’ mese, paramese, and so on. A second set of names for the pitches – their potestates – reflected their origin in modal theory: for the Dorian mode these names were identical to those of the positiones; for the others they were shifted by varying amounts. Wallis correlated each mode with a two-octave scale from A to aa, supplying the number of flats or sharps required to produce the correct set of intervals. The feasibility of such a correlation relied on the tacit assumption that the diatonic genus alone was being considered: as indicated throughout the Appendix, Wallis believed that Ptolemy’s tense diatonic genus provided a correct description of modern musical practice. Wallis’s procedure should be compared with a modern discussion of the Greek system such as that in GMW, pp. 19–23 or West, Ancient Greek Music, pp. 160–89.

148

John Wallis: Writings on Music

49 Wallis used the English word ‘flat’ in the Latin text. 50 Wallis used the English word ‘sharp’ in the Latin text. See also West, Ancient Greek Music, p. 191, n. 2. 51 One aspect of ancient music theory not discussed by Wallis is the analysis of what is nowadays termed ‘voice quality’. The locus classicus for this topic is Aristides Quintilianus, De musica, Bk 2, §§11–14: see Thomas J. Mathiesen, Aristides Quintilianus: on music in three books, with translation, with introduction, commentary and annotations (New Haven and London, 1983); Andrew Barker (ed.), Greek Musical Writings II: harmonic and acoustic theory (Cambridge, 1989); see also William Bedell Stanford, The Sound of Greek: studies in the Greek theory of euphony (Berkeley, 1967), pp. 51–66. This omission on Wallis’s part is curious, given his own preoccupation with sound symbolism in his grammatical work (see David Cram, ‘John Wallis’s English Grammar (1653): breaking the Latin mould’, Beiträge zur Geschichte der Sprachwissenschaft 19 (2009): 177–201) and his documented familiarity with both the tract by Aristides and contemporary discussions of sound symbolism that made explicit reference to it (e.g. Marin Mersenne, Harmonie universelle, contenant la theorie et la pratiqve de la mvsiqve [Paris, 1636], vol. 2, p. 77). It would seem that Wallis wished to maintain as discrete a boundary between the disciplines of music and grammar as he did elsewhere between grammar, logic, and metaphysics (see David Cram, ‘The Changing Relations between Grammar, Rhetoric and Music in the Early Modern Period’, in Rens Bod et al. (eds.), The Making of the Humanities (Amsterdam, 2010), pp. 263–82). 52 Wallis transliterated the word πεττεία (or πεσσεία), which seems to be used here with the very rare sense of ‘repetition of the same note’ (it was normally the name of a game like draughts or backgammon), for which Liddell and Scott, Greek–English Lexicon, cite Cleonides 14. 53 Compare the discussion of this topic – in which Wallis would reach substantially the same conclusion, though at greater length – in Wallis to Fletcher, 18 Aug. 1698 (pp. 224–8 below). 54 Once again (see p. 144 n. 3 above), Wallis took Euclid to be the author of Cleonides’ Harmonic introduction. 55 Euclid, Sectio canonis, proposition 15 showed that a fourth is less than two and a half tones, but did not specify by how much. 56 Compare the discussion of the Pythagorean scale, its ‘semitone’ and its less clearly specified adherents in Wallis to Oldenburg, 14 May 1664, fols 1v, 5r–6r (pp. 52–4 above). 57 Ptolemy, Harmonics 74; GMW, pp. 349–50. 58 See e.g. Marin Mersenne, Harmonie universelle (Paris, 1636), livre second des dissonances, Proposition 3 (pp. 118–21, with the table on pp. 119–20). 59 Porphyry, Commentario, ch. 3; Wallis, Operum mathematicorum volumen tertium, pp. 213–54. Compare Wallis’s more cautious or at least more ambiguous remarks on p. 168 (p. 113, n. 33 above). 60 See Introduction, pp. 10–12 above, for Wallis’s views on this subject expressed in the 1693 version of his paper on nodes of vibration. 61 This discussion of the coincidence theory of consonance (see Introduction, pp. 49–58 above), and much of what follows, seems indebted to Wallis to Oldenburg, 14 May 1664: see fols 3r–8v (pp. 49–58 above) and notice in particular the closely similar sequence of diagrams which appeared there.

The Harmonics of the Ancients compared with Today’s

149

62 This description of strokes coming into coincidence strongly recalls North, A philosophical essay of musick directed to a friend, p. 8; the brief reference to the alternation of strokes may suggest that Wallis had seen Pietro Mengoli, Speculationi di musica (Bologna, 1670); see p. 140 n. 71 below. 63 Wallis attempted to account for the special status of the octave by pointing to the fact that while an octave doubled was still a consonance (because the ‘multiple’ ratio 2 : 1 remains a ‘multiple’ ratio when multiplied by itself), the same was true for no other consonance (cf. Euclid, Sectio canonis, propositions 1–2, 4–5). 64 The procedure of dividing into near-equals, perhaps the main addition to Wallis’s account of the construction of the just scale since 1664, was taken from Ptolemy, Harmonics 1.7; see also 1.15. 65 This was the first time Wallis had named Descartes as a source for his ideas: see Introduction, p. 7 above. 66 Wallis used the Latin participle denominata, which denoted the ‘denomination’ of the ratio in full, not just its denominator in the modern sense. 67 This discussion of the division of the fifth makes it clear that Wallis was thinking of his ratios as ratios of string lengths: notes a, c, e have lengths 6, 5, 4 units. Compare the division of the octave, above, and of the major third and the fourth, below, where an interpretation as ratios of frequencies was left open, and Wallis to Oldenburg, 14 May 1664, where frequencies seem to have been intended throughout. See Introduction, pp. 4–5, 15 above. 68 Cf. Wallis to Oldenburg, 14 May 1664, §13 (pp. 55–6 with n. 35 there). 69 Schisma was Descartes’s term for the discrepancy 81/80 between two forms of the same pitch arising from the difference between a major and a minor tone. 70 See ‘A Question in Musick’, p. 83 (pp. 209–10 below), where this suggestion was developed a little further. 71 Although Ptolemy did indeed make remarks to this effect, this passage is irresistibly reminiscent of Pietro Mengoli’s 1670 discussion of the relative ease to the soul of various combinations of operations of multiplication and division: Speculationi di musica, pp. 54–8; see Benjamin Wardhaugh, ‘The Logarithmic Ear: Pietro Mengoli’s mathematics of music’, Annals of Science 64 (2007): 327–48, at pp. 335–6. 72 Compare Wallis to Oldenburg, 14 May 1664, fol. 8v and table (pp. 58–9 above). 73 The two diagrams which follow illustrate the names of consonances: perfect consonances are the fourth, fifth, octave, eleventh, twelfth, fifteenth, eighteenth, nineteenth, and twenty-second. Imperfect consonances are the thirds, sixths, tenths, thirteenths, seventeenths and twentieths, each existing in a minor (marked with a ') and major form. 74 See Wallis’s remark on p. 176 (p. 131 above) that the ratio ‘of the major tone, 9/8, [differs] from the ratio 10/9, by the ratio 81/80; which is so small that the ear can hardly distinguish it from the unison’: there it was part of an argument for the use of reason rather than sense in the study of harmonics. 75 On the status of the fourth, see Cohen, Quantifying Music, pp. 94–7. 76 This argument was probably taken from Mersenne: see Harmonie universelle, Livre premier des consonances, proposition 30, p. 77; cf. Cohen, Quantifying Music, pp. 105–6. In the present, melodic, context it is not clear that it really makes sense. 77 See Wallis to Oldenburg, 14 May 1664 (p. 48–9 above): perhaps Wallis was thinking of the phenomenon he described there in which the strength of sympathetic resonance depended on the relationship between the two strings in question.

John Wallis: Writings on Music

150

LATIN TEXT [p. 153] Veterum Harmonica ad Hodiernam comparata [The Definition of Music] MUSICAM (μουσικὴν,) seu, ut loquitur Aristoxenus, p.  1. (περὶ μέλους ἐπιστήμην) de Cantu scientiam, vel, ut Aristides Quintilianus, p.  5. (ἐπιστήμην μέλους καὶ τῶν περὶ μέλους συμβαινόντων) scientiam cantus, eorumque quæ circa cantum contingunt; latiori sensu intelligebant Veteres Græci, quam quo plerique nunc dierum vocem eam intelligimus. Quod ex Aristoxeno, p. 1. (editionem intellige Meibomianam, Amstelodami 1652; & similiter de aliis Musicis ab ipso editis,) aliisque, discimus; & Meibomii Notis, tum ad hunc, tum ad Euclides Introductionema Musicam, p. 41. qui (ex Porphyrio) illius partes numerat Harmonicam, Rhythmicam, Metricam, Organicam, Poeticam, Hypocriticam: Et Johanne Gerardo Vossio, qui (cap. 22. de Scientiis Mathematicis,) partes ejus facit Harmonicam, Rhythmicam & Metricam; Atque Alypius similiter, p. 1. Aristoxenus, p. 32. tribus illis subjungit Organicam: Idemque Vossius (cap. 4 de quatuor Artibus Popularibus) tribus Musicæ Practicæ formis, Harmonicæ, Organicæ, & Hydraulicæ; quartam subjungit Orchestriam, seu Saltatoriam. Maximeque ex Aristide Quintiliano, p. 6. qui aliorum definitionem τέχνην πρέποντος ἐν φωναῖς καὶ κινήσεσιν, (Artem decori in vocibus & motibus,) ampliat ipse in hanc, γνῶσιν τοῦ πρέποντος ἐν σώμασιν καὶ κινήσεσι, (cognitionem decori in corporibus & motibus:) cujus item plerasque partes p. 8. enumerat; &, in sua Musica, percurrit ipse. Quam hodie Musicam potissimum dicimus, dixerunt illi potius (ἁρμονικὴν,) Harmonicam; quæ (ut Aristox. p. 1.) de Systematis & Tonis, (συστημάτων τε καὶ τόνων) potissimum agit. Versatur ea circa (φωνὴν) Vocem, prout hæc secundum Acumen & Gravitatem (ὀξύτητα καὶ βαρύτητα) consideratur. Ptolemæus, p. 1. (editionis nostræ) ampliori significatu, ψόφον dicit, de quocunque Sonitu; & Harmonicam definit, potentiam perceptivam, τῶν ἐν τοῖς ψόφοις, περὶ τὸ ὀξὺ καὶ τὸ βαρὺ, διαφορῶν, differentiarum in Sonitibus secundum Acutum & Grave. [Pitch] Vocem in certo aliquo Acuminis aut Gravitatis gradu constitutam, Sonum (φθόγγον) vocabant; quem definit Nicomachus, p. 7. φωνῆς ἐμμελοῦς ἀπλατῆ τάσιν, vocis ad cantum aptæ tensionem, latitudinis expertem: Aristides, p. 9. τάσιν μελῳδικὴν appellat, (tensionem melodicam:) & Ptolemæus, p.  17. φθόγγος ἐστὶν, inquit, ψόφος, ἕνα καὶ τὸν αὐτὸν ἐπέων τόνον. Et Aristoxenus, p. 15. (φωνῆς πτῶσις ἐπὶ μίαν τάσιν, ὁ φθόγγος,) Sonus, est Vocis casus in unam tensionem. Et Gaudentius similiter, p. 3. Nonnunquam & (τόνος) tonus dicitur, seu certus vocis Tenor, (ut Ptol. p. 16.) παρ᾽ ἓν εἶδος τὸ τῆς τάσεως εἰλεμμένος, una tensionis specie sumptus.  1682: Notis, tum ad Euclidis Introductionem

a

The Harmonics of the Ancients compared with Today’s

151

Sed hæc vox πολύσημος est, ut quæ aliter alibi significat. Et prior illa (φθόγγος) speciatim aliquando dicitur de Sonis illis seu Chordis, certo numero definitis, in Systemate Perfecto numeratis: ut sunt Proslambanomenos, & qui sequuntur. Vocem prout ab acutiori ad graviorem sonum, aut ab hoc ad illum, transit, (κινεῖσθαι) Moveri dicunt; & quidem (κατὰ τόπον κινεῖσθαι) moveri in Loco. Quodque inter unum & alium acuminis aut gravitatis gradum interjacet, (τόπον) Locum vocant, seu spatium per quod movetur vox. Ut apud Aristoxenum, p. 3, 8, 9, 10; Aristidem, p; 8, 9; Nicomachum, p. 3, 4. Gaudentium, p. 2, 3. Euclidem, p. 2. Duplicemquea itidem, faciunt, hunc in loco motum (τῆς κατὰ τόπον κίνησιν) Alterum (συνεχῆ) Continuum (dum Vox locum aliquem continue percurrit, nullibi interea se sistens;) Alterum (διαστηματικὴν) Disjunctum, seu Intervallis distinctum, (dum Vox ab una in aliam stationem transiliens, nunc hic, nunc illic, aliquantisper se sistit.) Illum volunt esse (τῶν διαλεγομένων) Loquentium seu Differentium; quam κίνησιν vocant λογικὴν (Rationalem interpretatur Meibomius; malim ego Sermocinalem:) Hunc, (τῶν μελῳδούντων) Modulantium; vocantque (κίνησιν μελῳδικὴν) motum Melodicum seu cantui accommodum: (Quippe, qui hoc faciunt, non λέγειν disserere, sed ᾄδειν canere dicuntur:) Atque Hunc (non Illum) hic considerandum docent. Et Ptolemæus, p. 16, 17. similiter, ψόφους ἀνισοτόνους (sonitus inæquitonos) distinguit in συνεχεῖς (continuos) & διωρισμένους (discretos:) illosque (ἁρμονικῆς ἀλλοτρίους) ab Harmonica alienos; hos, (οἰκείους) hujus Proprios, esse docet. Aristides Quintilianus, p. 6. duobus dictis Motibus, tertium interponit Medium (κίνησιν μέσην,) qualis est recitantium Poema. Quod ipsum à Boethio citatum deprehendo cap. 12. lib. I. sub Albini nomine; Quasi idem fuerit [p. 154] Albinus ille, atque Aristides hic: vel hunc Latine exscripserit ille. Albinum utique, ut Scriptorem Latinum, citat (ex Cassiodoro) Gerardus Vossius, cap. 60. de Scientiis Mathematicis; & Boethius cap. 26. lib. I. A Graviori in Acutiorem locum, Vox transiens (ἐπιτείνεσθαι) Intendi dicitur; ab Acutiori in Graviorem, (ἀνίεσθαι) Remitti: Motusque ille seu Transitus (ἐπίστασις) Intensio dicitur; hic, (ἄνεσις) Remissio: Quodque Intensione sit, (ὀξύτης) Acumen dicitur; quod Remissione, (βαρύτης) Gravitas: Eam autem vocis Stationem (μονὴν καὶ στάσιν φωνῆς) quæ sive Intensione sive Remissione acquiritur, (τάσιν) Tensionem seu vocis Tenorem vocant. Sic Aristoxenus, p. 10, 11, 12, 13; & Aristides, p. 8, 9. Gaudentius, p. 2, 3. Euclides, p. 2. Hæc autem Acuti & Gravis Distensio (διάτασις,) utut sua natura interminabilis (sive Magnitudinem sive Parvitatem spectemus) videatur: Usum tamen si consideremus, omnino terminandam & desiniendam volunt; (sive ad id quod Sonat, sive, ad id quod Judicat, relatio habeatur; Vocem scilicet, & Auditum.) Quippe nec vox Humana (aut etiam Organica) Distensionem vocis aut immense Magnam, aut supra modum Exiguam,b exhibere potest; nec Auditus de illa judicare. Aristox.  1682: Duplicemq;  1682: modum exiguam,

a

b

John Wallis: Writings on Music

152

p. 13, 14. Nicom. p. 4, 5 (nec multo aliter Ptolemæus, p. 15, 16. nisi quod hic Auditus, quam Vocis, terminos ampliores faciat.) Adeoque haud minorem Distensionem in usum recipit Aristoxenus, p. 14, 20. quam est minima Diesis, quam facit ille Toni quadrantem; (& similiter Aristides, p. 13.) & vix Majorem, quam quæ ad Dis-dia-pason & dia-tessaron extenditur; aut etiam ad Dis-dia-pason & dia-pente, (non autem ad Tris-dia-pason,) si vox Humana spectetur: aut, ad Tris-dia-pason, si spectentur alia Organa. Nempe, si unius ejusdemque hominis Vox spectetur; aut unum idemque Organum, puta, una eademque Tibia, Fistulave. Nec tamen negatur Aristoxenus, p. 21. etiam ad Ter, aut Quater Dia-pason, aut etiam ultra, extendi posse Vocis Locum, si plura comparentur: puta, si vox Puerorum aut Fœminarum acutissima, cum Virorum gravissima, comparetur; aut Tibiarum intensiorum acutissima, cum gravissima remissiorum. [Intervals] Quod duobus Sonis (φθόγγοις) non eandem Tensionem (τάσιν) habentibus (respectu habito ad Acumen & Gravitatem) terminatur, Intervallum (διάστημα) vocat Aristoxenus, p. 15, 16. (aliique.) Eodem fere sensu quo Locum (τόπον) ante dixerat; (sed ut plurimum strictiori sensu usurpatur τάσις pro strictiori Soni sensu.) Estque, inquit, quasi Tensionum differentia; seu Locus duobus Sonis interjectus Sonorum capax, (διαφορά τις τάσεων· καὶ τόπος δεκτικὸς φθόγγων, ὀξυτέρων μὲν τῆς βαρυτέρας τῶν ὁριζουσῶν τὸ τάσεων διάστημα, βαρυτέρων δὲ τῆς ὀξυτέρας·) qui graviores sint quam intervallum illud terminantium acutior, acutiores quam terminantium gravior. Quodque ex pluribus Intervallis componitur, Systema (σύστημα) vocat. Intervallorum autem varias enumerat Discrepantias, p. 16. Quippe, secundum Magnitudinem, alia aliis majora sunt, aut minora. Item alia consona, alia dissona, (σύμφωνα, διάφωνα.) Alia item composita, alia incomposita, (σύνθετα, ἀσύνθετα.) Alia item quæ ad unum, alia quæ ad aliud Genus (γένος) spectant; puta, Genus Diatonicum, Chromaticum, aut Enarmonium. Alia denique Rationalia, alia Irrationalia, (ῥητὰ, ἄλογα.) Eademque facit Systematum discrimina; (nisi quod Systema Incompositum, nullum sit:) sed & alia (Systematum) discrimina subjungit; Nempe, (situm quod spectat) alia sunt inter se Disjuncta, alia Conjuncta, alia Mixta, (διεζευγμένα, συνημμένα, μικτά.) Item, Systema Continuum, & Præposterum, (συνεχὲς, ὑπερβατόν.)a Item, Simplex, Duplex, Multiplex, (ἁπλοῦν, διπλοῦν, πολαπλοῦν.) Aliaque istiusmodi, p. 17. Nec multo aliter Aristides Quintilianus, p. 12, 13, 14, 15, 16. Gaudentius, p. 4, 5. Euclid. p. 8, 18. [Melody: The Consonances] Ex Sonis & Intervallis fit (μέλος) Cantus. Non modo Sermocinalis ille (λογῶδες) qualis in Sermocinando auditur (nam habet & Sermo suum Cantum  1682: ὑπερβατόν.) ; 1699 lacks ); MS correction in Bodleian Savile Gg 31 inserts closing bracket. a

The Harmonics of the Ancients compared with Today’s

153

& Accentum;) sed (τὸ ἡρμοσμένον μέλος) Cantus Harmonicus seu Modulatus. Nec tamen ex quibuslibet utcunque sumptis, (nam & quod ἀνάρμοστον est, & minime Harmonicum, Sonis & Intervallis constare potest,) sed quæ aptam σύνθεσιν habeant, & Compositionem debitam, pro cujusque Cantus ratione. Aristox. p. 18. Quippe Soni, invicem comparati, sunt (pro Intervallorum ratione) vel Concinni dicti (ἐμμελεῖς) quasi ad cantum Apti, vel Inconcinni (ἐκμελεῖς) ad cantum Inepti, adeoque hinc rejiciendi. Sic Ptolemæus, p. 18. aliique. Ex Concinnis, alios Consonos (συμφώνους,) alios: Dissonos (διαφώνους) vocant: Illos quidem auribus gratiores; hos vero, utut non plane ineptos, minus tamen gratos. Consonorum species, dictas item Consonantias (συμφωνίας,) enumerabant; simpliciores quidem atque imperfectiores duas; Dia-tessaron & Dia-pente, (quas aurium tantum judicio æstimabant ambas Aristoxenei; Pythagorei vero, illam ratione Sesquitertia terminabant, sive ut 4 ad 3; hanc ratione Sesquialtera, seu ut 3 ad 2:) atque ex binis his compositam (perfectiorem) Dia-pason (cui itaque Pythagorei rationem Duplam assignant, propter 4/3 × 3/2 = 2:) &, magis adhuc compositas, Diapason & dia-tessaron, (cujus ratio 8/3 = 4/3 × 2;) Dia-pason & dia-pente, (cujus ratio Tripla, propter 3/2 × 2 = 3;) & Dis-dia-pason, (cujus ratio Quadrupla, propter 2 × 2 = 4:) & siquas alias velit quis adjicere; ut Dia-tessaron, vel Dia-pente, cum binis pluribusve Dia-pason compositas. Ptol. cap. 5. lib. 1. aliique. Unam tamen ex his, nempe Dia-pason cum dia-tessaron,a pro Consonantia non admittunt Pythagorei; eo quod ratio eius neque sit Multiplex neque Superparticularis (quales illi solas pro consonantiis admittendas volunt;) sed quam admittendam tamen contendit Ptolemæus (ex-[p. 155]ceptione illa posthabita) cap. 7. lib. 1. Eo præsertim argumento, quod Dia-pason, cuivis additum Consono, Consonum faciat. (Quod & Aristoxenus ante statuerat, p. 20, 45.) Ptolemæus porro (ibidem) præ cæteris Consonantiis, præfert Diapason, & Dis-dia-pason, (aut ex pluribus adhuc Dia pason compositas,) quas peculiari nomine (ὁμοφώνιας) Unisonantias appellat, vel (ὁμόφωνα) Unisona. Alii ἀντίφωνα vocant (Unisonorum nomine Aequitonis reservato, Aristid. p. 12. Gaudent. p.  11. Iidemque παράφωνα vocant Dia-pente, itemque Dia-pason & dia-pente,b (aut etiam Dia-pente cum pluribus Dia-pason compositum:) utpote potiora quam Dia-tessaron, aut hoc cum uno pluribusve Dia-pason compositum; quæ generali tantum nomine vocant σύμφωνα. Sic Bryennius, in Meibomii ad Gaudentium Notis, p. 35. Psellus & Theo Smyrnæus (ibid. p. 36.) Dia-tessaron, itemque Dia-tessaron & dia-pason, pariter ac Dia-pente, & Dia-pente cum dia-pason, Paraphona vocant; sed Dia-pason, & Dis-dia-pason, Antiphona. Gaudentius tamen p. 77. per Paraphona intellegit, ea quæ sunt Consonis & Dissonis intermedia, ut Ditonum, Tritonum, & horum simila.  1682: Dia-tessaron,  1682: Dia-pente,

a

b

154

John Wallis: Writings on Music

[Dissonances] Duarum vero primarum Consonantiarum (Dia-tessaron & Dia-pente) differentiam, Tonum vocant: quem Aurium judicio æstimant Aristoxenei; Pythagorei vero, ratione Sesqui-octava definiunt; (propter 3/2 = 4/3 × 9/8.) Sed quem in Consonantiarum censum non admittunt vel hi vel illi: ut nec ejus partes, aut etiam Ditonum, Tritonum; aliasve (præter ante memoratas) compositiones in Cantum admissas: Quas (ἐμμελεῖς) Sonos Concinnos dicere contenti, aut etiam (ἐμμελίας) Concinnitates; Consonantias non dicebant: Sive, quæ sunt minores quam Dia-tessaron, ut Diesis, Hemitonium, Tonus, Triemitonium, Ditonum; sive, quæ sunt consonantiis interjectæ, ut Tritonum, Tetratonum, Pentatonum, & horum similia: sic Euclides, Introductionis Harmonicæ, p. 8, 13. atque ad eundem sensum alii omnes. Sunt autem Soni (φθόγγοι) absolute sumpti, in quolibet Loco (τόπῳ) innumeri, nedum numero infiniti, (quod Aristoxenus diserte afferit p. 26; & Aristides, p. 9; Euclid, p. 2. & res ipsa loquitur;) ut, in omni continuo, Puncta. Adeoque & (his interjecta) Intervalla (διάστηματα) itidem Innumera. [The Pitches in Use] Verum, ex eis omnibus, eos seligunt Musici, κατ᾽ ἐξοχην Sonos dicendos, quos (φθόγγους ἐμμελεῖς) Concinnos, seu Cantui aptos existimant: (quo sensu φθόγγους & τάσεις contradistinguit Aristides p.  9.) Et similiter, Diastemata, speciatim intelligenda sunt, de eis quæ sunt huiusmodi Sonis interjecta. Tales in unoquoque Genere, (Diatonico, Chromatico, & Enarmonio,) Sonos Octodecim, numerabant Veteres. Qui hoc ordine ab Euclide (Introductionis Harmonicæ p. 3, 4, 5, 6, 7,) recensentur: Προσλαμβανομένος· ὑπάτη ὑπατῶν· παρυπάτη ὑπατῶν· λιχανὸς ὑπατῶν (ἐναρμόνιος, χρωματικὴ, διάτονος·) ὑπάτη μέσων· παρυπάτη μέσων· λιχανὸς μέσων (ἐναρμόνιος, χρωματικὴ, διάτονος·) μέση[·] τριτη συνημμένων· παρανήτη συνημμένων· (ἐναρμόνιος, χρωματικὴ, διάτονος·) νήτη συνημμένων· παραμέση[·] τρίτη διεζευγμένων· παρανήτη διεζευγμένων (ἐναρμόνιος, χρωματικὴ, διάτονος·) νήτη διεζευγμένων· τρίτη ὑπερβολαίων· παρανήτη ὑπερβολαίων (ἐναρμόνιος, χρωματικὴ, διάτονος·) νήτη ὑπερβολαίων. (Et similiter fere apud Aristidem Quintilianum p. 9. 10. aliosque.) Eademque nomina Latinis etiam ut plurimum retinentur: Proslambanomenos; hypate hypaton; &c. Nec multo aliter apud Nicomachum p. 27. & Gaudentium p. 11, & 18, recensentur: Proslambanomenos; Hypate hypaton; Parypate hypaton (enarmonios, chromatice, diatonos;) Lichanos hypaton (enarmonios, chromatice, diatonos;) Hypate meson; Parypate meson (enarmonios, chromatice, diatonos;) Lichanos meson (enarmonios, chromatice, diatonos;) Mese; Trite synemmenon (enarmonios, chromatice, diatonos;) Paranete synemmenon (enarmonios, chromatice, diatonos;) Nete synemmenon; Paramese; Trite diezeugmenon (enarmonios, chromatice, diatonos;) Paranete diezeugmenon (enarmonios, chromatice, diatonos;) Nete diezeugmenon; Trite hyperbolæon (enarmonios, chromatice, diatonos;) Paranete hyperbolæon (enarmonios, chromatice, diatonos;) Nete hyperbolæon.

The Harmonics of the Ancients compared with Today’s

155

Ubi videmus, non tantum Lichanos, & Paranetas, sed etiam Parypatas & Tritas, in triplici differentia positas, pro triplici Genere, (Enarmonio, Chromatico, & Diatonico:) Quod quidem omnino par est, cum & harum soni, Mobiles sint, pariter atque illarum. Cur autem hoc (de Parypatis & Tritis) omiserit Euclides, aliique; non video. [Two Philological Matters] Sed obiter moneo, locum illum Gaudentii, pag. 11. ver. 10, quem mancum existimo; sic, ni fallor, restituendum esse, οἷον παρυπάτη ἐναρμόνιος, [καὶ παρυπάτη χρωματικὴ, καὶ παρυπάτη διάτονος, καὶ λιχανὸς ἐναρμόνιος,] καὶ λιχανὸς χρωματικὴ, καὶ λιχανὸς διάτονος· cui congruit quod habet idem pag. 18. v. 21. Nempe; cum modo dixerat, Sonos mutabiles, esse, Parypates, Lichanos, Tritas, & Paranetas; & propterea, ad nomen commune, adsumendum esse quod est cujusque generis proprium; exemplum gratia (λιχανὸς μέσων ἐναρμόνιος, λιχανὸς μέσων χρωματικὴ, λιχανὸς μέσων διάτονος,) Lichanos meson enarmonios, lichanos meson chromatice, lichanos meson diatonos; subjungit, (ὁ δὲ αὐτὸς λόγος καὶ περὶ τῶν παρανητῶν τε, καὶ παρυπατῶν καὶ τρίτων,) idem dicendum est de Paranetis, Parypatis, & Tritis. Monendum item Lichani vocem non raro subticeri; & Paranetes similiter: ut cum, pro Lichanos meson enarmonios, dicitur simpliciter, Meson enarmonios, sed subaudita voce Lichanos: & in reliquis similiter. Item in voce Proslambanomenos (generis Masculini) respicitur vox φθόγγος· in reliquis, Hypate, Mese, &c. vox χορδή. Adeoque reputanda est ἡ λιχανὸς, fœminini generis; & similiter ἐναρ-[p. 156]μόνιος, διάτονος, prout hic occurrunt. (Est utique ὁ καὶ ἡ ἐναρμόνιος, & ὁ καὶ ἡ διάτονος.) Sed apud Bryennium, (& hunc, ni fallor, solum) habetur προσλαμβανομένη. Item ὑπάτη, παρυπάτη, νήτη, παρανήτη, pro Nominibus Substantivis habentur (per appositionem cum voce χορδή·) ut patet ex Genitivis pluralibus circumflexis ὑπατῶν, παρυπατῶν, νητῶν, παρανητῶν· sed μέσων, τρίτων, συνημμένων, διεζευγμένων, ὑπερβολαίων, pro Adjectivis. Quod ad Cap.  5. lib. 2. Ptolemæi monuimus. [The Pitches in Use] Eadem, quæ supra, recenset nomina (pro genere Diatonico) Martianus Capella (Edit. Meibom. p. 179.) adjunctis item Latinis nominibus à se formatis, (quam accurate in omnibus, non dixerim.) Nempe προσλαμβανόμενος, Adquisitus: ὑπάτη ὑπατῶν, Principalis principalium: παρυπάτη ὑπατῶν, Subprincipalis principalium: ὑπατῶν διάτονος, Principalium extenta: ὑπάτη μέσων, Principalis mediarum: παρυπάτη μέσων, Subprincipalis mediarum: μέσων διάτονος, Mediarum extenta: μέση, Media: τρίτη συνημμένων, Tertia conjunctarum: συνημμένων διάτονος, Conjunctarum extenta: νήτη συνημμένων, Ultima conjunctarum: παραμέση, Prope-media: τρίτη διεζευγμένων, Tertia divisarum: διεζευγμένων διάτονος, Divisarum extenta: νήτη διεζευγμένων, Ultima divisarum: τρίτη ὑπερβολαίων, Tertia excellentium: ὑπερβολαίων διάτονος, Excellentium extenta: νήτη ὑπερβολαίων, Ultima excellentium.

