Computational musicology in Hindustani music 9783319114712, 9783319114729, 3319114719

Table of contents :
An Introduction to Hindustani Music --
The Role of Statistics in Computational Musicology --
Introduction to Rubato: The Music Software for Statistical Analysis --
Modeling the Structure of Raga Bhimpalashree: A Statistical Approach --
Analysis of Lengths and Similarity of Melodies in Raga Bhimpalashree --
Raga Analysis Using Entropy --
Modeling Musical Performance Data with Statistics --
A Statistical Comparison of Bhairav (A Morning Raga) and Bihag (A Night Raga) --
Seminatural Composition --
Concluding Remarks.

Citation preview

Computational Music Science

Soubhik Chakraborty Guerino Mazzola Swarima Tewari · Moujhuri Patra

Computational Musicology in Hindustani Music

Computational Music Science

Series Editors Guerino Mazzola Moreno Andreatta

More information about this series at http://www.springer.com/series/8349

Soubhik Chakraborty • Guerino Mazzola • Swarima Tewari • Moujhuri Patra

Computational Musicology in Hindustani Music

Soubhik Chakraborty Swarima Tewari Department of Applied Mathematics Birla Institute of Technology (BIT), Mesra Ranchi, Jharkhand India

Guerino Mazzola School of Music University of Minnesota Minneapolis, MN USA

Moujhuri Patra Dept. of Computer Applications Netaji Subhash Engineering Coll (NSEC) Kolkata, West Bengal India

ISSN 1868-0305 ISSN 1868-0313 (electronic) Computational Music Science ISBN 978-3-319-11471-2 ISBN 978-3-319-11472-9 (eBook) DOI 10.1007/978-3-319-11472-9 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014957648 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Computational musicology is the fruit of two factors that were brought to florescence in the twentieth century: modern mathematics and computer technology. The mathematical contribution can be attributed to an incredible expansion of the mathematical concept architecture, reaching far beyond simple numbers and functions. The culmination of this development can be concretized in the theory of topoi that was initiated by Alexander Grothendieck and ultimately unites geometry and logic in a revolutionary restatement of what is a space, namely, that concepts are understood as being points in a conceptual space. Mathematical music theory has drawn substantially from topos theory, as has become evident with the publication of The Topos of Music (Mazzola 2002). A concrete consequence of this development has been a computational description and modeling of fundamental topics of music theory: harmony, rhythm, melody, counterpoint, performance, and composition. But it became evident very soon that such computational approaches could only be related to existing musical works with powerful computational tools, much as modern physics cannot be developed without impressive experimental devices, such as particle accelerators and their computational background machinery. In fact, a melodic analysis of a one-page composition can easily imply billions of comparisons of motivic units. This suggests a future musicology that might move in the direction of big science when it comes to understanding major works in music, be it in the Western classical score-driven tradition, in the Indian raga tradition, or in free improvisation. This is why music software has been developed to calculate quantitative results that reflect the theoretical models of computational musicology. The RUBATO software (Mazzola and Zahorka 1994) was one of the first tools that offered comprehensive analytical machinery for computational harmonic, rhythmical, melodic, and performance-theoretical investigations. It is not by chance that such investigations were first conducted in collaboration with a statistician (Beran and Mazzola 1999) since experimental science cannot be realized without statistical methods. Statistics in musicology has become a fascinating new field of research (Beran 2004). These investigations have revealed significant relations between the v

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analytical structure of classical Western compositions and the tempo curves of human performances (Mazzola 2002). In this sense, we are proud of having supported the opening of a path to a deeper understanding of the great Indian raga tradition, which is not score driven but builds on a deep oral canon of gestural creation and communication (Rahaim 2012). This tradition would be difficult to analyze in precise terms without mathematical music theory, its technology, and the statistical methodology of experimental research. Chapter 1 gives an introduction to Indian music, with special emphasis on Hindustani classical music and its critical comparison with Western classical music; it was written by the first two authors. In the critical comparison the authors even contradicted themselves, but neither view can be discarded. This chapter will immensely help music enthusiasts who have knowledge of Western classical music but are new to Indian music. Chapter 2 talks about the role of statistics in computational musicology; it was written by the first author. Chapter 3 describes RUBATO, the music software for statistical analysis; it was written by the second author. Chapters 4–6 teach us how to analyze a musical structure using a statistical approach; they were written by the third author and the first author. In particular, Chap. 4 deals with modeling, Chap. 5 with melodic similarity and lengths, and Chap. 6 with entropy analysis. These three chapters explain how and why it becomes important to bring out some general features of a musical piece (in this case, a raga) structurally without bringing the style of the artist into play. This style, however, provides additional features demanding further statistical analysis, and consequently the problem of analyzing a musical performance is addressed in Chaps. 7 and 8. Chapter 7 is focused on modeling, wherein the strength of the statistical approach lies; it was written by the fourth author and the first author. Chapter 8 gives a statistical comparison of a morning raga (Bhairav) and a night raga (Bihag) using RUBATO; it was written jointly by the first three authors. Raga-based songs are important in promoting Indian classical music among laymen. Chapter 9, written by the third author and the first author, explains how the concept of seminatural composition, using a Markov chain of first order, can help a music composer in obtaining the opening line(s) of a raga-based song using Monte Carlo simulation. Once this opening line is obtained, the song can be completed by any intelligent composer. It is the opening line that is crucial, and this is where musical plagiarism comes into play. Chapter 10 summarizes the first author’s practical experience of presenting the science of music together with the art of music on stage with professional artists, and it provides the scheme and motivation for doing so. It also briefly explains why it is important to achieve success in computational musicology in order to achieve success in music therapy. This book, written with the sole objective of promoting computational musicology in Indian music, is primarily aimed at teaching how to do music analysis in Indian music, although most of the concepts are applicable in other genres of music as well. It assumes that the musical data is already available either from text

Preface

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(structure) or from audio samples (performance). Thus, this is not a book that teaches you how to acquire the musical data using signal processing. Consequently, several aspects of music analysis involving signal processing such as raga identification and tonic (Sa) detection had to be left out, and we have provided references for these. One reason is that while there are many good books available on musical signal processing (see, e.g., Roads et al. 1997; Klapuri and Davy 2006), there were no books on computational musicology in Indian music. Most of the works are available as research papers, and, apart from those that are published online, they are not accessible unless you or your institute has a subscription for the journal concerned. Hopefully this book will meet some of the requirements of a music analyst interested in Indian music. A second reason is that we had to consider the overall size of this book. However, musical signal processing is an interesting area of music research, and we promise to write a book on music information retrieval (MIR) in the context of Indian music in the near future, in which we would deal extensively with musical signal processing. Most of the issues that could not be addressed here would be taken up then. The authors are grateful to their respective family members, friends, colleagues and other well wishers for the moral support they received during the manuscript preparation. Special thanks go to Ronan Nugent of Springer for doing an excellent editorial handling and to K. SheikMohideen for the technical issues involved during the production stage. Ranchi, India Minneapolis, MN, USA Ranchi, India Kolkata, India July 30, 2014

Soubhik Chakraborty Guerino Mazzola Swarima Tewari Moujhuri Patra

References J. Beran, Statistics in Musicology (Chapman & Hall, New York, 2004. J. Beran, G. Mazzola, Analyzing musical structure and performance – a statistical approach. Stat. Sci. 14(1), 47–79 (1999) A. Klapuri, M. Davy (Eds.), Signal Processing Methods for Music Transcription (Springer, New York, 2006. G. Mazzola, O. Zahorka, The RUBATO Workstation for Musical Analysis and Performance. Proceedings of the 3rd ICMPC, ESCOM, Lie`ge, 1994. G. Mazzola et al., The Topos of Music (Birkhaeuser, Basel, 2002. M. Rahaim, Musicking Bodies (Wesleyan University Press, Middletown, 2012. C. Roads, S.T. Pope, A. Piccialli, G.D. Poli (Eds.), Musical Signal Processing. Studies on New Music Research (Routledge/Taylor and Francis, London, 1997.

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Contents

1

An Introduction to Indian Classical Music . . . . . . . . . . . . . . . . . . 1.1 A Critical Comparison Between Indian and Western Music . . 1.2 Terminologies Used in Hindustani Classical Music . . . . . . . . . 1.2.1 Raga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Notation Used in Describing Ragas . . . . . . . . . . . . . . 1.3 Systematic Presentation of Ragas . . . . . . . . . . . . . . . . . . . . . 1.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 2 5 5 6 9 12 13

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The Role of Statistics in Computational Musicology . . . . . . . . . . . . 2.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Similarity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Rhythm Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Entropy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Multivariate Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . 2.6 Study of Varnalankars Through Graphical Features of Musical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Link Between Raga and Probability . . . . . . . . . . . . . . . . . 2.8 Statistical Pitch Stability Versus Psychological Pitch Stability . . . 2.9 Statistical Analysis of Percussion Instruments . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 17 18 18 19 19 20 20 21 22 23

Introduction to RUBATO: The Music Software for Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The Overall Modularity . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Frame and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The RUBETTE® Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 MetroRUBETTE® . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 MeloRUBETTE® . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 HarmoRUBETTE® . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 28 29 30 33 35

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3.2.4 PerformanceRUBETTE® . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 PrimavistaRUBETTE® . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 45 51

Modeling the Structure of Raga Bhimpalashree: A Statistical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Getting the Musical Data for Structure Analysis . . . . . 4.3 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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53 53 54 55 55 56 58 59 60

Analysis of Lengths and Similarity of Melodies in Raga Bhimpalashree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Statistical Analysis of Melody Groups . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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61 61 61 64 64

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Raga Analysis Using Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Mean entropy of raga Bhimpalashree . . . . . . . . . . . . . . 6.2 Discussion: Information on a Possible Event E with P(E) ¼ 0 . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 66 66 67 68