156

John Wallis: Writings on Music

[Notation and Solmization] Pro singulis autem hisce Chordis, Sonisve; habuerunt Græci (in singulis generibus; Enarmonio, Chromatico, Diatonico;) suos cujusque Characteres seu Notas (σημεῖα) quibus designarentur. Quod ex Bacchio, Aristide, Boethio, aliisque liquet; sed præsertim ex Alypio, & Gaudentio. De eisdem videatur item Vossius, (aliique ab illo citati) de Artibus Popularibus cap. 4. § 10; & de Scientiis Mathematicis, cap. 22. § 14. His Notis restituendis, bonam operam impendit Meibomius, in suis præsertim ad Alypium Notis; apud quem habentur conspiciendæ. Atque has quidem Notas callere, magna pars erat Praxeos Musicæ; quo possent auditam Cantilenam notis describere, & descriptam canere. Quam Adnotandi artem, vocat Aristoxenus (p. 39.) παρασημαντικήν· & Meibomius ad Alypium, p. 66. σημειοτικήν. Pro Notis hisce; habemus, in hodierna Musica, (pro Genere saltem Diatonico,) Scalam (quam vocant) Guidonianam; ut quam invenerit Guido Aretinus Abbas, circum Annum Christi, ut docet Joh. Ger. Vossius, (cap 22. de Scientiis Mathematicis) 1070; seu (quod ipse præfert, cap. 4. de Artibus Popularibus) 1028, ex Sigeberto. Hanc Manum vocant alii, (quia in manus forma, memoriæ gratia, solebant eam aliquando depingere; quod apud Mersennum aliosque videre est:) Aut etiam Gamm-ut, (sumpto nomine ab ejus voce prima, seu gravissima.) Qua nempe Lineis aliquot (ad instar Scalæ gradatim ascendentibus) & Spatiis interjectis, designant loca Sonorum (φθόγγων) singulorum; eo ordine quo à Gravi in Acutum ascenditur. Ut aliis ad hoc Notis non sit opus, quam ipso Situ. Quippe quæ porro conspicitur, in nostra Musica, notarum varietas: (puta pro Larga, Longa, Brevi, Semibrevi, &c.) non Tonum, sed Tempus spectat, secundum quod vox aliqua Productius Contractiusve canenda sit: atque ad Musicæ partem Rhythmicam pertinet, quæ est de Rhythmo seu Modulatione secundum Tempus, breve & longum; non ad partem Harmonicam, de qua jam agitur, quæ est de modulatione secundum Tonum, acutum & gravem. Quo autem melius percipiatur, quomodo inter se consentiunt, Hodierna Scala, & Veterum Diagramma, (pro Genere Diatonico, quo solo jam utimur;) utrumque junctim exhibemus. Et quidem in Veterum Diagrammate, Sonos quos vocant Mobiles (linea punctata) à Stantibus seu quiescentibus (recta continue notatis) distinximus; & pariter, in hodierna Scala, Sonos qui in Spatio (ut loquuntur,) ab eis qui in Linea, notari solent. Utrobique observatis Intervallis, sive Tonicis sive Hemitonicis, in Genere Diatonico censeri solitis. Atque harum quidem linearum (quot opus est) aliquot, puta quaternas aut quinas, (sive ex imis, sive ex summis, sive ex intermediis, prout cujusque cantus ratio postulat,) depingunt (omissis reliquis;) quarum unam aliquam (puta, F fa ut, aut c sol fa ut, aut g sol re ut,) peculiari nota (quam Clavem vocant) designant (ut innotescat, quam Scalæ lineam ea referat;) reliquæque, ex situ ad hanc, æstimantur. Non autem ita terminatur (his cancellis) vel Veterum Diagramma, vel nostra Scala, ut non ultra possit vel in Grave, vel in Acutum procedi. Sed, quemadmodum in Scala nostra, post septem (ascendendo) literas A, B, C, D, E, F, G, (quæ totidem Chordas seu Voces designant,) eædem secundo, & tertio, & (si opus est) pluries

The Harmonics of the Ancients compared with Today’s

157

recurrunt, (quas repetitiones, quo invicem distinguantur, mutata literæ forma solent distinguere;) & similiter in Grave (descendendo) ordine inverso; (ubi, pro G diverso, Græcum Γ solent pingere; unde toti Scalæ nomen factum est Γ ut, seu Gamma ut:) literæque similes (puta A a, aut G g, &c.) Dia-pason terminant: Ad eundem modum, in Veterum Diagrammate, postquam a Proslambanomeno (in acutum ascendendo) ad Neten hyperbolæon perventum est, (quæ Chorda, eadem censetur cum recurrentis in altum diagrammatis Proslambanomeno,) recurrunt, ut prius, Hypate hypaton, Parypate hypaton, &c: & similiter, (descendendo in Grave) infra Proslambanomenon (quæ eadem, est Nete hyperbolæon infra positi Diagrammatis) recurrunt si opus est (ordine inverso) Paranete hyperbolæon, Trite hyperbolæon, &c. (ut in cap. 11. lib. 2 Ptolemæi:) similiaque chordarum nomina, (puta Mese, Mese; aut Paramese, Paramese; aut quæ pro eadem habentur, Proslambanomenos, Nete hyperbolæon, &c.) Dis-dia-pason continent. [p. 157] Voces quæ (in Scala Guidoniana) post literas ascriptæ sunt; desumuntur ex Hymno quodam Ecclesiatico (Pauli cujusdam, Diaconi Romani,) pro S. Johanne Baptista; quem habet Vossius (cap 4. de quatuor Artibus Popularibus,) aliique. UT queant laxis   REsonare fibris MIra gestorum   FAmuli tuorum, SOLve pollutis   LAbiis reatum. (Mersennus, aliique, his Sapphicis, addunt Adonium, Sancte Johannes; Alstedius, O pater alme: sed qui, ad præsentem scopum redundat.) Hinc enim sex hemistichiorum syllabas initiales desumpsit, ut, re, me, fa, sol, la, pro totidem chordis continue sequentibus, puta Γ A B C D E: Quæ recurrunt (absoluto Dia-pason) in G a b c d e; iterumque in g aa bb cc dd ee. Sed omnino deest vox septima, pro sedibus F f, voce destitutis; (nam, quod ibidem habeantur fa, ut, id est alterius Syzygiæ:) & similiter ubique, sive ante vocem ut; sive post vocem la. Quod mirum est Guidonem non vidisse & præcavisse. Recentiores aliqui (inter quos Mersennus) supplent vocem Si, aut huic similem. Potuissent (ex Adonii initio) vocem Sa supplere. [The Modes] Harum vocum usus, hic est: Re mi continent (quem vocant Græci) Tonum diazeuticum, (quo disjunguntur duo Tetrachorda, & qui utrivis additus facit Pentachordum, seu Dia-pente:) quibus respondent, in situ naturali, (qui est Ptolemæi Tonus Dorius, & nostra Scala dura,) Mese & Paramese; adeoque (quæ inde distant Dia-pason) Proslambanomenos & Hypate hypaton: Hoc est, in scala nostra, a b, & A B; (posita nimirum voce mi in B, b, bb.) Atque huic casui respondet Hexachordum illud Ut re mi fa sol la, (quod in Scala septies occurrit) in loco primo, quarto, & septimo posi-[p. 158]tum: (quippe hi casus tres, pro uno habentur; recurrentibus, absoluto Dia-pason, eisdem vocibus.) Ubi autem mutatione secundum Tonos (ut loquuntur Græci,) seu translata mi voce in aliam sedem, cantus mutatur; re & mi sunt, quæ Græcis dicuntur potestate, mese

John Wallis: Writings on Music

158

& paramese, (quamcunque sedem illæ occupant,) itemque Proslambanomenos & Hypate hypaton, potestate. Puta, posita b molli (ut loquuntur) in B, b, bb; transfertur mi ad E, e; & cantatur Hexachordum illud prout in locis secundo & quinto recurrit: Qui est Ptolemæi Tonus (seu Modus) Mixolydius, in quo Mese & Paramese potestate, sunt (positione) in paranete & nete diezeugmenon; hoc est, in scala nostra, in d e. Sin ponatur item alia b mollis in E, e; quo transferatur mi ad a aa; cantatur illud Hexachordum ut in locis tertio & sexto: Qui est Ptolemæi Modus Hypolydius, quo Mese & Paramese, potestate, sunt in Lichano meson & mese, positione; hoc est, in scala nostra, in G, a. Atque hos quidem tres casus (nec plures) innuit Scala Guidoniana. Sed recentiores post illum Musici, alias adhuc Clavium signaturas addiderunt; quarum ope, vocem mi in quavis chorda (prout opus erit) canendam insinuent; quæ cæteris Ptolemæi Modis seu Tonis respondeant: (Ut ad Cap. 11. lib. 2. Ptolemæi ostendimus.) Nempe, posita tertia b molli in a, transfertur inde mi in d; qui est Ptolemæi Modus Lydius. Item, posita Acuti nota in F fa ut, cui naturaliter (ut loquuntur) convenit vox fa, mutatur hæc in mi, fitque vocis mi sedes in F, f; qui est Ptolemæi Modus Hypodorius. Sed hinc transfertur in C, c, cc, si etiam hic reperiatur similis Acuti nota; qui est Ptolemæi Modus Phrygius. Atque hinc denique transfertur mi in Γ, G, g,a si & hic porro conspiciatur eadem Acuti nota; qui est Ptolemæi Modus Hypophrygius. Quæ omnia, citato loco, depinximus. [Hard b, soft b, and Modern Solmization] Harum (pro Molli & Acuto) notarum origo, hæc est. In Græcorum Musica, cum à Proslambanomeno (in acutum ascendendo) perventum erat ad Mesen, (hoc est, ad nostram a la mi re,) ambiguum erat an ad tetrachordum (Synemmenon) Conjunctarum; an ad (Diezeugmenon) Disjunctarum tetrachordum procederetur: Adeoque, an chorda proxima (quæ est b fa b mi) hemitonio (pro trite synemmenon) an tono integro (pro paramese) acutior foret. Priori casu, chorda illa Gravior esset futura (hemitonio) quam casu posteriori, hoc est (ut loqui amabant) Mollior: atque hoc, quam illo, tanto Acutior; hoc eft, Intensior, seu Durior. Priori itaque casu signabant chordam illam litera b rotunda, quam b mollem vocabant, quo casu (ab a la mi re) ascendebatur Hemitonio per vocem fa; dicique solet Scala mollis: Posteriori casu, signabant chordam illam litera b quadrata, (quæ sensim corrupta est in hanc fere notam #)b quam b duram vocabant; diciquec solet Scala dura; (aut etiam, quod nunc fere fit, omittebant eam signaturam; quoniam hæc reputatur Scala naturalis, adeoque signatura non indigere;) quo casu, Tono ascenditur per vocem mi. (Hunc fit, quod ea chorda binominis est, b fa, b mi:d nempe b mollis canitur voce fa: qua innuitur intensio per Hemitonium: b dura,e voce mi; qua   1682: G, g, gg   1682 represents this symbol with a non-serif X. c  1682: diciq; d  1682: b mi e  1682: b dura a

b

The Harmonics of the Ancients compared with Today’s

159

innuitur intensio per Tonum integrum.) Tandem vero, quæ ad solam b fa b mia chordam distinguendam destinatæ primum fuerant notæ, Mollis & Dura; jam cuivis chorda adhibitæ tantundem præstant: Nempe, quæ chorda secus foret Tono acutior quam proxime subjecta; adhibita nota Mollis (hoc est, Gravis,) Hemitonio deprimitur, adeoque non nisi Hemitonio fit acutior quam proxime subjecta: contra vero quæ secus esset Hemitonio tantum acutior quam quæ est proxime subjecta; adhibita nota Duræ (hoc est, Acutæ,) Hemitonio acuitur, adeoque jam fit integro Tono acutior quam proxime subjecta. Nostrates vero Musici, nunc dierum, omissis vocibus ut re, solis utuntur mi fa sol la. Nempe, posita mi in sede sua sic ut innuit clavium Signatura, sursum canitur fa sol la, fa sol la, (post quas iterum recurrit mi;) deorsum vero la sol fa, la sol fa, (redeunte iterum mi.) Quarum quidem vox, fa, reputatur hemitonio acutior quam proxime subjecta; reliquæ, tono integro. Verum, qui subtilius loquuntur, voci sol assignant tonum majorem (in ratione sesquioctava;) voci la, tonum minorem (in ratione sesquinona;) adeoque voci fa, pro Hemitonio, assignant rationem sesquidecimamquintam, (quæ tetrachordi rationem sesquitertiam compleat;) sed voci mi assignant omnes rationem sesquioctavam. Qui vero (neglecta differentia tonorum majoris & minoris) assignant indiscriminatim tonis omnibus rationem sesquioctavam, iidem hemitonio assignant rationem limmatis (λείμματος) quæ est 256 ad 243: vel (crassius loquendo) toni semissem, (à quo limma non multum abest.) Atque sic fere loquuntur Musici practici omnes, qui præter Tonum, & Dimidium toni, (quæque ex his componuntur,) non alia censere solent intervalla. [The Divisions of the Tetrachord] Sed, dimissa aliquantisper Scala nostra; redeo ad Græcorum Diagramma. Ubi Octodecim, quas dixi, in unoquoque genere, chordæ, non semper sunt totidem Soni, sed totidem Nomina, seu Potestates (δυνάμεις·) quippe horum Sonorum alii cum aliis nonnunquam reputantur coincidere; præsertim in genere Diatonico; (utpote Paranete synemmenon & Trite diezeugmenon, item Nete synemmenon & Paranete diezeugmenon, in Diagrammate jam exposito:) et vix alibi sunt Octodecim distincti, quam in Genere Enarmonio. Nominum autem rationem, rem omnem ab origine repetentes, ordine exponemus. Ex Sonis hisce (Chordisve eisdem accommodatis) Diastemata & Systemata (ἐμμελῆ) Concinna, seu cantui accommoda, (ut ante dictum est,) formabant veteres: quorum alia Consona, alia Dissona (σύμφωνα, διάφωνα,) nominabant; prout erant Sonis consonis, dissonisve (ὡρισμένα) [p. 159] finita seu terminata. Aristox. p. 16, 17. Aristid. p. 13, 14. Illud autem (διάστημα) diastema, (seu intervallum aut distantiam) quam reliquarum quasi mensuram faciunt, (præsertim qui Aristoxenei dicebantur, qui per Differentias magis, quam per Rationes Intervalla metiebantur,) Tonum (τόνον) dixerunt; (non quo sensu certum aliquem vocis Tenorem innuit; sed quo  1682: b mi

a

John Wallis: Writings on Music

160

certam quandam sonorum differentiam significat;) quanta scilicet est (secundum magnitudinem) duarum primarum Consonantiarum (Dia-tessaron & Dia-pente) differentia; (ἡ τῶν πρώτων συμφώνων κατὰ μέγεθος διαφορά·) Aristox. p. 21, 46; Cujus (inquit) dimidium & pars tertia & pars quarta, cantari potest; non autem intervalla his minora. Quanta autem sit hæc distantia, auribus dijudicandum permittunt Aristoxenei (non ultra de hoc soliciti:) Pythagorei vero (qui potius per rationesa seu Proportiones metiebantur intervalla,) Ratione Sesquioctava (λόγῳ ἐπογδόῳ) determinabant. (Quippe, cum Rationem Dia-pente posuerint Sesquialteram, seu ut 3 ad 2; & Rationem Dia-tessaron, sesquitertiam, seu ut 4 ad 3; ut ante dictum est: Harum Differentiam, quæ sit Toni, colligebant Sesquioctavam; puta, cum Soni unius ad alterius acumen sit ut 9 ad 8: propter 4/3 ) 3/2 ( 9/8. Sic Euclides, de Sectione Canonis, p. 30; Ptolemæus cap. 7. lib. 1. aliique.) Primum systema Consonum, seu primam Consonantiam, (τῶν συμφώνων διαστημάτων πρῶτον καὶ ἐλάχιστον,) fecerunt (τετράχορδον) Tetrachordum, quam & (διὰ τεσσάρων) Dia-tessaron consonantiam vocabant, omnium Minimam; (quæ autem sit omnium Maxima, definire nequeunt;) Sic Aristox. p. 20, 21, 22 aliique (ubi pro μέγεθος p. 20, v. 14 rescribendum esse μέγιστον, recte monet Meibomius in Notis; & res ipsa loquitur.) Hanc Consonantiam (Dia-tessaron) Sonis quatuor (unde nomen sortitur) in tria intervalla (invicem continua) distinguebant. Non quidem Aequaliter; sed neque in omnibus Generibus Similiter: Sed (aurium judicio docti) in intervalla inæqualia dividebant: & quidem aliter in Genere Enarmonio dicto; aliter in Chromatico; aliter in Diatonico. Atque secundum hanc variam Tetrachordi divisionem, tria (quæ vocabant) Genera (γένη) distinguebant; Enarmonium, Chromaticum, Diatonicum, (ἐναρμόνιον, χρωματικὸν, διατονικόν·) Sic Aristox. p. 21, 22. aliique. Tetrachordi cujusque, seu Dia-tessaron, Extremi duo Soni (qui totum terminabant) Stantes, Immobiles, seu Quiescentes dicebantur (ἑστῶτες, ἀκινητοὶ, ἠρεμοῦντες·) Intermedii duo, Mobiles (κινητοὶ seu κινούμενοι·) Eo quod illorum quidem, tum ab invicem Distantia (quam duobus tonis cum semisse, definiebant Aristoxenei; Pythagorei, ratione sesquitertia;) tum Sedes in integro Systemate; stabiles sint, eædemque in Generibus omnibus; (adeoque ἠρεμεῖν quiescere dicebantur:) Horum vero, tum à se invicem, tum ab extremis distantiæ; sedesque in integro Systemate; erant, pro variis Generibus, aliæ atque aliæ; (adeoque κινεῖσθαι, moveri dicebantur;) & quidem pro diverso horum in Tetrachordo situ, varia dicebantur (γένη) Genera: variæque in his generibus Species, quas χρόας (colores) vocitabant. Sic etiam Euclides, p. 10. Item, Alypius, p. 2. τῶν φθόγγων οἱ μέν εἰσιν ἑστῶτες καὶ ἀκλινεῖς· οἱ δὲ κινούμενοι· ἑστῶτες μὲν οὖν λέγονται, ὅτι ἐν ταῖς τῶν γενῶν διαφοραῖς οὐ μεταπίπτουσιν· κινούμενοι δὲ, ὅτι ἐν ταῖς τῶν γενῶν διαφοραῖς μεταπίπτουσιν εἰς ἑτέρας τάσεις. Tetrachordi vocem seu chordam Gravissimam vocabant Hypaten (ὑπάτην) Supremam; Acutissimam vero Neten (νήτην seu νεάτην) Ultimam seu Imam; (quod & Hen. Stephanus agnoscit ad vocem νήτη, qui sic ultimam seu imam dici docet,  1682: Rationes

a

The Harmonics of the Ancients compared with Today’s

161

ad differentiam τῆς ὑπατης & Paraneten, παρανήτην, ut imæ proximam:) Quippe qui primi hæc imponebant nomina (contra quod jam facimus) Grave pro Summo habuerunt, & Acutum pro Imo. Et sic Nicomachus, p.  6. ὁ βαρύτατος φθόγγος ὑπάτη ἐκλήθη, ὕπατον μὲν τὸ ἀνώτατον· κατώτατος δὲ νεάτη, καὶ μὲν νέατον τὸ κατώτατον· adeoque in Heptachorda Lyra, Hypaten ascribit Saturno; & Neaten, seu contracte Neten, Lunæ. Et Boethius ubique, in sua Musica, sonos Graviores in schematis Summo locat, Acutiores in Imo. Hinc (quæ post occurrent) Hyperhypate, & Hyper-mese (ὑπερυπάτη, ὑπερμέση,) sunt proxime Graviores quam Hypate, & Mese. (Ut frustra si qui ὕπατον quasi ὑπόατον dictum vult: cum Homerus certe, quasi ὑπέρτατον posuerit; cum Jovem appellare soleat ὕπατον κρείοντων.) Sed, quo rem dissimulent; postquam Grave pro Imo haberi coeperit & Acutum pro Summo, (prout nunc fieri solet,) Latini (post Martianum Capellam; & Severinum Boethium,) Hypaten, Principalem interpretantur (potius quam Supremam,) & Parypaten, Subprincipalem. Et Aristides Quintilianus p. 10. per Primam potius exponit, quam Supremam; ὑπάτη ὑπατῶν, ὅτι τῶν πρώτου τετραχόρδου πρώτη τίθεται· τὸ μὲν πρῶτον, ὕπατον ἐκάλουν οἱ παλαιοί· παρυπάτη δὲ, ἡ παρ᾽ αὐτὴν κειμένη· Et paulo post, παρανῆται καλοῦνται, διὰ τὸ πρὸ τῆς νήτης κεῖσθαι·ἐπὶ δὲ ταύταις ἡ νήτη, τουτέσθιν ἐσχάτη· νέατον δὲ ἐκάλουν τὸ ἔσχατον οἱ παλαιοί. Ubi (dissimulata significatione Summi & Imi) ὕπατον & νῆτον interpretatur Primum & Ultimum. Voces intermedias (quæ & mobiles erant) vocabant Parhypaten (παρυπάτην) Subsupremam seu Subprincipalem (ut quæ erat ad Hypaten proxima;) & Paraneten (παρανήτην) Penultimam, ut quæ τῇ νήτῃ (ultimæ) adjacebat. ὑπάτη παρυπάτη παρανήτη νήτη

Suprema. Subsuprema. Penultima. Ultima.

la fa sol la

Atque sic quidem dicebantur Tetrachordi voces dum solitarium erat: ut in Mercurii (quæ dicitur) Lyra: quam (τετράχορδον) quatuor chordarum fuisse tradit Boethius, cap. 20, lib. 1. [p. 160] aliique. Et sic fere (nisi quod, pro Paranete, eodem sensu Lichanon dixerit, sitque locus alias mendosus;) Nominum ordinem (tanquam notissimum et receptissimum) pro quovis Tetrachordo, recenset Aristoxenus, p.  22. τῶν δὲ συγχορδιῶν, πλειόνων τε οὐσῶν, τὴν εἰρημένην τάξιν τῶν διὰ τεσσάρων κατεχουσῶν, καὶ ὀνόμασιν ἰδίοις ἑκάστης αὐτῶν ὡρισμένης· μία τίς ἐστιν ἡ μέσης [lego νήτης] καὶ λιχανοῦ καὶ παρυπάτης καὶ ὑπάτης, σχεδὸν γνωριμωτάτη τοῖς ἁπλομένοις μουσικῆς. Quod sic reddo: Cum plures sint Chordarum complexiones, dictum Dia-tessaron Ordinem continentes, (scilicet, quod jam modo dixerat, extremos duos sonos stabiles, duosque mobiles intermedios,) sintque illarum singulæ suis nominibus finitæ, (puta Tetrachordum Supremarum, Mediarum, Conjunctarum, Disjunctarum. &c:) unus quidam est (omnium communis) Ordo, (nimirum, Netes, Lichani, Parypates, & Hypates,) iis omnibus, qui Musicam vel leviter attigerunt, notissimus. Notat ad hunc locum

John Wallis: Writings on Music

162

Meibomius, (quo Syntaxis constet) vel legendum ὅτι μία, vel (quod sequor) γνωριμώτατα, potius quam (quod erat in aliis codicibus) γνωριμώτατον· sed non animadvertebat (quod ego quidem existimo) μέσης, mendose scriptum esse, pro νήτης, (tum hic, tum mox iterum.) Quippe de uno omnium Ordine, quam de uno præ cæteris Tetrachordo, potior ibidem erat habenda ratio. Verum, siquis lectioni alteri mordicus adhærere malit, quam meæ de mendo conjecturæ; ut de uno (Meson) Tetrachordo, quasi præ cæteris notissimo, hæc dixerit Aristoxenus; (non de uno omnium Ordine:) non tanti res est, ut velim anxius contendere: quoniam, etiam sic, hoc unum instar omnium proponitur; cum sit de omnibus eadem ratio. Atque huic quidem favere videtur, quod habet Aristoxenus, p. 46. ubi Tetrachordum à Mese ad Hypaten (nempe Meson hypaten) quasi aliorum exemplar exponit: ut etiam in eadem ipsa pag. 22. Postquam autem Tetrachordum unum, solitarium esse desierit; & plura componi coeperant Tetrachorda: pro Parypate (subsuprema,) dicebatur, in quibusdam tetrachordis, Trite (nempe, τρίτη ἀπὸ νήτης) Tertia ab ultima, seu Antepenultima: ina aliis vero, pro Paranete (penultima) dicebatur Lichanos (λιχανὸς,) Index; sive quod Indice digito pulsaretur ea Chorda, (quod innuit Aristides, p. 10) sive (quod ego potius putaverim) quod Index sit (ille sonus) situ suo; num, & quatenus, Molle sit genus aut Intensum: Quippe quo Remotior est, à tetrachordi voce acutissima, Lichanos, eo Mollius censetur Genus; quo Propior, eo Intensius: Quod docet Ptolemæus cap. 3. lib. 2. p. 54 quippe de sonis his (acutissimo, eique proximo,) quos φθόγγους ἡγουμένους vocat, verba faciens, (eorumque ab invicem intervallo,) subdit, οἵ τινες ποιοῦσι τὰς ἐπὶ τὸ μαλακώτερον ἢ τὸ συντονώτερον παραλλαγάς· qui faciunt (Generum) discrimina, in Mollius Intensiusve. Utut sit; Triten in aliis Tetrachordis, idem valere atque in aliis Parypaten; item in aliis Lichanon, & Paraneten in aliis; res ipsa loquitur. Hinc est, quod Tritæ dicantur παρυπατοειδεῖς· & Paranete, λιχανοειδεῖς· atque in aliis similiter: Utpote cum Tritarum & Parypatarum eadem sit ratio; item Lichanorum & Paranetarum. Et similiter, qui sunt cujusvis Tetrachordi soni gravissimi, vel Hypatæ, vel saltem Hypatoides, vocantur. Boeth. p. 11 [Systems of the Two Tetrachords] Post Mercurii (aut cujuscunque demum fuerit) quatuor chordarum lyram (pro quatuor elementorum numero,) facta est Orphei Lyra (ἑπτάχορδος) septem chordarum (pro numero Planetarum.) Sunt tamen, qui docent, Mercurium ipsum, Lyram (quam à superioribus acceperat Quadrichordem, Mercurii Lyram dictam) Septichordem fecisse; talemque Orpheum (à Mercurio traditam) accepisse. Nicom. p. 29. Sed (ne simus de Inventore primo nimis soliciti) de hoc convenit; Lyram hanc Septichordem (Orphei dictam,) ex duobus Tetrachordis conjunctis fuisse constitutam, ita quidem ut Gravioris vox Acutissima, fuerit Acutioris Gravissima; illudque dixerunt tetrachordum Hypaton (ὑπατῶν,) supremarum; hoc vero,  1682: In

a

The Harmonics of the Ancients compared with Today’s

163

Neton (νητῶν,) ultimarum. Ne vero chorda illa, utrique Tetrachordo communis, Binominis foret, (Supremarum ima, & Imarum suprema;) hanc Mesen (μέσην,) Mediam appellabant: Graviorisque tetrachordi gravissimam (non tam Hypaten hypaton, quam simpliciter) Hypaten; &, huic proximam Parypaten: item Acutioris acutissimam (non tam Neten neton, quam simpliciter) Neten; & huic proximam Paraneten: Gravioris autem quæ fuerat Paranete (quod nomen propter netes nomen in meses mutatum, jam esset incommodum) ne foret innominis, novo nomine dicebatur Lichanos, (aut etiam Hypermese; Nicom. p. 7. & Meibom. ad illum p. 57.) Acutiorisque quæ fuerat Parypate (propter hujus Hypaten similiter in Mese immersam) dicebatur promiscue vel Paramese (utpote post mesen proxima) vel Trite (utpote, à nete, tertia.) Quod ex Boethio (ibidem) discimus; & Nicomacho, p. 6,7. Qui Hypaten chordam ascribit Saturno (Planetarum supremo;) Neten, Lunæ (Planetarum infimo;) reliquisque reliquas suo ordine: Nisi quod Mercurium Veneri præponat; quod itaque, quasi mendum fuerit, mutandum insinuatur, p. 33. (sed & pariter insinuatur, aliis placere, ut, contrario ordine, Gravissima Lunæ ascriberetur, Acutissima Saturno, pariterque de intermediis.)

[p. 161] Talisque jam conspicitur, integri Diagrammatis ante positi, ea pars quæ est ab Hypate meson ad Neten synemmenon: aut etiam ab Hypate hypaton ad Mesen: item à Paramese ad Neten hyperbolæon. Post hanc successit (posthabito Planetarum numero) Pythagoræ Lyra (ὀκτάχορδος) chordarum octo. Qui, ex Heptachordo retentis fere ut prius nominibus, primus omnium (ut docet Nicomachus. p. 7.) inter duo quæ prius erant Conjuncta tetrachorda, Tonum interposuit Diazeucticum: qui utrivis tetrachordo additus compleret (πεντάχορδον) Pentachordum; (cujus extremæ

164

John Wallis: Writings on Music

consonarent Dia-pente;) utrique vero, (pro Heptachordo) Octachordum constituit, cujus extremæ chordæ exhiberent Dia-pason (Consonantiarum præcipuam) quæ in Heptachordo non audiretur. (Cujus extremæ, ne quidem erant Consonæ.) Adeoque, quæ prius fuerant una chorda (Mese dicta) jam factæ sunt duæ; nempe, Hypaton nete, nunc ut ante Mese dicta, (ut quæ à medio necdum multum recesserit:) & Neton hypate, jam dicta Paramese, (quod prius fuerat Trites nomen:) Quæ à Mese distat Tono integro; à Trite, Hemitonio. Quodque ante fuerat Neton synemmenon (conjunctarum) tetrachordum, jam factum est tetrachordum (disjunctarum) neton diezeugmenon.

Talis jam conspicitur, ea pars Diagrammatis, quæ est ab Hypate meson, ad Neten diezeugmenon. Ubi inter duo tetrachorda (Meson & Diezeugmenon) Tonus interjicitur (à Mese ad Paramesen,) qui, utrivis Tetrachordorum conjunctus, Pentachordum efficit. Adeoque ab Hypate meson, ad Mesen, est Dia-tessaron; ad Paramesen, Dia-pente: Itemque à Nete diezeugmenon, ad Paramesen, est Diatessaron; ad Mesen, Dia-pente. Atque ab Hypate meson, ad Neten diezeugmenon, est Dia-pason. Et quidem, si, cum Meson tetrachordo, continuetur tetrachordum (Conjunctarum) Synemmenon; habetur (antiquorum) Heptachordum: si cum eodem Meson tetrachordo, sumatur, post tonum interjectum, tetrachordum (Disjunctarum) Diezeugmenon; habetur (Pythagoræ) Octachordum. Unde etiam patet ratio, cur, duorum Tetrachordorum, alterum Synemmenon (conjunctarum,) alterum Diezeugmenon (divisarum, seu disjunctarum) dicatur: quoniam, nimirum, cum tetrachordo Meson, illud conjunctum sit; hoc ab eodem (interposito Tono) Disjunctum. Horum duorum (sive in antiquorum Heptachordo, sive in Octachordi Pythagoræ) Tetrachordorum; alterum Hypaton (quod gravius erat,) alterum Neton (quod acutius,) ipsos dixisse non dubito. Et quidem, in Heptachordo, neton

The Harmonics of the Ancients compared with Today’s

165

Synemmenon, (ultimarum conjunctarum) in Octachordo, neton Diezeugmenon (ultimarum Disjunctarum.) Quæ quidem nomina (συνημμένων νητῶν, & διεζευγμένων νητῶν,) Gaudentius p. 8, 9, 10, perpetuo retinet. (Neque putaverim, quod sentit Meibomius, id mendum esse.) Alii plerumque (& Gaudentius aliquando) omissa voce Neton (ut quæ tuto abesse posset, sed subintellecta,) dicunt simpliciter Synemmenon, & Diezeugmenon: eadem ratione qua (curte loquentes) dicere item solent, Hypaton diatonos, Meson diatonos, &c. pro quo integre dicendum erat (& passim dicitur) Lichanos hypaton diatonos, Lichanos meson diatonos, &c. Sed interea temporis, ut videtur (sive à Terpandro, sive à Timotheo Milesio, sive à quopiam alio,) adjecta erat (chordis septem ante receptis) Octava chorda; non quidem duobus tetrachordis intermedia (ut in Pythagoræ Lyra) sed utrique in grave: quæ vel (ὑπερυπάτη) Hyper-hypate dicebatur (ut Boeth. lib. 1. cap. 20.) utpote, quam Hypate, gravior; aut Proslambanomenos, sive (ut Bryennius passim) Proslambanomene, tanquam (præter duo tetrachorda) extrinsecus Assumpta, quo compleretur Dia-pason Consonantia (ut quæ in Septichorde Lyra, ex duobus conjunctis Tetrachordis, non comparebat:) retentis, ut prius, Septichordis Lyræ nominibus: ad hanc formam,

Atque hanc passim adhibet Bryennius, (lib. 2. sect. 4. lib. 3. sect. 1. & alibi) adeoque Tonos seu Modos suos septem, paulo aliter disponit quam suos septem Ptolemæus: Quæ tamen ipsi dicitur (Πυθαγόρου ὀκτάχορδος λύρα) Pythagoræ Lyra octachordis; sicut illa altera, absque hac assumpta chorda, (ἀρχαιότροπος ἑπτάχορδος λύρα ἑρμοῦ) antiquæ formæ Septichordis lyra Mercurii: (lib. 1. sect. 1.) Quodque vocant Systema perfectum, quindecim chordarum, conflatum esse vult ex istiusmodi binis Lyris invicem compostis; quarum alteram (à Proslambanomeno ad Mesen) Graviorem vocat (Πυθαγόρου ὀκτάχορδον λύραν βαρυτέραν·) alteram Acutiorem (Πυθαγόρου ὀκτάχορδον λύραν ὀξυτέραν·) & quidem, tum Gravioris Neten, tum Acutioris Proslambanomenon, (ut quæ une sint eademque chorda,) communi nomine Mesen vocari. Et similiter, Heptachordum [p. 162] illud, Hypate hypaton & Mese terminatum, vocat Mercurii Lyram antiquam septichordem Graviorem, (Ερμοῦ τρισμεγίστου ἀρχαιότροπον βαρυτέραν λύραν ἑπτάχορδον·) quodque à Paramese & Nete hyperbolæon terminatur, Mercurii

166

John Wallis: Writings on Music

Lyram Septichordem Acutiorem, (Ερμοῦ τρισμεγίστου ἑπτάχορδον λύραν ὀξυτέραν.) Sed redeo ad priorem illam Octachordi dispositionem, (quæ potior videtur; quamque in Tonis suis, seu Modis, ordinandis sequitur Ptolemæus, cap. 8, 9, 10, 11. lib. 2. atque alibi passim;) quæ Tonum habet Diazeucticum, duobus Tetrachordis interjectum. Atque hoc quidem, pro ea ætate, Systema perfectum habebatur: indeque hanc Octo chordarum compagem, non δι᾽ ὀκτὼ, pro chordarum numero, (prout quatuor, & quinque chordarum compages dixerunt Dia-tessaron, & Dia-pente;) sed Dia-pason, quasi chordarum omnium. Quod docet Ptolemæus, cap. 4. lib. 2, & cap. 1. lib. 3. & Aristoxenus, p. 16. [Scales of Three or Four Tetrachords] Postea vero, inter duo tetrachorda (hypaton & neton,) medium interposuerunt recentiores, Meson (μέσων) tetrachordum: Ut jam, non tam chorda Mese, quam tetrachordum Meson (mediarum,) sit medio loco positum. Nempe, qua ratione, in Heptachordo, mediam chordam dixerant Mesen (mediam;) eadem &, in Trium Tetrachordorum systemate, tetrachordum Medium dixerunt Meson (μέσων) Mediarum: Retentis Supremarum & Infimarum (seu ultimarum) nominibus, ut prius. (Per Supremum, illud hucusque intelligentes, quod erat Gravissimum; per Infimum, seu ultimum, illud quod erat Acutissimum: contra quam nos hodie facimus.) Adeoque ex Heptachordo & Octachordo (prout Neton tetrachordum Conjunctum erat vel Disjunctum, hoc est Synemmenon vel Diezeugmenon,) fecerunt (interposito jam alio tetrachordo). Decachordum & Hendecachordum. Et quidem hoc Meson (mediarum) tetrachordum, præsertim respicit extrema duo in systemate conjuncto, Hypaton (supremarum) & Synemmenon (ultimarum Conjunctarum,) quod diserte dicit Aristides, p.  10. τῶν μέσων τετράχροδον· τοῦτο γὰρ μόνον μεταξὺ θεωρεῖται τῶν τε ὑπατῶν καὶ τῶν συνημμένων. Sed nec male ad (νητῶν διεζευγμένων) Disjunctarum tetrachordum referri potest (utut non præcise in medio duorum;) atque sic Boethius, cap. 20. lib. 1.

The Harmonics of the Ancients compared with Today’s

167

Jamque bis dicitur Hypate, parypate, lichanos, addito primum hypaton, deinde meson: quæ, cum Mese, faciunt duo tetrachorda invicem conjuncta; quibus Hypate meson (quæ eadem est & hypaton nete) communis est. Post mesen vero, Neton tetrachordum vel est Conjunctum (ut in Heptachordo) completque lyram (δεκάχορδον) decem chordarum, (ut, in præmisso Diagrammate, ab Hypate hypaton ad Neten Synemmenon:) vel Disjunctum (ut in Octachordo) completque lyram (ἑνδεκάχορδον) undecim chordarum, (ut, in præmisso Diagrammate, ab Hypate hypaton ad Neten diezeugmenon.)

168

John Wallis: Writings on Music

Tandem vero, quo Cantus ambitum porro ampliarent, (retentis quæ jam fuerant tetrachordorum nominibus, Hypaton, Meson, & Neton.) Post Neton (ultimarum) tetrachordum (sive conjunctarum, sive disjunctarum,) aliud adhuc (in Acutum) adjiciebant tetrachordum, Hyperbolæon (ὑπερβολαίων) dictum, (aut etiam, ut Gaudentius, p. 9, 10. νητῶν ὑπερβολαίων) quasi Extravagantium, exorbitantium, seu excedentium, (quippe, quæ sunt ultra Ultimas recte dixeris extravagantes,) aut mitiori vocabulo (quo & plerique utuntur) Excellentium. (Quod quidem Hyperbolæon tetrachordum, est cum neton synemmenon tetrachordo, Disjunctum; sed cum neton Diezeugmenon tetrachordo, Conjunctum.) Quæ quidem nomina, Neton Synemmenon, neton Diezeugmenon, & neton Hyper-[p. 163]bolæon. habentur integra (ut jam dictum est) apud Gaudentium, p. 8, 9, 10, 18. Et quidem apud Euclidem legebatur νητῶν συνημμένων, νητῶν διεζευγμένων, νητῶν ὑπερβολαίων, antequam Meibomius vocem νητῶν expunxerit; ut monet ipse Meibomius in Notis suis ad illum locum, pag. 6. Idemque apud Porphyrium conspicitur, in ejus ad Ptolemæum Commentario. Sed apud alios plerumque, omisso (sed subintellecto) Neton, dici solent (simpliciter) Synemmenon, Diezeugmenon, & Hyperbolæon, tetrachorda. Postremo omnium, (cum unus adhuc deesset tonus ad interum Dis-diapason,) additus est in gravi Proslambanomenos (sonus ascitus seu acquisitus;) uno tono gravior quam erat Hypate hypaton. Ut jam tandem Mese (quæ sic primitus dicta fuerat, quod fuerit in Heptachordo Media) redeat (ex postliminio) in suum situm medium; justo Dia-pason distans, tum a Proslambanomeno, tum à Nete Hyperbolæon (bina Dia-pason jam dirimens, ut pridem bina Tetrachorda.) Totumque sic completur Ptolemæi Systema perfectum (τέλειον) quindecim chordarum. Ut in prius exposito Diagrammate. [The Perfect System] Quis autem (in tota compage) hanc aut illam Chordam primus introduxerit, expedire non valeo; cum neque id sat certum sit, neque hac in re consentiant Authores: Horum aliquot Nomina habemus apud Nicomachum, p. 29, 35. & Boethium, cap. 20. lib. 1. Quid autem quisque fecerit, non est ut simus admodum solliciti. Constat igitur, integrum Systema perfectum, Tetrachordis quatuor, & binis Tonis: (Quæ simul complent duo Dia-pason; Nempe, alterum à Proslambanomeno ad Mesen; alterum à Mese ad Neten Hyperbolæon.) Quatuor, inquam, Tetrachordis; (quæ singula nos canimus, ab ima voce la vel mi, ascendendo in acutum per fa sol la:) Nempe; vel bis binis conjunctis (Hypaton & Meson; itemque Diezeugmenon & Hyperbolæon;) ut in Systemate quod Disjunctum vocant: Vel, ternis conjunctis, (Hypaton, Meson, & Synemmenon,) unoque singulari, (Hyperbolæon;) ut in Systemate quod Conjunctum vocant, (propter Tetrachordum tertium, cum Mese, adeoque cum binis primoribus Conjunctum.) Et binis, inquam, Tonis; (qui, ab ima voce la, ascendunt in acutum, per integrum semper tonum sesquioctavum, voce mi;) quos Diazeucticos vocant, aut Diazeuxes: Nempe, altero à Proslambanomeno ad Hypaten hypaton, (quam vocant Diazeuxin graviorem;) altero à Mese ad

The Harmonics of the Ancients compared with Today’s

169

Paramesen (quam vocant Diazeuxin acutiorem) in Systemate Disjuncto; vel (hujus loco) in Systemate Conjuncto, (si saltem hoc ad quartum Tetrachordum continuetur) à Nete synemmenon ad Neten Diezeugmenon. Quod diserte asserit Boethius, p. 10. (qui tamen p. 18. Systema conjunctum, non nisi ad Dia-pason & dia-tessaron extendit.) Et Bacchius, pag. 10. Sunt quidem omnino Tetrachorda quinque; & Toni tres: sed non quæ simul adhibentur. Nam, quando adhibetur tetrachordum Diezeugmenon, adeoque Diazeuxis à Mese ad Paramesen; omittitur Synemmenon tetrachordum, tonusque à Nete synemmenon ad Neten diezeugmenon: Et, vice versa, ubi hæc habentur; omittuntur illa. Item, in Systemate Conjuncto, vix aut ne vix memorare solent Hyperbolæon tetrachordum, tonumque huic præfixum; sed, quasi hæc non essent, systema terminant in Nete synemmenon: Systema facientes (non Quindecim, sed) Undecim chordarum. Atque jam non sufficiet (quo designetur chorda) Hypaten, Parypaten, & Lichanon dicere, (ut prius;) sed dicendum porro, in quo Tetrachordo; num Hypaton, an Meson: Neque sufficiet, Triten, Paraneten, & Neten dicere; sed, cujus item Tetrachordi; Synemmenon, Die-zeugmenon, an Hyperbolæon. Hinc igitur oriuntur (atque ob causas jam dictas) Chordarum, in Græcorum Systemate, Nomina & Dispositiones. Potuissent autem non incommode (postquam duarum vocum redierit necessitas pro designandis chordis) jam rejici (quæ quo id vitaretur introductæ fuerant) voces Lichanos & Trite; atque ex postliminio reduci Paranete & Parypate: ut (pro Licbanis) diceretur Hypaton paranete, & Meson paranete; itemque (pro Tritis,) Synemmenon parypate, Diezeugmenon parypate, Hyperbolæon parypate: (nec enim minus commode dicitur Paranete hypaton, quam Lichanos hypaton; & sic de cæteris:) Sed mos obtinuit ut retineantur ante-introducta vocabula; utut cessante causa qua fuerant introducta. [The Genera and Their Species] Sed ad Dia-tessaron revertimur:a Pro cujus varia divisione, varia quæ vocant Genera (ut dictum est) oriuntur. Quippe, Genus est ποιὰ τεττάρων φθόγγων διαίρεσις, Eucl. p. 1. vel ποιὰ τετραχόρδου διαίρεσις καὶ διάθεσις, Gaudent. p. 5. hoc est, Qualiter dividitur & disponitur Tetrachordum, ejusve soni quatuor. Ut autem unum aliquod, sic (in eodem Genere) reliqua omnia Tetrachorda dividuntur. Genera (γένη) communi omnium consensu statuuntur tria; (ἐναρμόνιον, χρωματικὸν, & διατονικὸν·) Enarmonium, Chromaticum, & Diatonicum, Aristox. p. 19, 44. Gaudent. p. 5. Eucl. p. 8, 9, 10. Sed non ita consentiunt, in Speciebus, sub his generibus enumerandis; quas χρόας colores dicunt (χρόα δέ ἐστι γένους εἰδικὴ διαίρεσις, Eucl. p. 10.) Neque in Tetrachordi, pro suis Generibus, divisione; quod alii aliter, in singulis Generibus dividunt. Aristoxenus integrum Dia-tessaron (quod Tonis duobus cum semisse constare statuit) sic distribuit, p. 24, &c.   MS correction in Bodleian Savile Gg 31: revertimus:

a

John Wallis: Writings on Music

170

In Genere Enarmonio (γένει ἐναρμονίῳ) Intervallum ab Hypate ad Parypaten, itemque à Parypate ad Lichanon (seu Paraneten) Diesin facit; & quidem (δίεσιν τεταρτημόριον) Diesin Quadrantalem; ut quam Toni partem quartam, seu tres duodecimas, continere vult; vocatque Diesin enarmoniam, δίεσιν ἐναρμόνιον, atque omnium minimam; Quo nullum canitur minus intervallum: Aristox. p. 46. Inde vero ad Neten, Ditonou; & quidem (δίτονον ἀσύνθετον) Di-[p. 164]tonum incompositum; seu duorum tonorum Intervallum indivisum. Adeoque, posito Tono, partium duodecim, utraque Diesium erit partium trium; & Ditonum, partium viginti quatuor; & simul omnia, partium triginta: puta 3 + 3 + 24 = 30. In genere Chromatico molli (μαλακῷ χρωματικῷ) Ab Hypate ad Parypaten; itemque ab hac ad Lichanon, Diesin facit Trientalem (τριτημόριον) seu quatuor duodecimas (δωδεκατημόρια) unius Toni: Indeque ad Neten, Duodecimas viginti duas; (hoc est, intervallum indivisum ex tono, hemitonio, & diesi trientali constans: seu ex 12 & 6 & 4 duodecimis:) Nempe 4 + 4 + 22 = 30. In Chromatico Hemiiolio (ἡμιολίῳ) Ab Hypate ad Parypaten; indeque ad Lichanon; Diesin facit hemiolion, (nempe Sesquialteram Dieseos Harmonicæ;) adeoque duodecimarum quatuor cum semisse, seu 41/2: Indeque ad Neten;a duodecimas viginti & unam; (hoc est intervallum indivisum ex septem diesibus quadrantalibus; seu ex Tono & Hemitonio & Diesi quadrantali constans; hoc est ex duodecimis 12 & 6 & 3:) Nempe 41/2 + 41/2 + 21 = 30. In Chromatico Tonico seu Toniæo (τοναίῳ) Ab Hypate ad Parypaten; itemque ab hac ad Lichanon; Hemitonium facit, (partium duodecimarum sex:) Adeoque, ab hac ad Neten, Triemitonion (τριημιτόνιον) seu Sesquitonum, (hoc est, intervallum indivisum ex Tono & Hemitonlo constans; seu duodecimis 12 & 6, hoc est 18:) Nempe 6 + 6 + 18 = 30. In Diatonico molli (διατονικῷ μαλακῷ) Ab Hypate ad Parypaten, Hemitonium, seu 6 duodecimas: Ab hac ad Lichanon, toni dodrantem, seu hemitonium cum diesi quadrantali, seu intervallum indivisum ex tribus diesibus quadrantalibus compositum; hoc est, 9 duodecimas: Adeoque, ab hac ad Neten, 15 duodecimas (seu intervallum indivisum ex quinque diesibus [p. 165] quadrantalibus, seu ex tono & diesi quadrantali constans.) Nempe 6 + 9 + 15 = 30. In Diatonico Intenso (συντόνῳ) Ab Hypate ad Parypaten, Hemitonium facit: A Parypate ad Lichanon; itemque ab hac ad Neten; Tonum. Nempe 6 + 12 + 12 = 30. Eadem habet Genera Euclides, pag. 10, 11. & sic divisa. Nec multo aliter Gaudentius, p. 5, 6. Atque hæc ipsa Genera, ad Aristoxeni mentem, recitat Ptolemæus, cap. 12. lib. 1. sed numeris duplicatis, (posito nempe Tono, partium 24; adeoque duobus cum semisse, partium 6o; scilicet ne fractione opus sit in Chromate Hemiolio:) Nempe

 1682: Neten,

a

The Harmonics of the Ancients compared with Today’s

171

Aristoxeni Enharmonium, 6 + 6 + 48 = 60 Chroma molle, 8 + 8 + 44 = 60 Chroma hemiolion, 9 + 9 + 42 = 60 Chroma tonicum, 12 + 12 + 36 = 60 Diatonum molle, 12 + 18 + 30 = 60 Diatonum intensum, 12 + 24 + 24 = 60 Sunt itaque, ad Aristoxeni mentem, Genera (ut apud alios) Tria; sed Colores (χρόαι) Sex, (quasi totidem Species sub his Generibus:) Generis Enarmonii, Unus; Chromatici, Tres; & Diatonici, Duo. Videtur autem (in his Generibus) Aristoxeni tempore, aut temporibus eo superioribus, Genus Enarmonium maxime fuisse celebratum, Aristox. p. 2. & Meibom. ad eum p.  76, 77. sed quo pertingere, difficile censebatur; quo itaque pauci tunc utebantur, Aristox. p. 23. Meibom. 92. Gaudentii tempore, solum fere Diatonicum in usu fuit, Gaudent. p. 6. Ptolemæi temporibus, omnia Diatonica genera, & Chroma intensum; sed non item Enarmonium, & molliora Chromatica, Ptol. cap. 16. lib. 1. Nostra vero ætate, vix aut ne vix aliud quam Diatonum intensum; aut quod huic suppar sit. [The Ranges of Various Sounds] Locus itaque in quo movetur Parypate (à parypaton gravissima ad acutissimam) est Diesis Enarmonia, Aristox. p. 23, 47. Eucl. p. 10. Nempe à Parypate gravissima, (quæ diesi quadrantali distat ab hypate) ad acutissimam (quæ ab hypate distat hemitonio, hoc est duabus hujusmodi diesibus,) est una diesis quadrantalis, vel Enarmonia; per quem locum vagatur Parypate; nempe, à fine primæ Dieseos quadrantalis, ad finem secundæ, seu Hemitonii. Locus in quo movetur Lichanos, seu Paranete, est Tonus: Quippe à Mese, seu Nete, non minus distat quam Tono, (ut in genere Diatonico intenso;) nec plus quam Ditono (ut in Enarmonio;) adeoque per Toni locum vagatur, Aristox. p. 22, 46. (ubi, pro τρόπος modus, legendum τόπος locus p. 22. lin. 25.) Eucl. p. 10. Suntque hi loci, non quidem Communicantes invicem (ut alter in alterius terminos penetret;) nec Disjuncti, (ut, inter utrumque locum, aliquid interveniat;) sed Continui; ubi alter desinit alter incipit: estque primi (ab Hypate) hemitonii finis, terminus utrique loco communis; (Parypaton intensissima, & remississima Lichanon.) Atque hoc est, quod vult Aristoxenus, p. 23. οὐ γὰρ ἐπαλλάττουσιν οἱ τόποι, ἀλλ᾽ ἔστιν αὐτῶν πέρας ἡ συναφή· (quod nescio an editor satis intellexerit:) Sic enim sequitur, ὅταν γὰρ ἐπὶ τὴν αὐτὴν τάσιν ἀφίκωνται ἥ τε παρυπάτη καὶ ἡ λιχανὸς, ἡ μὲν ἀνιεμένη, ἡ δὲ ἐπιτεινομένη, πέρας ἔχουσιν οἱ τόποι. καὶ ἔστιν, ὁ μὲν ἐπὶ τὸ βαρὺ, παρυπάτης· ὁ καὶ ἐπὶ τὸ ὀξὺ, λιχανοῦ (deleo τε καὶ παρυπάτης· quod, si placet, mox restituas, viz.) καὶ περὶ τούτων μὲν [λιχανοῦ τε καὶ παρυπάτης] οὕτως ὡρίσθω· περὶ τῶν δὲ κατὰ γένη τε καὶ τὰς χρόας, λεκτέον. Quæ sic reddo; Neque enim loci illi (parypates & lichani) alternantur: sed communi termino connectuntur. Ubi enim in eandem tensionem pervenerint parypate & lichanos (illa quidem intensa; hæc, remissa;) terminum communem habent eorum loci. Est

172

John Wallis: Writings on Music

utique, qui in grave est, locus parypates; qui in acutum, lichani locus. Atque de his quidem (lichani & parypates locis) sic esto definitum. De eis quæ Genera & Colores spectant, jam dicendum. Quoties autem, Tetrachordi, Duo diastemata minora (ab Hypate ad Lichanon) sunt (simul sumpta) minus quam Tertium indivisum (à Lichano seu Paranete ad Neten) Spissum (πυκνὸν) vocatur; (quod in quatuor primoribus contingit; Enarmonio & tribus Chromaticis:) cui opponitur (ἄπυκνον) Non-spissum (Martianus Capella Rarum vocat,) ubi hoc non contingit; ut in Generibus Diatonicis; ubi sunt, aut Reliquo simul-æqualia (ut in Molli Diatono,) vel (ut in Diatono intenso) Reliquo simul-majora. Quippe; illic, spissius seu arctius connectuntur tres chordæ (Hypate, Parypate, & Lichanos;) hic, laxius seu minus spisse, Aristox. p. 24 Qui autem Soni chordæve possunt Spissum ingredi (nempe Hypate, Parypate, Lichanos, aut quæ sunt harum instar, in Tetrachordo quolibet) πυκνοὶ dicuntur; Nempe, Hypatæ (& ὑπατοειδεῖς omnes) βαρύπυκνοι· Parypatæ (& παρυπατοειδεῖς, hoc est Tritæ) μεσόπυκνοι· Lichani (λιχανοειδεῖς, hoc est Paranetæ) ὀξύπυκνοι· (Quippe soni illi, in Spissi parte gravi; isti, in media; hi, in acuta parte reperiuntur:) Qui vero Spissum ingredi nunquam possunt, ἄπυκνοι. Sunt igitur (Sonorum octodecim potestate) Barypycni quinque, Hypate hypaton, Hypate meson, Mese, Paramese, Nete hyperbolæon; (ut quæ sunt ὑπατοειδεῖς in Tetrachordis Hypaton, Meson, Synemmenon, Diezeugmenon, & Hyperbolæon:) Mesopycni totidem, Parypate hypaton, Parypate meson, Trite synemmenon, Trite diezeugmenon, Trite hyperbolæon; (ut [p. 166] quæ sunt, in eisdem Tetrachordis, παρυπατοειδεῖς) Totidemque Oxypycni; Lichanos hypaton, Lichanos meson, Paranete synemmenon, Paranete diezeugmenon, Paranete hyperbolæon; (ut quæ sunt, in eisdem Tetrachordis, λιχανοειδεῖς·) Apycni, Tres; Proslambanomenos, Nete synemmenon, Nete hyperbolæon; (ut quæ spissum nunquam ingrediuntur.) Suntque Apycni & Barypycni, Stantes omnes: Mesopycni & Oxypycni, soni Mobiles. Arist. p. 12. Alypius, p. 2. Atque hæc quidem sunt Genera, secundum Aristoxenum, ejusque sequaces, Aristoxeneos dictos; qui per Intervalla symphonias metiebantur; & Dia-tessaron statuebant duorum Tonorum cum semisse. [The Pythagorean Approach] Pythagoras, ejusque sequaces, Pythagorei dicti, posthabita Aristoxeni methodo per Intervalla symphonias æstimandi, easdem per sonorum ad invicem Rationes designabant: &, speciatim, Dia-tessaron consonantiam statuebant, non, quando Soni Distarent invicem per Tonos duos cum semisse; sed, quando ad invicem essent in Ratione sesquitertia. Adeoque, pro tribus Intervallis, quae simul addita conficerent Tonos duos cum semisse; quærebant tres Rationes quæ (compositæ) Rationem constituerent sesquitertiam, seu 4/3. Et prout hoc aliis atque aliis modis fieri posset commode, sic alia atque alia Genera definiebant. Ptol. cap. 9, lib. 1. Nec tamen ad hoc Rationes indifferenter omnes admittebant, pro sonis continue sequentibus designandis; sed Multiplices, & Superparticulares; non autem

The Harmonics of the Ancients compared with Today’s

173

(nisi coacte) rationes Superpartientes, & multo adhuc minus quæ sunt (ἄλογοι, ἄῤῥητοι,) numeris Ineffabiles. Ptol. cap. 5, 15, 16. lib. 1. Hujusmodi divisiones tetrachordi, plurium Musicorum, simul exhibet Ptolemæus, cap. 14. lib. 2. & speciatim Archytæ, cap. 13. lib. 1. Et suam ipsius cap. 15, 16. lib. 1. Nempe

De quibus omnibus, quod judicium fecerit Ptolemæus; &, qua ratione sua instauraverit, cæterisque prætulerit; ab ipso fuse disputatum est, cap. 12, 13, 14, 15, 16. lib. 1. [Mutation] Dividuntur autem, in eodem Genere, omnia Tetrachorda seu Dia-tessaron similiter; nempe, in Enarmonio, Enarmonice omnia; in Chromatico, Chromatice; in Diatonico, Diatonice omnia; & in singulis sub quoque genere (χρόαις) coloribus, pro sua cujusque ratione omnia. Si quando secus contingit; puta, ut Tetrachordum aliud Enarmonice, aliud Chromatice vel Diatonice dividatur; aut etiam aliud secundum Diatonum molle, aliud secundum Diatonum intensum; (& de reliquis similiter:) dicitur hæc (μεταβολὴ κατὰ γένος) Mutatio secundum Genus; nempe transitus de Genere in Genus. Eucl. p. 20. Quæ quidem Commutatio, seu de Genere in Genus transitio, facienda est (ut Concinne fiat) in uno aliquo ex Sonis Stabilibus, (Tetrachordum terminantibus;) quippe si, in eodem Tetrachordo, sumeretur Parypate secundum genus unum, & Lichanos secundum aliud, Inconcinna foret ea commutatio. Eucl. p. 21. Quod ipsum observandum est in aliis item Commutationibus; puta secundum Tonum, secundum Systema, secundum Melopœiam, aut cantus naturam; de quibus suo loco dicetur, Eucl. ibid.

174

John Wallis: Writings on Music

Nam Quatuor, inquit, modis sumitur vox (μεταβολὴ) Mutatio. Nempe, Mutatio secundum Genus (μεταβολὴ κατὰ γένος) de qua jam agitur: Mutatio secundum Tonum (κατὰ τόνον) de qua post dicetur: Mutatio secundum Systema (κατὰ σύστημα) utputa cum ex Systemate disjuncto (διεζευγμένῳ) in conjunctum (συνημμένον) transitur; vel ex hoc in illud; (quam aliam non esse post dicetur, quam transitum de Tono Dorio in Mixolydium, vel contra:) & Mutatio secundum Melopœiam (κατὰ μελοποιΐαν) utputa, quando ex More seu Genio Musices magis Dilatato (ἤθοις διασταλτικοῦ) quo signatur Magnificentia, virile animi Robur, & actiones Heroicæ, & Affectus hisce congrui; ut in Tragœdiis, & quæ huc accedunt; transitur in (συσταλτικὸν) Contractiorem, quo status animi humilior, magisque effœminatus insinuatur; ut in affectibus amatoriis, lamentis, miserationibus, & horum similibus: vel in Pacatiorem (ἡσυχαυτικός) qui tranquillum, liberalem, & pacatum animi statum insinuat; tic in Hymnis, Encomiis, Consultis, & horum similibus: Aut, vice versa. Eucl. p. 21. Hanc (κατὰ γένος μεταβολὴν) Mutationem secundum Genus, fusius exponit Ptolemæus, cap. 16. lib. 1. & cap. 15. lib. 2. Docetque, quæ cum quibus apte commutentur Genera; Transitumque illum de Genere in Genus, faciendum esse in ipso Tono, Diazeuctico, qui disjungit ea duo Tetrachorda (quorum acutius sit unius generis, & gravius alterius,) & utrique generi communis est. Sed tota hæc, de variis Generibus, doctrina jam fere desuevit. Quippe jam, per multa se-[p. 167]cula, unicum habemus Genus Musicum usu receptum. Quod sive sit Aristoxeni Diatonum Intensum, per Tonos & Hemitonia, saltem crassius loquendo; Sive Ptolemæi Diatonum Ditonicum, per Tonos & Limmata (λείμματα) quæ ab Hemitoniis exiguo differunt; (cui soli suam accommodavit Sectionem Canonis Euclides; quem plerique omnes ad Boethium usque, indeque ad nostra fere tempora, secuti sunt Musurgi, seu Musici practici:) Sive Ptolemæi Diatonum Intensum, per Tonos majores, minores, & quasi-hemitonia; (de quo qui subtilius loquuntur, nonnumquam disceptant:) non est ut his loci subtilius disputemus, (id post facturi:) Quippe hæc tantillo inter se differunt, ut aurium judicio vix aut ne vix distingui possint; &, crassius loquentibus, pro eodem haberi solent; & subtilior disputatio speculationem magis praxin spectat. [The Principle Consonances and their Compounds] Atque hactenus de Tetrachordo diximus, seu Dia-tessaron; quam Primam vocant Consonantiam; pro cujus varia divisione, varia quæ diximus oriuntur Genera. Quam duorum Tonorum cum semisse faciunt Aristoxenei, Pythagorei vero Ratione Sesquitertia definiunt. Sin Tetrachordo (sic diviso) unus adjungatur (infra suprave) Tonus integer; fit (πεντάχορδον) ex Quinque chordis constans systema, seu, (διὰ πέντε) Dia-pente. Quam Secundam vocant Consonantiam. Quam itaque Trium tonorum cum semisse faciunt Aristoxenei, (propter 21/2 + 1 = 31/2:) Pythagorei vero ratione definiunt sesquialtera, (propter 4/3 × 9/8 = 3/2.) Ex duobus his (Tetrachordo & Pentachordo) composìtum, Octachordum (ὀκτάχορδον) conficiunt; chordarum Octo systema. (Quippe chordarum una, est

The Harmonics of the Ancients compared with Today’s

175

Tetrachordo & Pentachordo communis; adeoque omnium numerus, Octo; non, Novem.) Estque hæc Tertia Consonantia. Quam Tonorum Sex faciunt Aristoxenei, (propter 21/2 + 31/2 = 6,) seu potius, Tonorum Quinque cum Duobus Hemitoniis, (propter duo per se Hemitonia:) Pythagorei vero ratione Dupla definiunt, (propter 4 /3 × 3/2 = 2/1.) Vocant autem, non (δι᾽ ὀκτὼ) Di‑octo (pro chordarum numero; quod fit in Dia‑tessaron, & Dia‑pente;) sed Dia‑pason (διὰ πασῶν Per-omnes;) utpote quod (σύστημα τέλειον) systema Perfectum censebant antiquiores: (Aristid. p. 16. Ptol. cap. 4. lib. 2.) Quam tamen Appellationem Ptolemæus (ibidem) nonnisi Disdia‑pason Systemati concedit. Hanc consonantiam (quam quasi pro Unisono habent; Sonosque qui illam continent, instar unius Potestate soni;) consonantiæ cuivis (Dia‑tessaron, Dia‑pente, aut sibi ipsi,) semel (vel pluries) additam, efficere item volunt consonantiam; (quod huic soli consonantiæ peculiare statuunt.) Aristox. p. 20, 45. Eucl. p. 13. Ptol. cap. 6. lib. 1. Quo potissimum nomine disputat (ibidem) Ptolemæus contra Pythagoreos; qui, Dia‑pason & dia‑tessaron, consonantiarum numero exemptam vellent, eo quod ratione contineatur, 8 ad 3, (propter 2/1 × 4/3 = 8/3) quæ neque est Multiplex, neque Superparticularis, (quas solas consonantiis convenire volunt;) sed Superpartiens; nempe (ut loquuntur) Dupla superbipartiens tertias, (propter 8/3 = 22/3.) Ptol. cap. 5, 6. lib. 1. Quartam itaque Consonantiam faciunt, Dia-pason & dia‑tessaron, (Undecim chordarum systema;) quam Tonorum octo cum semisse faciunt Aristoxenei, (propter 6 + 21/2 = 81/2,) sin mavis, Tonorum Septem cum tribus Hemitoniis: Pythagorei ratione 8 ad 3 definiunt, (propter 4/3 × 2/1 = 8/3.) Quintam Consonantiam faciunt Dia‑pason & dia-pente, (chordarum Duodecim;) Tonorum (secundum Aristoxeneos) Novem cum semisse, (propter 6 + 31/2 = 91/2,) vel Tonorum Octo cum tribus Hemitoniis: Ratione (secundum Pythagoreos) Tripla, (propter 2/1 × 3/2 = 3/1.) Sextam Consonantiam faciunt, Dis‑dia-pason, chordarum Quindecim;) Tonorum (secundum Aristoxeneos) duodecim, (propter 6 + 6 = 12,) seu Decem tonorum cum quatuor Hemitoniis: Ratione (secundum Pythagoreos) Quadrupla, (propter 2/1 × 2/1 = 4/1.) Atque hactenus enumerando procedit Ptolemæus, cap. 4. lib. 1. terminos Dis‑dia‑pason systematis (quod ipsi Perfectum systema dicitur) non excedens: Non quod plures non sint, (patitur enim natura rei processum in infinitum,) sed quod hæ sibi impræsentiarum sufficere videbantur; reliquæque, ultra has, essent (potestate) non nisi eædem repetitæ. Alii Septimam adjiciunt Consonantiam, Dis-dia-pason & dia‑tessaron, (chordarum Octodecim;) Tonorum (secundum Aristoxeneos) Quatuordecim cum semisse, (propter 6 + 6 + 21/2 = 141/2;) aut Tonorum duodecim cum Quinque Hemitoniis: Ratione (secundum Pythagoreos) 16 ad 3, (propter 4/1 × 4/3 = 16/3.) Aut etiam Octavam Consonantiam; nempe Dis‑dia‑pason cum diapente; (chordarum Novendecim;) Tonorum (secundum Aristoxeneos) Quindecim cum semisse, (propter 6 + 6 + 31/2 = 151/2) aut Tonorum tredecim cum quinque Hemitoniis: Ratione (secundum Pythagoreos) sextupla, (propter 2/1 × 2/1 × 3/2 = 6/1.)