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Modeling Musical Performance Data with Statistics . . . . . . . . . . . . 7.1 Statistics and Music . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Time Series Analysis and Music . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Time Series Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Goal of Time Series Analysis . . . . . . . . . . . . . . . . . . . . 7.3 Autoregressive Integrated Moving Average . . . . . . . . . . . . . . . 7.3.1 Autoregressive Process . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Moving Average Process . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 ARIMA Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Modeling Musical Data . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Analysis of Bihag . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Identification of ARIMA ( p, d, q) Models . . . . . . . . . . . . . . . . 7.4.1 Autoregressive Components . . . . . . . . . . . . . . . . . . . . . 7.4.2 Moving Average Components . . . . . . . . . . . . . . . . . . . 7.4.3 Mixed Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 ACFs and PACFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Estimating Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 70 70 70 71 71 71 72 72 73 73 74 74 74 75 76

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Model Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1 Ljung–Box (Q) Statistic for Diagnostic Checking . . . . 7.8 Modeling: Finding Fitted Model . . . . . . . . . . . . . . . . . . . . . . 7.9 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9.1 Results for Night Raga Bihag . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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76 77 77 77 78 78 79

A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 IOI Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Duration Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 RUBATO Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 RUBATO Analysis of Bhairav . . . . . . . . . . . . . . . . . . . . . . . 8.5 RUBATO Analysis for Raga Bihag . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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81 83 84 84 85 89 94 95

9

Seminatural Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 97 . 97 . 99 . 99 . 100 . 101

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Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8

Chapter 1

An Introduction to Indian Classical Music

The origins of Indian classical music lie in the cultural and spiritual values of India and go back to the Vedic Age (Sam Veda). Even in those times, music was handed down orally from the guru (teacher) to the shishya (disciple). The art was called sangeet and included vocal music, instrumental music, and dance. The great sages who dwelt in ashramas (hermitages) imparted instruction to their students who lived with them on the premises. The art of music was regarded as holy and heavenly. It not only gave aesthetic pleasure but also induced a joyful religious discipline. Devotional music was intended to take man towards God and give him an inner happiness and self-realization. Subsequently this art branched off into three separate streams: vocal music (geet), instrumental music (vadya), and dancing (nritya). Many of us often confuse between Indian classical music and Hindustani classical music. In fact, Indian classical music is divided into two distinct streams, Hindustani and Carnatic or North Indian and South Indian classical music. Hindustani classical music mainly evolved in North India around the thirteenth and fourteenth centuries A.D. It owes its development to the religious music, as well as popular and folk music, of the time. Carnatic music is also known as Karnataka Sangitam which was developed in South India around the fifteenth and sixteenth centuries. It drew on existing popular forms of music, and probably it also retained the influence of ancient Tamil music. Emotion and devotion are the essential characteristics of Indian music from the aesthetic side. From the technical perspective, we talk about melody and rhythm. See also http://www.esikhs.com/articles/indian_classical_music_&_sikh_ kirtan.pdf Raga is the nucleus of Indian classical music, be it North Indian (Hindustani) or South Indian (Carnatic). A raga may be defined as a melodic structure with fixed notes and a set of rules that characterize a certain mood conveyed by performance (Chakraborty et al. 2009a). The “certain mood” refers to the emotional content that is typical of the raga. The rules (like how a particular note or note combination should be used in the raga, the note sequence allowed in ascent or descent, etc.) help © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_1

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1

An Introduction to Indian Classical Music

in building the raga mood and are not meant to handicap the artist. In fact, the artist has infinite freedom to express himself/herself despite these rules! Although it may be possible to associate a part of the emotional content of the raga with an identifiable human emotion such as sadness (“karuna rasa”), it hardly tells the full story. Musical emotion, when it comes to ragas, is typically a characteristic in its own right, and even two ragas that apparently seem to evoke sadness (like the ragas Bageshree and Shivaranjani) are actually quite different. It is something like saying that a mango and a guava when ripe are both sweet (commonality), and yet a mango taste is typical and different from a guava taste (diversity). It is this diversity that signifies the emotional content (or rasa) of Indian classical music, not just the commonality. In other words, it is fine clubbing the aforesaid two ragas as of karuna rasa with the acknowledgement that the “Bageshree-rasa” of Bageshree is different from the “Shivaranjani-rasa” of Chakraborty (2010). Suvarnalata Rao (2000) has given a helpful insight into the raga–rasa theory from an acoustical perspective, but it should be understood that what kind of acoustic stimulus leads to what kind of emotional changes in the brain is a subject matter of psychophysics of music, which is a branch of psychology (not physics!). We refer the interested reader to the classic text on this by Roederer et al. (2008).

1.1

A Critical Comparison Between Indian and Western Music

Over the years, Indian classical music has evolved into a complex musical system. It has some main points of difference from Western music. Western music is polyphonic, which means that it depends on the resonance of multiple musical notes occurring together. In contrast, Indian classical music is essentially monophonic. Here, a melody or sequence of individual notes is developed and improvised upon, against a repetitive rhythm. In Western classical music, a performer strictly abides by a written composition. In contrast, in Indian classical music, the performer improvises the composition rendered. A Western classical concert is never performed extempore; it will be always prepared and rehearsed several days before the concert. The percussion is never as prominent as in Indian classical music. To say more, in European classical music, percussion has always been placed on a sidetrack. The European system tonic never changes. This means the first note of the scale (saptak) C or do (Sa) will always be of the same pitch. On the Indian subcontinent, the pitch of the tonic is changed according to the chosen instrument or voice but still will be called Sa. One reason that Indian music sounds unfamiliar to the Westerner is that the major–minor tonal system is not used. Harmony, and specifically the major–minor tonal system, has been the basic organizing principle in Western music—classical, folk, and popular—for centuries. In this system, a piece of music is in a certain key, which means it uses the notes of a particular major or minor scale. The harmonies

1.1 A Critical Comparison Between Indian and Western Music

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developed using those notes are an integral, basic part of the development and form of the music. Most of the complexity of Western music lies in its harmonies and counterpoint. The music of India does not emphasize harmony and does not feature counterpoint. In fact, most Indian classical music features a single melody instrument (or voice) accompanied by drone and percussion. There is no counterpoint and no chord progression at all. Instead, the interest and complexity of this music lie in its melodies and its rhythms. Western music divides an octave into the 12 notes of the chromatic scale. But most pieces of music mainly use only seven of these notes, the seven notes of the major or minor key that the piece is in. Indian music also has an octave divided into 12 notes. These 12 notes are called swaras; they are not tuned like the notes of the chromatic scale. Also similarly to Western music, only seven notes are available for any given piece of music. But there are important differences, too. Western scales come in only two different “flavors”: major and minor. The two are quite different from each other, but any major key sounds pretty much like any other major key, and any minor key sounds basically like every other minor key. This is because the relationships between the various notes of the scale are the same in every major key, and a different set of relationships governs the notes of every minor key. The sevennote thaats of Indian music, on the other hand, come in many different “flavors.” The interval pattern varies from one thaat to the next, and so the relationships between the notes are also different. There are ten popular thaats in North Indian music and many more in the South. Although the first note of an Indian scale is often given as C, Indian thaats and ragas are not fixed in pitch; any raga may actually begin on any pitch. The important information about each thaat and raga “scale” is the pattern of intervals, the (relative) relationship between the notes, not absolute fundamental frequencies. Making for even more variety, a piece of Indian classical music may not even use all seven of the notes in the thaat. The music will be in a particular raga, which may use five, six, or all seven of the notes in the thaat. And a thaat can generate more than just three ragas (one pentatonic, one hexatonic, and one full raga). Kalavati raga (C, E-flat, G, A, and B-flat) and Shivaranjani raga (C, D, E-flat, G, and A), for example, are two different pentatonic ragas derived from Kafi thaat. Thus, there are hundreds of ragas available, and a competent Indian musician is expected to be able to improvise many of them. Furthermore, the raga is not just a collection of the notes that are allowed to be played in a piece of music. There are also rules governing how the notes may be used; for example, the notes used in an ascending (arohi) scale may be different from the notes in a descending (awarohi) scale. Some notes may be considered “main pitches” in the raga, while others are used in a more ornamental way typical of the raga. The raga may even affect the tuning of the piece. Those who are particularly interested in modes and scales may notice that there is a rough correlation between some Indian thaats and the Western church modes. For example, the pattern of intervals in Asavari is similar to that of the Aeolian mode (or natural minor scale), and that of Bilawal is similar to the Ionian mode (or major scale). Some thaats do not correlate at all with the Western modes

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(e.g., Purvi and Todi), but others that do include Bhairavi (similar to Phrygian mode), Kafi (Dorian), Kalyan (Lydian), and Khamaj (Mixolydian). Even for these, however, it is important to remember the differences between the traditions. For example, not only is Asavari used in a very different way from either Aeolian mode or the natural minor scale; the scale notes are actually only roughly the same, since the Indian modes use a different system of tuning. The tuning of modern Western music is based on equal temperament; the octave is divided into 12 equally spaced pitches. But this is not the only possible tuning system. Many other music traditions around the world use different tuning systems, and Western music in the past also used systems other than equal temperament. Medieval European music, for example, used just intonation, which is based on a pure perfect fifth. The preferred tuning system of a culture seems to depend in part on other aspects of that culture’s music, its texture, scales, melodies, harmonies, and even its most common musical instruments. For example, just intonation worked very well for medieval chant, which avoided thirds, emphasized fifths, and featured voices and instruments capable of small, quick adjustments in tuning. But equal temperament works much better for the keyboard instruments, triadic harmonies, and quick modulations so common in modern Western music. In India, the most common accompaniment instrument (as ubiquitous as pianos in Western music) is the tanpura. (There are several alternative spellings for this name in English, including taanpura and tambura.) This instrument is a chordophone in the lute family. It has four very long strings. The strings are softly plucked, one after the other. It takes about 5 s to go through the four-string cycle, and the cycle is repeated continuously throughout the music. The long strings continue to vibrate for several seconds after being plucked, and the harmonics of the strings interact with each other in complex ways throughout the cycle. The effect for the listener is not of individually plucked strings. It is more of a shimmering and buzzing drone that is constant in pitch but varying in timbre. And the constant pitches of that drone are usually a pure perfect fifth. Assuming tuning in C (actual tuning varies), two of the strings of the tanpura are tuned to middle C and one to the C an octave higher. The remaining string is usually tuned to a G (the perfect fifth). (If a pentatonic or hexatonic raga does not use the G, this string is tuned instead to an F. The pure perfect interval is still used however, and you may want to note that a perfect fourth is the inversion of a perfect fifth.) So a just intonation system based on the pure fifth between C and G (or the pure fourth between C and F) works well with this type of drone. Pure intervals, because of their simple harmonic relationships, are very pleasing to the ear and are used in many music traditions. But it is impossible to divide a pure octave into 12 equally spaced pitches while also keeping the pure fifth. So this brings up the question: where exactly are the remaining pitches? The answer, in Indian music, is: it depends on the raga. Indian music does divide the octave into 12 swaras, corresponding to the Western chromatic scale. Also, just as only seven of the chromatic notes are available in a major or minor scale, only seven notes are available in each thaat. But because just intonation is used, these notes are tuned differently from Western

1.2 Terminologies Used in Hindustani Classical Music

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scales. For example, in Western music, the interval between C and D is the same (one whole tone) as the interval between D and E. In Indian tuning, the interval between C and D is larger than the interval between D and E. Using the simpler ratios of the harmonic series, the frequency ratio of the larger interval is about 9/8 (1.125); the ratio of the smaller interval is 10/9 (1.111). (For comparison, an equal temperament whole tone is about 1.122.) Western music theory calls the larger interval a major whole tone and the smaller one a minor whole tone. Indian music theory uses the concept of a shruti (microtone), which is an interval smaller than the intervals normally found between notes, similar to the concept of cents in Western music. The major whole tone interval between C and D would be four shrutis; the minor whole tone between D and E would be three shrutis. In some ragas, some notes may be flattened or sharpened by one shruti, in order to better suit the mood and effect of that raga. So, for tuning purposes, the octave is divided into 22 shrutis. This is only for tuning; however, for any given thaat or raga, only 12 specifically tuned notes are available. The 22 shrutis each have a specific designation, and the intervals between them are not equal; the frequency ratios between adjacent shrutis range from about 1.01 to 1.04. In spite of the fact that these tunings are based on the physics of the harmonic series, Indian music can sound oddly out of tune to someone accustomed to equal temperament, and even trained Western musicians may have trouble developing an ear for Indian tunings (C. S. Jones, Indian Classical Music, Tuning and ragas, http:// cnx.org/content/m12459/1.6/). In the next section, we shall describe a raga and some related terms in greater details. Remark The comparison between Indian and Western music given above is only for a general firsthand understanding and is not valid in every situation. Western music has known many different harmonic systems, in the medieval music and also in modern jazz; the modes (ionian, dorian, etc.) were more important than the major–minor reduction. Also, even if one plays in one tonality, say B-flat major, many other notes are allowed, already, with Bach or Beethoven. So the seven notes are only a reference system. The tuning is also a problem in Western music. Twelve-tempered tuning is frequent, especially for pianos. But violin and other strings use just fifth very often, causing conflicts with pianos. It is also important to see that microtonal systems are well known in Western music, for example, Alois Haba’s music for quarter, third, fifth, etc., tones.

1.2 1.2.1

Terminologies Used in Hindustani Classical Music Raga

The heart of Hindustani classical music is the raga. As mentioned earlier, a raga may be defined as a melodic structure with fixed notes and a set of rules characterizing a certain mood conveyed by performance (Chakraborty et al. 2009a).

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The following are the characteristics of the raga. A raga consists of a fixed set of five or more musical notes (so this is one of the rules!). Ragas (in Sanskrit it is known as color or passion) are supposed to evoke various moods in the listener. In Hindustani music, there are certain ragas which are specific to different season or time of the day. Monsoon ragas belong to the Malhaar group, though they are mainly performed during the rains, while morning ragas, such as Bibhas and Bhairavi, and night ragas, such as Kedar and Malkauns, are suitable for rendition in morning or night, respectively. We also have afternoon ragas like Bhimpalashree and evening ragas like Yaman. The Hindustani and Carnatic classical music systems usually have different ragas. There are some ragas which are similar but use different names in both systems. Others have similar names but different in their actual form. Also, Hindustani classical music classifies ragas into ten parent raga groups called thaats, as organized by Vishnu Narayan Bhatkhande in the early 1900s. The Carnatic system, on the other hand, depends on an older classification having 72 parent raga groups.

1.2.2

Notation Used in Describing Ragas

Notation is the art of describing musical ideas in written characters or symbols. Indian classical music has seven basic notes and is called shudh (natural or pure) swara. They are shadja, rishabha, gandhara, madhyama, panchama, dhaivata, and nishada. In short form, they are known as Sa, Re, Ga, Ma, Pa, Dha, and Ni. This group of Indian notes is called a saptak (seven notes of diatonic scale). There are three types of saptak: – Mandra/mandar (lower octave) – Madhya (middle octave) – Tara (higher octave) In addition, there are four komal (soft or flat) notes (Re, Ga, Dha, and Ni) and one teevra/teebra (sharp) note (Ma), thus making a total of 12 notes in the chromatic scale. These five notes are called modified (vikrita) notes. In this book the notes (swara) used in representing the ragas are: Small letter indicates a komal swara as komal Re is r, etc.,1 and capital letter for sudh swara as sudh Re is R, etc. (see footnote 1), with only exception of m as tivra Ma (as M represents sudh Ma). Further, a note in italics, normal, and bold font stands for the notes in lower octave, middle octave, and higher octave, respectively.

1

Other notations e.g. for indicating a meend (glide) or a grace note (kan swar) have not been used in the book. The interested reader can see Jairazbhoy (1995)

1.2 Terminologies Used in Hindustani Classical Music

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Further the ragas are described with following terminologies: – – – – – – –

Thaat (parent scale) Jati (class) Aroha (ascent) and avaroha (descent) Vadi (most important note) and samvadi (second most important note) Peshkash (rendition) Rasa (aesthetic joy or emotion) Pakad (catch or grip of the raga) Let us take a look at each of these musical terms.

– Thaat This is a method of grouping of ragas according to the specific notes used. Two ragas using the same notes will be placed in the same thaat even if their melodic structures, mood, and emotions are different, for example, Bhimpalasi and Bageshree ragas both belong to the Kafi thaat. The ten thaats in Hindustani classical music are Kafi, Bilaval, Purvi, Asavari, Todi, Khamaj, Kalyan, Bhairav, Marwa, and Bhairavi. For more details, see www.chandrakantha.com Some ragas like Ahir Bhairav and Charukeshi do not fall into any of these thaats. – Jati Jati literally means “caste.” Just as there are castes in any community in India, there are three castes or classes of raga. There is Arava/Audava/Oudava, pentatonic (five notes); Sharva/Shadava, hexatonic (six notes); and Sampoorna, perfect heptatonic (seven notes). Thus an Aurabh-Sampoorna raga means five distinct notes are allowed in ascent, seven in descent. – Aroha and avaroha They depict the sequence of permissible notes in ascent and descent, respectively. They help in characterizing the mood of the raga. – Vadi swara This is the most important or dominating note in a raga, which is a sort of key to the unfolding of its characteristics. As it is the pivotal note, it is played very prominently or repeatedly. In it lies the particular rasa of that raga. It also determines the time for the singing of the raga. If the vadi swara falls in the first half (Sa to Pa), then the raga is called Poorvang Pradhan. If it falls in the second half (Ma to Sa), then the raga is called Uttarnga Pradhan. Ma and Pa are common to both the halves. If the vadi swara is Ma or Pa, expert’s guidance is needed to decide whether the raga is Poorvang or Uttarnga Pradhan. – Samvadi swara This is the second important note in the raga, after the vadi note. Its position is at an interval of fourth or fifth from the vadi note. It has the same status as a minister in relation to a king represented by the vadi note. – Peshkash Classical musicologists have assigned a specific time to the performance of a raga. This has been based on the types of swara (notes) used in a particular raga.

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Table 1.1 Musical features (structural) of four ragas of four different timings Raga 4: Bhimpalashi

Musical feature

Raga 1: Bhairav

Raga 2: Bihag

Raga 3: Kedar

Thaat Arohan (ascent)

Bhairav S r G M Pd N S

Bilaval NSGMPNS

Avarohan (descent)

SNdPMGrS

Pakad (note assembly giving a catch of the raga) Vadi and samvadi swars (most important and second most important notes) Jati (raga group according to number of distinct notes allowed in ascent and descent)

G M P d, P d M P, G M r S

S N D P m G, M GRS N S, G M P, G MG

d, r

G, N

Kalyan S M G P, m P, D P, N D S S N D P, m P D P, M G M R S S M, M P, D P M, M G M, R S M, S

SnDPMgR S n S M, M g, P M, g, M g R S M, S

SampoornaSampoorna (7 distinct notes allowed in ascent; 7 in descent) rMPd

AurabhSampoorna (5 distinct notes allowed in ascent; 7 in descent) GPN

SarabhSampoorna (6 distinct notes allowed in ascent; 7 in descent) SMP

AurabhSampoorna (5 distinct notes allowed in ascent; 7 in descent) gMPn

Restful 5 a.m.–8 a.m. (first raga of the morning) Uttaranga pradhan (second half more important)

Restful 9 p.m.–12 p.m.

Restful 6 p.m.–9 p.m.

Restful 1 p.m.–3 p.m.