John Wallis: Writings on Music

176

Ultra non extendebant Vocis Locum (φωνῆς τόπον·) saltem citra Tris-dia‑pason terminabant, (sive vocem Humanam spectemus, sive Organicam.) Aristox. p. 20, 45. Euclid. p. 13. Saltem si unum aliquod Instrumentum spectemus, uniusve Hominis vocem. Quippe, in diversis Tibiis Fistulisve, aut etiam Pueri vel Mulieris voce cum voce viri comparata, non negat Aristoxenus, p. 21. quin ultra distendi possit vocis ambitus (διάστασις,) puta ad Ter Quaterve Dia-pason, aut etiam ultra. Atque eadem eft Scalæ Guidoniæ dictæ, distensio; (quod notat ad Aristoxenum Meibomius, p. 90. & ad Euclidem, p. 45. Nempe, ab A re ad ee la, est Dis-dia‑pason cum dia‑pente; & quidem à Γ ut ad ee la, (unico adhuc tono major,) est saltem citra tris-dia‑pason. (Sic utique moderanda puto Meibomii verba, locis [p. 168] citatis; qui diserte affirmat, minus caute, eandem esse magnitudinem Scalæ Guidonianæ, nempe Dia-pente & dis-dia-pason; item Guidonianis maximum Systema, quod Scalam magnam perfectam vocant, est Icosachordum, seu viginti sonorum; id est, Dis-dia- pason & dia-pente: cum sit revera uno tono major.) Quæ autem sit omnium Maxima Consonantiarum, definiri nequit; potest enim Consonum, per ipsius Cantus naturam (ut dictum est) in infinitum augeri; Aristox. p. 20. Et quidem Dissonum, tum in infinitum augeri, tum in infinitum minui; cum nullum sit Intervallum absolute Minimum, (ut nec Linea, absolute brevissima,) Aristox. p. 20, 46. sed, quod cani possit minimum, esse volunt Diesin Enarmoniam (ut supra dictum est) quam Toni quadrantem faciunt. Hæc autem quæ diximus Intervalla, hodierni nostri Musici, aliis nominibus indicare solent: Nempe, Dia-tessaron vocant Quartam, (quod sic intelligendum est: nempe, Extremarum illud terminantium chordarum alteram, esse, à reliqua, Quartam inclusive sumptam:) Dia-pente (simili sensu) Quintam vocant; Diapason, Octavam; Dia-pason & dia-tessaron, Undecimam; Dia-pason & dia-pente, Duodecimam; Dis-dia-pason, Decimamquintam; Dis-dia-pason & dia-tessaron, Decimamoctavam; Dis-dia-pason & dia-pente, Decimamnonam; Tris-dia-pason (quae est Nona Consonantia) Vigesimamsecundam; atque in reliquis similiter. [Pythagorean and Aristoxenean Views of Interval] Cum vero, in superioribus, sæpius inciderit mentio, Consonantias (adeoque & Dissonantias) aliter ab Aristoxeneis, aliter à Pythagoreis, æstimatas esse: quippe illi per Intervalla (subductione æstimanda;) hi, per Rationes (divisione inquirendas,) easdem Consonantias designant: Puta; Dia-tessaron, duorum sonoruma cum semisse, dixerit Aristoxenus; Pythagoras vero, in ratione Sesquitertia, (& in reliquis similiter:) Opportunum erit, hic loci, utriusque sententiam plenius exhibere. Pythagoras scilicet; quique ab eo dicti sunt Pythagorei; Sonum ex percussi Aeris motu fieri reputantes; pro varia motus tremuli celeritate, varium Acuminis gradum oriri autumabant: & quidem, pro dupla celeritate, duplum acuminis gradum; quem esse faciunt Consonantiæ Dia-pason; Atque in reliquis similiter. Adeoque consonantiæ Dia-pason assignabant rationem Duplam; Dia-pente, Sesquialteram; Dia-tessaron, sesquitertiam: reliquasque hinc calculo derivabant;  1699: tonorum; MS correction in Bodleian Savile Gg 31: sonorum

a

The Harmonics of the Ancients compared with Today’s

177

Multiplicationis & Divisionis ope, sive Rationum compositione. Eucl. p. 23. Boeth. c. 30. lib. 1. Aristoxenus vero, ejusque sequaces, nimiam hanc esse subtilitatem rati, ad Aurium judicium provocabant, primaque inde principia desumenda volebant, unde deducerentur reliqua. Frustra esse, clamitantes, eos qui Rationum subtilitates sectabantur; quas & Phænomenis sæpe contrarias esse, aurium judicio comprobatum esse asserebant: (Ideo forsan, quod ex Pythagoreis aliqui non tam caute, quam par erat, rationes nonnunquam assignaverant; quod & causatur aliquando Ptolemæus.) Sensus itaque judicium, Principii loco habet Aristoxenus, indeque Demonstrationes deducit suas; secus quam Geometræ, qui ab Intellectu incipiunt. Sic enim loquitur, p. 33. Ο μὲν γὰρ γεωμέτρης οὐδὲν χρῆται τῇ τῆς αἰσθήσεως δυνάμει. – τῷ δὲ μουσικῷ σχεδόν ἐστιν ἀρχῆς ἔχουσα τάξιν ἡ τῆς αἰσθήσεως ἀκρίβεια. – τῇ μὲν ἀκοῇ κρίνομεν τὰ τῶν διαστημάτων μεγέθη· τῇ δὲ διανοίᾳ θεωροῦμεν τὰς τούτων δυνάμεις. Quippe, inquit, Geometra Sensus viribus non habet opus. – Musicus autem, subtilitatem Sensus, quasi principii loco habet. – Auditu enim Intervallorum magnitudines judicamus: Intellectu vero eorundem demum Potestates contemplamur. A Sensuum itaque Phænomenis Demonstrationes deducit suas; aliosque perstringit (Pythagoreos scilicet) ut nimium subtiles, qui secundum Numerorum Proportiones determinant Intervalla. Καὶ τούτων [διαστημάτων] ἀποδείξεις πειρώμεθα λέγειν ὁμολογουμένας τοῖς φαινομένοις. Οὐ καθάπερ οἱ ἔμπροσθεν· οἱ, ἀλλοτριολογοῦντες, καὶ τὴν μὲν αἴσθησιν ἐκκλίνοντες, ὡς οὖσαν οὐκ ἀκριβῆ· νοητὰς δὲ κατασκευάσαντες αἰτίας· καὶ φάσκοντες λόγους τέ τινας ἀριθμῶν εἶναι, καὶ τάχη πρὸς ἄλληλα, ἐν οἷς τό τε ὀξὺ καὶ βαρὺ γίνεται· πάντων ἀλλοτριωτάτους λόγους λέγοντες, καὶ ἐναντιωτάτους τοῖς φαινομένοις, pag. 32. (Quæ & ab Eruditissimo Johanne Gerardo Vossio, cap. 19. De scientiis Mathematicis, citantur; ut ab Aristoxeno contra Pythagoreos dicta.) Intervallorum, inquit, seu Diastematum Demonstrationes allatum ibimus a (sensuum) phænomenis deductas. Non, ut qui ante nos fuerunt (Pythagorei:) Qui, aliena prorsus hunc negotio proferentes, Sensum declinant ut non satis accuratum; causasque texunt à ratione depromptas: Dicuntque Numericas quasdam esse Proportiones, & Celeritatum adinvicem Rationes, in quibus Acutum & Grave consistunt: Rationes ita proferentes, ab hoc negotio maxime alienas, ipsisque Phænomenis plane contrarias. Nec multo aliter pag. 43, 44. à Sensum Phænomenis sumenda docet Principia & Postulata (ἀρχὰς & ὁμολογούμενα,) unde reliqua demonstrentur; non, extra subjectæ materiæ limites progrediendo, à natura Vocis, motuve Aeris petenda. Primam itaque Consonantiam, deprehensam auribus gratam, vocabant Diatessaron; proximam ad hanc, aurium judicio pariter deprehensam, vocabant Diapente. Has, quasi communi aurium consensu comprobatas (plerisque hominum, ne dicam omnibus, eadem auditu intervalla, ut consona aurique grata æstimantibus,) tanquam Cognitas sumebant. Harumque (ut cognitarum) differentiam (quanto scilicet hæc quam illa sit acutior, graviorve,) Tonum vocabant: quem itaque ut cognitum, & aurium judicio dignoscendum, sumebant.

178

John Wallis: Writings on Music

Hunc igitur (Tonum inquam) sumebant quasi Mensuram notam (aurium consensu comprobatam) ad quam exigenda essent cætera Intervalla Musica; puta, Hemitonia, Dieses, aliaque minora intervalla, ut Toni Semisses, Trientes, Quadrantes, &c: majora vero, tanquam ex [p. 169] Tonis aliquot, certisve toni partibus, composita: Et, speciatim, Dia-tessaron, esse Duorum tonorom cum semisse: Unde & reliqua intervalla (Additionis & Subductionis ope) calculo colligebant. Puta, Dia-pente, Trium cum semisse (ut quod, ex constructione, tono major est quam Dia-tessaron;) Dia-pason, tonorum sex, (utpote quod, aurium judicio, ex utrisque conflatur:) Et sic de cæteris. An vero (unde reliqua dependent) Dia-tessaron sit præcise duorum tonorum cum semisse; an eo proxime accedat, tantillo inde distans; dubitanter aliquando loqui videtur Aristoxenus. Utpote, pag. 24. An Dia-tessaron à minorum intervallorum aliquo (Hemitonio, vel Diesi,) mensuretur; an vero sit omnibus incommensurabile: (puta, an Hemitonium dictum, sit toni pars præcise dimidia; adeoque intervalli Dia-tessaron pars quinta; & Diesis Enarmonia, sit toni pars præcise quarta; & Chromatica, ejusdem pars tertia, &c.) Et, p. 28. An Ditonum sit minimæ Dieseos præcise Octuplum; an exiguo minus (μικρῷ τινι παντελῶς καὶ ἀμελῳδήτῳ ἔλαττον.) Et similiter, p. 46. ubi, Aliud esse vult, Toni partem tertiam quartamve sumere; aliud, tonum in tria quatuorve æqualia divisum canere; quasi non præcise de parte tertia quartave intellectum vellet, sed quod huic proxime accedat. (Nisi potius illud hic velit, posse quidem tertiam quartamve partem præcise sumi; sed non tres continue tertias, quatuorve quartas: qui sensus non est incommodus.) Sed hæc forte ab eo dicta non sunt, quasi de hoc dubitaverit ipse; sed quod hoc nondum demonstraverit (quod post demonstraturus foret, p. 56.) adeoque impræsentiarum hoc tantum postulat; nempe, vel tantundem esse, vel ab eo saltem pauxillo differre. [An Aristoxenian Test] Ad hanc itaque formam, probationem instituit suam; Nempe, (postquam definiendo determinaverat, Toni magnitudinem eam esse qua Dia-pente superat Diatessaron,) probatum it, p. 56. ipsius Dia-tessaron magnitudinem, esse, duorum tonorum cum semisse, (auditus Judicium pro Postulato habens,) his fere verbis; (nisi quod mendis scateat locus, quæ in Notis suis, sublatum it Meibomius;) Nimirum, Sumpto (auditus judicio) Dia-tessaron puta A B; auferatur, ab utroque ejus extremo, Ditonum; ut AC, BD: (quod quomodo fieret, auditus auxilio, ante dixerat, p. 55.)

Aequalia itaque erunt (propter æqualia ab æqualibus dempta) residua, AD, BC. Sumantur demum (aurium judicio,) ab horum terminis, D, C, duo Dia-tessaron,

The Harmonics of the Ancients compared with Today’s

179

(à termino graviori, in acutum; ab acutiori, in grave;) DE, CF: Quorum itaque excessus (ultra terminos primitus sumpti Dia-tessaron) AF, BE, erunt (ob causam modo insinuatam) invicem æquales, (ipsisque AD, BC; quibus item Ditona, à Dia-tessaron, superantur.) His itaque præparatis; Explorentur denique, ad Sensus judicium, extremi termini sic inventi, F, E. Qui quidem si dissoni deprehendantur; non erit, Dia-tessaron, duorum tonorum & semissis: sin deprehendantur consoni, (quod, ut Phænomenon experimento comprobatum præsumit, p. 24.) puta Diapente; manifestum est, ex duobus tonis cum semisse constare. Quam consequentiam sic probat. Cum enim FC sit Dia-tessaron; & FE, Dia-pente; hujusque super illud excessus CE, Tonus; atque, hic in B bisectus: erit BC (excessus Dia-tessaron supra Ditonum) toni semissis; adeoque Dia-tessaron, duorum tonorum cum semisse. Quodque modo ut Lemma sumpserat, nempe, extremos sonos FE, modo consoni sint, non aliam esse Consonantiam quam Dia-pente; sic probat. Nam & Majus est intervallum hoc quam Dia-tessaron, (quod patet, propter excessum utrinque additum, primitus sumpto Dia-tessaron:) Et Minus, quam Dia-pason; quod sic probat. Quod enim ex binis illis fit excessibus, est saltem Ditono minus; (quippe Dia-tessaron, minus quam tono, superat Ditonum; in confesso enim est apud omnes, Dia-tessaron majus quidem esse tonis duobus; sed minus tribus:) adeoque multo minus quam Dia-pente, (quod, cum Dia-tessaron, complet Dia-pason,) ut certum sit FE non esse Dia-pason. Si itaque consonum sit FE intervallum; cum majus sit quam Dia-tessaron, & minus quam Dia-pason; necesse est, esse Diapente; ut quæ sola intercedit magnitudo consona (sensuum judicio) inter Diatessaron & Dia-pason. Hoc ipsum Aristoxeni argumentum, summatim recitat Ptolemæus, cap. 10. lib. 1. Atque ostendit, quam sit infirmum. Quippe, in tam exigua discrepantia, quanta est (de qua quæritur) Hemitonii à Limmate, (nempe parte Limmatis 1/128 proxime,) impossibile est (etiam ipsis non distitentibus) ut possit auris tam accurate distinguere. Et quidem, si tantundem errare possit auris una vice; multo magis id fiat, in toto illius Argumenti processu; ubi ter sumendum sit (aurium judicio) Dia-tessaron, & bis Ditonon, idque in contrarias partes. Ut non sit, in tam subtili negotio, aurium judicio standum. Et Euclides ipse, cætera Aristoxenus, (qui per totam Introductionem Harmonicam Aristoxeni vestigiis presse insistit,) in Sectione Canonis, demonstrat Limma (quo Dia-tessaron superat Ditonum) minus esse quam Hemitonium; sed, sumpto Pythagoreorum principio. Et similiter Aristides Quintilianus (Aristoxeneus & ipse) initio libri 3. Forte nec ipse Aristoxenus (qui dubitanter, ut diximus, aliquando loquitur; suamque, ut vidimus, demonstrationern Hypothetice concludit, Si deprehendantur consoni, &c.) aliud voluit, quam, Intervallum illud, quo Dia-tessaron superat Ditonum, esse (quatenus saltem id Auditu æstimare possumus) Hemitonium.

180

John Wallis: Writings on Music

[The Pythagorean Alternative] Pythagorei contra Rationibus æstimabant So-[p.  170]norum ad invicem relationes. Quod Ptolemæus docet, lib. 1. cap. 5. & sequentibus: & quibus sumptis principiis id fecerint. Et quamquam id minus caute in aliquibus factum fuisse à quibusdam notet; id methodo tamen (quam ipse probat) imputandum negat; & quod ab aliis incaute peccatum fuerat, emendare satagit; quodque probe constitutum erat, confirmat. Neque aurium judicium refugit, sed inde sua comprobat. Summa rei huc redit. Sonos invicem comparatos, magis minusve consonos dissonosve, hoc est, auribus gratos vel ingratos, reputabant, pro Acuminum inter se ratione. Adeoque potiores Rationes, potioribus Consonantiis accommodabant. Et, speciatim, Rationem duplam 2 ad 1 (utpote, post æqualitatem, potissimam) Consonantiæ Dia-pason, omnium (post æquitonium saltem) gratissimæ: Sesquialteram, 3 ad 2; consonantiæ Dia-pente: Sesquitertiam, 4 ad 3; consonantiæ Dia-tessaron: &, quidem, sicut ex consonantiis Dia-pente & Diatessaron componitur Dia-pason, (quarum major sit Dia-pente:) sic ex Rationibus Sesquialtera & Sesquitertia, (quarum major est Sesquialtera,) componitur ratio Dupla; (propter 3/2 × 4/3 = 2/1.) Adeoque Tono, quo Dia-pente superat Diatessaron, rationem assignabant Sesquioctavam; (propter 4/3 × 9/8 = 3/2.) Quæ fusius prosequitur, lib. 1, cap. 5 & 7. Quæ omnia, quatuor numeris (ad hoc minimis) exponi solent. 12. 9. 8. 6. Quippe 12 ad 6, est Dia-pason; 9 ad 6 (itemque 12 ad 8) Dia-pente; 8 ad 6 (itemque 12 ad 9) Dia-tessaron; 9 ad 8, Tonus. Atque, ex paucis hisce positis, reliqua colliguntur. Hoc autem eo maxime confirmat Ptolemæus, quod, si Chordarum longitudines (cæteris paribus) sint in ratione dupla, auribus exhibebunt Dia-pason; si in Sesqui-altera, Dia-pente; in Sesquitertia, Dia-tessaron; in Sequioctava, Tonum: & in reliquis similiter. (Ut Sonorum Gravitates semper habeantur Longitudinibus chordarum proportionales: adeoque Acumina, in ratione contraria; ipsisque Longitudinibus ἀντιπεπονθότα, Reciprocantia.) Quod fuse prosequitur, cap. 8 & 11. adhibito Monochordo Canone. Ex his positis; facile convellitur eorum doctrina qui Dia-tessaron ponunt duorum tonorum cum semisse; Dia-pente, trium cum semisse; Dia-pason, sex tonorum; & quæ hinc consequuntur. Quod ostendit ille cap. 10, 11. Et, pro Hemitonio, substituendum (quod dicitur) Limma (λεῖμμα) hoc est, Residuum, (quod nempe restat, dempto Ditono ex Dia-tessaron;) in ratione 256/243; quod est, Hemitonio, aliquanto minus. Sed neque posse Tonum in duo plurave æqualia dividi (concinna, intellige,) & quæ sunt hujusmodi. Atque hic, credo, modus (Sonorum discrepantias, per Rationes, non Intervalla, æstimandi,) est, qui Ptolemæo (cap.  2. lib. 1.) κανὼν ἁρμονικὸς, Canon Harmonicus dicitur; (ad quem subtilius exigantur sonorum discrimina, quam quo possit Sensus pertingere:) atque inde est quod, qui sic loquebantur, Canonici (κανονικοὶ) dicerentur; alii (μουσικοὶ) Musici. Quanto autem potior sit hæc methodus quam Aristoxenea (per Intervalla sonos æstimandi) pluribus ostendit Ptol. cap. 9. lib. 1. Adeoque, quæ διαστήματα

The Harmonics of the Ancients compared with Today’s

181

vocat Aristoxenus, quique cum eo loquuntur, (Distantias, seu Intervalla,) eadem Ptolemæus, vocem eam (ubi sua, non eorum, sensa eloquitur) ubique declinans, aliis appellat nominibus; puta, διαφορὰς differentias, ὑπεροχὰς excessus, μεγέθη magnitudines, εἴδη species, λόγους rationes, aut horum similibus; nunquam (quod memini) διαστήματα· saltem si ultima tria Capita (quod innuit Scholium ad initium cap. 14. lib. 3.) non fuerint ab ipso Ptolemæo scripta. Posteriores tamen Musici, ad nostra usque tempora, (præsertim Practici,) cum Aristoxeno plerumque loquuntur; per Tonos & Hemitonia metientes Intervalla: Sed, per Hemitonium, intelligunt Limma, (ut quod ab Hemitonio parum abest;) quod &, ubi opus est, diserte asserunt; quodque Limmati deest ad Tonum integrum, vocant Apotomen, in ratione 2187/2048: quod etiam vocari solet Hemitonium majus, sicut illud alterum Hemitonium minus. Sic Boethius, cap. 27 & 29. lib. 2. Horumque differentiam Comma vocant. Boeth. cap. 6. lib. 3. Qui autem Musicam Speculativam subtilius tractant, Sonorum discrepantias potius, ad mentem Pythagoreorum, per Rationes designant. Sed de his hactenus. [The ʻSpecies of Consonancesʼ] Proximum est, ut Species quas vocant consideremus; quas & Figuras vocant (εἴδη καὶ σχήματα) ejusdem alicujus Consonantiæ; puta, Dia-tessaron, Diapente, Dia-pason. Aristox. p. 6, 74. Eucl. p. 13, 14, 15, 16. Gauden, 18, 19, 20. Ptol. cap. 3. lib. 2. Quas ut melius intelligamus; Sumantur Duo Conjuncta Tetrachorda, secundum idem aliquod Genus (quodcumque demum fuerit) similiter divisa: Puta, ab Hypate Hypaton (potestate) ad (potestate) Mesen; Quæ nos quidem canere solemus, vocibus mi fa sol la fa sol la: Nempe ab ima seu gravissima voce mi, canimus (in acutum ascendendo) bis tria intervalla, quæ vocibus fa sol la, fa sol la, signamus. Certum est (ob tetrachorda, quod ponimus, similiter divisa) eandem (magnitudine) consonantiam esse, ab Hypate ad Hypaten, atque à Parypate ad Parypaten, eandemquea à Lichano ad Lichanon, itemque ab Hypate Meson ad Mesen. Cum enim æqualis sit magnitudo (propter æqualia tetrachorda similiter divisa) ab Hypaton-hypateb ad Parypaten, atque à Meson-hypate ad Parypaten; quod intermedium est, utrivis additum, tantumdem facit: puta ab Hypate-hypaton ad Hypaten-meson, atque à Parypate-hypaton ad Parypatenmeson: Atque, eadem ratione, à Lichano-hypaton ad Lichanon-meson; atque etiam ab Hypate-meson ad Mesen. Sed aliter disponuntur (ut patet) eorum membra constituentia: Quippe quæ (omnium minima magni-[p. 171]tudo) est in unius loco imo (puta quæ est ab Hypate ad Parypaten,) est in alterius loco summo; & de reliquis similiter. Atque hic diversus ordo, seu dispositio partium, est quæ dicitur Species, seu Figura. Sic enim Ptolemæus, cap. 3. lib. 2. εἶδός ἐστι ποιὰ θέσις τῶν καθ᾽ ἕκαστον γένος ἰδιαζόντων, ἐν τοῖς οἰκείοις ὅροις, λόγων· Qualiter, seu quo ordine, disponuntur, intra suos terminos, peculiares  1682: eandemq;  1682: Hypaton-Hypate

a

b

182

John Wallis: Writings on Music

seu characteristicæ cujusque generis Rationes. Characteristicas vero quas vocat in quoque genere Rationes (λόγους ἰδιάζοντας) sunt, in Dia-pason, & Dia-pente, tonus Diazeucticus (quem nos voce mi signamus;) in Dia-tessaron, duorum sonorum (ἡγουμένων) præcedentium, hoc est, Acutissimarum; (quam nos signamus voce la.) Est enim ubi ἡγούμενον, præcedens, de Acuto dicitur; & ἑπόμενον, sequens, de Gravi: ut apud Prolemæum hoc loco, & cap. 12, 13, 14, 15, 16. lib. 1. & alibi passim: Quamquam alias, etiam apud eosdem scriptores, censetur Grave præcedere, & Acutum sequi, ut Ptol. cap.  5. lib. 2. ubi πρὸ ante, de graviori dicitur; & κατὰ post, de acutiori. Et Aristides, p. 10, 11. Gravissimum, ὕπατον dictum vult, quasi πρῶτον primum; & Acutissimum, νῆτον seu νέατον, quasi ἔσχατον ultimum. Quod moneo, ut caveatur Lectoribus, quo quis loco sensus sumendus sit. Tot autem sunt in quaque harum Consonantiarum species, quot sunt Intervalla, seu Rationes, quibus componi censetur. Adeoque, in Dia-tessaron, (propter quatuor chordarum tria intervalla) Tres species; in Dia-pente (ob similem rationem) Quatuor species; in Dia-pason, Septem. Species autem Prima, in singulis, ea dicitur, quæ habet rationem seu vocem suam (ἰδιάζουσαν) Characteristicam, (hoc est, la in Dia-tessaron, & mi in Diapente & Dia-pason,) in loco (ad acutum) primo; Secunda, quæ secundo; Tertia, quæ tertio; & sic deinceps. Adeoque, in Dia-tessaron, Prima species est quae continetur sonis (ut loquuntur) βαρυπύκνοις, (ab Hypate ad Hypaten,) quam nos canimus (ab ima voce, in acutum ascendendo,) vocibus fa sol la: Secunda, μεσοπύκνοις, (à Parypate ad Parypaten,) vocibus sol la fa: Tertia, ὀξυπύκνοις (à Lichano ad Lichanon,) vocibus la fa sol. Quæ autem species esset ab Hypate meson ad Mesen (canenda vocibus fa sol la,) non alia est à Prima, (quippe eodem ordine recurrunt omnia ab Hypate-hypaton ad Hypaten meson, atque ab hac ad Mesen:)

The Harmonics of the Ancients compared with Today’s

183

adeoque Quartam speciem non constituit, sed Primam repetit. Vocem imam, unde ascenditur, in schemate non accensemus (quoniam hic, non tam Sonos, quam Rationes seu Intervalla signamus,) saltem alio charactere scribimus. Sumantur deinde (quo habeantur Dia-pente) duo Tetrachorda Disjuncta (interposito tono diazeuctico, quem nos voce mi signamus;) puta, ab Hypatemeson ad Neten-diezeugmenon: Ubi, post primum fa sol la, canimus mi, atque tum demum alterum fa sol la. Sunt itaque in Consonantia Dia-pente (ut facile est colligere ex jam dictis de Dia-tessaron) species seu figuræ Quatuor (prout ejus vox Characteristica mi, sit in loco ad acutum primo, secundo, tertio, quartove;) vocibus (post imam) canendæ, 1. fa sol la mi. 2. sol la mi fa. 3. la mi fa sol. 4. mi fa sol la.

Quæ autem proxime expectanda esset species, in c d e f, canenda (post vocem imam mi) vocibus fa sol la fa; non esset species consonantiæ Dia-pente, sed, quæ dici solet, falsa quinta, propter tonum Diazeucticum, voce mi signandum, non intervenientem; nisi, mutata (ut loquuntur) Clave, (posita Duri nota in f,) vox alias fa futura, transeat in mi. (Nam vox mi in imo posita, unde ascenditur, innuit tonum diazeucticum infra se positum, adeoque extra hoc Pentachordum.) Sumantur denique, (pro speciebus Consonantiæ Dia-pason,) duo conjuncta Dia-pason: Puta à Proslambanomeno ad Neten hyperbolæon; canenda vocibus la, mi, fa sol la, fa sol la, mi, fa sol la, fa sol la. In quibus reperientur (modo quo prius) septem species Dia-pason. Ultra quas si progredi velimus, recurret (octavæ loco) Prima: ut supra in Dia-tessaron. Et, si retrocedatur; quæ chordis A a terminaretur, non alia erit quam iterata Septima. Nam à Proslambanomeno ad Mesen, eadem recurrunt omnia, atque eodem ordine, quo à Mese ad Neten hyperbolæon. [p. 172]

184

John Wallis: Writings on Music

Atque hinc est (nempe, quia duobus Dia-pason opus est, quo istius Consonantiæ Species omnes habeantur,) quod Dia-pason non pro Perfecto Systemate habeat Ptolemæus (quod censuerunt Antiquiores,) sed Dis-dia-pason. Cap. 4. lib. 2. Utut enim, post simplex Dia-pason, iidem redeant (potestate) soni; (quod ipse contendit cap. 8, 9. lib. 2. ubi de Tonis seu Modis agitur, eorumque numero:) Non tamen unius Dia-pason ambitu habentur ipsius omnes Species; quod fieri oportere arguit in Systemate Perfecto. Atque hæ quidem Dia-pason Species septem, sua singulæ sortiebantur nomina: Prima dicta est Mixolydia; Secunda, Lydia; Tertia, Phrygia; Quarta, Doria; Quinta, Hypolydia; Sexta, Hypophrygia; Septima, tum Locrensis, tum Hypodoria, & Communis. Eucl. p. 15, 16. Gaudent. p. 28. Bacchius, p. 18, 19. [The Modes] Tandem, de Modis seu Tonis dicendum restat. (Sic Alypius, p. 2. τοὺς λεγομένως τρόπους τε καὶ τόνους.) Quippe τόνος Tonus, quadruplici saltem sensu occurrit, (quod notat Euclides, p. 19.) 1. pro Sono; ut cum ἑπτάτονος φόρμιγξ, heptatonos cithara, de ea dicitur quæ septem Sonos seu chordas habet. 2. pro Intervallo; ut cum Dia-pente à Dia-tessaron Tono differre dicitur; (puta ratione Sesquioctava;) aut Mesen à Paramese Tono distare. 3. pro vocis Loco quodam; ut qui toni, Dorius, Phrygius, Lydius, &c. dicuntur; de quo sensu jam agitur; (eodem sensu ab aliis dicuntur Modi.) 4. pro Tensione, ut ubi quis ὀξυτονεῖν, βαρυτονεῖν, ἢ μέσῳ τῷ

The Harmonics of the Ancients compared with Today’s

185

τῆς φωνῆς τόνῳ κεσχρῆσθαι dicitur: acuto, gravi, aut medio Tono (seu tenore aut tensione) vocis uti. Modus itaque seu Tonus, prout hic sumitur, denotat vocis Locum, non quo una vox, sed quo tota vocum series seu systema canitur; acutiorem puta gravioremve. Utpote, prout (apud nos) mi canitur, nunc in b fa b mi, nunc in e la mi, nunc in a la mi re, &c. Sic, apud illos, verbi gratia, Paramese Potestate, (quod tantundem est atque nostrum mi,) posita erat, nunc in Paramese (positione), nunc in (positione) Nete-diezeugmenon, nunc in Mese, &c. Hoc est, Sonus qui pro cantus ratione vicem sustinet Parameses, eo acumine profertur quo alias proferri solet vel ipsa Paramese, vel Nete diezeugmenon, vel Mese, &c: Reliquique totius cantus soni suum sortiuntur acumen, prout earum ad hunc ratio postulat. Modi itaque, seu Toni hoc sensu sumpti, sunt quidem (potentia) numero infiniti, sicut & Soni (quod docet Ptolemæus, cap. 7. lib. 2.) Quippe, ut Soni, sic & Sonorum congeries, moveri possunt in Acumen vel Gravitatem, plus minusve, varietate innumerabili; cum sit (ut ante dictum) vocis Locus quilibet, Sonorum innumerabilium capax, (ut Linea, Punctorum:) &, prout Punctum, sic Linea integra, in locos transferri potest majori minorive spatio distantes in infinitum; atque Sonus pariter, & Sonorum congeries: ut Tonorum numerus, hac ratione, definiri non possit. Actu tamen, & quod ad usum spectat; definiri solet Tonorum multitudo, & quidem ab aliis aliter; sive Tonorum extremorum distantiam spectemus; sive Tonorum intra eos terminos numerum; sive qua ratione aut quo intervallo inter se distent illi Toni. Tonos, ab origine, Tres tantum statuerunt Veteres, (ut docet Ptolemæus, cap. 6. lib. 2. aliique:) Dorium, Phrygium, & Lydium; quorum quisque à proximo distaret tono, hoc est ratione sesquioctava, (adeoque extremi duo, tonis duobus:) Nempe, Modus Phrygius erat, quam Dorius, tono acutior; pariterque Lydius quam Phrygius; adeoque Lydius quam Dorius, tonis duobus. Aristoxenus (referente Euclide, p. 19.) Tonos constituit Tredecim; qui sic enumerantur, Hypermixolydius, (qui & Hyperphrygius vocatur;) Myxolydii duo, acutior & gravior, (quorum acutior, qui ordine est secundus, etiam Hyperiastius vocatur; gravior vero, etiam Hyperdorius;) Lydii duo, acutior, & gravior (qui & Æolius appellatur;) Phrygii duo, quorum alter gravis (qui & Jastius,) alter acutus; Dorius unus; Hypolydii duo, acutior, & gravior (qui & Hypoæolius vocatur); Hypophrygii duo, (quorum gravior quoque appellatur Hypoastius;) Hypodorius. Ex quibus gravissimus [p. 173] est Hypodorius; qui vero continue sequuntur à gravissimo ad acutissimum, hemitonio se invicem superant. Alii Quindecim statuerunt Tonos, hemitonio inter se continue distantes: Sic Alypius, p. 2. & ad hunc Meibomius; quod & nos ex illo repetimus ad Ptolemæi cap. 11. lib. 2. quos illic enumeravimus. Nempe, tredecim jam enumeratis, præmittuntur in acutum duo, Hyperlydius, & Hyperæolius. Nempe hoc volunt: Proslambanomenon Toni Hypodorii (omnium gravissimi) Sonum esse volunt gravissimum quem possit vel humana vox proferre, vel auris distincte judicare; (adeoque graviores omnes non tam Sonos esse volunt, quam

186

John Wallis: Writings on Music

Bombos aut incondita Murmura:) Proslambanomenon Hypoiastii (seu gravioris Hypophrygii) hemitonio acutiorem faciunt quam Hypodorii, (adeoque Hypaten hujus, hemitonio acutiorem, quam Hypaten illius; & sic de cæteris;) ut medio loco sit Proslambanomenos Hypoiastii, inter Hypodorii Proslambanomenon & Hypaten-hypaton: Acutioris Hypophrygii Proslambanomenon, hemitonio adhuc acutiorem faciunt; adeoque tono integro acutiorem quam Hypodorii, cum cujus itaque Hypate-hypaton coincidet. Et sic de cæteris, ut videre est in tabella ad calcem cap. 11. lib. 2. Ptolemæi. Contra hos (qui Tonos seu Modos sic augent per Hemitonia) disputat Ptolemæus, cap. 7, 8, 9, 10, 11. lib. 2. Docetque, Modorum variorum usum, non in hunc solum finem introductum, ut acutior graviorve sit totius Cantus tenor: Quippe huic sufficeret Cantoris vox acutior graviorve; aut Musici Organi ad hosce tenores accommodatio. Et quidem, si huic soli attenderetur, posset quis, Tonis Quindecim jam enumeratis, pro arbitrio plures addere, quatenus Humanæ Vocis (διάτασις) distentio, aut etiam Organicæ, permittit: Aliosque intermedios quotvis pro arbitrio interponere. Sed eo potissimum fine fuisse introductum, ut in ipsius Cantus curriculo transitus fiat de modo in modum; quam vocant (μεταβολὴν κατὰ τόνον) mutationem secundum Tonum: (quod à nostris sit, mutata clavis signatura, adhibitis Mollis & Duri notis.) Adeoque prospiciendum esse, non tam ut æqualibus inter se distent intervallis (puta, toni, hemitoniive,) quam ut transitio fiat per Consonantias, (puta Diatessaron, aut Dia-pente, aut etiam Dia-pason:) Neque ut in proximum semper Tonum fiat transitus, (qui Tono vel Hemitonio distet,) sed sæpius (& quidem gratius) in alium qui per consonantiam aliquam (puta Dia-tessaron, Dia-pente, aut Dia-pason) inde distet. Tonum autem qui Dia-pason ab aliquo distat, non tam pro alio habendum esse Tono, quam pro eodem continuato, aut quasi-eodem: ea ratione qua Sonos Dia-pason differentes pro eodem cel quasi-eodem habendos esse supra dictum erat. (Quippe idem omnino est atque, in hodierna Musica, si dicamus mi vocem ponendam in B mi, an in b fa b mi, aut bb fa bb mi: quod Modum non mutat: nam si horum alterum fiat, reliqua pariter facienda supponimus, si eo cantus pertingat.) Tonos autem qui plusquam Dia-pason differunt (puta Dia-pason & dia-tessaron, aut Dia-pason & dia-pente,) pariter habendos (eadem de causa) acsi tanto differrent quantus est excessus ille supra Dia-pason (puta Dia-tessaron, aut Dia-pente.) Adeoque citra Dia-pason terminandos esse extremos Tonos (quippe, qui eousque, aut ultra excurrunt, alii censendi non sunt à citra positis;) Et propterea Tonos distinctos ponendos esse docet omnino Septem; Myxolydium, Lydium, Phrygium, Dorium, Hypolydium, Hypophrygium, & Hypodorium. Totidemque admittit hodierna Musica, pro varia Clavis signatura; quod ad Ptol. cap. 11. lib. 2. ostendimus.

The Harmonics of the Ancients compared with Today’s

187

(Glareanus tamen justo volumine, quod Δωδεκάχορδον vocat, pro Modis Duodecim contendit; quem videbis.a Sed nondum erat, illius ætate, in usum reductum Ptolemæi genus Dia-tonum Intensum, quod postea revocavit Zarlinus; adeoque ad Aristoxeneorum modum loquebantur omnes; in Tonum, Tonum & Hemitonium, seu Limma, dividentes intervalla; Modosque per Hemitonia augebant.) [The Modes and the Modern Key Signatures] Hanc autem methodum adhibet Ptolemæus in Tonis suis septem designandis, (quam & nostri secuti sunt in mutandis Clavium signaturis:) Quæ quo melius intelligatur, respiciendum hic erit Schema quod habetur ad Ptol. cap. 11. lib. 2. Primum omnium facit (qui est omnium medius) tonum Dorium; qui Mesen potestate (adeoque & Paramesen, quæ est nostrorum mi,) habeat in situ naturali; hoc est, (ut loquitur ille) Mesen potestate (μέσην δυνάμει) in Mese positione (μέσῃ θέσει) adeoque Paramesen potestate, in Paramese positione; (hoc est, ut nos loquimur, mi in b fa b mi:) & in cæteris similiter. Cui convenit, apud nos, Clavis nude posita, absque Mollis aut Duri nota. Secundo; Tonum sumit isto acutiorem Dia-tessaron; qui itaque potestate Mesen habeat (dia-tessaron acutiorem) in Dorii Paranete-diezeugmenon; adeoque Paramesen (quæ est nostra mi) in Nete diezeugmenon; hoc est, (ut nos loquimur) mi in e la mi: quem Tonum vocat Mixolydium. In hunc finem nos, posita Mollis nota (a Flat) in b fa b mi (quo mi excludatur inde,) remittimus mi ad e la mi. Atque hinc alibi concludit (cap. 6. lib. 2) Systemate quod Conjunctum vocant (à Proslambanomeno ad Neten-synemmenon) opus non esse: utpote quod satis suppleatur facto transitu (in Mese) de modo Dorio in Mixolydium. Quippe jam, post duo conjuncta Tetrachorda in Dorio (ab Hypate-hypaton ad Mesen, hoc est à B mi ad a la mi re,) sequitur [p. 174] tertium in Mixolydio, ab hujus Hypate-meson (quæ est in Dorii Mese) ad Mesen; hoc est, ab a la mi re ad d la sol re: ut jam tria sint conjuncta tetrachorda; nimirum à B mi (quæ est Dorii Hypate-hypaton) ad d la sol re (quæ est Mixolydii Mese.) Tertio; cum non possit ille alterum Dia-tessaron inde in acutum sumere, quin excedat illud Dia-pason in cujus medio posuerat Dorii Mesen; sumit (ejus loco) Dia-pente in grave, (quod æquipolleat sumpto Dia-tessaron in acutum; propter sonos, sic sumptos, Dia-pason invicem differentes, quasi pro eodem habendos:) Cujus itaque toni Mese sit (Dia-pente gravior quam Mixolydii) in Mixolydii Lichano-hypaton; hoc est, in Dorii Lichano-meson; adeoque Paramese in Dorii mese; hoc est, ut nos loquimur, mi in a la mi re: Quem tonum vocat Hypolydium. In hunc finem nos (post Mollis notam ante positam in b fa b mi) secundam Mollis notam ponimus in e la mi, (quo etiam hinc excludatur mi,) adeoque mi remittimus ad a la mi re.   1682 and 1699: videsis

a

188

John Wallis: Writings on Music

Quarto; cum hinc in grave non possit ille, vel Dia-pente vel Dia-tessaron sumere, quin ante dictum Dia-pason excedat; Tonum sumit qui sit (quam Hypolydius) Dia-tessaron acutior; quem Lydium vocat: Cujus itaque Mese sit in Hypolydii Paranete-diezeugmenon; adeoque Paramese in ejusdem Nete-diezeugmenon; hoc est, in Dorii Paranete-diezeugmenon; hoc est, ut nos loquimur, mi in d la sol re. Nos in hunc finem (post duas ante-positas in b & e) tertiam Mollis notam ponimus in a la mi re: quo mi (inde exclusa) transeat in d la sol re. Quinto; ut ante à Dorio sumptus est Mixolydius, Dia-tessaron Acutior: sic, ab eodem Dorio, sumit (in grave) Hypodorium, Dia-tessaron graviorem quam est Dorius: Cujus itaque Mese est in Dorii Hypate-meson; adeoque Paramese (quæ estl nostra mi,) in Dorii Parypate meson; hoc est, ut nos loquimur, mi in F fa ut. Nos, ut hoc significemus, (omissis Mollis notis,) ponimus Duri notam (a Sharp) in F fa ut; ut, quæ secus hemitonio elevanda foret, & proferenda fa; jam tono integro (à subjecta chorda) elevetur, & proferatur mi; unde propterea chorda proxime acutior hemitonio distet, adeoque fa dicenda; redeatque tandem mi (absoluto Diapason) in superiori f fa ut acuta. Sexto; ab Hypodorio sic posito, cum sumi non possit aliud in grave Diatessaron, quin extra dictum Dia-pason transeatur; sumitur (quod eodem recidit, utpote Dia-pason inde distans,) Dia-pente acutior, tonus Phrygius: Cujus itaque Mese sit in Hypodorii Nete-diezeugmenon; hoc est, in Dorii Paramese; adeoque illius Paramese in Dorii Trite-diezeugmenon; hoc est, ut nos loquimur, mi in c fa ut. Quod ut fiat (præter ante-positam Duri notam in F fa ut,) ponitur altera Duri nota in c fa ut: quo, quæ secus à subjecta chorda elevanda erat hemitonio, & proferenda fa; jam tono integro elevetur, & dicatur mi; unde igitur deest ad d sol re ut, non nisi hemitonium; quæ propterea est proferenda fa; redeunte iterum mi, sive in cc sol fa supra, sive in C fa ut infra. Septimo; à Phrygio sic posito, sumitur (Dia-tessaron distans in grave) Hypophrygius: Cujus itaque Mese est in Phrygii Hypate-meson; hoc est, in Dorii Parypate-meson; adeoque Paramese (quæ est nostra mi) in Dorii Lichano meson; hoc est, ut nos loquimur, mi in G sol re ut. Quod ut fiat, (manentibus reliquis,) ponenda est tertia Duri nota in G sol re ut; ut quæ secus (propter F fa ut ante factam acutam) hemitonio esset elevanda, & proferenda fa; jam elevetur tono integro & mi dicatur; adeoque a la mi re (à G sol re ut acuta, distans hemitonio) dicatur fa; redeunte mi in g sol re ut supra, & infra in Г ut. His positis; inde colligitur (ut cap. 10.) Toni Mixolydii à Lydio, distantium Limma (seu crassius loquendo Hemitonium;) hujus à Phrygio, Tonum; hujus à Dorio, Tonum; Dorii ab Hypolydio, Limma; Hypolydii ab Hypophrygio, Tonum; hujusque ab Hypodorio, itidem Tonum. Tandem concludit; Hos septem Tonos sufficere, nec locum esse pluribus: Eo quod omnes (intra dictum Dia-pason) chordæ jam occupatæ sint. Quippe cum omnes chordæ, ab Hypate-meson ad Paraneten-diezeugmenon (inclusive) sint alicujus Toni mese, nulla suppetit quæ Mese sit toni alicujus his intermedii. Verbi gratia; cum Hypodorii Mese potestate, sit positione Hypate-meson; & Hypophrygii, Parypate-meson; neque ulla his interjaceat Chorda; nulla suppetit

The Harmonics of the Ancients compared with Today’s

189

quæ sit intermedii Toni mese, qui (ut supponit Aristoxenus) dicatur Hypoiastius, aut Hypophrygius gravior. Quodque de Mese dicitur, similiter dicatur, de Paramese, quæ est nostris mi. Hoc est; (ut cum nostris loquamur,) per Modos jam positos, vox mi jam occupavit om-nes chordas. Quippe, in Hypodorio, habetur mi in F, (adeoque in f quæ inde distat Dia-pason:) In Hypophrygio, habetur in G, (adeoque in Г & in g quæ inde distant Dia-pason:) In Hypolydio, habetur in a, (adeoque in A & aa:) In Dorio, habetur in b, (.adeoque in B & bb:) In Phrygio, habetur mi in c, (adeoque in C & cc:) In Lydio, habetur in d, (adeoque in D & dd:) In Mixolydio denique, habetur in e, (adeoque in E & ee.) Nulla itaque suppetit chorda qua (pro alio quovis Tono, qui non cum horum aliquo coincidat,) locetur mi. Sunt qui, septem his, octavum addunt Hypermixolydium; sed, quem alium non esse ab Hypodorio (unde distat Dia-pason) docet ibidem Prolemæus. Atque de Tonis seu Modis, hactenus. [Melopoetics] Atque .jam universam fere veterum Harmonicum absolvimus; Cujus septem partes enumerat Euclides, p. 1. περὶ φθόγγων, περὶ διαστημάτων, περὶ γενῶν, περὶ συστημάτων, περὶ τόνων, περὶ μεταβολῆς, περὶ μελοποιίας. De Sonis, de Intervallis, de [p. 175] Generibus; de Systematis; de Tonis; de Commutationibus; & de Melopœia. (Et similiter Alypius, p. 1. Aristox. p. 34, 35, 36, 37, 38.) Quæ jam expedivimus omnia, saltem si Melopœiam excipias, quam cantus Compositionem hodie dicimus. Huc autem spectant quæ de Consonantiis supra dicta sunt: Quippe, in vocum sequela, Consonantiores sunt minus Consonis (cæteris paribus) præferendæ. Ut enim in literis componendis qui fiant voces, non quævis quamvis apte sequitur, sed delectu habito: pariter &, in compositione cantus, non quodvis intervallum post quodvis rite sequitur; sed, prout cantus natura postulat. Aristox p. 27. Neque hic jam spectator, Quid in Diagrammatis, aut (quod jam dicimus) Scalæ, constructione apte fiat; Utputa, quod post duas Dieses, aut duo Hemitonia (puta, in Genere Chromatico, vel Enarmonio) Tertium, in acutum, continue poni non possit; sed, quod, in acutum, sequi potest minimum, est id quod hisce deest ad Dia-tessaron; nec minus, in grave, quam Tonus: Quam (potius) Quæ continue sequi possint in Cantus processu intervalla, sive in Acutum, sive in Grave, tum per Sonos in scala continue proximos, tum utcunque interruptos: Utputa, quos post sive Spissi, sive Non-spissi, intervallum, (quale est ab Hypate ad Lichanon in quovis genere,) minus non canendo sequi possit intervallum, in acutum, quam quod est residuum ad Dia-tessaron; nec in grave, quam Tonus. Aristox. p. 27, 28, 29. iterumque p. 52, 53, 54. Et hujusmodi complura habentur per totum fere Aristoxeni librum tertium: Utputa, Quod Spissum Spisso appositum non canatur; neque totum, nec quidem pars ejus, p. 62. Quod duo Ditona deinceps non ponantur, p. 63. Neque in Harmonia & Chromate, duo Toni, p. 64. Neque etiam in Diatono molli; sed secus in Diatono Intenso; ubi tres continue sequi possunt Toni, non plures. p. 65. Aliaque istiusmodi multa. Quæ omnia ex hoc fere postulato deducit, Quod (in toto Diagrammate) vel

190

John Wallis: Writings on Music

quartus à dato sonus debeat Dia-tessaron; vel quintus, Dia-pente: prout vel nempe intercedit, vel non intercedit, tonus Diazeucticus, p. 58, 59. Quibus positis, inde concludit reliqua. Huc etiam spectant Ductuum, Motuumve vocis, sive in Acutum, sive in Grave, appellationes variæ: ut sunt ἀγωγὴ, πλοκὴ, πεττεία, τονή· Ductus, Plexus, Pettia, Extensio; &c. Quorum nominum significata habeantur apud Euclidem, p.  22. Arist. p.  20. Bryen. lib. 3. sect. 10. Item ἔκλυσις, ἐκβολή· Dissolutio, Projectio: Item ἄνεσις, ἐπίτασις, μονὴ, στάσις· Remissio, Intensio, Mansio, Statio: Item συναφὴ, διάζευξις, ὑποδιάζευξις, ἐπισυναφὴ, ὑποσυναφὴ, παραδιάζευξις, ὑπερδιάζευξις: Item πρόληψις, ἔκληψις, προλημματισμὸς, ἐκλημματισμὸς, μελισμὸς, πρόκρουσις, ἔκκρουσις, προκρουσμὸς, ἐκκρουσμὸς, κομπισμὸς, τερετισμὸς, διαστολή. aliaque istiusmodi nomina, quae habentur apud Bacchium, p. 9. 11, 12, 13, 14, 19, 20, 21, 22. et apud Bryennium, lib. 3 sect. 3. &c. Sed quibus his immorandum non sentio. Superest, de Veterum Melopœia monendum, Simplicem eam fuisse, & (quantum quidem ego persentio) nonnisi Unius (ut jam loquimur) Vocis; ut, qui in ea fuerit concentus, in sonorum sequela spectaretur; quem nempe faceret sonus antecedens aliquis cum sequente. Secundum illud Aristoxeni, p.  38, 39. ἐκ δύο γὰρ τούτων ἡ τῆς μουσικῆς σύνεσίς ἐστιν, αἰσθήσεώς τε καὶ μνήμης· αἰσθάνεσθαι μὲν γὰρ δεῖ τὸ γενόμενον· μνημονεύειν δὲ τὸ γεγονός· κατ᾽ ἄλλον δὲ τρόπον οὐκ ἔστι τοῖς ἐν τῇ μουσικῇ παρακολουθεῖν. Ex duobus enim hisce, Musices intellectus constat, Sensu scilicet & Memoria: Quandoquidem sentire oportet, quod fit; Memoria vero retinere, quod est factum: Alio modo, ea qui in Musica fiunt, consequi non licet. Quippe, ex collatione soni præteriti, cum eo qui jam auditur, concentum persentiscimus. Ea vero, quæ in hodierna Musica conspicitur, Partium (ut loquuntur) seu Vocum duarum, trium, quatuor, pluriumve inter se Consensio, (concinentibus inter se, qui simul audiuntur, sonis,) Veteribus erat (quantum ego video) ignota. Quamquam enim tale quid innuere videantur quæ apud Ptolemæum occurrunt, cap 12. lib. 2. voces aliquot, ἐπιψαλμὸς, σύγκρουσις, ἀναπλοκὴ, καταπλοκὴ, σύρμα, καὶ ὅλως ἡ διὰ τῶν ὑπερβατῶν φθόγγων συμπλοκή· (quæ desiderari dicit, præ aliis instrumentis, in Monochordo Canone, eo quod manus percutiens unica sit, nec possit distantia loca simul pertingere:) quæ faciunt ut plures aliquando chordas una percussas putem: Id tamen rarius factum puto, in unis aut alteris subinde sonis; non in continuis (ut aiunt) Partibus, (ut sunt, apud nos, Bassus, Tenor, Contra-tenor, Discantus, &c;) altera alteri succinente; aut etiam in Divisionibus (ut loquuntur) seu Minuritionibus cantui tardiori concinentibus. Quorum ego, in Veterum Musica, vix ulla vestigia (haud certa saltem) deprehendo. Adeoque omnino mihi persuadeo, neque Veterum Musicam accuratiorem nostra fuisse; neque prodigiosos illos effectus (qui memorari solent) in hominum animos (puta ab Orpheo, Amphione, Timotheo, &c. præstitos,) olim obtigisse; nisi per audacem satis Hyperbolen ab Historicis enarratos dicas; vel id ob summam Musices raritatem (magis quam præstantiam) apud imperitam plebem contigisse.

The Harmonics of the Ancients compared with Today’s

191

At hoc interim facile concesserim; cum id sibi solum fere proponant hodierni Musici, ut animum oblectent; potius quam (quod affectasse videntur Veteres) ut affectus huc illuc trahant: fieri omnino potest, ut, in movendis affectibus, ipsi quam nos peritiores fuerint. Adde quod eorum Musica simplicior, uniusque Vocis, non ita prolata verba obscurabat, ut nostra magis composita: unde fiebat ut (verbi gratia) Tragica Verba, cum Gestu Tragico, Tragico Carmine. Sonoque Tragico prolata (quæ omnia componebant eorum Musicam,) [p. 176] non mirum si Tragicos Affectus concitabant, (haud secus quam apud nos in agendis Tragœdiis;) Pariterque in cæteris affectibus. Quæ autem Melopœi nostri de Compositione præcepta tradunt, sive Cantuum simplicium, sive plurium concinentium; adeoque de sonorum concentu sive in sequela, sive simul auditorum; (qua de re omnino pauca sunt quæ à Veteribus accepimus:) neque jam libet fusius prosequi, neque patitur ea brevitas quam præsentis instituti ratio postulat. Atque de his hactenus. [The Modern Scale] Tandem superest, quam in hunc locum retulimus, discutienda Quæstio. De Generibus quæ supra memorata sunt, eorumque Coloribus, supra diximus, Unicum ex omnibus jam à multis seculis usu receptum esse; quod & Diatonicum esse apud omnes convenit: (omissis Enarmonio, Chromaticis omnibus, reliquisque Diatonicis.) Dubitatur autem, num hoc sit Aristoxeni quod dicitur Diatonum Intensum; an quod Ptolemæo dicitur Diatonum Ditonicum, an ejusdem Ptolemæi Diatonum Intensum. Nempe, dum à mi vel la in gravi, Dia-tessaron cantamus in acutum, vocibus fa sol la, totidem intervalla notantibus; Quanta sint horum intervalla singula. Prima sententia est Aristoxeni; qui (cum Dia-tessaron statuat ex Tonis duobus cum semisse constare) intervallum fa (gravissimum) Hemitonium esse vult, reliqua duo (sol la) Tonos integros. Quod est Aristoxeni Diatonum Intensum; Atque sic fere loquuntur, ad hunc usque diem, Musici omnes; saltem ubi crassius loquuntur. Verum, qui subtilius rem disceptant, per Hemitonium, intellectum volunt, non præcise Toni Semissem, sed quod eo sit aliquanto minus. Atque hoc demonstravit Euclides (cætera Aristoxeneus) nescio an omnium primus: qui (admissis Pythagoreorum principiis, de Sonorum intervallis, per Rationes æstimandis; & quidem Tono, in ratione Sesquioctava;) Utut, in Introductione Harmonica, Tonos & Hemitonia (cum Aristoxeneis) memoret; in Sectione tamen Canonis, id quod post duos Tonos restat ad Dia-tessaron, ostendit, Hemitonio minus esse; quod λεῖμμα (limma) deinde dici coepit: in ratione 256/243. Quippe, si Dia-tessaron contineat tonos duos & semissem; Dia-pason (hoc est, bis Dia-tessaron cum Tono,) continebit Tonos Sex. At Dia-pason (cujus ratio est Dupla) minus est quam Sex Toni. Nam ratio Sesquioctava, sexies composita, plus est quam Dupla. Dia-pason igitur minus est quam Sex toni; & Dia-tessaron, minus quam duo cum semisse.

John Wallis: Writings on Music

192

/8 × 9/8 ×9/8 ×9/8 ×9/8 ×9/8 = 531441/262144 > 524288/262144 = 2/1. Proxima igitur est eorum sententia, qui, pro Tono Tono & Hemitonio, substituunt Tonum & Limma: & quidem, siquando Hemitonium dicunt, intellectum volunt Limma; quod ab Hemitonio parum abest. Adeoque voce fa, signatum volunt Limma; & vocibus sol la, duos Tonos: Hoc est 256/243 × 9/8 × 9/8 = 4/3. Estque hoc, Ptolemæi Diatonum Ditonicum. Sed jam ante Ptolemæum, ab Euclide indicatum. Eratque etiam Eratosthenis Diatonicum, ut supra dictum est. Atque sic ad nostra fere tempora senserunt Musici. Ipseque Ptolemæus, utut alia tradat Genera Diatonica, neque hoc tamen rejicit; reliquis qui ita locuti sunt omnibus, hac in re, morem gerens, potius quam suis ipsius adhærens placitis; quod ipse docet cap. 16. lib. 1. Atque sic Boethius Tetrachordum dividit; & post eum, Guido Aretinus, Faber Stapulensis, Glareanus, aliique: donec (exeunte superiore seculo,) Zarlinus (nescio an omnium primus, jam ante annos centum circiter) Kepplerus item, aliique, resumendum censuerunt Ptolemæi Diatonum Intensum. (Utut contra Zarlinum disputet Vicentius Galilæus in suo de Musica Dialogo.) Tertia itaque sententia est, eorum qui, Ptolemæum secuti, pro Hemitonio vel Limmate, substituunt rationem (16/15) sesquidecimam-quintam; (quod & Hemitonium dicunt ipsi;) & pro Tonorum altero (quos ambos ante posuerant alii in ratione 9/8) Tonum Minorem dictum, in ratione 10/9. Adeoque Dia-tessaron componunt ex rationibus 16/15 × 9/8 × 10/9 = 4/3. Voce fa rationem innuentes 16/15; voce sol, 9/8; voce la, 10/9. Quod est Ptolemæi Diatonum Intensum: Idemque est Didymi Diatonicum, nisi quod hic (mutato ordine) habeat 16/15 × 10/9 × 9/8 = 4/3. Cumque 16/15 Hemitonium majus dixerint; Hemitonium minus ponebant in ratione 25/24; ut quæ, cum 16/15, complet tonum minorem 10/9 = 16/15 × 25/24; estque differentia Tertiæ majoris (ut loquuntur,) & Tertiæ minoris. (Mersennus alia duo addit Hemitonia; Nempe in ratione 135/128 quæ cum 16/15 complet 9/8 tonum majorem; & 27/25 quæ cum 25/24 complet 9/8 tonum item majorem.) Cum vero hæc quæ diximus Genera, tantillo inter se distent, ut Aurium subtilitas ea vix aut ne vix valeat distinguere, (quippe ratio 16/15 à ratione Limmatis 256 /243, itemque Toni majoris ratio 9/8 à ratione 10/9, differunt ratione 81/80, quæ tantilla est ut eam auris ab æquitonio ægre distinguat:) non tam Sensu quam Ratione judicandum erit, utra sententia potior æstimanda sit; (quippe Sensus, ad utramvis permittendam, non erit admodum difficilis.) Ratio autem sententiæ posteriori impense favet. 9

[Pythagorean Theory Placed on a Physical Basis] Ut rem igitur ab origine repetam: Non displicet Pythagoreorum sententia, qui, dum Sonum ex percussi æris tremore ortum volunt, Acuminis causam Physicam à tremuli motus hujusce celeritate deducunt; adeo ut, quo celerius spissiusque fiunt (vibrantis puta chordæ) itus reditusque, eo acutior fiat Sonus; quo tardius rariusque, eo gravior. Sic, ex Pythagoreorum sententia, Porphyrius (in suis ad Ptolemæi Harmonica commentariis,) aliique. Atque hinc est, quod chordarum æque tensarum, juxta positarum, altera ad alterius motum tremit; [p. 177] utpote

The Harmonics of the Ancients compared with Today’s

193

quæ, contemporales motus facere, sua quasi sponte sit apta; adeoque, levissimo tactu concitati æris incitata, similem tremorem subit. Et quidem vel hoc, vel aliud quid quod hujus instar sit, ponendum videtur pro Acuminis & Gravitatis sonituum causa Physica. Et consequenter ad hoc, (aut ad illud quicquid demum fuerit quod hujus vice substituendum erit, de quo non libet acriter hic disputare,) indidem petenda erit Consonantiæ & Dissonantiæ causa item Physica. Puta, quo frequentius duarum chordarum (juxta percussarum) itiones & reditiones coincidunt, eo gratius auribus concinunt illæ chordæ; quo rarius (adeoque discrepant frequentius) eo minus grate. Hinc duas, dicas, chordas Æquitonas, omnium mollissime aures ferire; quoniam (dato quod vel simul incipiant vel aliquando saltem coincidant) perpetua erit (quamdiu perseverant) vibrationum in utrisque coincidentia. Quod si, propter diversa initia aut aliunde, non fiat coincidentia, erit saltem perpetua similisque alternatio: aut etiam (quod non est improbabile) mutuis ictibus (mediante communi aere) percussæ, adeoque suis utriusque motibus impeditæ, sensim, post paucos ictus, in concursum coeunt seu simultaneum tremorem. Æquitonæ. 1/1.

Proxime post Æqualitatem, sequitur ratio Dupla; (sicut, post æquitonas, illæ quæ Dia-pason sonant, quas Octavas vocant: utpote quarum Celeritas, vel Acumen saltem, est in ratione dupla.) Puta, cum secunda quæque celerioris vibratio, cum tardioris singulis coincidat; adeoque ex tribus vibrationibus (duabus scilicet in una chorda, tertiaque in reliqua,) duæ coincidant. Dia-pason. Octavæ. 2/1.

Post illas, Quintæ dictæ, quæ Dia-pente sonant; quarum Acumen sive Celeritas aut Frequentia reditionis est in ratione sesquialtera, seu ut 3 ad 2: Quum tertiæ quæque celerius vibrantis coincidant cum tardius vibrantis singulis secundis; ut ex quinis vibrationibus (ternis in una, & binis in altera chorda,) duæ coincidant. Dia-pente. Quintæ. 3/2.

Tandem Quartæ, Dia-tessaron sonantes; in ratione sesquitertia, seu ut 4 ad 3: ut coincidentia sit in Septenis quibusvis (duarum chordarum) vibrationibus. Dia-tessaron. Quartæ. 4/3.

194

John Wallis: Writings on Music

Quæ quidem omnes dici solent Consonantiæ; suntque eo ordine consonantiores (veI Aurium judicio) quo minoribus numeris signetur earum ratio; unde & vibrationum coincidentia sit frequentior. Post omnes autem longo intervallo sequitur Tonus; cui ratio competit sesquioctava, seu ut 9 ad 8: adeoque, post unam coincidentiam, non alia redit nisi factis (in duabus chordis) vibrationibus septemdecim; puta, in una, Novem; in altera, Octo. Tonus. 9/8.

Num autem hæc sit Acuminis Sonituum vera causa Physica, (quod non sine ratione supponunt Pythagorei, & post illos alii:) sitque porro gradus Acuminis frequentiæ vibrationum (in dato tempore) proportionalis, (quod an hactenus experimento satis fuerit comprobatum, haud scio;) nolim ergo pertinaciter affirmare: At certe Sonorum gravitatem chordarum longitudini (cæteris paribus) proportionalem esse, (adeoque, Acumen longitudini reciproce proportionale) à Ptolemæo jam olim ostensum est, (cap. 8. lib. 1.) & comprobant phænomena: easque rationes auribus esse gratiores (cæteris paribus) quæ minoribus numens signantur; puta, 1/1 quam 2/1; & hæc quam 3/2; hæc autem quam 4/3; & de cæteris pariter. Quod præsenti sufficit negotio. Secundum hanc hypothesin; Post Æquitonium (quod an Consonans, an Idem-sonans dicendum sit, disputant de nomine aliquando aliqui,) Dia-pason Consonantiam simplicissimam & perfectissimam statuunt; cujus Soni sint in ratione dupla, ut 2 ad 1. Quam & tam arctam volunt esse Consonantiam, ut, qui Dia-pason distant soni, pro eisdem habeantur, vel quasi-eisdem; ὁμόφωνοι dicti Ptolemæo; &, nostris, Unisoni. (Eo quod ratio Multipla bis composita, multiplam faciat; non autem alia quævis, puta superparticularis aut superpartiens, bis composita, faciat vel multiplam vel superparticularem; quas solas rationes Consonantiis, saltem simplicibus, appropriabant.) Adeoque Dia-pason & dia-tessaron pro eadem quasi consonantia habent cum simplici Dia-tessaron; similiterque Dia-pason & diapente, cum simplici Dia-pente; atque in reliquis similiter. Quod tum Ptolemæus, tum (eo antiquior) Aristoxenus (locis supra citatis) tradiderunt. De Intervallis itaque quæ majora sint quam Dia-pason, non est ut hic simus soliciti; (upote quæ rationes semper sortiantur duplas earum quas habent earundem excessus supra Dia-pason; verbi gratia, cum ratio Consonantiæ Dia-tessaron sit 4 /3; Dia-pason & dia-tessaron erit [p. 178] ejusdem dupla 8/3, & sic de cæteris:) Adeoque sufficit Dia-pason considerare, eaque Intervalla quæ sint eo minora, & in quæ Dia-pason dividitur. Tale Dia-pason esto, verbi gratia, quod est in Scala nostra, ab E ad e; quod canimus Vocibus la, fa sol la, mi, fa sol la: cujus itaque extremi soni sunt in ratione dupla, 2 ad 1.

The Harmonics of the Ancients compared with Today’s

195

Hanc autem rationem 2 ad 1, cum in duas æquales (numeris effabiles) partire non possent, (quippe nulla numeris effabilis ratio, potest bis posita duplam præstare) in duas proxime-æquales (numeris quam fieri potest minimis signabiles) dividebant; Nempe, pro numeris 2 ad 1, duplos ponebant (in eadem ratione) 4 ad 2, quibus intercederet unus medius 3, rationem expositam in duas dividens 4. 3. 2. puta 4/3 × 3/2 = 4/2 = 2/1. Adeoque Dia-pason la la (ab E ad e) dividebant in intervalla duo, Dia-pente & Dia-tessaron; in rationibus 3/2 & 4/3: Nempe, vel in la mi (ab E ad b) & mi la (à b ad e,) ut Dia-pente sit in parte graviori, & Dia-tessaron in acutiori; vel in la la (ab E ad a) & la la (ab a ad e) ut Dia-tessaron sit in graviori, & Diapente in acutiori; (cantando la mi la, vel la la la:) prout scilicet tonus intermedius la mi (in ratione 9/8) vel cum graviori vel cum acutiori Dia-tessaron (in ratione 4/3) sumatur; quippe 4/3 × 9/8 = 3/2. Adeoque, ex totius Dia-pason sonis Octo, saltem Quatuor (vocibus la la mi la signati) sunt suis inter se rationibus definiti; puta, ut numeri 12, 9, 8, 6; Dia-pason in duo Dia-tessaron cum interjecto Tono dividentes.