Purvanga pradhan (first half more important)

Purvanga pradhan

Purvanga Pradhan

Nyas swars (stay notes in the raga) Prakriti (nature) Time of rendition

Anga

Kafi nSgMPnS

Certain ragas can be sung during the morning hours, some in the afternoon, some in the evening, and some late at night. The 24 h of the day and night have been divided into eight parts called pahar, four of the day and four of the night. Each period consists of about 3 h. The first pahar of the day is from 6 a.m. to 9 a.m., the second pahar from 9 a.m. to 12 noon and so on. The first pahar of the night is from 6 p.m. to 9 p.m., the second pahar from 9 p.m. to 12 midnight, and so on. – Rasa This reflects the type of emotions, for example, Karun rasa depicts sadness. – Pakad This gives the catch of the raga through a note assembly in sequence. Table 1.1 compares the structural features of the four ragas Bhairav, Bihag, Kedar, and Bhimpalashree of four different timings as per suitability of rendition while Table 1.2 compares Indian and Western notation.

1.3 Systematic Presentation of Ragas

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Table 1.2 Comparison of Indian and corresponding Western notation C S

Db r

D R

Eb g

E G

F M

F# m

G P

Ab d

A D

Bb n

B N


12 S


11 r


10 R


9 g


8 G


7 M


6 m


5 P


4 d


3 D


2 n


1 N

0 S

1 r

2 R

3 g

4 G

5 M

6 m

7 P

8 d

9 D

10 n

11 N

12

13

14

15

16

17

18

19

20

21

22

23

Western notation Indian notation (lower octave) Numbers for pitch Indian notation (middle octave) Numbers for pitch Indian notation (higher octave) Numbers for pitch

Abbreviations: The letters S, R, G, M, P, D, and N stand for Sa, Sudh Re, Sudh Ga, Sudh Ma, Pa, Sudh Dha, and Sudh Ni, respectively. The letters r, g, m, d, and n represent Komal Re, Komal Ga, Tibra Ma, Komal Dha, and Komal Ni, respectively. Normal type indicates that the note belongs to middle octave; italics implies that the note belongs to the octave just lower than the middle octave, while a bold type indicates it belongs to the octave just higher than the middle octave. Sa, the tonic in Indian music, is taken at C. Corresponding Western notation is provided in this table. The terms Sudh, Komal, and Tibra imply, respectively, natural, flat, and sharp

Note 1: According to the renowned musicologist Suresh Chandra Chakraborty, Bhairav although called the Adi-raga being the first raga of the morning is not older than Bhairavi. The “Komal Re” (r) of Bhairav is actually ati komal (a microtone between Sa and Komal Re) and also andolita (oscillating) in a way that helps create the raga mood. The “Komal Dha” (d) is also andolita but only komal, not ati komal. In performance, it is necessary to give a meend (glide) from M to r and also from S or N to d (Chakraborty 1965). Note 2: The numbers representing pitch will be used for structure analysis of ragas later. The advantage of a structure analysis is that we can make some general comments about the raga statistically using these numbers without bringing the style of the artist into play. The assignment of the number 0 to the natural C of the middle octave (and accordingly the rest of the assignments follow) is followed by many music analysts. See, for example, Adiloglu et al. (2006).

1.3

Systematic Presentation of Ragas

The following points are important in presenting a raga systematically: • Varna Alankar In shastras, two types of alankars (ornaments) have been mentioned: shabdalankars and varnalankars. The latter refers to musical ornaments through which notes are decorated, or, in other words, they describe how a note sequence is rendered in a raga. There are four types of such melodic movements: asthai (level), which means staying on a note; aroha, the ascending motion of the notes

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•

•

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An Introduction to Indian Classical Music

in a raga; avaroha, the descending motion of the notes; and sanchari which is the mixed movement of the note. The transitory and non-transitory pitch movements between notes measure varnalankars. In some cases, the aroha and avaroha may differ in the type of notes, as, for example, Khamaj raga uses N in aroha and n in avaroha, or in the number of notes applied, as, for example, Jaunpuri raga uses five notes in aroha and seven notes in avaroha. Alaap It is the unfolding of the essence and the pattern of a raga with a word like AA or RE or NA and with emphasis on the notes of vadi and samvadi. The alaap is essential for the training of the voice. It is also a sort of invocation or prelude to the raga and is helpful in creating the particular rasa (feeling) and ethos of the raga. The rhythm is inherent in the alaap, flowing from the improvisation, and yet it does not belong to any fixed rhythm with a definite sam (closing of rhythmic variations). The singer, through the alaap, displays the transcendent nature of both melody and rhythm. Sometimes, the alaap is done in parts called angas. Anga literally means a limb that is part of the alaap. The angas would be from S to vadi, S to vadi and samvadi, S to vadi to samvadi to S, and then in the descending order. Alaap is slow in its movement and must stick to the rhythmic angas. The musician moves gradually shaping the raga according to his own talent and feeling. Bandish Bandish is a composition (vocal or instrumental) fixed in a rhythmic pattern. While alaap is the revelation of the raga, bandish is its design or display. Asthai and Antra Asthai is the basic and opening part of the raga. It is repeated throughout the alaap. Asthai brings together melody, rhythm, and tempo. It has definite form and is repeated from time to time. It offers the raga a framework, a skeleton for the performer to fill in with his improvisation. It generally moves in the lower tetrachord. Antra is the second section of the raga which generally relates to the upper tetrachord. It is fixed composition and complements of the asthai. Both asthai and antra hold an important place in the singing of khayal, dhrupad, and dhamar raga. Taan It means the exemplification or patternization of the raga. To practice the swaras of a raga in rhythmic patterns with different permutations is called taan. It can be one-half, one-fourth, and one-eighth of the asthai. The difference between alaap and taan is that of laya (tempo). Alaap is done in slow sequence, while taan is played in quick sequence like a fast sweep. The idea is to extend the basic melody. Taan requires a lot of practice and lot of attention to vadi and samvadi swaras. Nowadays taan is often used as an ornament (alankar). Laya (tempo) has an important place in taan. There are four varieties of taans: – Shudh or sapat taan: In this taan, the swaras (notes) are played in sequential order, in the direct aroha and avaroha style.

1.3 Systematic Presentation of Ragas

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– Koot taan: In this taan swaras are played in any order not in the usual S R G M P D N sequence. – Misra taan: This taan is a mixture of shudh and koot taans. The aroha is in shudh taan, while the avaroha is in koot taan. – Bol taan: When taan uses the words of the geet or song, the words so used are called bol taan. • Gamak Gamak is a kind of grace or embellishment or ornament of a melody. It may consist of a graceful jerk on a note or notes gliding over a note like a shadow or some other way in which the musical piece becomes graceful and attractive. Though taan comes close to gamak, it produces varieties of swara designs, but not designs of euphony. Gamak consists of various ways of touching and inflecting individual notes. • Taal Vocal music, instrumental music and dance rely on rhythm for its effect on the audience. Taal is the means of measurement of time in music or dance. Rhythm is the breaking up of time in small units. Time is cut into pieces at certain regular intervals. Literally taal means the palm of the hand; the time is measured by the clapping of hands (tali) or beats of drums or sticks. Taal is divided into two halves, i.e., Bhari (full) starting with sam and khali (empty). So taal is an organization of rhythms or different beats in certain groupings which are smaller units of matras. These rhythmic units repeat themselves in cycles. The drummer has to produce the spoken syllable indicating the position of the hand on the drum. The permutations related to taal are as follows: – Laya (tempo) The tempo of the rhythm or the duration of pace or speed is called laya. It is regular spacing of time. Laya has three different types: vilambit, madhya, and Drut laya. Vilambit laya: Slow tempo of the rhythm is called vilambit laya. The following are some of the taals of vilambit laya: ektal, chartal, jhumra, and tilwara. Madhya laya: Medium tempo of the rhythm is called madhya laya. It can be compared to the ticking of about half second of the clock. Some of the taals of madhya lays are teental, jhaptal, dadra, and kehrva. Drut laya: This is doubly quicker in tempo than the madhya laya. The taals of drut laya are the same as in madhya taal, the difference being that they are done quicker. Each beat lasts for about one-quarter second. Tarana and chota khayal use drut laya. – Matra The unit of measuring taal is matra. The matra is determined in length by the pace of the overall rhythm. Each taal has a number of matras, as, for example, dadra has six matras. The number of matras does not change in

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vilambit, madhya, or drut laya. Only the tempo or the time sequence becomes slower in vilambit, average in madhya, and faster in drut laya. The number of matras makes a taal, while the tempo or speed determines the types of laya. – Avartan One cycle of the matras of a taal is called avartan. For example, dadra has six matras; as soon as six matras are completed, we have done one avartan (cycle) of dadra tala. – Theka The playing of one avartan of a taal on the tabla is called theka. It includes the repetition of sound syllables (bol) to form rhythmic phrases. For example, jhaptaal has ten matras: – Jhaptaal Beats Words

1 DHI þ

2 NA

3 DHI

4 DHI

5 NA

6 TI 0

7 8 NA DHI

9 DHI

10 NA

• Sam The matra for which a taal begins is called the sam or gur. For example, in the above jhaptal and sooltaal, the cross (+) stands for sam. It is the first beat of the taal. There is an emphasis on the sam, by which it is recognized as different from the other matras. Some musicians indicate the sam either by a shake of the head or a beat on the knee. It is like the pivot, and all the rhythmic variations must close on the sam. • Khali After the sam, the next importance is khali, meaning “empty” or “blank.” The symbol for khali is zero (0). Generally khali marks the commencement of the second half of the cycle (avartan). • Tali Besides the sam, the matras on which the time beat falls is called tali (clapping).

1.4

Remarks

1. For an analysis of Indian percussion (tabla) performance, the reader is referred to Chakraborty et al. (2009b). 2. Joep Bor of the Rotterdam Conservatory of Music defined Raga as “tonal framework for composition and improvisation” (see Bor et al. 1999). 3. Nazir Jairazbhoy (1995), chairman of UCLA’s department of ethnomusicology, characterized ragas as separated by scale, line of ascent and descent, transilience, emphasized notes and register, and intonation and ornaments.