196

John Wallis: Writings on Music

Atque hactenus ni fallor (nec credo ulterius) processit Pythagoras; Nempe ut hæc quatuor intervalla, Dia-pason, Dia-pente, Dia-tessaron, & Tonum, rationibus 2 /1, 3/2, 4/3, & 9/8 determinaret, Et tale quidem fuisse Mercurii Tetrachordum, tradit Boethius cap. 20. lib. 1. & (post eum) Zarlinus, Vol. 1. par. 2. cap. 1. Sed non ita quatuor reliquos Sonos (vocibus fa sol fa sol signatos) vel definivit Pythagoras, vel de eis dividendis apud alios convenit: hoc est, de Tetrachordo utrovis in tria intervalla dividendo. Tetrachordum quidem hujusmodi in Genere Diatonico (ut reliqua superius tradita omittam) dividunt plerique (post Euclidem), in Tonum, Tonum, & Limma: ita nempe ut intervallum, voce fa signatum (puta, ab E ad F, vel b ad c) sit ubique Limma, in ratione 256/243; reliqua omnia, Toni integri, in ratione 9/8; (ut sæpius dictum est.) Sed hoc admodum incommode factum est. Nam; præterquam quod 256/243, sit ratio neque multipla, neque superparticularis, sed superpartiens, (nempe 113/243 supertredecupartiens ducentesimas quadragesimas tertias,) quod, quam fieri possit, in sonis saltem continue proximis refugiunt: (Euclid. p. 24. Ptol. p. 34.) Id porro accedit incommodi, quod tam magnis numeris signetur, ut, post unam vibrationum coincidentiam, non iterum coincidant nisi peractis (in binis chordis) vibrationibus 499. Quod ἐκμελέστατον videatur, & valde inconcinnum: Cum tamen, auditis mi fa vel la fa sonis proximis, nihil horridum aut ingratum auribus audiatur. Sed &, post sesquitertiam 4/3 = 11/3, plures sequuntur rationes superparticulares ante sesquioctavam (quæ est Toni,) puta sesquiquarta, sesquiquinta, sesquisexta, sesquiseptima, (5/4, 6/5, 7/6, 8/7,) quas omnes hæc Tetrachordi divisio negligit ut inconcinnas, dum illam Limmatis 256/243 retineat. Item; intervallum mi sol, vel la sol, Triemitonium dictum, (quas Tertias minores vocant,) à b ad d, vel E ad G, rationem habebit (secundum hanc divisionem) 32/27 256 /243 × 9/8: quod tamen Tono (cujus ratio 9/8) concinnius esse, testatur Auris, = adeoque rationem exigere minoribus numeris exponendam. Similiter; intervallum fa la Ditonum dictum (quas Tertias majores vocant,) ab F ad a, vel c ad e, rationem habebit 81/64 = 9/8 × 9/8; quæ minus concinna est, non modo quam Toni, sed quam (modo tradita) Tertiarum Minorum: cum tamen, vel aurium judicio, consonantius sit hoc intervallum, quam vel Tonus, vel Triemitonium. Cum itaque tantis incommodis urgeatur hæc tetrachordi divisio (quod est Euclidis & Eratosthenis Genus Diatonicum, & Ptolemæi Diatonum Ditonicum;) multo potior est (quam & recentiores amplectuntur, Zarlinus, Kepplerus, Mersennus, Cartesius, aliique) quam postremo loco nominavimus, quod est Ptolemæi Genus Diatonum Intensum. [The Modern Scale Constructed by Ratios] His itaque, intervallum mi (intellige, à sono proxime graviori,) qui est Tonus diazeucticus, est (ut apud omnes alios) Tonus major, in ratione Sesquioctava 9 /8; intervallum fa (quod [p. 179] pro Hemitonio censetur) est in ratione Sesquidecimaquinta 16/15; sol indem, in ratione Sesquioctava 9/8, Tonus major: sed la, Tonus minor dictus, in ratione sesquinona 10/9.

The Harmonics of the Ancients compared with Today’s

197

Atque hæc quidem tetrachordi divisio eodem modo colligitur, quo totius Dia‑pason in Dia-pente & Dia-tessaron, seu in duo Dia-tessaron cum interjecto Tono. Quippe cum Dia-pente la la, interjecta mi, (puta, ab a ad e,) sit (ut sæpius dictum est) in ratione sesquialtera, seu 3/2, nec possit hæc (ut nec alia ulla quæ non sit a numero quadrato denominata) in duas æquales (numero effabiles) dividi: dividatur (ut pridem dupla) in proxime-æquales (λόγους παρίσους) numeris quam minimis signatas: nempe, pro 3, 2, sumptis (in eadem ratione) eorum duplis 6, 4, (quo possit medius intercedere 5,) tres numeri 6, 5, 4, rationem expositam (3 ad 2, seu 6 ad 4) diriment in duas, 6/5 × 5/4 = 3/2. Similiterque, Dia-pente la la (ab a ad e) in duo dividetur intervalla proxime-æqualia, la fa (ab a ad c) in ratione 6/5 (quas Tertias Minores vocant,) & fa la (à c ad e) in ratione 5/4, quas vocant Tertias Majores. Cumque sit la mi (tonus diazeucticus) in ratione 9/8, erit (quæ reliqua est) mi fa, in ratione 16/15; propter 9/8 × 16/15 = 6/5, vel 9/8) 6/5 (16/15. Vel etiam, cum sit mi la (Dia-tessaron) in ratione 4/3, & fa sol in ratione 5/4; si hanc inde demas, relinquetur mi fa in ratione (ut prius) 16/15; propter 5/4) 4/3 (16/15, seu 5/4 × 16/15 = 4/3. Similiter; cum sit fa la (ut modo dictum) in ratione 5/4; quæ (ob causam aliquoties dictam) dividi non possit in duas æquales (numeris effabiles;) duplicatis (ut prius) numeris 5, 4, medioque interposito, 10, 9, 8: habentur duæ proxime-æquales 10/9 × 9/8 = 10/8 = 5/4. Quarum altera respondeat intervallo fa sol, altera intervallo sol la. Et quidem major duarum 9/8, secundum Didymum (ut supra dictum) assignatur intervallo sol la; minor autem 10/9 intervallo fa sol. Contra autem, Ptolemæus, & Recentiores, intervallo fa sol rationem 9/8 assignant (vocantque Tonum Majorem;) intervallo sol la rationem 10/9, vocantque Tonum Minorem. Sed &, siquando res postulat, utrumvis facias. Nempe, prout Dia-pason dividitur in duo Dia-tessaron cum interjecto Tono, qui vel huic vel illi additus faciat Diapente: Sic fa la Ditonum dividatur in duos Tonos minores (in ratione 10/9) cum interjecto Commate (in ratione 81/80) quod utrivis (prout res ferat) additum, faciat (vel hunc vel illum) Tonum majorem.

Cur autem intervallo potius sol la, quam fa sol, tonum minorem assignaverint, hanc potissimam fuisse causam censeo; Nempe, quum (quod in Dia-pente contingit) tres Toni continue sequuntur (vocibus fa sol la mi contenti) quorum unus sit Tonus Minor, (in ratione 10/9) reliqui duo Majores (in ratione 9/8;) Si tonus ille minor sit medio loco positus (in sol la) faciet hic cum utrovis extremorum

198

John Wallis: Writings on Music

Ditonum concinnum (in ratione 5/4, propter 9/8 × 10/9 = 5/4 utrobique:) Sin, medio loco positus, sit tonus Major; faciet quidem hic cum altero extremorum rationem (ut prius) 5/4; sed, cum reliquo, Ditonum inconcinnum, in ratione 81/64, (propter 9/8 × 9/8 = 81/64.)

Neque tamen tam inviolabile hoc est, quin possit (siquando opus est) Tonus major pro minore substitui (quo Schisma, quod vocant, vitetur,) aut vice versa. Quippe prout, in cantus ordine, siquando Tonus requiratur eo loci ubi contingit Hemitonium, aut contra; alterum pro altero substituendum esse, appositis Duri Mollisve notis indicatur: Pariter, siquando, in Toni Minoris sede, requiratur tonus Major, aut contra, (quo cantus fiat concinnior,) substituendus erit pro altero alter. Utputa ubi continue sequuntur sol la fa sol la, censeri solent extremi soni pro vera Dia-pente consonantia; sed, ut hoc fiat, pro altero duorum minorum, substituendus est tonus major; secus enim commate 81/80 deficit à justa Dia-pente. Et similiter alibi prout res postulat.

Nec tamen hic solet (ut in casu altero, de Tono & Hemitonio,) apposita Nota aliqua insinuari; (sed potius vox non monita id præstabit:) Non enim cum tanta accuratione procedit (quam retinemus) Guidoniana Notatio. Quippe Guidonis tempore tonus major à minore non distinguebatur; ut nec quanta pars toni præcise fuerit quod Hemitonium diceretur; Sed, Tonos ab Hemitoniis distinguere contenti;

The Harmonics of the Ancients compared with Today’s

199

num majores an minores fuerint toni, non erant soliciti; nec quanta præcise toni pars fuerit Hemitonium, sed saltem ut sit prope-dimidium: Et, siquid curiosius quæreretur, Auris ope sanandum esset. Quodque de hac notatione utentibus dictum est; id à fortiori intelligendum erit de Orga-[p. 180]nopœis; ut qui his minutis accurate prospicere nullo modo possunt; sed, pro vero, vero-proximum sumere necesse habent. Sed hæc obiter. Dia-pason igitur sic divisum (quod est Ptolemæi Genus Diatonum Intensum,) in hodierna Musica, sic conspicietur.

[Extensions of the Modern System] Adhiberi posset similis item divisio Intervalli Dia-tessaron mi la, (à b ad e,) qualem ipsi Dia-pente jam accommodavimus: Nempe, duplicatis numeris 4, 3, medioque interposito; ut numeri 8, 7, 6, exhiberent rationes 8/7 × 7/6 = 8/6 = 4/3: adeoque vox nova interponeretur inter fa & sol, (quæ diceretur fa acuta vel sol mollis,) quæ cum altera duarum mi la rationem faceret 8/7, cum altera 7/6. Item triemitonium mi sol, aut la fa, (à b ad d, vel ab a ad c,) similiter posset dividi, rationibus 12/11 × 11/10 = 12/10 = 6/5: interposita voce nova inter fa & sol quæ cum altera duarum mi, sol, rationem habeat 12/11, cum altera 11/10; itemque inter la mi, quæ ad alteram duarum la, fa, similiter habeat rationem 12/11, ad alteram 11/10. Atque hæ quidem, aliæque plures, (non modo duplicatis numeris, sed & triplicatis, quo interponerentur duo numeri, ratioque in ternas divideretur,) adhibendæ forent divisiones, pluresque interponendæ voces, si resumenda essent

200

John Wallis: Writings on Music

Veterum Genera Enarmonia, Chromatica, variaque Diatonica, quæ jam per multa secula desueverunt. Sed hodierna Musica, eo subtilitatis pervenire, non affectat; neque aut Vox, aut Auris, tam minuta facile distingueret, prout nos saltem nunc dierum existimamus. Et quamvis non negem quin aliquando forsan, puta in c intensa aut d remissa, (notâ Duri Mollisve, affectis,) hujusmodi sonum aliquem subtilius feriat lingua (auribus adjuta) quam scriptæ notæ designant; nostri tamen hodiernæ praxeos Musicæ magistri, saltem Organopœi, qui c intensam & d remissam pro eodem habere solent, id saltem his Duri Mollisque notis insinuatum volunt, Proferendum esse sonum quendam qui sit inter c & d quasi medius; nec scrupulosius inquirunt quam præcise habeat ad sonos adjacentes rationem; Tonos & Semitonia (crassius intellecta) numerare contenti. Quum itaque alicubi tandem sistendum erit (neque enim res patitur ut in infinitum procedant hujusmodi divisiones,) divisionibus ante traditis contenti, non ultra procedunt, hodierni Musici, in Dia-pason dividendo. Quamquam enim rationes 7/6 & 8/7 (quas negligunt,) potiores videantur quam (quas admittunt) 9/8 & 10/9 (utpote minoribus numeris signatæ;) considerandum tamen est, quod numerus 7 (qui in rationibus prioribus designandis occurrit) sit numerus primus seu incompositus; sed numeri 8, 9, 10, (quibus signantur rationes posteriores) sint numeri (ex minoribus) compositi, (puta 8 = 2 × 2 × 2; 9 = 3 × 3; 10 = 2 × 5) qui singuli minores sint quam 7; adeoque quos, in rationibus æstimandis, mens promptius persentiscat. Quippe expeditius est, & quod animus promptius assequitur, (verbi gratia,) in duo æqualia quidquam partiri, iterumque in duo, (ut pars quarta habeatur;) quam semel in tria, (quo habeatur pars tertia,) utut 3 quam 4 sit minor numerus: Itemque, primum in duo, iterumque in duo, & tertio tandem in duo, (quo habeatur pars Octava;) quam (quo habeatur pars Septima) una vice in Septem: (Similiterque; Duplicare, iterumque Duplicare; quo habeatur Quadruplum: quam una vice Triplicare; quo habeatur Triplum: pariterque Duplicare, iterumque Duplicare, & tertio Duplicare; quo habeatur Octuplum: quam una vice Septuplicare, quo habeatur Septuplum.) Quod notavit Ptolemæus (in hoc negotio) pag. 3. Atque ob hanc causam, hodierni Musici, dum rationes 7/6, 8/7, (nedum 11/10, 12/11, &c.) negligunt, (ut quæ numeros primos 7, 11, aut his majores postulant quo designentur,) quas in Musicam receperunt Veteres; admittunt tamen (non inepte) rationes 9/8, 10/9, 16/15 &c. majoribus quidem numeris designatas, sed qui à minoribus componuntur. Nullam utique rationem jam admittunt, nisi quæ designantur numeris primis 1, 2, 3, 5, aut horum inter se compositis. Hinc item est, quod intervallum Dis-dia-pason (cujus ratio 4/1) consonantius reputetur quam Dia-pason & dia-pente (cujus ratio est 3/1) utut numerus 3 minor sit quam 4; quoniam 4 (= 2 × 2) est ex minoribus primis compositus. Simileque in aliis non raro sit judicium. Quanquam, in præferendis Consonantiis, non semper eadem omnium est sententia. Ex his, demum (jam traditis) sonorum continue proximorum rationibus, habentur (componendo) sonorum utcunque remotorum rationes. Quarum specimen exhibemus in sonis singulis ad Dis-dia-pason usque: Quod est Ptolemæi Systema

The Harmonics of the Ancients compared with Today’s

201

Perfectum, à Proslambanomeno, ad Neten hyperbolæon: Hoc est, in Scala nostra, ab A re ad aa la mi re in alto. Rationes autem sic expositæ, quanto concinniores sint, quam si (secundum alios) divideretur Dia-tessaron in Tonum, Tonum, & Limma, 9/8 × 9/8 × 256/243 = 4 /3; facile censebitur, si secundum hanc formam colligerentur rationes omnes sonorum inter se omnium ad Dis-dia-pason usque, prout nos eas secundum alteram divisionem jam collegimus. [p. 181] Sonorum in Dis-dia-pason Rationes inter se.

John Wallis: Writings on Music

202

Ex his in Dia-pason (puta à mi ad mi,) censeri solent rationes præcipuæ. 1 ad 16 10 9 6

1 15 9 8 5

5 4 3 8 5 16 9 15 2

4 3 2 5 3 9 5 8 1

Æquitonium. Hemitonium dictum, Secunda minor; mi fa Tonus minor, Secunda media; sol la. Tonus major, Secunda major; la mi, fa sol. Triemitonium, (a nonnullis dictum Semiditonum,)Tertia minor; mi sol, la sol. Ditonum, Tertia major; fa la. Dia-tessaron, Quarta; mi la. Dia-pente, Quinta, la mi. Hexachordum minus, Sexta minor; mi sol. Hexachordum major, Sexta major; fa la. Heptachordum minus, Septima minor; mi la. Heptachordum medium, Septima media; la sol Heptachordum majus, Septima major; fa mi. Dia-pason, Octava; mi mi.

Quarum secundæ & septimæ omnes haberi solent pro Dissonantiis; reliquæ pro Consonantis; & quidem Æquitona, Ocatava, Quinta, & Quarta, (cum earum post Octavam replicationibus,) pro Perfectis; Tertiæ & Sextæ minores majoresque, cum suis replicationibus, pro consonantiis imperfectis. Consonantiæ perfectæ. 1

4 11 18

5 12 19

8 15 22

6' 13' 20'

6 13 20

Imperfectæ. 3' 10' 17'

3 10 17

Præter has, habetur ex accidenti, (Dissonæ quidem omnes,)

The Harmonics of the Ancients compared with Today’s

203

Quæ oriuntur, vel ab Hemitonio in Toni locum assumpto, seu contra; (quæ adscitis Duri Mollisque notis sanari solent:) vel à Tono minori pro majore sumpto, seu contra; quæ (cum sit tantum commatis 81/80 differentia) ab humana voce (etiam non monita) emendari solent in prolatione, (vel dissimulari;) in instrumentis, [p. 182] res sic attemperari solet, ut tantilla differentia vix percipiatur. Ut autem veræ Quarta & Quinta complent Octavam seu Dia-pason: Sic falsæ Quartæ cum falsis Quintis; itemque Tertiæ cum Sextis, & Secundæ cum Septimis respective sumptis: puta Tertia major cum Sexta minore; & pariter de cæteris. His omnibus (si ultra Dia-pason procedatur) accensendæ sunt eædem (respective) rationes cum Dupla (semel, bis, aut pluries) compositæ: hoc est, eadem intervalla (semel, bis, aut pluries) composita cum Dia-pason: Dicique solent, eorundem Replicationes. Quod autem Tertias, & Sextas, pro Consonantiis habeant, cum bona quidem ratione factum censeo, sed praeter omnium Veterum authoritatem. De Quarta quidem dubitaverunt Recentiorum aliqui, (contra quos Andreas Papius Gandensis, ante centum annos, peculiari tractatu disputavit;) Sed contra omnium Veterum sententiam. De hac quidem cum Dia-pason composita, non ita erant Veteres unanimes; quia scilicet Dia-pason cum dia-tessaron rationem exigit 8/3 = 4/3 × 2, quæ est ratio neque Multiplex neque Superparticularis; (Quam tamen exceptionem repudiat Ptolemæus, ut supra dictum; nostrique negligunt, in Sexta majore, minoreque:) At, simplex Dia-tessaron, veteres omnes unanimi consensu Consonantiis annumerabant. Verum quidem est, Quartam solitariam, in cantuum Compositione, rarius admitti: At hoc non ideo fit, quod hæc per se non sit Consonantia; sed quia Quintæ (ex altera parte) prævalentia (quæ ad hanc sonat Dia-pason, & in hac quasi subauditur,) Quartam quasi obumbrat, redditque minus ideoneam; adeoque parcius adhibitam. Qua methodo, quove ordine, adhibendæ sunt in Melopœia (quas enumeravimus) Consonantiæ, & cum Dissonantiis admiscendæ, (qua de re parce admodum egerunt Veteres, Recentiores copiosius;) non est præsentis instituti fusius enarrare. Qui plura cupit, eos adeat qui rem Musicam plenius & ex professo tradiderunt. Sed neque libet his disquirere, quo ordine pro dignitate sua censendæ sint Consonantiæ; Puta, an Dia-pente cum dia-pason (cujus ratio sit 3/1;) potior sit quam simplex Dia-pente (cujus ratio 3/2;) aut etiam quam Dis-dia-pason (cujus ratio 4/1;) quod minoribus numeris contineatur; quodque paucioribus vibrationibus redeat coitio: Vel contra; eo quod simplicior videatur Consonantia Dia-pente, quam Dia-pason & dia-pente; sitque 4, utut numerus major, ex minoribus tamen primis compositus, puta 2 in 2, quorum utervis minor est quam 3. Similiterque, An rationes 7/6, 8/7, (utpote minoribus numeris contentæ;) an 9/8, 10/9, 16/15, (utpote quarum numeri à minoribus primis componuntur,) potiores sint. Et quæ horum sunt similia. Qua de re Mersennum videas, lib. 4. prop. 18, 22, &c.

204

John Wallis: Writings on Music

Convenit utique, cæteris paribus, Rationem numeris minoribus explicabilem, potiorem esse quam quæ majoribus: Item, in qua frequentius occurrit vibrationum coitio, quam in qua rarius: Item, ubi numerus ex minoribus primis componitur, quam ubi ex majoribus. Addo etiam (quod criterium est ex natura, magis quam ex imaginatione nostra,) in qua duarum chordarum altera percussa, majorem infert reliquæ tremorem. Verum, ubi horum indiciorum alterum alteri contrariatur, utrum præferendum sit; non eadem omnium est sententia; neque ego me arbitrum interpono. Pariter de aliis dicendum est, quorum fusior disquisitio in brevi tractatiuncula non est expectanda. F I N I S.

Chapter 4

Notice of Wallis’s Edition of Ptolemy’s Harmonics in the Philosophical Transactions, January 1683 Editorial Note Wallis’s edition of Ptolemy’s Harmonics (see Chapter 3) had appeared in 1682 and was announced in a brief note in Philosophical Transactions 13/143 (January 1683): 20–21. As noted in the Introduction,1 Robert Plot is probably the most likely candidate to have written it. Text [p. 20] ΚΛΑΥΔΙΟΥ ΠΤΟΛΕΜΑΙΟΥ ΑΡΜΟΝΙΚΩΝ ΒΙΒΛΙΑ Γ.a CLAUDII PTOLEMÆI Harmonicorum Libri Tres. Ex Codd. Mss. Undecim,2 nunc primum Græce Editi. Johannes Wallis, ss. Th. D. Geometriæ Professor Savilianus Oxoniæ, Regiæ Societatis Londini Sodalis, Regiæque Majestati a Sacris; Recensuit, Edidit, Versione & Notis illustravit, & Auctarium adjecit. Oxonii, e Theatro Sheldoniano, A.D. 1682. In Quarto.b THis work having been never before Printed in Greek, and but very imperfectly in Latine, by Anton. Gogavinus of Graves, above a 100 Years since,3 when good skill in the Greek Tongue was more rare: our learned Professor took it for a Task well agreeing with his Province, to give it us in a more perfect Edition. For which purpose, he hath most diligently compared the Manuscript Copies, for restoring and perfecting the Greek Text. And adjoyned a new Latine Translation of the whole: together, with Notes on the Text. Therein rectifying many Mistakes of the Transcribers; especially in the Numbers; which (by Calculation of the whole anew, according to the mind of Ptolemy, declared in the Text) he hath restored to their Integrity.

a

  ‘Claudius Ptolemy’s Three Books on Harmonics’.   ‘Claudius Ptolemy’s Three Books on Harmonics. Now for the first time edited in Greek, from 11 manuscript codices. Collated, edited, illustrated with translation and notes, and with additions, by John Wallis, D.D., Savilian Professor of Geometry at Oxford, F.R.S., and chaplain to his Majesty. At Oxford, at the Sheldonian Theatre, 1682. Quarto.’ b

206

John Wallis: Writings on Music

This Work of Ptolemy, gives an Account of the Nature of Sounds in general, but especially of those which are Musical; as of the several Sorts of Tones, and their Ratio’s one to another;4 with other Particulars. Shewing also, how Harmony may be fitly compared to the Motions of the Humane Soul; and those of Cœlestial Bodys. And is the more considerable, as being not only the best of all the Greek Musicians; but as it also gives an Account of the rest; wherein they agreed, or disagreed, one with another; and upon what Principles. There is also added an Appendix, by the Doctor, Containing a brief Account of the Ancient Harmonicks, according to the different Sects of the Authors, compared one with another; and with the Musick of this Age: Shewing [p. 21] how and wherein the Greek Musick agreed or differ’d from ours; And how the several particulars thereof, are, for the most part, retained in ours, but very differently expressed; and what in the one, answers to what in the other. There is herein, a short Collection of all or most of what occurs material, in the several Greek Authors on this Subject; as well those Published by Meibomeus, as others yet remaining amongst us in Manuscript; all Methodically digested, and brought into a Narrow Compass, that the Reader may have at once a view of the Greek and Modern Musick, compared together. A Work which we doubt not, will be very acceptable to those who are willing to look into the Speculative Part of Musick. He hath now also, in the Press at London, a Treatise of Algebra, Historical and Practical; Shewing its Original and Progress in several Ages, and the several steps whereby it hath advanced to the height at which now it is. Whereof a further Account is intended, when it is Printed.5 Notes 1 See p. 17 above. 2 On the manuscripts which Wallis had collated, see Introduction, p. 12 above. 3 On Gogava’s translation, see Introduction, pp. 12, 16 above. 4 Evidently ‘tones’ were ‘pitches’ in this sentence, otherwise they could not bear ‘ratios’ to one another. 5 The ‘Treatise of Algebra’ would appear as A treatise of algebra, both historical and practical (London, 1685).

Chapter 5

‘A Question in Musick’, Philosophical Transactions, March 1698 Editorial Note This, the first of Wallis’s three papers on musical subjects in the Philosophical Transactions for 1698, dealt with the placement of frets on a viol. By contrast with the two papers which followed it, we know nothing about the circumstances which prompted its composition. A comparison with Wallis’s earlier writings on the theory of tuning (Chapters 1, 3 above) shows considerable development in detail. As discussed in the Introduction, this text also displays certain novelties compared with Wallis’s sources.1 No autograph is known; the text printed here is that of Philosophical Transactions 20/238 (March 1698): 80–84. Text [p. 80] A Question in Musick lately proposed to Dr. Wallis, concerning the Division of the Monochord, or Section of the Musical Canon: With his Answer to it. Question. TAKE a String of any Musical Instrument, and divide the same into two equal Parts, and stop the String there; it shall be an Eighth, which consists of twelve Semi-tones. Hence it appears, that the Frets are nearer to one another toward the Bridge, and wider toward the Nut or Head of a Viol. And that they decrease or proceed in a Geometrical Proportion.2 Quære, How is it possible, from the foresaid Hypothesis, to divide the other 11 Semi-tones,3 in their due Proportion, and to demonstrate the same. And whether the other Distances assigned by Simpson in his Compendium of Musick (and Chapter of Greater and lesser Semi-tones) are demonstrable from the said Hypothesis.4 [p. 81]

208

John Wallis: Writings on Music

Answer.

March 5. 1697/8. WHAT Method is used by Simson (in the Book mentioned) to divide a String or Chord, I know not: Nor have I the Book at Hand to consult.5 That a String open (or at its full Length) will sound (what we call) an Octave (or Diapason) to that of the same String stopped in the Middle (or at half its Length) is very true. And hence it is that we commonly assign, to an Octave, the Duple Proportion (or that of 2 to 1:) because such is the Proportion of Lengths (taken in the same String) which give those Sounds.6 And (upon a like Account) we assign to a Fifth (or Diapente) the Sesqui-alter Proportion (or that of 3 to 2:) And, to a Fourth (or Dia-tesseron) the Sesquitertian (or that of 4 to 3:) And to a Tone (which is the Difference of a Fourth and Fifth) The Sesqui-octave (or that of 9 to 8:) Because Lengths (taken in the same String) in these Proportions, do give such Sounds. And (universally) whatever Proportion of Lengths (taken in the same String, equally stretched) do give such and such sounds; such Proportions (of Gravity) we assign to the Sounds so given.7 But when an Eighth (or Octave) is said (in common Speech) to consist of 12 Hemi-tones, or 6 Tones; this is not to be understood according to the utmost Rigour of Mathematical Exactness, (of such 6 Tones, as what they call the Diazeuctick Tone, or that of la mi, which is the Difference of a Fourth and Fifth;) but, as exact enough for common Use. For Six such Tones (that is, the Proportion of 9 to 8 Six Times repeated) is somewhat more than that of an Octave (or the Proportion of 2 to 1:) And, consequently, such an Hemi-tone, is somewhat more than the Twelfth Part of an Eighth, or Octave, or Dia-pason.8 But the Difference is so little, that the Ear can hardly distinguish it: And therefore (in common Speech) it is usual so to speak. And, accordingly, when we are directed to take the Lengths (for what are called the 12 Hemi-tones) in Geometrical Proportion: it is to be understood (not, to be so in the utmost Strictness, but) to be accurate enough for common Use; for [p. 82] placing the Frets on the Neck of a Viol, or other Musical Instrument; wherein a greater Exactness is thought not necessary. And this is very convenient, because (thus) the Change of the Key (upon altering the Seat of mi) gives no new Trouble, for this doth indifferently serve any Key; and the Difference is so small, as not to offend the Ear. But those who choose to treat of it with more exactness, go this way to work. Presupposing the Proportion for an Octave (or Dia-pason) to be that of 2 to 1; they divide this into Two Proportions; not just equal (for that would fall upon surd 9 Numbers, as of √2 to 1;) but near-equal (so as to be expressed in small Numbers.)10 In order to which, instead of taking 2 to 1, they take (the double of these Numbers) 4 to 2; (which is the same Proportion as before;) and interpose the middle Number 3. And, of these Three Numbers, 4, 3, 2, that of 4 to 3, is the Proportion for a Fourth (or Dia-tesseron.) And that of 3 to 2, the Proportion for a Fifth (or Dia-pente.) And these Two put together, make up that of an Octave (or

‘A Question in Musick’

209

Diapason,) that of 4 to 2, (or 2 to 1.) And the Difference of those Two, that of a Tone or 9 to 8. As will plainly appear in the ordinary Method of Multiplying and Dividing Fractions. That is, 4/3 × 3/2 = 4/2 = 2/1. And 4/3 ) 3/2 ( 9/8

Thus, in the common Scale (or Gam-ut) taking an Octave, in these Notes, la, fa sol la, mi, fa sol la; suppose, from E to e, (placing mi, in b fa ba mi; which is called the Natural Scale;) the Lengths for the Extremes la la, an Octave, are as 2 to 1, or 12 to 6. Those for la la (in la fa sol la) or mi la (in mi fa sol la) a Fourth, as 4 to 3, or 12 to 9, or 8 to 6. Those for la mi (in la fa sol la mi) or la la (in la mi fa sol la) a Fifth, as 3 to 2, or 12 to 8, or 9 to 6. Those for la mi, the Diazeutick-Tone (or difference of a Fourth and Fifth,) as 9 to 8. So have we for those Four Notes la la mi la, their Proportionate Length in the Numbers 12 9 8 6. [p. 83] Then, if we proceed in like manner, to divide a Fifth (or Dia-pente) la fa sol la mi, or la mi fa sol la, or the Proportion of 3 to 2, into near-equals, (taking double Numbers in the same Proportion, 6 4; and interposing the middle Number 5;) of these Three Numbers, 6 5 4; that of 6 to 5, is the Proportion of a lesser Third (called a Tri hemitone, or Tone and half,) as la fa (in la mi fa.) And that of 5 to 4, is the Proportion of the Greater Third (commonly called a Ditone, or Two Tones,) as fa la (in fa sol la) which Two put together, make a Fifth, as 3 to 2; That is 6/5 × 5/4 = 6/4 = 3/2. And their difference is as 25 to 24: That is 6/5 )b 5/4 ( 25/24. So have we for these 3 Notes la fa la, their proportionate Lengths in Numbers, as 6 5 4.  Original has a black letter b, although a  sign would be expected.  Original lacks ).

a

b

210

John Wallis: Writings on Music

In like manner, if we divide a Ditone (or Greater Third) as fa la (in fa sol la) whose Proportion is as 5 to 4 (or 10 to 8) into Two near-equals (by help of a middle Number 9;) then have we (in these Three Numbers 10 9 8) that of 10 to 9, for (what they call) the Lesser Tone: And that of 9 to 8, for (what they call) the Greater Tone. But, whether fa sol shall be made the Lesser (as 10 to 9) and sol la the Greater, (as 9 to 8;) or, This the Lesser (as 10 to 9) and That the Greater (as 9 to 8) or sometime This and sometime That, as there is occasion, (to avoid what they call a Schism;) is somewhat indifferent: For, either way, the Compound will be, as 5 to 4; and the Difference (which they call a Comma) as 81 to 80. That is, 9/8 × 10/9 = 10/9 × 9/8 = 10/8 = 5/4. And 10/9 ) 9/8 ( 81/80.11 Lastly, If from that of the Tri-hemitone (or Lesser Third) la mi fa; whose Proportion is as 6 to 5; we take that of the Tone la mi (which is the Difference of a Fourth and Fifth) as 9 to 8; There remains for the Hemi-tone mi fa (or la fa) that of 16 to 15. That is 9/8 ) 6/5 ( 48/45 = 16/15. Or, the Tri-hemitone (or Lesser Third) whose Proportion is as 6 to 5; may be divided into Three Near-equals, (by taking Triple Numbers, in the same Proportion, 18 15; and interposing the Two Intermediates, 17 16;) which will therefore be as 18 to 17, and as 17 to 16, and as 16 to 15; That is, 18/17 × 17/16 × 16/15 = 18/15 = 6/5.12 Where also the Greater Tone, who’s Proportion is as 9 to 8, or 18 to 16, is divided into its Two Near- equals (commonly called Hemi-tones,) that of 18 to 17, and that of 17 to 16: That is, 18/17 × 17/16 = 18/16 = 9/8. [p. 84] And the Lesser Tone, that of 10 to 9, or 20 to 18, may be in like manner divided into that of 20 to 19, and that of 19 to 18: That is, 20/19 × 19/18 = 20/18 = 10/9. Which Divisions of the Greater and Lesser Tone, answer to what is wont to be designed by Flats and Sharps.13 So that (by this Computation,) of these Eight Notes, la, fa sol la, mi, fa sol la; their Proportions stand thus; that of la fa (or mi fa) is as 16 to 15. That of fa sol as 10 to 9, and that of sol la as 9 to 8: (or else that of fa sol as 9 to 8, and that of sol la as 10 to 9,) That of la mi as 9 to 8.14 And if either of the Tones (Greater or Lesser) chance to be divided (by Flats or Sharps) into (what they call) Hemi-tones, their Proportions are to be such as is already mentioned. There may be a like Division of a Fourth (or Dia-tessaron) into Two Nearequals: And of some others of these, into Three Near-equals. Which might be of use for (what they were wont to call) the Chromatick and Enarmonick Musick. But those Sorts of Musick, having been long since laid aside, there is now no need of these Divisions, as to the Musick now in use. Notes 1 See pp. 20–22 above. 2 The word ‘hence’ seems to claim for this second paragraph a relationship which as it stands it does not possess with the first. Possibly the questioner intended to say

‘A Question in Musick’

211

more in his first paragraph: for example, that the halved string could be halved again to produce a pitch another octave higher. This would have shown both that equal intervals required smaller stretches of string at higher pitches (‘that the Frets are nearer to one another toward the Bridge’) and that this decrease was ‘in a Geometrical Proportion’ (that is, in a sequence governed by equality of ratios). 3 The thought which seems to be intended here might be more clearly expressed by the words ‘to place the other 11 Frets’. 4 The reference was to Simpson, A Compendium of Practical Musick, pp. 102–9. 5 Simpson in fact adopted the position that ‘all our Musical Intervals arise from the division of a Line or String into equal Parts’; that is, the arithmetical division of intervals. Wallis had specifically criticised this procedure: ‘Remarks on the Proposal to perform musick, in perfect and mathematical proportions’, p. 39. 6 Wallis used ‘sounds’ in the sense of ‘intervals’. 7 Cf. Wallis to Oldenburg, 14 May 1664, fol. 2v (pp. 48–9 above); Wallis, ‘The Harmonics of the Ancients’, pp. 176–7 (pp. 132–3 above), where the quantification of ‘gravity’ was by the frequency of vibrations, not the lengths of strings. 8 As in his earlier writings, Wallis assumed that the ‘tone’ commonly referred to in musical discussions particularly by practitioners was the ‘diazeuctick’ (or Pythagorean) tone with ratio 9 : 8 and showed that six of them make more than an octave. Here he omitted any statement that half of such a tone is an unmusical ratio. See Wallis to Oldenburg, 14 May 1664, fol. 1v (pp. 47–8 above); Wallis, ‘The Harmonics of the Ancients’, p. 176 (pp. 130–31 above). 9 Literally ‘deaf’ or ‘dumb’ numbers, but the word also bore the meaning of mathematically incommensurable (cf. Wallis to Oldenburg, 14 May 1664, §7 (p. 54, n. 29)). 10 Cf. Wallis, ‘The Harmonics of the Ancients’, p. 178 (p. 134, n. 64) on the procedure of division into near-equals. 11 Cf. Wallis to Oldenburg, 14 May 1664, §13 (p. 56, n. 35), where the matter was somewhat indifferent; Wallis, ‘The Harmonics of the Ancients’, p. 179 (p. 137 above), where it was less so. 12 Cf. Wallis, ‘The Harmonics of the Ancients’, p. 180 (pp. 139–40 above), where the possibility of dividing intervals into three near-equals was raised but not pursued. 13 See Introduction, pp. 18–22 above on the relationship of this new set of intervals to the work of Salmon, Simpson, and Holder, and Wallis’s remarkable claim that this was what was meant by flats and sharps. 14 In other words, in the scale E–F–G–A–B–C–D–e, the scale steps had, respectively, the ratios 16 : 15, 10 : 9, 9 : 8, 9 : 8, 16 : 15, 10 : 9, 9 : 8, with the possibility of reversing the order in the two places where 10 : 9 was followed by 9 : 8. (In fact, the thirds E–G and G–B would both be unusable unless the first such change was carried out; the second would improve the third B–D at the expense of the third D–f.)