References

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4. The two streams of Indian classical music, Carnatic music and Hindustani music, have independent sets of ragas. There is some overlap, but more “false friendship” (where raga names overlap, but raga form does not). In North India, the ragas have been primarily categorized into ten thaats or parent scales (by Vishnu Narayan Bhatkhande, 1860–1936); South India uses an older and even more systematic classification scheme called the melakarta classification, with 72 parent (melakarta) ra¯gas. Overall there is a greater identification of raga with scale in the south than in the north, where such an identification is impossible. Ragas in North Indian music system follow the “law of consonances” established by Bharata in his Natyashastra, which does not tolerate deviation even at the shruti (microtone) level. As ragas were transmitted orally from teacher to student, some ragas can vary greatly across regions, traditions, and styles. Many ragas have also been evolving over the centuries. There have been efforts to codify and standardize raga performance in theory from their first mention in Matanga’s Brihaddeshi (c. tenth century) (source: Wikipedia 2013). 5. For a comprehensive treatise on Indian music, the interested reader is referred to The Oxford Encyclopedia of the music of India by Mahabharati et al. (2011) in three volumes. The reader is also referred to a recent article on Hindustani music in the context of computational musicology by Rao and Rao (2014) where one can find several other interesting aspects of Indian music, e.g., intonation. It is not the pitch but the pitch contours of notes that is of interest. 6. Readers interested in tonic (Sa) detection in Indian music are referred to the article by Gulati et al. (2014).

References K. Adiloglu, T. Noll, K. Obermayer, A paradigmatic approach to extract the melodic structure of a musical piece. J. New Music Res. 35(3), 221–236 (2006) V.N. Bhatkhande, Hindustani Sangeet Paddhati (Marathi/Hindi), vol. 1–4 (Popular Prakashan, Mumbai, 1995) J. Bor, S. Rao, W.V. Meer, J. Harvey, The Raga Guide (Nimbus Records, Charlottesville, 1999), p. 181 S.C. Chakraborty, Raga Rupayan (Bengali) (General Printers and Publishers, Kolkata, 1965) S. Chakraborty, On identifying a common goal between musicians and scientists. Georgian Electron. Sci. J. Musicol. Cultur. Sci. 1(5), 3–11 (2010) S. Chakraborty, R. Ranganayakulu, S. Chauhan, S.S. Solanki, K. Mahto, A statistical analysis of Raga Ahir Bhairav. J. Music Meaning 8, sec. 4, Winter (2009a), http://www. musicandmeaning.net/issues/showArticle.php?artID¼8.4 S. Chakraborty, R. Ranganayakulu, S.S. Solanki, A. Patranabis, D. Ghosh, R. Sengupta, A.K. Dutta, N. Dey, Shadowing along the Maestro’s rhythm. Electron. Musicol. Rev. XII (2009b)

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An Introduction to Indian Classical Music

S. Gulati, A. Bellur, J. Salamon, H.G. Ranjani, V. Ishwar, H.A. Murthy, X. Serra, Automatic tonic identification in Indian art music: approaches and evaluation. J. New Music Res. 43(1), 53–71 (2014) http://en.wikipedia.org/wiki/Raga. Accessed 17 June 2013 N.A. Jairazbhoy, The Ra¯gs of North Indian Music (Popular Prakashan, Mumbai, 1995), p. 45 S. Mahabharati, The Oxford Encyclopedia of the Music of India, vol. 1–3 (Oxford University Press, Oxford, 2011) S. Rao, Acoustical Perspective on Raga-Rasa Theory (Munshiram Manoharlal Publishers, New Delhi, 2000) S. Rao, P. Rao, An overview of Hindustani music in the context of computational musicology. J. New Music Res. 43(1), 24–33 (2014) J. Roederer, The Physics and Psychophysics of Music: An Introduction, 4th edn. (Springer, New York, 2008)

Chapter 2

The Role of Statistics in Computational Musicology

Music and mathematics are “school-mates,” having been related since the time of the ancient Greeks [Benson, 2007]. In contrast, music and statistics may be called “college-mates,” as they have only been linked after significant progress in computer technology and the availability of digitized scores (musical notation). [Soubhik Chakraborty, Review of the book Music and Probability by D. Temperley, The MIT Press, 2007, published in Computing Reviews (ACM), Mar 04, 2009]

Music analysis, broadly speaking, can be divided into commonality analysis (“what is common?”) and diversity analysis (“what is special?”). It is worth pointing out here that a statistician is essentially a commonality expert in the sense that the philosophy of statistics is to summarize and average and make inferences which are true on the whole and which describe a process rather than an individual entity. Fortunately, there are issues in music where this traditional mindset of the statistician finds a support. For example, a collection of recordings of the same artist if analyzed statistically will definitely reflect certain common features having to do with the style of the artist. But the statistician must realize that every single music piece will have something special to offer. Fortunately, again, there are issues even in statistics where the statistician does take an individual observation seriously—as in the case of an outlier or influential observation, for example. There is a whole literature in statistics to deal with outliers. When an outlier comes, the traditional philosophy of summarizing and averaging is brushed aside. The statistician goes after this individual influential observation exploring how it came and what it signifies. The case of outliers is an exception in statistics. It is the very grammar in music as music is a work of art! If the statistician can use his experience and mindset of handling outliers (regarding every musical piece as a musical outlier) along with his commonality expertise, he can be a very effective music analyst. Similar point of view has been expressed by Nettheim who has © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_2

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also provided a good bibliography of statistical applications in musicology (Nettheim 1997). For a sound statistical treatment of musical data, see Beran and Mazzola (1999). Additionally, we acknowledge the contributions from Meyer (1989), Snyder (1990), Winkler (1971), Wilson (1982), Todd and Loy (1991), and Morehen (1981). It is an irony that computational musicology in Indian classical music is still lagging behind the progress in Western classical counterpart, although we do appreciate the efforts of Castellano et al. (1984), Chordia and Rae (2007), and Sinha (2008) among others. We hope this book will provide some food for thought in that direction. Statistics is a useful tool both for analyzing a musical structure and quantitative assessment of a musical performance. The former helps in revealing features of a musical piece in general, while the latter brings the style of the artist into consideration as well in performing the musical piece. In performance data, our approach for structure analysis (see Chap. 4) again works with the additional job of note detection using signal processing. Performance data has other features demanding additional analysis. For example, Chakraborty et al. (2009a) have used inter-onset-interval graphs to study the rhythmic properties of notes in a vocal rendition of raga Ahir Bhairav. Rhythmic and melodic properties of notes can also be studied using RUBATO. Chakraborty et al. (2008) have given the first use of this software in North Indian music. RUBATO has been extensively covered in Chap. 3 of this book, and its use in Indian classical music is explained with illustrations in two ragas Bhairav (which is the first raga of the morning) and Bihag (a night raga) in Chap. 8 of this book which is devoted to performance analysis. Further literature on ragas can be found in Jairazbhoy (1971). Western art music (WAM) readers who are new to Indian music are referred to Jones (http://cnx.org/content/m12459/1.6/). Here are some fundamental idea of the musical terms: • Raga A raga, the nucleus of Indian classical music, is a melodic structure with fixed notes and a set of rules characterizing a certain mood conveyed through performance (Chakraborty et al. 2009a) • Swar Sa, Re, Ga, Ma, Pa, Dha, and Ni • • • • •

Vadi: The most important note in a raga Samvadi: The second most important note in a raga Anuvadi: The important but not vadi or samvadi, note in a raga Alpvadi: The unimportant or weak note in a raga Vivadi: The non-permissible note in a raga

• Melody A sequence of notes “complete” in some musical sense. For example, {Sa, Sa, Re, Re, Ga, Ga, Ma, Ma, Pa} is a melody in raga Kafi. • Segment A sequence of notes which is a subset of melody but is itself incomplete. For example, {Sa, Sa, Re, Re} is a segment in raga Kafi.

2.1 Modeling

17

• Length The number of notes in a melody or its segment. • Significance of a melody The product of the length of the melody and the number of times it occurs in the musical piece. This formula is recommended for monophonic music (single melody line) such as Indian music. For polyphonic music (multiple melody lines), one can use the formula used by Adiloglu et al. (2006). • Shape The difference of successive pitches of the notes in a melody or a segment. Here are some of the ways statistical analysis is helpful in music appreciation from a scientific domain:

2.1

Modeling

One of the strengths of statistics lies in modeling. This is because of the following reasons: 1. Although these models are subjective and biased, we can at least make the data objective as far as possible. Also, behind these models stand some beautiful mathematical theorems, and they are unbiased. 2. The true model could be both complex and unknown. This is particularly true in music where even the composer does not know which model is actually generating the composition! Moreover, the true model may contain multiple parameters related to music theory, the training and background of the artist, the genre of music, and even the place of performance and the audience, and we do not have an explicit idea as to how exactly (in what functional form) these parameters enter the model. Statistical models are, in contrast, approximate models that use fewer parameters to capture the phenomenon generated by these complex unknown true models. Although approximate, it is possible to verify the goodness of fit of these models as well as control the errors in them. See Klemens (2008). In the light of (1) and (2), modeling a musical structure or a musical performance has been a coveted research area in computational musicology. There are three fundamental steps in statistical modeling: deciding which model to fit, estimating the parameters of the chosen model, and verifying the goodness of fit of this model. We all know that statistics can be broadly divided into two categories: descriptive and inferential. In statistical modeling, both are involved—as we first describe a pattern (through modeling) and then infer about its validity. Two types of models are used in statistics: probability models and stochastic models. Through a probability model, we can tell the probability of a note or a note combination but cannot predict the next note. Through a stochastic

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model, we can predict (make an intelligent guess of) the next note, given the previous. Apart from modeling, there are various other areas in which statistics is applicable in music analysis. They are briefed as follows:

2.2

Similarity Analysis

Two melodies are in translation if the correlation coefficient r of their shapes equals +1. Two melodies are in inversion if the correlation coefficient r of their shapes equals
1. Two melodies are called different if the correlation coefficient of their shapes approaches 0. Thus, correlation coefficient here is a measure of similarity between melodies. • Significance of correlation coefficient of two melodies can be tested using t-test. • The calculation of correlation coefficient r and how to test its significance are explained next. Significance of a correlation coefficient can be tested using t-test. If r is the value of correlation coefficient and n be the number of pairs of observations (here pffiffiffiffiffiffiffiffiffi ðn
2Þ p successive differences), we calculate the statistic t ¼ r ffiffiffiffiffiffiffiffiffiffi . If jtj exceeds the 2 ð1
r Þ

table value of t at 5 % level of significance (say) and (n
2) degrees of freedom, then the value of r is significant at 5 % level otherwise insignificant. Here it is assumed that the n pairs are coming from a bivariate normal distribution. The formula ðx;yÞ for r is covariance fsdðxÞsdðyÞg where sd ¼ standard deviation. Covariance (x, y) can be computed rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   hn x  xo iffi Σð x  y Þ and simix  y sd ð x Þ ¼ þ Σ x  x easily as

f g. f g n n Σð x Þ Σð y Þ larly for sd( y). x ¼ n and y ¼ n . Since the t has (n
2) degrees of freedom, we must have n > 2. This means melodies of lengths at least 4 (whose shapes will have at least n ¼ 3 successive pitch values) can be compared for similarity.

2.3

Rhythm Analysis

The notes are said to be in rhythm if the inter-onset times between successive notes is equal. This can be easily detected by an inter-onset-interval (IOI) graph where the term onset refers to the point of arrival time of a note (the idea also applies to beats in a percussion instrument). If mean IOI is less, it implies that notes have come more rapidly in the recording. If standard deviation is less, it implies that there is more rhythm in the notes.

2.5 Multivariate Statistical Analysis

2.4

19

Entropy Analysis

If P(E) is the probability of an event, the information content of the event E is defined as I(E) ¼
log2(P(E)). Events with lower probability will signal higher information content when they occur. If X is a random variable, the information content of X is also random which we denote by I(X). Its mean value is called entropy. It should be emphasized that entropy is measuring surprise (not meaning!). The mean value of I(X) called its entropy, denoted by H(X), so that we have   H ðXÞ ¼
Σpj log2 pj where the summation is over j ¼ 1 to n. Chakraborty et al. (2011) have introduced entropy in raga analysis.

2.5

Multivariate Statistical Analysis

Multivariate statistical analysis consists of cluster analysis, principle component analysis, discriminant analysis, multidimensional scaling, etc., which have applications in music. For details, we refer the reader to Prof. Jan Beran’ book Statistics in Musicology (Chapman and Hall/CRC, 2004). In the words of Prof. Beran: “The primary aim of descriptive statistics is to summarize data by a small set of numbers or graphical displays, with the purpose of finding typical relevant features. An in-depth descriptive analysis explores the data as far as possible in the hope of finding anything interesting. This activity is therefore also called “exploratory data analysis” or data mining. Observations in music often consist of vectors. Consider, for instance, the tempo measurements for Schumann’s Tra¨umerei. In this case, the observational units are performances, and an observation consists of a tempo “curve” which is a vector of n tempo measurements x(ti) at symbolic score onset times ti (i ¼ 1, . . ., p). The main question is which similarities and differences there are between the performances. Principal component analysis (PCA) provides an answer in the sense that the “most interesting,” and hopefully interpretable, projections are found. Discriminant analysis, often also referred to under the more general notion of pattern recognition, answers the question of which category an observed item is most likely to belong to. A typical application in music is attribution of an anonymous composition to a time period or even to a composer. Other examples are discussed below. A prerequisite for the application of discriminant analysis is that a “training data set” is available where the correct answers are known. In discriminant analysis, an optimal allocation rule between different groups is estimated from a training sample. The type and number of groups are known. In some situations, however, it is neither known whether the data can be divided into

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homogeneous subgroups nor how many subgroups there may be. How to find such clusters in previously ungrouped data is the purpose of cluster analysis. In music, one may, for instance, be interested in how far compositions or performances can be grouped into clusters representing different “styles.” In some situations data consist of distances only. These distances are not necessarily Euclidian so that they do not necessarily correspond to a configuration of points in a Euclidian space. The question addressed by multidimensional scaling (MDS) is in how far one may nevertheless find points in a hopefully low-dimensional Euclidian space that have exactly or approximately the observed distances. The procedure is mainly an exploratory tool that helps to find structure in distance data” (Beran 2004).

2.6

Study of Varnalankars Through Graphical Features of Musical Data

It is held that the emotions of the raga, in Indian classical music, depend not only on the note combinations, as are typical of the raga, but also on other features like the pitch movements taking place between two successive notes, the note duration, the pause between the notes, etc. The shastras have categorized alankars into varnalankar and shabdalankar. The varnas include sthayi (stay on a note), arohi (ascent or upward movement), awarohi (descent or downward movement), and sanchari (mixture of upward and downward movement). For further details, refer to http://www.itcsra.org/alankar/alankar.html.

2.7

The Link Between Raga and Probability

When an artist is rendering a raga, it is not possible to say definitely which note will come next. One can, still, assign a probability for a particular note to come. Now, let us fix our attention to a particular note, say, Sa (or Do), the tonic. One question of interest is: at every instance, does Sa maintain the same probability of coming? The same question can be raised for every other note permissible in the raga. If this probability is fixed, our model is multinomial, otherwise quasi-multinomial. Additionally, independence of notes overall is also required, but this is weaker than mutual independence and hence, as we shall see, generally fulfilled. If the performer is a novice and can use a wrong note (vivadi or varjit swar) by mistake, such a possibility W (for wrong) should also be kept to make the list of possible notes exhaustive. One then asks for W’s probability as well! This is not required here as we are analyzing an accomplished vocalist. Since a note can be vadi, samvadi, anuvadi, alpvadi, or vivadi with respect to a raga, it is logical that its probability is raga dependent. For example, Pa being alpvadi in Bageshree will have a

2.8 Statistical Pitch Stability Versus Psychological Pitch Stability

21

Table 2.1 Information content of Pancham swar (Pa) in five different ragas

Raga

Relative occurrence of Pancham swar out of first 100 notes ¼ p

Classification of Pancham swar in the concerned raga

Information content ¼
log2( p)

Bageshree Malkauns Kafi Bhupali Desh

2/100 ¼ 0.02 0/100 ¼ 0 18/100 ¼ 0.18 19/100 ¼ 0.19 19/100 ¼ 0.19

Alpvadi Vivadi Vadi Anuvadi Samvadi

5.6439 Infinitya 2.4739 2.3959 2.3959

a

Sometimes a vivadi swar is used at a “peak point” in a performance to create romanticism (unrestricted beautification) in contrast to classicism (disciplined beautification). Well, this only proves that an event with zero probability can be possible! In such a case, the probability of a vivadi swar, theoretically zero, will have some nonzero though very small value empirically. Accordingly, entropy will be calculated (Tewari and Chakraborty 2011). For more on romanticism and classicism debate, see the views of Dr. Vanamala Parvatkar, a renowned vocalist of Banaras Gharana (school of music) and formerly Head, Faculty of Performing Arts (vocal), Banaras Hindu university, Varanasi documented in Chakraborty et. al (2010).

small probability, but in Kafi the same note has a high probability (one musical school holds Pa as vadi in Kafi; another school holds Komal Ga as vadi; even if Pa is not a vadi, definitely it is anuvadi). Again, Pa is vivadi for Malkauns, and hence its probability is zero (Chakraborty et al. 2009a). This will be clear from Table 2.1. In the book Music and Probability (MIT Press, 2007), David Temperley has investigated music perception and cognition from a probabilistic point of view with an extensive usage of Bayesian techniques. The most important note statistically speaking is not necessarily the one having the highest probability but one having a high probability that is maintained consistently. This concept of statistical pitch stability is different from Krumhansl’s psychological pitch stability based on note duration. Both concepts can be combined to give rise to psycho-statistical pitch stability by ranking the notes differently using the two contrasting concepts and then giving the notes the average rank (Tewari and Chakraborty 2011). Note, however, that probability does not directly determine the decision process of the artist (this decision process is deterministic as music, even extempore, is planned and not random) but still is important from an analyst or a listener’s point of view especially in a raga performance where scores (musical notation) are not fixed (Chakraborty et al. 2009c).

2.8

Statistical Pitch Stability Versus Psychological Pitch Stability

An empirical definition of probability implies that probability is relative frequency in the long run. This is so as it is in the long run that this relative frequency stabilizes which fluctuates initially. A musical note is statistically stable if it maintains its relative frequency of occurrence in the musical piece consistently. From this, (1) it seems that the most important note in a raga is not necessarily one

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which is used most often (as this could well be a stay note or nyas swar like Sudh Re in Yaman) but one whose relative frequency, apart from being high (though not necessarily highest), stabilizes faster in a short period of time. Lesson (1) leads to lesson (2) that controversies which still exist among Indian musicians as to which notes should be Vadi and Samvadi in certain ragas like Bageshree and Shankara can perhaps be resolved now. In Bageshree, the Vadi and Samvadi Swars are Sudh Ma and Sa according to some experts, whereas others say they are Sudh Dha and Komal Ga. Similarly in raga Shankara, they should be Sudh Ga and Sudh Ni according to some, while others say they should be Pa and Sa, respectively. The other two lessons are that (3) if one has a recording of longer duration, it still makes sense to sample a small subset from it for analysis—the advantage with a longer recording, however, is that there can be several such samples from different positions reflecting perhaps different moods of the performer in rendering the raga, and (4) we need to have a recording of sufficiently longer duration to get more precise values of probabilities of the less important notes (as they take some time to settle). This concept of statistical stability is different from the concept of psychological stability of Krumhansl based on note duration (Castellano et al. 1984). For example, the tonic Sa is always a psychologically stable note. So is Pa except in ragas where it is vivadi or alpvadi. Statistical stability is a good tool for detecting the VadiSamvadi scientifically in case of conflict, while psychological stability detects the nyas swars or stay notes. Since the Vadi-Samvadi are generally also nyas swars, we recommend Krumhansl’s technique to reduce the search space and then investigate with statistical stability (Tewari and Chakraborty 2011).

2.9

Statistical Analysis of Percussion Instruments

Musicologists regard handclapping and African drumming as a motor activity where a hand or an arm is raised and then dropped (to strike the instrument in the latter case). According to Jay Rahn, one possible mechanism for the tacit motor mediation of attack points of onsets is the peaking of the gesture at the temporal midpoint of the two sounds. He calls the sequence of midpoints of the onsets of a rhythm the shadow of the rhythm. There is a beautiful piece of mathematics associated with the shadow rhythm as the question “what happens to the sequence when we continue to perform this operation on every rhythm resulting from the shadow of another?” turns out to be a fascinating geometrical problem. In the paper (see Chakraborty et al. 2009b and the references cited therein), we take a close look at the shadow graphs for sets of interval 10 s each considering the first 3 min of a 7-min tabla recording played by a renowned Indian musician. This is followed by metric and melodic analysis of the percussion performance. In melodic analysis, the timbre parameters (tristimulus 1, 2, and 3, spectral brightness, odd and even parameters, irregularity, spectral centroid, and spectral inharmonicity) are being calculated from the spectrum of sound signals of tabla. The most important features

References

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for the melodic properties are spectral inharmonicity, irregularity, brightness, and centroid. The pitch of each stroke and the attack time for each stroke were calculated. The correlation between the parameters was studied. N. B. There are many more recent publications of using statistics and other quantitative and/or empirical methods in musicology. For instance, see Luke Windsor’s “Data collection, experimental design and statistics in music research” (in Empirical Musicology, edited by Clarke and Cook, Oxford University Press, 2004) and Eerola, T. (2012), Computational modeling of emotions conveyed by music. Topics in Cognitive Science, 4(4), 607–624, etc. Other contributors include Geraint A. Wiggins and Marcus T. Pearce, and we refer the reader to The Oxford Handbook of Computer Music edited by Roger T. Dean, Oxford University Press, USA, 2009.

References K. Adiloglu, T. Noll, K. Obermayer, A paradigmatic approach to extract the melodic structure of a musical piece. J. Music Res. 35(3), 221–236 (2006) D. Benson, Music: A Mathematical Offering (Cambridge University Press, New York, 2007) J. Beran, Statistics in Musicology (Chapman and Hall/CRC, Boca Raton, FL, 2004) J. Beran, G. Mazzola, Analyzing musical structure and performance – a statistical approach. Stat. Sci. 14(1), 47–79 (1999) M.A. Castellano, J.J. Bharucha, C.L. Krumhansl, Tonal hierarchies in the music of north India. J. Exp. Psychol. Gen. 113(3), 394–412 (1984) S. Chakraborty, S.S. Solanki, S. Roy, S.S. Tripathy, G. Mazzola, A statistical comparison of performance two ragas (dhuns) that use the same notes, in Proceedings to the International Symposium on Frontiers of Research in Speech and Music (FRSM2008), held at Sir C. V. Raman Centre for Physics and Music, Jadavpur University, Kolkata on 20–21 Feb 2008, pp. 167–171 S. Chakraborty, R. Ranganayakulu, S. Chauhan, S.S. Solanki, K. Mahto, A statistical analysis of raga Ahir Bhairav. J. Music Meaning 8, sec. 4 (Winter 2009a), http://www.musicandmeaning. net/issues/showArticle.php?artID¼8.4 S. Chakraborty, R. Ranganayakulu, S.S. Solanki, A. Patranabis, D. Ghosh, R. Sengupta, A.K. Dutta, N. Dey, Shadowing along the maestro’s rhythm. Revista eletronica de musicologia (Electron. Musicol. Rev.) XII (2009b) S. Chakraborty, M. Kumari, S.S. Solanki, S. Chatterjee, On what probability can and cannot do: a case study in Raga Malkauns. J. Acoust. Soc. India 36(4), 176–180 (2009c) S. Chakraborty, K. Krishnapryia, K. Loveleen, S. Chauhan, S.S. Solanki, K. Mahto, Melody revisited: tips from Indian music theory. Int. J. Comput. Cognit. 8(3), 26–32 (2010) S. Chakraborty, S. Tewari, G. Akhoury, S.K. Jain, J. Kanaujia, Raga analysis using entropy. J. Acoust. Soc. India 38(4), 168–172 (2011) P. Chordia, A. Rae, Raag recognition using pitch-class and pitch-class dyad distributions, in Proceedings of International Conference on Music Information Retrieval, 2007 http://www.itcsra.org/alankar/alankar.html N.A. Jairazbhoy, The rags of North India: Their Structure and Evolution (Faber and Faber, London, 1971) C.S. Jones, Indian classical music, tuning and ragas, http://cnx.org/content/m12459/1.6/ B. Klemens, Modeling with Data: Tools and Techniques for Scientific Computing (Princeton University Press, Princeton, NJ, 2008)

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L.B. Meyer, Style and Music: Theory, History, and Ideology (University of Pennsylvania Press, Philadelphia, PA, 1989) (Re statistics: 57–65) J. Morehen, Statistics in the analysis of musical style, in Proceedings of the Second International Symposium on Computers and Musicology, Orsay (CNRS, Paris, 1981), pp. 169–183 N. Nettheim, A bibliography of statistical applications in musicology. Musicol. Aust. 20, 94–106 (1997) P. Sinha, Artificial composition: an experiment on Indian music. J. New Music Res. 37(3), 221–232 (2008) J.L. Snyder, Entropy as a measure of musical style: the influence of a priori. Assumptions Music Theory Spectr. 12, 121–160 (1990) D. Temperley, Music and Probability (MIT Press, Cambridge, MA, 2007) S. Tewari, S. Chakraborty, Linking raga with probability. Ninad J. ITC Sangeet Res. Acad. 25, 25–30 (2011) P.M. Todd, D.G. Loy, Music and Connectionism (MIT Press, Cambridge, MA, 1991) S.R. Wilson, R. Sue, Sound and exploratory data analysis, in Compstat Conf. Part I: Proc. in Compu. Stat., ed. by H. Caussinus, P. Ettinger, R. Tomassone (Physica, Vienna, 1982), pp. 447–450 W. Winkler, Statistical analysis of works of art. Bull. Int. Stat. Inst. 44(2), 410–415 (1971)

Chapter 3

Introduction to RUBATO: The Music Software for Statistical Analysis

The most rigorous test of the efficiency of theories in modern cognitive science is the production of a working computer program whose external behaviour mimics that to be explained. John Sloboda (1985)

Summary RUBATO® is a metamachine designed for representation, analysis, and performance of music. It was developed on the NEXTSTEP environment during two SNSF grants from 1992 to 1996 by the author and Oliver Zahorka (Mazzola and Zahorka 1993–1995, 1994; Mazzola et al. 1995b, 1996; Zahorka 1997a, b). From 1998 to 2001, the software was ported to Mac OS X by Jo¨rg Garbers in a grant of the Volkswagen Foundation. RUBATO®’s architecture is that of a frame application which admits loading of an arbitrary number of modules at run-time. Such a module is called RUBETTE®. There are very different types of Rubettes. On the one hand, they may be designed for primavista, compositional, analytical, performance stemma, or logical and geometric predication tasks. On the other, they are designed for subsidiary tasks, such as filtering from and to databases, information representation, and navigation tasks, or else for more specific subtasks for larger “macro” Rubettes. A RUBETTE® of the subtask type is coined OPERATOR and implements, for example, what we have called performance operators in section Mazzola et al. (2002, 44.7). The RUBATO® concept also includes distributed operability among different peers. This software is conceived as a musicological research platform and not a hard-coded device; we describe this approach. Concluding this chapter, we discuss the relation between frame and modules.

3.1

Architecture

In the original concept of RUBATO® (Mazzola 1993), we had defined RUBATO® as being a software for analysis and performance, divided into two submodules: one for “structuring” a score and the other for “shaping” this score. This meant that © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_3

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.1 The info panel of the Mac OS X version of RUBATO®

structuring would yield analytical structures, whereas the other would yield a shaped performance transformation, alimented by analytical data from the structuring process. In the course of the software development, we learned that no data model for music objects known to the developers at that time would be sufficient for all requirements of a comprising music analysis and performance. This led to the concept of denotators and forms, as realized under the title of the “PrediBase” database management system (DBMS) of the first RUBATO® implementation in 1994, as described in (Zahorka 1995). At that time, it became clear that under such a universal data model, RUBATO® would split into an application framework comprising the “PrediBase” DBMS and a series of dynamically loadable software modules as implemented in the Objective C language of those NEXTSTEP OS driven NeXT computers. According to the diminutive convention for modules, such a module was coined “RUBETTE®.” In 1996, at the end of a grant of the Swiss National Science Foundation where RUBATO® was realized, three analytical Rubettes and one for performance have been developed1, which will be described in Sect. 3.2. The PerformanceRUBETTE® is connected to five OperatorRubettes (at those times still named “OPERATORS”). The PrimavistaRUBETTE® takes care of paratextual score predicates. In 2001, this software has been ported (and improved in many data management aspects) to Mac OS X by Jo¨rg Garbers and is now available as an open-source project on the internet (Mazzola et al. 1996) or on the CD appended to this book; see Mazzola et al. (2002, p. xxx). The RUBETTE® screenshots in Sect. 3.2 are all taken from this version of RUBATO®. Figure 3.1 shows the info panel of the Mac OS X version. However, the version included in the book’s CD-ROM is the latest version before the book went into production, whereas the screenshots are somewhat older. We hope that the reader will excuse us for this slight asynchronicity.