Chapter 6

Letter to Samuel Pepys, June 1698 Editorial Note Wallis’s second musical paper in the Philosophical Transactions of 1698 concerned Renatus Harris, and may have arisen originally from his challenge to his fellow organ-builder ‘Father Smith’.1 As Wallis recounts, Harris approached Pepys, who directed him to Wallis, and after some ‘little discourse’ with him Wallis wrote back to Pepys in a letter dated 27 June, which appeared in Philosophical Transactions 20/242 (July 1698): 249–56.2 Wallis’s letter was in fact a rather hurried survey of certain strategies for approaching the issue of keyboard tuning, and it did not add a great deal to the information that had already appeared in his letters (Chapter 1 above) or in print (Chapters 3 and 5 above). It is perhaps more important for its evidence of a renewed (or continuing) interest in the world of practical music; it should therefore be read in conjunction with his letters to Oldenburg of March 1677 (Chapter 2 above), where Wallis had approached a different practical topic with a similar seeming disregard for the knowledge that practitioners might be able to bring to the situation. In this case, however, we have the evidence of the letter itself that Wallis had been involved in a ‘little discourse’ with Harris, and thus that he was not writing in complete ignorance of the practitioners’ agenda. No autograph of this letter is known; the text printed here is that which appeared in the Philosophical Transactions. Text [p. 249] A Letter of Dr. John Wallis to Samuel Pepys Esquire, relating to some supposed Imperfections in an Organ. Mr. Harris an Organ-maker (whom I find, by the little discourse I had with him, to be very well skilled in his profession) was lately with me, as by direction from you, to ask my opinion about perfecting an Organ, in a point wherein he thinks it yet Imperfect. ’Tis an honour you please to put upon me, to think my opinion considerable in a thing wherein I am so little acquainted as that of an Organ. I do not pretend to be perfectly acquainted with the Structure of an Organ, its several Parts, and the Incidens thereunto; Having never had Occasion and

John Wallis: Writings on Music

214

Opportunity to inform my self particularly therein. And, for the same reason, many of the Words, Phrases, Forms of Speech, and Terms of Art, which are familiar to Organists and Organ-makers, are not so to me. Which therefore I shall wave; (For till we perfectly understand one anothers Language, it is not easy to speak intelligibly;) and apply my self directly to what is particularly proposed. This (I take it) is evident; That each Pipe in the Organ is intended to express a distinct Sound at such a Pitch; That is, in such a determinate Degree of Gravity or Acuteness; or (as it is now called) Flatness or Sharpness. And the Relative or Comparative Consideration of Two (or more) such Sounds or Degrees of Flatness and Sharpness, is the ground of (what we call) Concord and Discord; that is, a Soft, or Harsh, coincidence.3 Now, concerning this, there were amongst the Ancient Greeks, Two (the most considerable) Sects of Musicians: the Aristoxenians, and the Pythagorians.4 They both agreed thus far; That Dia-tessaron and Dia-pente, [p. 250] do together make up Dia-pason; that is (as we now speak) a Fourth and Fifth do together make an Eighth or Octave: And, the Difference of those two (of a Fourth and Fifth) they agreed to call a Tone; which we now call a Whole note. Such is that, (in our present Musick,) of La Mi, (or as it was wont to be called, Re Mi.) For La fa sol la, or Mi fa sol la, is a perfect Fourth: And La fa sol la mi, or La mi fa sol la, is a perfect Fifth: The Difference of which, is La mi.5 Which is, what the Greeks call, the Diazeuctick Tone; which doth Dis-join two Fourths (on each side of it;) and, being added to either of them, doth make a Fifth. Which was, in their Musick, that from Mese to Paramese; that is in our Musick, from A to B: supposing Mi to stand in B fa b mi, which is accounted its Natural position. Now, in order to this, Aristoxenus and his Followers, did take, that of a Fourth, as a Known Interval, by the judgement of the Ear; and, that of a Fifth, likewise; And consequently, that of an Octave, as the Aggregate of both; and that of a Tone, as the Difference of those Two. And this of a Tone (as a known Interval) they took as a common Measure,6 by which they did estimate other Intervals. And accordingly they accounted a Fourth to contain Two Tones and an half; a Fifth to contain Three Tones and an half; and consequently an Eighth to contain Six Tones, or Five Tones and two Half-tones. And it is very near the matter, though not exactlya so. And at this rate we commonly speak at this day; supposing an Octave to consist of Twelve Hemitones, or Half-notes. (Meaning thereby, somewhat near so many half-notes:) But, when we would speak more Nicely, we do not take those supposed Half-notes to be exactly Equal, or each of them just the Half of a Full-note, such as is that of La-mi. Pythagoras and those who follow him, not taking the Ear alone to be a competent Judge in a case so nice; chose to distinguish these, not by Intervals,  Original has ex-exactly, across a line break.

a

Letter to Samuel Pepys, June 1698

215

but by Proportions. And accordingly they accounted that of an Octave, to be, when the degree of Gravity or Acuteness of the one Sound to that of the other, is Double, or as 2 to 1; that of a Fifth, when it is Sesqui-alter, or as 3 to 2; that of a Fourth when Sesqui-tertian, or as 4 to 3. Accounting That, the Sweetest proportion, which is expressed in the Smallest Numbers; and therefore (next to the Unisone) that [p. 251] of an Octave, 2 to 1; then that of a Fifth, 3 to 2; and then that of a Fourth, 4 to 3. And thus, that of a Fourth and Fifth, do together make an Eighth; For 4/3 × 3 /2 = 4/2 = 2/1 = 2. That is, four thirds of three halves, is the same as four halves, that is Two. Or (in other words to the same sense) the proportion of 4 to 3, compounded with that of 3 to 2, is the same with that of 4 to 2, or 2 to 1. And, consequently, the Difference of those Two, which is that of a Tone or Full-Note, is that of 9 to 8. For 4/3 ) 3/2 ( 9/8; that is, three halves divided by four thirds, is nine eights; or, if out of the proportion of 3 to 2, we take that of 4 to 3; the Result is that of 9 to 8.7 Now, according to this Computation, it is manifest, That an Octave is somewhat less than Six Full-notes. For (as was first demonstrated by Euclide, and since by others) the Proportion of 9 to 8, being six times compounded, is somewhat more than that of 2 to 1. For 9/8 × 9/8 × 9/8 × 9/8 × 9/8 × 9/8 = 531441/262144, is more than 524288/262144 = 2/1. This being the Case; they allowed (indisputably) to that of the Dia-zeuctick Tone (La mi,) the full proportion of 9 to 8; as a thing not to be altered; being the Difference of Dia-pente and Dia-tessaron, or the Fifth and Fourth. All the Difficulty, was, How the remaining Fourth (Mi fa sol la) should bea divided into three parts, so as to answer (pretty near) the Aristoxenians Two Tones and an Half; and might, altogether, make up the proportion of 4 to 3; which is that of a Fourth or Dia-tessaron. Many attempts were made to this purpose: And, according to those, they gave Names to the different Genera or Kinds of Musick, (the Diatonick, Chromatick, and Enarmonick Kinds,) with the several Species, or lesser Distinctions under those Generals. All which to enumerate, would be too large, and not necessary to our business.8 The first was that of Euclide (which did most generally obtain for many ages:) Which allows to Fa sol, and to Sol la, the full proportion of 9 to 8; And therefore to Fa sol la (which we call the greater Third,) that of 81 to 64. (For 9 /8 × 9/8 = 81/64.) And, consequently, to that of Mi fa (which is the Remainder to a Fourth) that of 256 to 243. For 81/64 ) 3/4 ( 256/243; that is, if out of the proportion of 4 to 3, we take that of 81 to 64, the Result is that of 256 to 243. To this they gave the name of Limma (λεῖμμα) [p. 252] that is, the Remainder (to wit, over and above Two Tones.) But, in common discourse (when we do not pretend to speak nicely, nor intend to be so understood) it is usual to call it an Hemitone or Half Original has de.

a

John Wallis: Writings on Music

216

Note (as being very near it) and, the other, two Whole-Notes. And this is what Ptolemy calls Diatonum Ditonum, (of the Diatonick kind with Two full Tones.) Against this, it is objected (as not the most convenient Division,) that the Numbers of 81 to 64, are too great for that of a Ditone or Greater Third: Which is not Harsh to the Ear; but is rather Sweeter than that of a single Tone, who’s proportion is 9 to 8. And in that of 256 to 243, the Numbers are yet much greater. Whereas there are many proportions (as 5/4, 6/5, 7/6, 8/7,) in smaller numbers than that of 9 to 8; of which, in this division, there is no notice taken.9 To rectify this, there is another Division thought more convenient; which is Ptolemy’s Diatonum Intensum (of the Dia-tonick Kind, more Intense or Acute than that other.)a Which, instead of Two Full tones for Fa sol la; assignes (what we now call) a Greater and a Lesser Tone; (which, by the more nice Musicians of this and the last Age, seems to be more embraced;)10 Assigning to Fa sol, that of 9 to 8 (which they call the Greater Tone:) and to Sol la, that of 10 to 9, (which they call the Lesser Tone:) And therefore to Fa la (the Ditone or Greater Third) that of 5 to 4. (For 10/9 × 9/8 = 10/8 = 5/4.) And consequently, to Mi fa (which is remaining of the Fourth) that of 16 to 15. For 5/4 ) 4/3 ( 16/15. That is; if out of that of 4 to 3, we take that of 5 to 4, there remains that of 16 to 15. Many other waies there are (with which I shall not trouble you at present) of dividing the Fourth or Dia-tessaron, or the proportion of 4 to 3, into three parts, answering to what (in a looser way of Expression) we call an Half-note, and two Whole-notes. But this of 16/15 × 9/8 × 10/9 = 4/3, is that which is now received as the most proper. To which therefore I shall apply my discourse. Where 16/15 is (what we call) the Hemitone or Half-note, in Mi fa; 9/8 that of the Greater-Tone, in Fa sol; and 10/9 the Lesser-Tone, in Sol la. Onely with this addition; That each of those Tones, is (upon occasion) by Flats and Sharps (as we now speak) divided into two Hemitones or Half-notes: Which answers to what by the Greeks was called Mutatio quoad Modos (the change of Mood;) and what is now done by removing Mi to another Key. Namely 9/8 = 18/16 = 18 /17 × 17/16; and 10/9 = 20/18 = 20/19 × 19/18.11 [p. 253] Thus, by the help of Flats and Sharps (dividing each Whole-note, be it the Greater or the Lesser, into two Half-notes, or what we call so,) the whole Octave is divided into Twelve Parts or Intervals (contained between Thirteen Pipes) which are commonly called Hemitones or Half-notes. Not, that each is precisely Half a Note, but somewhat near it, and so called. And I say, by Flats and Sharps; For sometime the one, sometime the other, is used. As, for instance, a Flat in D, or a Sharp in C, do either of them denote a Midling Sound (tho’ not precisely in the Midst) between C and D; Sharper than C, and Flatter than D. Accordingly; supposing Mi to stand in B fa b mi (which is accounted its Natural seat) the Sounds of each Pipe are to bear these proportions to each other, viz.b  Original has other..)  The lengths of some braces have been adjusted in this diagram to make Wallis’s intentions clearer. a

b

Letter to Samuel Pepys, June 1698

217

And so in each Octave successively following. And if the Pipes in each Octave be fitted to sounds in these proportions of Gravity & Acuteness; it will be supposed (according to this Hypothesis) to be perfectly proportioned. But, instead of these successive proportions for each Hemi-tone; it is found necessary (if I do not mistake the practise) so to order the 13 Pipes (containing 12 Intervals which they call Hemitones) as that their Sounds (as to Gravity & Acuteness) be in Continual Proportion, (each to its next following, in one and the same Proportion;)12 which, all together, shall compleat that of an Octave or Dia-pason, as 2 to 1. Whereby it comes to pass, that each Pipe doth not express its proper Sound, but very near it, yet somewhat varying from it; Which they call Bearing.13 Which is somewhat of Imperfection in this Noble Instrument, the Top of all. [p. 254] It may be asked, Why may not the Pipes be so ordered, as to have their Sounds in just Proportion, as well as thus Bearing?14 I answer, It might very well be so, if all Musick were Composed to the same Key, or (as the Greeks call it) the same Mode. As, for instance, if, in all Compositions, Mi were alwaies placed in B fa b mi. For then the Pipes might be ordered in such proportions as I have now designed. But Musical Compositions are made in great variety of Modes, or with great diversity in the Pitch. Mi is not always placed in B fa b mi; but sometimes in E la mi; sometimes in A la mi re, &c. And (in summe) there is none of these 12 or 13 Pipes but may be made the Seat of Mi.15 And if they were exactly fitted to any one of these cases, they would be quite out of order for all the rest. As, for instance; If Mi be removed from B fa b mi (by a Flat in B) to E la mi: Instead of the Proportions but now designed, they must be thus ordered;16

218

John Wallis: Writings on Music

Where ’tis manifest, that the removal of mi doth quite disorder the whole series of Proportions. And the same would again happen, if mi be removed from E to A (by another Flat in E.) And again if removed from A to D. And so perpetually. But the Hemitones being made all Equal; they do indifferently answer all the positions of Mi (though not exactly to any:) Yet nearer to some than to others. Whence it is, that the same Tune sounds better at one Key than at another.17 It is asked, Whether this may not be remedied; by interposing more Pipes; and thereby dividing a Note, not only (as now) into Half-notes, but into Quarter-notes or Half-quarter-notes, &c. I answer; It may be thus remedied in part; (that is, the Imperfection might thus be somewhat Less, and the Sounds somewhat nearer to the just Proportions:) but it can never be exactly true, so long as their Sounds (be they never so [p. 255] many) be in continual proportion; that is, each to the next subsequent in the same Proportion. For it hath been long since Demonstrated, that there is no such thing as a just Hemitone practicable in Musick, (and the like for the division of a Tone into any number of Equal parts; three, four, or more.) For, supposing the Proportion of a Tone or Full-note, to be 9/8 (or, as 9 to 8;) that of the Half-note must be √9 to √8 (as the Square-root of 9 to the Square-root of 8; that is, as 3 to √8, or 3 to 2√2,) which are Incommensurable quantities. And that of a Quarter-note, as √qq9 to √qq8, (as the Biquadrate root of 9, to the Biquadrate root of 8,) which is yet more Incommensurate. And the like for any other number of Equal parts. Which will therefore never fall-in with the Proportions of Number to Number.18 So that this can never be perfectly adjusted for all Keys (without somewhat of Bearing) by multiplying of Pipes; unless we would for every Key (or every different Seat of Mi) have a different Set of Pipes, of which this or that is to be used, according as (in the Composition) Mi is supposed to stand in this or that Seat. Which vast number of Pipes (for every Octave) would vastly increase the Charge. And (when all is done) make the whole impracticable.19 These are my present thoughts, of the Question proposed to me, and upon these grounds. You will please to excuse me for the trouble I give you of so long a Letter. I thought it necessary, to give a little intimation of the Ancient Greek Musick compared with what is now in practise; which is more the same than most men are aware of: though the Language be very different. But I was not to be large in it. Those who desire to know more of it; may see my thoughts more at large, in that Appendix which I have added at the end of my Edition of Ptolemy’s Harmonicks in Greek and Latin. The two Eminent Sects amongst them, the Aristoxenian and the Pythagorian, differ much at the same rate as doth the Language of our ordinary practical Musicians, and that of those who treat of it in a more Speculative way. Our Practical Musicians talk of Notes and Half-notes, just as the Aristoxenians did; as if the Whole Notes were all Equal; and the Half-notes likewise each the just Half of a Whole Note. [p. 256] And thus it is necessary to suppose in the Pipes

Letter to Samuel Pepys, June 1698

219

of an Organ; which have each their determinate Sound and not to be corrected, in their little Inequalities, as the Voice may be by the guidance of the Ear. But Pythagoras, and those who follow him found (by the Ear) that this Equality of Intervals would not exactly answer the Musical Appearances, in Concords and Discords: just as our Organists and Organ-makers be now aware; that their Pipes at equal Intervals do not give the just desired Harmony, without somewhat of Bearing, that is, of some little variation from the just Sound. The Pythagorians, to help this, changed the notion of Equal Intervals into that of due Proportions. And this is followed by Zarline, Keppler, Cartes, and others who treat of Speculative Musick in this and the last Age.20 And though they speak of Notes and Half-notes (in a more gross way) much as others do, yet declare themselves to be understood more nicely. And though our present Gam-ut takes no notice of this little diversity; yet, in Vocal Musick, the Ear directs the Voice to a more just proportion. And, in String Musick, it may in like manner be helped by straining and slackening the Strings, or moving the Frets. But, in Wind Musick, the Pipes are not capable of such correction; and therefore we must be content with some little irregularity therein; that so they may tolerably answer (though not exactly) the different Compositions according to the different placing of Mi in the Gam-ut. Now the Design of Mr. Harris seems to be this; either (by multiplying intermediate Pipes) to bring the Organ to a just Perfection: Or else (if that cannot be done) to rest content with the little Imperfection that is; which though, by more Pipes, it may be somewhat abated, yet cannot be perfectly remedied. And in this I think we must acquiesce. I am SIR Yours to serve you John Wallis, Oxford June 27. 1698. Notes 1 See Introduction, pp. 22–3 above. 2 The letter is also printed in Samuel Pepys, Private Correspondence and Miscellaneous Papers, ed. J.R. Tanner (London, 1926), vol. 1, p. 155. 3 Wallis’s use of the word ‘coincidence’ here is his only glance in these publications of 1698 at the ‘coincidence’ theory of consonance: see Wallis to Oldenburg, 14 May 1664, fol 2 (pp. 48–9 above); Wallis, ‘The Harmonics of the Ancients’, p. 177 (pp. 132–3 above); Introduction, pp. 3–4 above. 4 With this very brief account of the Aristoxenian and Pythagorean positions compare Wallis, ‘The Harmonics of the Ancients’, pp. 158–66 (pp. 90–109 above). 5 The table in ‘A Question in Musick’ (p. 209 above) describes the same scale (of white

220

John Wallis: Writings on Music

notes on E) and is of some help in following Wallis’s quite compressed exposition. 6 ‘Common measure’ is a mathematical concept: X is a common measure for Y and Z if both Y and Z can be expressed as whole-number multiples of X: see Euclid, Elements 10, def. 1. (Strictly it is the half-tone which functions as a common measure in what follows). The idea that the tone could be used to measure the other intervals evokes the idea of intervals as defined by the subdivision of the octave into equal parts: cf Wallis, ‘The Harmonics of the Ancients’, p. 159 (p. 91 above); Wardhaugh, Music, Experiment and Mathematics, pp. 34–43; Barker, The Science of Harmonics, pp. 23–30. 7 Cf. Wallis to Oldenburg, 14 May 1664, §6 (pp. 52–3 above); Wallis, ‘The Harmonics of the Ancients’, p. 178 (pp. 134–5 above). 8 See Wallis, ‘The Harmonics of the Ancients, pp. 163–4 (pp. 105–6 above). 9 See Wallis to Oldenburg, 14 May 1664, §§6, 14 (p. 53, pp. 56–7 above); Wallis, ‘The Harmonics of the Ancients’, p. 180 (pp. 139–40 above); id. ‘A Question in Musick’, p. 84 (p. 210 above). 10 Our only evidence for the source of Wallis’s belief that ‘the more nice Musicians’ had ‘embraced’ major and minor tones is a letter to him from Thomas Salmon in 1684, reporting that the bass viol player Jacques Paisible adopted them in his performances: see Wardhaugh, Thomas Salmon, vol. 2, pp. 25–6, 46. 11 On this subdivision of the tones, see Wallis, ‘A Question in Musick’, pp. 83–4 (p. 210 above); Introduction, pp. 20–21 above; on changes of mode, see Wallis, ‘The Harmonics of the Ancients’, p. 158 (p. 89 above); compare the account of the moving of mi in Wallis to Oldenburg, 14 May 1664, §17 (p. 59 above). 12 By ‘Continual Proportion’ Wallis meant that the ratio between pairs of adjacent pipes should always be the same, in other words that the instrument should be equal-tempered. 13 The term ‘bearing’ seems to have been used by instrument tuners to mean any slight detuning from a pure interval: see Wardhaugh, Thomas Salmon, vol. 2, p. 77. 14 ‘as well as’ seems to be a mistake for ‘instead of’. 15 Cf. Wallis, ‘The Harmonics of the Ancients’, p. 174: ‘with the modes just established the note mi appeared on every string’ (p. 127 above), which may be the source of what otherwise seems the rather remarkable claim that mi could be placed in any of the octave’s twelve pitches. Compare Birchensha’s list of signatures with up to seven flats in Field and Wardhaugh, John Birchensha, p. 104; see also Christopher Simpson, Chelys / The Division-Violist (2nd edn; London, 1667), p. 16, where only eight keynotes were listed; Christopher D.S. Field, ‘Jenkins and the Cosmography of Harmony’, in Andrew Ashbee and Peter Holman (eds), John Jenkins and his Time (Oxford, 1996), pp. 1–74. 16 In the following diagram the letter B denotes modern B flat rather than modern B natural as in the previous diagram. 17 Wallis seems to have nodded here. The use of equal temperament has the opposite of the effect described: it does not make a tune sound better in one key than other; rather, it makes it sound exactly the same in every key. The fact that ‘the same Tune sounds better at one Key than at another’ is therefore evidence that equal temperament was not (always) being used in practice. 18 Cf. Wallis to Oldenburg, 14 May 1664, §7 (p. 53 above); Wallis, ‘A Question in Musick’, p. 82 (p. 208 above); only the division of the tone into two equal parts had been discussed before, whereas here Wallis considered its division into any number of equal parts.

Letter to Samuel Pepys, June 1698

221

‘Incommensurable’ is a mathematical term indicating that a pair of quantities has no common measure (see p. 54 n. 29 above) and therefore that their relationship cannot be expressed as a ratio of whole numbers. It is not a matter of degree, but in Euclid, Elements 10 it was extended to provide such concepts as ‘commensurable in square’ for quantities whose squares are commensurable: it was perhaps of this that Wallis was thinking when he wrote ‘yet more Incommensurate’. Wallis’s notation √qq and his term ‘Biquadrate root’ denote a fourth root. 19 Wallis was perhaps thinking here of Thomas Salmon’s 1688 proposals for the tuning of viols, in which for each key a different fingerboard was supplied, to be attached to the instrument as required. Wallis had suggested in his ‘Remarks on the Proposal to perform musick, in perfect and mathematical proportions’ printed with that book that a more practicable scheme would instead use movable frets: Salmon, A Proposal to Perform Musick, in Perfect and Mathematical Proportions, p. 41. 20 This is one of the longest lists of modern musical sources Wallis ever provided: see also Introduction, pp. 5–7 above. As in those earlier texts the works to which Wallis referred were Gioseffo Zarlino’s Istitutioni harmoniche, the Compendium musicæ (Utrecht, 1650) of René Descartes, and Johannes Kepler’s Harmonices mundi (Linz, 1619). All of these works did indeed espouse the just intonation; the absence of the names of those English authors who had done so – Francis North, Thomas Salmon, William Holder – may have been deliberate.

Chapter 7

Letters to Andrew Fletcher, August 1698 Editorial Note During August 1698 Wallis wrote two letters – the second a short postscript to the first – to ‘Mr. Andrew Fletcher’ on musical subjects. As suggested in the Introduction, this was probably the political theorist Andrew Fletcher of Saltoun.1 They are his last known writing of any size on the theory of music. These letters differ from all of Wallis’s other published pieces on music in dealing not with the technical aspects of tuning nor with the mechanical effects involved in the production of musical sound, but with music’s ethical effects. They should perhaps be compared with the references to the effectiveness and the nature of music in contemporary sermons2 and with the brief account of the same subject in ‘The Harmonics of the Ancients compared with Today’s’ (Chapter 3).3 Wallis’s original letters to Fletcher are not known to survive: both are preserved in copies Wallis sent to Hans Sloane, secretary of the Royal Society, intending that they should be printed in the Philosophical Transactions (London, Royal Society, MS Early Letters W2, nos 77, 78). In fact only the first was printed (Philosophical Transactions 20/243 (August 1698): 297–303), and the compositor made numerous emendations of spelling and punctuation, the latter occasionally impairing the sense. He also occasionally misread Wallis’s difficult handwriting and on one occasion omitted a line of text. The present edition uses Wallis’s manuscripts as its basis: a few readings from the Philosophical Transactions are reported in the footnotes with the siglum PT; W denotes the manuscript reading. Three ‘errata’ Wallis conveyed to Sloane on 21 September (London, Royal Society, MS Early Letters W2, no. 79) are also reported in the footnotes.

John Wallis: Writings on Music

224

Text [London, Royal Society, MS Early Letters W2, no. 77] [fol. 1r]a A Letter of Dr John Wallis to Mr Andrew Fletcher, Concerning the Strange Effects Reported of Musick in Former Times,b ‹beyond what is› to be found in Later Ages. Oxford Aug. 18. 1698.

Sir

The Question you lately proposed to me (by a Friend4 of yours) concerning Musick; was not, Whence it comes to pass, that Musick hath so great an Influence or Efficacy on our Affections, Passions, Motions, &c: But, whence it is that these great Effects which are Reported of Musick in Former Times, (of Orpheus, Amphion, &c) are not as well found to follow upon the Musick of Later Ages.5 If that First had been the Question; Whence it is that Musick operates on our Fancies, Affections, Passions, Motions, &c; And, not Ours onely, but of other Animals, (For it is manifest, that Birds, & Beasts are affected with Musical ‹Notes›c, as well as Men:) And even asd things Inanimate; (For ’tis well known, that, of Two Unisone Strings, though at some distance, if One be struck, the Other will move:)6 I say; If this were the Question; I must, in Answere to it, have discoursed of the Nature of Sounds; produced by some Subtile Motions in the Air; propagated & continued to the Ear, & Organs of Hearing; and thence communicated to the Animal Spirits; which excite sutable Imaginations, Affections, Passions, &c; & these attended with conformable Motions & Actions, And, according to the various Proportions, Measures, & Mixtures of such Sounds, there do arise various Effects in the Mind or Imagination, sutable thereunto.7 Thus; the Rough Musick of Drums and Trumpets, is apt to produce Courage & Fie[r]ceness in Martial Minds: and, more or less, according to the Degrees of Roughness. And sweeter Sounds of more sedate Musick, aree apt to excite softer Passions; &, of different Kinds & Degrees, according to the Slowness or Swiftness, Loudness or Calmness, Acuteness or Gravity, & the various Measures & Mixtures of such Musical Sounds. The Animal Spirits being apt to receive Impressions answerable to those subtile Motions, communicated to them from the Organs of Hearing. But the Question you move, is onely of the Comparative Effects of Musick, ‹Reported to have been› in the days of Old; beyond what appears upon that of Later Ages.  Endorsement: Read Aug: 31: 1698 / Entd. LB. 7. 227. / Phil. Trans. 243.  Deleted: which are not c  Deleted: sounds d  Corrected to on in the ‘errata’ of 21 September. e  Deleted: apte[r] a

b

Letters to Andrew Fletcher, August 1698

225

In Answere to which, there are many things to be considered. 1. I take it for granted, That much of those Reports, is highly Hyperbolical, & next door to Fabulous: According to the Humour of those Ages, termed by Historians, Tempus Mythicum, (the Fabulous Age.) For (whatever may be thought of Men, Beasts, & Birds.) no man can think that the Trees & Stones did Dance after their Pipe.8 And, even in more modest times, the Poetical Stories of Olympus, Atlas, and other Mountains, reaching up to Heaven; are much beyond what is now to be found in those parts, where they are sayd to have been; and many Mountains now well known (as the Alps, the Apennines, the Pika of Tenariff,) are much higher than their Atlas or Olympus. And their Famed Tyber, is but a Ditch, compared with our Thames. And like Abatements we must allow to the Hyperbolical Elogies of their Musick. 2. We must consider, That Musick (to any tolerable Degree) was then (if not a New, at lest) a Rare Thing; which the Rusticks, on whom it is reported to have ‹had› such Effects; had never heard before. And on such, a little musick [fol. 1v] will do great Feats. As we find, at this day; a Fidd[l]e or a Bag-pipe, among a company of Countrey Fellows & Wenches (who never knew better,) or at a Country Morricsb Dance; will make them skip & shake their Heeles notably.9 (And the like, heretofore, to a Shep-herds Reed or Oaten-Pipe.) And, when some such thing ‹happened› amongst those Rusticks of Old; That, with somewhat of Hyperbole, would make a great Noise. 3. We are to consider, that their Musick (even after it came to some good degree of Perfection) was much more Plain & Simple than Ours now a-days. They had not Consorts of Two, Three, Four, or more Parts or Voices: but one single Voice, or single Instrument, a-part;10 which, to a Rude Ear, is much more taking, than more Compounded Musick. And we find that a simple Jig, sung, or playd on a Fiddle or Bag-pipe, doth more Affect a company of Rusticks, than a sett of Violsc & Voices. For, That, is at a pitch not above their Capacity; whereas, This other, confounds it, with a great Noise, but nothing Distingabled to their capacity. Like some delicate Sawce, made-up of a mixture of many Ingredients; which may yield an agreeable Tast; but not so as to distinguish the Particular Relish of any one: But Hony or Sugar by itself, they could Understand & Relish with ae ‹more particular› Gusto. 4. We are to consider, That Musick, with the Ancients, was of a larger extent than what We call Musick now a-days. For Poetry and Dancing (or Comely Motion,) were then accounted parts of Musick, when Musick arrived to some perfection.11 Now we know that Verse of it self, if in good Measures, & affectionate Language, & this sett to a Musical Tune, & sung by a decent Voice, & accompanyed but with Soft Musick (instrumental) if any; such as not to Drown or   PT: Pike   PT: Morrice c   PT, W: Vials. d   PT: Distinguishable e  Deleted: better a

b

John Wallis: Writings on Music

226

Obscure the Emphatick Expressions, (like what we call Recitative Musick;) will work strangely upon the Ear, & move Affections sutable to ‹the› Tune & Ditty; (whether Brisk & Pleasant, or Soft & Pityfull, or Fierce & Angry, or Moderate & Sedate:) especially if attended with a Gesture & Action sutable. (For ’tis well known, that sutable Acting on a Stage, gives great Life to the Words.) Now all this together (which were all Ingredients in what they called Musick) must needs operate strongly on the Fansies & Affections of Ordinary People, unacquainted with such kind of Treatments. For, if the deliberate Reading of a Romance (when well penned) will produce Mirth, Tears, Joy, Grief, Pitty, Wrath, or Indignation, sutable to the respective Intents of it: much more would it so do, if accompanied with all those Attendants. 5. You will ask perhaps, Why may not all this be Now done, as well as Then?a I answere, No doubt it may; and, with like Effect. If an Address be made, in Proper Words, with Moving Arguments, in just Measures (Poetical or Rhetorical) with the Emphatick Wordsb sett in Signal places, pronounced with a Good Voice, & a True Accent, and attended with a Decent Gesture; and all these Sutably adjusted to the Passion, Affection, or Temper of Mind, particulary Designed to be produced, (be it Joy, Love, Grief, Pitty, Courage, or Indignation;) will certainly Now, as well as Then, produce great Effects upon the Mind. Especially, upon a Surprize, & where Persons are not otherwise Preingaged: And, if so managed as that you be (or seem to be) in Ernest; and, if not over-acted by apparent Affectation. 6. We are to consider, That the Usual Design, of what we now call Musick; is very different from that of thec Ancients. What we Now call Musick, is but what they called Harmonick; which was but one Part of their Musick, (consisting of Words, Verse, Voice, Tune, Instruments, [fol. 2r] and Acting;)12 and we are not to expect the same Effect of one Piece, as of the Whole. And, of their Harmonick at first, when we are told (by a great Hyperbole) that it did draw after it, notd Men onely, bute Birds, Beasts, Trees and Stones: This is no more (bating the Hyperbole,) but what we now see dayly in a Countrey-town; when Boys, & Girls, & Countrey-folk, run after a Bag-pipe or a Fidler, (especially, if they had never seen the like before;) of which we are apt (even now) to say, All the Town runs after the Fidler; or, The Fidler draws All the Town after him. Or, as when they flock about a Ballad-singer in a Fair; or, the Morrice-dancers at a Whitsund-Ale.13 And all their Hyp[erb]ole’s can signify no more but this; when their Musick was but a Reed, or an Open-pipe. 7. It’s true, that, when Musick was arrived to greater perfection, it was then applyed to particular Designs, of Exciting this or that particular Affection, Passion, or Temper of Mind; (as Courage, to Souldier[s] in the field; Love, in an Amorous   PT: that, corrected to then in the errata of 21 September.  W: Words Words; PT: Words, Words. Corrected to Words in the errata of 21 September. c  Deleted: Ancien[t] d  Deleted: onely e  W: by a

b

Letters to Andrew Fletcher, August 1698

227

Address; Tears & Pitty, in a Dolefull Ditty; Fury & Indignation, in a Fiercer Tune; and a Sedate Temper, when applyed to Compose or Pacify a Furious Quarel;) the Tunes & Measures being sutably Adapted to such Designes. 8. But such Designes as these, seem allmost quite Neglected in our present Musick. The chief Designe now, in our most Accomplished Musick, being, to please the Ear. When, by a sweet Mixture of different Parts & Voices, with just Cadences & Concords intermixed, a Gratefull Sound is produced, to Please the Ear; (as a Cooks well-tempered Sawce, doth the Palate;) which, to a Common Ear, is onely a Confused Noise of they know not what, (though somewhat Pleasing;) while onely the judicious Musician can Discern & Distinguish the just Proporti[ons]a (of Time & Tune) which makeb up this Compoundc Noise.d 9. ’Tis true, that, even this Compound Musick, admits of different Chara[cters;]e some is more Brisk & Airy; others, more Sedate & Grave; others, more Languide; as the different Subjects do require. But that which is most proper tof Excite particular Passions or Dispositions, is such as is more Simple & Uncompounded: Such as a Nurses languid Tune, lulling her Babe to sleep; or g a Continued Reading, in an Even Tone; or even the Soft Murmure of a little Rivulet, running upon Gravel or Pibbles, inducing a Quiet Repose of the Spirits: And, contrarywise, the Briskness of a Jig, on a Kit14 or Violine, exciting to Dance. Which are more Operative to such Particular Ends, than an Elaborate Composition of Full Musick. Which two, Differ as much, as that of a Cooks mixing a Sawce to make it Palatable; and, that of a Physician, mixing a Potion, for Curingh a particular Distemper, or Procuring a just Habit of Body (where yet, a little Sugar to Sweeten it, may not do amiss.) 10. To conclude then; If we aim onely at Pleasing the Eare, by a Sweet Consort; I doubt not but our Modern Compositions, may Equal, if not Exceed, those of the Ancients. Amongst whom I do not find any footsteps of what we call several Parts or Voices, (as Bass, Treble, Mean, &c sung in Consort.) answering each other to Compleat the Musick. But if we would have our Musick so adjusted, as to Excite particular Passions, Affections, or Temper of Mind, (as that of the Ancients is supposed to have done;) We must then imitate the Physician, rather than the Cook; and apply more Simple Ingredients, fitted to the Temper we would produce. For, in the Sweet Mixture of Compounded Musick, One thing doth so Correct Another, that it doth not Operate strongly any one way. And this I doubt ‹not› but a Judicious Composer may so Effect, that (with the help of such Hyperbole’s, as a

  Paper damaged.  Altered from makes c  Deleted: Musick d   PT lacks (of Time … Compound Noise. e   Paper damaged. f  Deleted: produce. g  Deleted: even h  Altered from Cure b

John Wallis: Writings on Music

228

those with which the Ancient Musick is wont to be sett-off) Our Musick may be sayd to do as great Feats, as any of theirs. a

I am, Sir, Your very humble Servant, John Wallis. [fol. 2v]b These For Dr Sloan, Secretary to the Royal Society, at Gresham-College London. [London, Royal Society Early Letters, W2, No. 78.] [fol. 1r]c Oxford Sept. 5. 1698.

Sir

I sent you lately the Copy of a Letter of mine to Mr Andrew Fletcher, relating to Musick. If you think it worth printing, you may adde to it this Post-Scrip. Post-script. Aug. 27. 1698.

Sir

If I did mis-take your Directions by D.G.15 and have over-done what was desired; pray excuse me that fault. If (as you now intimate) you intended particularly; How, what the Ancients called Musick, (under which you allow to be comprehended Poetry, and (d ‹Dancing, that is,› comely Gesturee & decent Actionsf,) might be of such good use in Education (as Aristotle in his Politicks, & Plato in some of his Works, so intimate;) influencing men to Virtue,g and directing their Morals & Conversation: I think it is sufficiently answered at numb. 5. 7.  Deleted: These are  Endorsement: A Letter from Dr. Wallis / concerning the strang Effects / of Musick in former times – / August 18th. 98. c  Endorsement: Entd LB. 11.2.303. d  Deleted: Dancing, or) e  Altered from Gestures f  Altered from Acting g  Deleted: and a

b

Letters to Andrew Fletcher, August 1698

229

10. For, if such an Address as is there mentioned at numb. 4. 5. be so managed as that the Discourse be Grave & sober; exciting to Virtue & discouraging Vice; and so Attended as is there intimated: No doubt but such Addresses would be strongly Operative that way. (Their Design therein being much the same with that of our Sermons & Religious Exhortations.) But if, in stead thereof, Lewdness & Immoralities be Represented on the Stage; and sett-off with all the Advantages & Incentives thereunto: it is not strange if the Effect be quite Contrary to that other. But the fault is not in the nature of Paranetick ‹or› Recitative Musick: but in the Mis-application of it to Contrary Purposes. Which I thought fit to adde, in answere to yours of Aug. 25. And am, Sir, Your very humble servant, John Wallis. [fol. 1v]a These For Dr Sloane, Secretary to the Royal Society, at Gresham College, London. Notes 1 See pp. 24–5 above. 2 See Introduction, p. 28 above. 3 See Wallis, ‘The Harmonics of the Ancients’, p. 175 (p. 130 above). 4 The only hint as to the identity of this ‘Friend’ is the ‘D.G.’ of the subsequent letter: see n. 15 below. 5 Orpheus charmed trees and animals (Metamorphoses 10.86–109, 144–5); Amphion the stones (see OCD, s.v. Amphion). 6 See Wallis to Oldenburg, 14 March 1677 (p. 70, n. 6 above). 7 Cf. Wallis to Oldenburg, 27 March 1677 (p. 73, n. 10 above); Wallis, ‘The Harmonics of the Ancients’, pp. 166–7 (pp. 110–11 above), where slightly more informative hints had been given about Wallis’s theory of sound. Since they were written, however, Newton had published his mathematical account of the sound wave in the Principia (Philosophiæ naturalis principia mathematica (London, 1687) bk 2, § 8), and Wallis’s views could well have changed. 8 See n. 5 above. 9 Compare Kepler’s assertion that ‘a peasant does not reason what geometric ratio one voice bears to another voice. And yet that external harmony of chords flows through the ears of the rustic into his mind and cheers the man’: J. Bruce Brackenridge and Mary Ann Rossi, ‘Johannes Kepler’s On the More Certain Fundamentals of Astrology.

 Endorsement: Dr. Wallis To Dr. Sloane / Concerning his musicall / observations & Instructions, / Oxford Sept. 5. 1698. / No. 59. a

230

John Wallis: Writings on Music

Prague 1601’, Proceedings of the American Philosophical Society 123 (1979): 85– 116, at p. 139. 10 See Wallis, ‘The Harmonics of the Ancients’, p. 175–6 (pp. 129–30 above). 11 Ibid., p. 153 (p. 77 above), where the parts of music are variously listed. 12 See n. 10 above. 13 Whitsund Ale: ‘a parish festival formerly held at Whitsuntide, marked by feasting, sports, and merry-making’ (OED, s.v. ‘Whitsun’, §1; see also under ‘Ale’, §3). 14 A kit was ‘a small fiddle, formerly much used by dancing masters’ (OED, s.v. ‘Kit, n.2’). 15 ‘D.G.’ cannot be identified with any confidence; one candidate is David Gregory (1659–1708), Savilian Professor of Astronomy.

Select Bibliography Adkins, C., and A. Dickinson, A Trumpet by any Other Name: a history of the trumpet marine (Buren: F. Knuf, 1991). Alsted, Johann Heinrich, Templum Musicum; or the Musical Synopsis of the Learned and Famous Johannes–Heinricus–Alstedius, being a Compendium of the Rudiments both of the Mathematical and Practical part of Musick, of which Subject not any Book is extant in our English Tongue, trans. John Birchensha (London: William Godbid for Peter Dring, 1664). Barbour, J. Murray, Tuning and Temperament: a historical survey (East Lansing: Michigan State College Press, 1951). Barker, Andrew (ed.), Greek Musical Writings II: harmonic and acoustic theory (Cambridge: Cambridge University Press, 1989). —— The Science of Harmonics in Classical Greece (Cambridge: Cambridge University Press, 2007). —— Scientific Method in Ptolemy’s ‘Harmonics’ (Cambridge: Cambridge University Press, 2000). —— ‘Words for Sounds’, in C.J. Tuplin and T.E. Rihll (eds), Science and Mathematics in Ancient Greek Culture (Oxford: Oxford University Press, 2002), pp. 22–35. Beeley, Philip, and Christoph J. Scriba (eds), The Correspondence of John Wallis (3 vols to date; Oxford: Oxford University Press, 2003–). —— and Siegmund Probst, ‘John Wallis (1616–1703): Mathematician and Divine’, in Luc Bergmans and Teun Koetsier (eds), Mathematics and the Divine: a historical study (Amsterdam: Elsevier, 2005), pp. 441–57. Benedetti, Giovanni Battista, Diversarum speculationum … liber (Turin: apud hæredem N. Beuilaque, 1585). Birch, Thomas, A History of the Royal Society of London (4 vols; London: A. Millar, 1756–7). Burney, Charles, A General History of Music, from the earliest ages to the present period (4 vols; London: for the author, 1776–89). Charleton, Walter, Physiologia Epicuro–Gassendo–Charletoniana: Or a Fabrick of Science Natural, Upon the Hypothesis of Atoms (London: Thomas Newcomb for Thomas Heath, 1654). Chilmead, Edmund, ‘De musica antiquâ Græcâ’, in Αρατου Σολεως Φαινόμηνα καὶ Διοσημεῖα. Θεωνος Σχόλια. … Accesserunt Annotationes in Eratosthenem et Hymnos Dionysii (Oxford: E Theatro Sheldoniano, 1672). Cohen, H. Floris, Quantifying Music: the science of music at the first stage of the scientific revolution, 1580–1650 (Dordrecht: D. Reidel, 1984).

232

John Wallis: Writings on Music

Cram, David, ‘The Changing Relations between Grammar, Rhetoric and Music in the Early Modern Period’, in Rens Bod et al. (eds), The Making of the Humanities (Amsterdam: Amsterdam University Press, 2010), pp. 263–82. —— ‘John Wallis’s English Grammar (1653): breaking the Latin mould’, Beiträge zur Geschichte der Sprachwissenschaft 19 (2009): 177–201. Descartes, René, Musicae compendium (Utrecht: Gisbertus à Zÿll and Theodorus ab Ackersdÿck, 1650). —— Renatus Des-Cartes Excellent Compendium of Musick and Animadversions of the Author, ed. and trans. anon. [ed. William Brouncker, trans. Walter Charleton] (London: Thomas Harper for Humphrey Moseley, 1653). Drake, Stillman (ed.), Discoveries and Opinions of Galileo (New York: Doubleday, 1957). Early English Books Online: . Early Modern Letters Online: . Eighteenth-Century Collections Online: . Feingold, Mordechai, The Mathematician’s Apprenticeship (Cambridge: Cambridge University Press, 1984). —— and Penelope M. Gouk, ‘An Early Critique of Bacon’s Sylva Sylvarum: Edmund Chilmead’s “Treatise on Sound”’, Annals of Science 40 (1983): 139–57. Fétis, François-Joseph, Biographie universelle des musiciens et bibliographie générale de la musique (2nd edn; 8 vols; Paris: Librairie de Firmin Didot Frères, Fils et Cie, 1866–8). Field, Christopher D.S., ‘Jenkins and the Cosmography of Harmony’, in Andrew Ashbee and Peter Holman (eds), John Jenkins and his Time (Oxford: Clarendon Press, 1996), pp. 1–74. —— and Benjamin Wardhaugh (eds), John Birchensha: writings on music (Farnham: Ashgate, 2010). Galilei, Galileo, Discorsi e dimostrazioni matematiche intorno è due nuove scienze (Leiden: appresso gli Elsevirii, 1638). Gouk, Penelope M., ‘Music in the Natural Philosophy of the Early Royal Society’ (Ph.D. thesis, Imperial College London, 1982). —— Music, Science and Natural Magic in Seventeenth-Century England (New Haven and London: Yale University Press, 1999). —— ‘Some English Theories of Hearing in the Seventeenth Century: before and after Descartes’, in Charles F. Burnett, Michael Fend, and Penelope M. Gouk (eds), The Second Sense: studies in hearing and musical judgement from antiquity to the seventeenth century (London: Warburg Institute, 1991), pp. 95–113. —— ‘Speculative and Practical Music in Seventeenth-Century England: Oxford University as a Case Study’, International Musicological Society Congress Report XIV (Bologna: IMS, 1990), pp. 199–205. Hall, A. Rupert, and Marie Boas Hall (eds), The Correspondence of Henry Oldenburg (13 vols: Madison, Milwaukee and London: University of Wisconsin Press, 1965–86).

Select Bibliography

233

Hawkins, John, A General History of the Science and Practice of Music (5 vols; London: for T. Payne and Son, 1776). Hearne, Thomas (ed.), Peter Langtoft’s Chronicle (2 vols; Oxford: printed at the Theater, 1925). Holder, William, A Discourse concerning Time (London: for L. Meredith, 1694). Hornblower, Simon, and Antony Spawforth (eds), The Oxford Classical Dictionary (3rd edn; Oxford: Oxford University Press, 1996). Hunter, Michael (ed.), The Boyle Papers: understanding the manuscripts of Robert Boyle (Aldershot and Burlington, VT: Ashgate, 2007). —— The Royal Society and its Fellows 1660–1700: the morphology of an early scientific institution (Chalfont St Giles: British Society for the History of Science, 1982). Jeans, Susi, ‘Renatus Harris, Organ-Maker, his Challenge to Mr Bernard Smith, Organ-Maker’, The Organ 61 (1982): 129–30. Kassler, Jamie C., The Beginnings of the Modern Philosophy of Music in England: Francis North’s A Philosophical Essay of Musick (1677) with comments of Isaac Newton, Roger North and in the Philosophical Transactions (Aldershot: Ashgate, 2004). Kepler, Johannes, Harmonices mundi libri quinque (Linz: sumptibus Godofredi Tampachii, 1619). Liddell, H.G., and R. Scott, Greek–English Lexicon (revd edn; Oxford: Clarendon Press, 1996). Lindley, Mark, Lutes, Viols and Temperaments (Cambridge: Cambridge University Press, 1984). Lloyd, Ll. S., ‘Music Theory in the Early “Philosophical Transactions”’, Notes and Records of the Royal Society of London 3 (1941): 149–57. Lundberg, Mattias (ed.), Studies on Marcus Meibom (Copenhagen, forthcoming). Mathiesen, Thomas J., Apollo’s Lyre: Greek music and music theory in antiquity and the Middle Ages (Lincoln: University of Nebraska Press, 1999). —— Aristides Quintilianus: on music in three books, with translation, with introduction, commentary and annotations (New Haven and London: Yale University Press, 1983). Matthew, H.C.G., and Brian Harrison (eds), Oxford Dictionary of National Biography: from the earliest times to the year 2000 (60 vols; Oxford: Oxford University Press, 2004; online edn, 2008, at ). Meibom, Marcus, Antiquae musicae auctores septem (Amsterdam: apud Ludovicum Elzevirium, 1652). Mengoli, Pietro, Speculationi di musica (Bologna: per l’herede del Benacci, 1670). Mersenne, Marin, Harmonicorum libri (Paris: sumptibus Guillielmi Baudry, 1636) —— Harmonie universelle (Paris: Pierre Ballard, 1636). Miller, Leta, and Albert Cohen, Music in the Royal Society of London (Detroit: Information Coordinators, 1987). Morley, Thomas, A Plaine and Easie Introduction to Practicall Musicke (London: P. Short, 1597).

234

John Wallis: Writings on Music

Neale, Katherine, From Discrete to Continuous: the broadening of number concepts in early modern England (Dordrecht: Kluwer Academic, 2002). Newton, Isaac, Philosophiæ naturalis principia mathematica (London: jussu Societatis Regiae ac typis Josephi Streater, 1687). North, Francis, A philosophical essay of musick directed to a friend (London: John Martyn, 1677). Pepys, Samuel, Private Correspondence and Miscellaneous Papers, ed. J.R. Tanner (London: Bell, 1926). Pesic, Peter, ‘Hearing the Irrational: Music and the Development of the Modern Concept of Number’, Isis 101 (2010): 501–30. Playford, John, A breefe introduction to the skill of musick for song & violl (London: Jo. Playford, 1654). Plot, Robert, The Natural History of Oxford-Shire (Oxford: printed at the Theater in Oxford, 1677). Poole, H. Edmund, ‘The Printing of William Holder’s “Principles of Harmony”’, Proceedings of the Royal Musical Association 101 (1974): 31–43. Reeve, M., ‘John Wallis, Editor of Greek Mathematical Texts’, in G.W. Most (ed.), Aporemata: Kritische Studien zur Philologiegeschichte, vol. 2: Editing Texts / Texte edieren (Göttingen: Vandenhoeck & Ruprecht, 1998), pp. 77–93. Sadie, Stanley (ed.), The New Grove Dictionary of Music and Musicians (2nd edn, 29 vols; London: Grove, 2001; online edn, 2007–10, at ). Salmon, Thomas, An Essay to the Advancement of Musick, by casting away the perplexities of different cliffs. and uniting all sorts of musick … in one universal character (London: J. Macock, to be sold by John Carr, 1672). —— A Proposal to Perform Musick, in Perfect and Mathematical Proportions … (London: J. Lawrence, 1688). —— ‘The Theory of Musick reduced to Arithmetical and Geometrical Proportions’, Philosophical Transactions 24 (1705): 2072–7, 2069. —— A Vindication of an Essay to the Advancement of Musick (London: A. Maxwell, to be sold by John Carr, 1672). Scala, Gail Ewald, ‘An Index of Proper Names in Thomas Birch, “The History of the Royal Society” (London, 1756–1757)’, Notes and Records of the Royal Society 28 (1974): 263–329. Scott, J.F., The Mathematical Work of John Wallis, D.D., F.R.S. (1616–1703) (London: Taylor and Francis, 1938). Scriba, Christoph J., ‘The Autobiography of John Wallis, F.R.S.’, Notes and Records of the Royal Society 25 (1970): 17–46. Simpson, Christopher, Chelys / The Division-Violist (2nd edn; London: William Godbid for Henry Brome, 1667). —— A Compendium of Practical Musick (London: William Godbid for Henry Brome, 1667). Stanford, William Bedell, The Sound of Greek: studies in the Greek theory of euphony (Berkeley: University of California Press, 1967).

Select Bibliography

235

Stedall, Jacqueline, A Discourse Concerning Algebra: English algebra to 1685 (Oxford: Oxford University Press, 2002). —— ‘John Wallis and the French: his quarrels with Fermat, Pascal, Dulaurens, and Descartes’, Historia Mathematica 39 (2012): 265–79. Stephen, Leslie and Sidney Lee (eds), The Dictionary of National Biography (63 vols; London: Smith, Elder & Co., 1885–1900). Stiles, Francis Eyles, ‘An Explanation of the Modes or Tones in the antient Græcian Music’, Philosophical Transactions 51 (1760): 695–773. Statuta Universitatis Oxoniensis (Oxford: E Typographeo Academico, 1857). Summers, Montague (ed.), The Complete Works of Thomas Shadwell (5 vols; London: Fortune Press, 1927). Tilmouth, Michael, ‘A Calendar of References to Music in Newspapers Published in London and the Provinces (1660–1719)’, Royal Musical Association Research Chronicle, 1 (1961). Turnbull, H.W. (ed.), The Correspondence of Isaac Newton (7 vols; Cambridge: Cambridge University Press for the Royal Society, 1959–77). Wallis, John, Johannis Wallis … De algebra tractatus historicus & practicus (Oxford: E Theatro Sheldoniano, 1693). —— Johannis Wallis S.T.D. … Operum mathematicorum volumen tertium (Oxford: E Theatro Sheldoniano, 1699). —— Johannis Wallisii … operum mathematicorum pars altera qua continentur de angulo contactus & semicirculi, disquisitio geometrica. De sectionibus conicis tractatus. Arithmetica infinitorum: sive de curvilineorum quadraturâ, &c (Oxford: typis Leon. Lichfield academiæ typographi, veneunt apud Octav. Pullein, 1656). —— A treatise of algebra, both historical and practical (London: Richard Davis, 1685). Ward, G.R.M. (ed.), Oxford University Statutes. Volume I. Containing the Caroline Code, or Laudian Statutes (London: William Pickering, 1845). Wardhaugh, Benjamin, ‘The Logarithmic Ear: Pietro Mengoli’s Mathematics of Music’, Annals of Science 64 (2007): 327–48. —— Music, Experiment and Mathematics in England, 1653–1705 (Farnham: Ashgate, 2008). —— (ed.), Thomas Salmon: writings on music (2 vols; Farnham: Ashgate, 2013). West, Martin L., Ancient Greek Music (Oxford: Clarendon Press, 1992). Wood, Anthony à, Athenae oxonienses. An exact history of all the vvriters and bishops who have had their education in the most ancient and famous University of Oxford … (2 vols: London: for Tho. Bennet, 1691–2). Zarlino, Gioseffo, Istitutioni harmoniche (Venice: n.p., 1558).

Index

Aldrich, Henry 28 Alsted, Johann Heinrich 16 Alypius 13, 77, 85, 86, 92, 108, 123, 124, 128, 150, 156, 160, 172, 184, 185, 189 Aristides Quintilianus 13, 28, 77, 78, 79, 80, 81, 82, 83, 85, 90, 93, 94, 99, 111, 117, 120, 150, 151, 152, 153, 154, 156, 159, 161, 162, 166, 175, 179, 182 Aristoxenean methods 14, 105–8, 112–19, 130, 170–72, 175–82, 190–91, 214, 218 Aristoxenus 14, 15, 28, 77, 150 Bacchius 85, 102, 123, 129, 156, 169, 184, 190 Benedetti, Giovanni Battista 6 Bernard, Edward 13, 18 Birchensha, John 3–4, 7, 16, 25, 41, 45, 62 Bisse, Thomas 28 Boethius 5, 7, 14, 53, 63, 79, 85, 92–5, 98–9, 101–2, 110, 113, 119, 130–131, 135, 151, 156, 161–3, 165–6, 168–9, 174, 177, 181, 192, 196 Boyle, Robert 3, 6, 8, 41–3, 69 Burney, Charles 21, 29 Capella, Martianus 85, 92, 107, 155, 161, 172 Chilmead, Edmund 13, 16 Cleonides, see ‘Euclid’ consonance, coincidence theory of 3–4, 6, 15, 19–20, 27, 48–52, 54–6, 132–3, 136, 143–4, 192–4, 196–7, 203–4 consonances classifications of 81, 152–3 species of 119–22, 181–4

Descartes, René 7, 16, 27, 136 dissonances 82, 153–4 ear

anatomy of 26–7 role and capabilities of 5, 15–16, 45, 113–15, 117, 131, 140, 176–80, 200, 208, 214, 219 equal temperament 24–5, 208, 217–19 ‘Euclid’ (i.e. Cleonides), Introductio 14, 77–9, 81, 83, 91–2, 101–3, 106–7, 109–10, 112, 117, 119, 123–4, 128–9, 150–54, 159–601, 168–76, 179–82, 184–6, 189–90 Euclid Elements 2 Sectio canonis 5, 7, 19, 45, 47–8, 58, 110–11, 113, 117, 130–31, 135–6, 173–4, 179–80, 191–2, 215 experiments, musical 8–11, 18, 48–9, 69–73 Fétis, François-Joseph 16, 29 Fletcher, Andrew 24–5, 223–4, 228 fourth, status of 143, 203 Galilei, Galileo 6 Galilei, Vincenzo 16, 65, 131, 192 Gaudentius 13, 78–9, 81–5, 97, 101–3, 106, 119, 123, 150–56, 165, 168–71, 184 genera 48, 102–11, 139, 168–75, 199–200, 210, 215 Glarean 10, 125, 186–7 Gogava, Antonio 12, 16, 27, 205 Gregory, David 25, 228 Guido 86, 138, 156, 198–9 Harris, René 22–4, 213, 219 Hawkins, John 16, 25, 28

238

John Wallis: Writings on Music

hearing, see ear Holder, William 21 intervals classification of 80, 82–3, 142–3, 152–4, 202–3 division of 4, 15, 20, 21, 22, 23, 50–59, 111–12, 134–7, 139–40, 174–6, 194–200, 208–10, 215 harmoniousness of 50–52, 54–6, 132–4, 140, 143, 192–5, 200, 203, 216 just intonation 4, 7, 15, 19, 21, 23, 54, 56, 63, 90, 131, 159, 191–2, 216–17 construction of 136–9, 196–200, 209–10 extensions of 4–5, 15, 20–22, 56–8, 61, 136, 139–41, 196–7, 199–201, 210, 217 Kepler, Johannes 6, 62–3, 131, 191–2 key, changes of 20, 23, 59–60, 89, 125, 158, 186–7, 217–18 key signatures, see sharp and flat signs Kircher, Athanasius 16 major third, division of 19, 60, 137–8, 197–9, 210 manuscripts Byzantine 25–6 Savilian 12–13 Marsh, Narcissus 6, 9–10, 27, 42 Meibom, Marcus 12–13, 26, 76–7, 85–6, 93, 112, 116, 150, 155–6, 161–2, 175–6, 178–9, 206 melody 128–30, 143, 189–91, 203 types of 81, 152–3 Mengoli, Pietro 16, 149 Mersenne, Marin 6, 10, 16, 63, 86–7, 156–7 modal system, construction of 126–7, 187–9 modes, modal theory 14, 29, 87, 89, 123–7, 157–8, 184–9 Moray, Robert 8 Morley, Thomas 7 music

and education 228–9 effects of 24–25, 130, 190–91, 223–9 nature of 77, 150, 225–6 see also experiments, musical; notation, musical; practitioners, musical; terminology, musical mutation (of mode or genus) 89, 109–10, 121, 125, 172–4, 186–7 Noble, William 9, 71 nodes of vibration 9–11, 69–72 North, Francis 11, 27 notation, musical 85–7, 89–90, 155–9 Oldenburg, Henry 2–3, 5–10 letters to 2–7, 15, 41–63, 69–73, 75 Pepys, Samuel 10, 23, 213 Perfect System 14, 78, 98, 99, 101, 111, 112, 113, 122, 140, 183–4 Pigot, Thomas 9–10, 71 pitch, division of 13–14, 78–80, 83, 105–7, 109, 112–13, 123–4, 150–152, 154, 170–173, 175–7, 184–6 Playford, John 7 Plot, Robert 9–10, 17, 26, 71, 73, 205 practitioners, musical, knowledge of 22–4, 110–11, 118–19, 173–5, 180–82, 213–14, 217–18 Pythagorean methods 14, 48, 82, 91, 108–9, 111–14, 116–19, 130–32, 134, 153–4, 159–60, 172–82, 190–95, 214–16, 218–19 see also scales, Pythagorean Royal Society 2–3, 5–8, 18, 45, 48, 69–70 Salmon, Thomas 17, 19, 25 relationship with Wallis 17–19, 22 scales construction of 4, 5, 14, 15, 20, 21, 22, 23, 24, 47–53, 56, 59, 87, 90–92, 94–5, 157, 159–63 Greek 96–102, 163–9 historical development of 97–9, 101, 135, 164–6, 168, 195–6 Guidonian 87, 112–13, 157, 175–7

Index irregular 24 modern, construction of 110–11, 173–5, 207–8, 214–16 Pythagorean 4–5, 15, 20, 53, 63, 131, 191–2, 215 see also just intonation, equal temperament semitone, size(s) of 4–5, 7, 20–23, 47–8, 53–5, 63, 115, 130–31, 135–6, 138, 177–8, 190–92, 195–9, 207–8, 210, 216–18 sharp and flat signs 7, 56–7, 59, 61, 89–90, 121, 125–7, 138, 140, 143, 186–9, 203, 210, 216 Simpson, Christopher 19–21, 207–8 Sloane, Hans 18, 24–5, 223, 228–9 solmization 87, 89–90, 119, 121–2, 135, 157–9, 181–4, 195–6 sound, nature of 27, 48–9, 73, 113, 132, 176–7, 192–3, 224–5 speech 26–7, 30, 81, 21, 78, 79 Stiles, Francis Eyles 29 sympathetic resonance 9, 11, 27, 70, 73, 132, 144, 192–3, 204 systems, classification of 81, 152–3 terminology, musical 13, 77–8, 83–4, 87, 92–5, 150–51, 154–5, 157, 160–63 Greek 97–102, 108, 110, 118, 120, 123–5, 126–9, 164–9, 172–4, 180–82, 184–90 Latin or English 85, 113, 124–127, 142, 155–6, 176–7, 186–9, 202

239

trumpet marine 10–11 Tyson, Edward 26–7 voice abilities of 57, 61 range of 80, 112, 138, 143, 152, 175–6, 198–9, 203 role of 48, 219 Voss, John Gerard 16, 77, 79, 85–7, 114, 150–51, 156–7, 177 Wallis, John biography of 2 contribution to Thomas Salmon’s Proposal 18, 21 as controversialist 2 knowledge of musical practice 18 legacy in musical scholarship 28 mathematical work of 2 as musical expert 7, 26, 29–30, 45–6 as philologist 12–13, 29, 84, 93–4, 155, 161–2, 205 on Porphyry and Bryennius 26, 29 on Ptolemy 12–17, 26–7, 75–206 reception of 27–9 as Savilian professor 2, 5, 12–13, 205 sources used by 5–7, 10, 14, 16–17, 41–2, 62–3 as theologian 2, 28 theory of sound 11–12 Wanley, Humphrey 25–6 Zarlino, Gioseffo 7, 16, 131, 191–2