For NEXTSTEP, RUBATO® as well as these Rubettes is available on the internet; see (Mazzola et al. 1996). The source code is GPL and is contained in the book’s CD-ROM; see Mazzola et al. (2002, p. xxx). 1

3.1 Architecture

27

In the following sections of this chapter we shall however not describe the Mac OS X implementation, we will rather expose the more advanced and flexible architectural principles of the ongoing Java-based implementation of the distributed RUBATO® version.

3.1.1

The Overall Modularity

Summary RUBATO® is a modular engine for metamachine rationales and because research is itself increasingly modular. Built upon the denotator language, the RUBATO® concept is fully modular; all parts that can be split into modules have been split in this way. The modularity of RUBATO® has two aspects: First, it shows a composition of the software from an arbitrary, a priori undetermined, number of functional units—the RUBETTE® modules. Second, the available Rubettes are a dynamic factor: According to the research progress, new Rubettes of any flavor may be added to the existing arsenal. Modularity is an old principle, in fact, the traditional disciplinarity of science preconizes modules of scientific activities which have or pretend to have a relative autonomy in knowledge production. What is new with respect to the traditional disciplinarity is that this modularity is a dynamical one; at any time new modules of knowledge processing may be added or old ones removed. Discipline becomes a task-driven decision instead of being a rigid preset splitting. Such a modularization can only work on the common ground of unrestricted cross-communication among any subgroup of knowledge modules. Without a common language ground, which in our case is the denotator and form data model, dynamic disciplinarity would inevitably collapse since a new module would require language modifications, adaptations, and extensions. To be clear, we view dynamic disciplinarity as the idealized version of inter- and transdisciplinarity. The unity of knowledge cannot be achieved without a temporary and task-driven compartmentation of research fields—grouping and regrouping is inevitable; there is no direct path to the unity of knowledge. This credo can however not be realized without a common language basis. Otherwise, language engineering frictions would paralyze any major effort of dynamical grouping of disciplines. Evidently, the present RUBATO® environment is limited to musical and musicological scopes. Here dynamic disciplinarity is not that utopic. After all, the denotators and forms are universal language approaches issued from music (ologic)al requirements. But it has been shown in (Ru¨etschi 2001) that denotators are not only a priori applicable to non-musical concept modeling but also in concrete cases such as geographic information systems. This suggests that dynamic disciplinarity could be realized on RUBATO® for modules of completely general scopes as long as the denotator data model is joined. This is one of the most intriguing vectors of future developments concerning the RUBATO® environment.

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

The principle of dynamic disciplinarity has its social form: a so-called collaboratory. According to Bill Wulf, this is “a ‘center without walls’ in which the nations researchers can perform their research without regard to geographical location, interacting with colleagues, accessing instrumentation, sharing data and computational resources, and accessing information in digital libraries” (Kouzes et al. 1996). To collaborate in this way requires adequate software platforms, and RUBATO® is precisely this type of software in the field of musicology.

3.1.2

Frame and Modules

Summary Modularity has been realized on the basis of a frame application which offers interfaces to an arbitrary number of modules. This is one of the technical core features in the realization of a metamachine. We describe its splitting interfaces and their functional positions. The RUBATO® platform consists of a number of installations of the software on different peers which may communicate via Java’s remote method invocation protocol (RMI, see SUN’s Java documentation on the Internet). For each peer, RUBATO® contains two layers: the RUBETTE® layer and the RUBATO® framework layer. The first contains a number of Rubettes which are autonomous Java applications that communicate with each other and with Rubettes of another peer exchanging denotators via RMI. These are instances of the denotator class. The class library on the RUBATO® framework layer contains corresponding basic Java classes for denotators, forms, diagrams of presheaves, and modules. It also contains other classes which provide Rubettes with the necessary routines. The concept of these libraries is that they should contain all classes and methods that are of general interest, while classes and methods with specific interest for a RUBETTE®’s functionality should be installed in that RUBETTE®. There are a number of mandatory Rubettes: The InfoRUBETTE® (with an “i” in Fig. 3.2) is the initialization RUBETTE®. It is automatically started when RUBATO® starts and informs the user about available peers and Rubettes on the distributed environment. The visualization of all Rubettes’ content and manipulation structures is managed by the PVBrowserRUBETTE®, whose functionality is to visualize any denotator in 3D space via Java3D classes; in Fig. 3.2, this RUBETTE® is represented by a lens symbol. The concept of this RUBETTE® has been described in Mazzola et al. (2002, 20). The advantage of this centralization is that no other RUBETTE® designer has to take care of graphical and other multimedia representations; every denotator is piped to the PVBrowserRUBETTE® in case a multimedia representation is required. And every such representation is uniform according to this Rubette’s visualization routines, which makes orientation much easier than individual design for every RUBETTE®. Nonetheless, as explained in Mazzola et al. (2002, 20), the flexibility of the Satellite form for multimedia objects allows an unlimited multiplicity of shape and behavior.

3.2 The RUBETTE® Family

29

FS

FS

PEER 3

PEER 2

DBMS

DBM S DTX

Rubettes

DTX

LoGeo P Prriim ma aV Viisstta a

Rubettes

Rubato

RMI

LoGeo

m V m V Pr i ma Viissttaa

Rubato

Class Libraries

Class Libraries

RMI

RMI

FS

PEER 1

DBMS

DTX

Rubettes PP m m aa VVViissttaa Pr iim

Rubato

LoGeo

Class Libraries

Fig. 3.2 The RUBATO® layers of Rubettes with denotator communication and the RUBATO®Framework layer with class libraries that are related to different Rubettes. Each peer has one such configuration. Different peers are interconnected via remote method invocation (RMI)

Two further Rubettes are devoted to the storage of denotators. The first, to the upper left of the RUBETTE® layer, we have the DenotexRUBETTE®. It takes care of the storage and editing of denotators which are given in the Denotex ASCII format. The second storage RUBETTE® is a filter to a SQL DBMS such that SQL databases can be transformed into denotators for RUBATO®. A last central RUBETTE® is the LoGeoRUBETTE®, as shown by a toothed wheel symbol to the right front of the RUBETTE® layer. It manages the logical and geometric operations on denotators (see Mazzola et al. 2002, 18.3.4) and can be used by any RUBETTE® for its specific needs. The methods of this RUBETTE® are encoded in the class library of the RUBATO® layer.

3.2

The RUBETTE® Family

Vo¨gel, Vieh und alles, was auf Erden kriecht, die lass heraus mit dir, dass sie sich tummeln auf der Erde und fruchtbar seien und sich mehren auf Erden. The Holy Bible, Genesis 8

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Summary We give an overview of the analytical MetroRUBETTE®, MeloRUBETTE®, HarmoRUBETTE®, the PerformanceRUBETTE®, and the PrimavistaRUBETTE®, which have been realized on the NEXTSTEP and then on the Mac OS X environment. Originally, the Rubettes as such were not the central concern of the RUBATO® project; this was rather to establish their collaboration and the realization of the whole transmission process from analytical data to the performance shaping operators of the PerformanceRUBETTE®. Each of these Rubettes was more an experimental prototype without the claim of a high-end tool in the specific domain. The interest in such experiments lies in the fact that when one starts the design of a RUBETTE®, it turns out that musicology and music theory do not offer any reasonable support, be it in conceptual, be it in operational aspects. The path from the given score to a specific analysis reveals an incredible complexity of what in musicology and music theory looks like an easy enterprise. For example, in the design of the HarmoRUBETTE®, the mere definition of what is a chord cannot be traced from traditional literature. Should we only look for local compositions of pitches that stem from notes with a common onset, or should one also consider durational aspects? The standard answer—or rather excuse—states that it depends on the particular context, and the context of the context, but this is no way out if one has to implement clear concepts and methods rather than rhetorics. So the design of a RUBETTE® is always a very good test of the validity of a model and of its adequacy with traditional fuzzy understanding. But it is also a test for the tradition: After all, who decides what is a good model for harmony? Here, the alternative between general speculative nonsense theories and concrete, but possibly non-sufficiently general implementation and operationalization, becomes dramatic. At least, one can hope that this confrontation will force everybody to rethink ill-defined approaches.

3.2.1

MetroRUBETTE®

Die Za¨hlzeiten (Schlagzeiten, rhythmische Grundzeiten) gewinnen unter allen Umsta¨nden erst reale Existenz durch ihre Inhalte. Hugo Riemann (Riemann 1903)

Summary The MetroRUBETTE® is an elementary analysis module which shows that seemingly simple approaches yield complex but informative results. We also make evident that operationalization of abstract concepts reveals unexpected insights into generically not foreseeable structures. The MetroRUBETTE® is built on the sober weight calculation which we developed in Mazzola et al. (2002, 21) in the frame of the maximal meter nerve topologies. The formula in example (Mazzola et al. 2002, 43) of section (Mazzola et al. 2002, 21.2) is realized, except that there is X no upper length limitation (i.e., it is put to 1). The mixed weight formula W ðxÞ ¼ W ðxÞ is also realized in X 2Sp ðxÞ i i

I

3.2 The RUBETTE® Family

31

Fig. 3.3 The 31-part score denotator deduced from Richard Wagner’s composition “Go¨tterda¨mmerung”

this RUBETTE®. The input is a score denotator (although still in the early shape of the PrediBase data model, which is the very special denotator form of a list of lists, which start from the simple form of strings). Figure 3.3 shows the 31-part score deduced from Richard Wagner’s composition “Go¨tterda¨mmerung.” This data is imported to the RUBETTE® and then evaluated according to the said formulas. For example, the weight of the union of all the onsets of the 31 parts is shown in Fig. 3.4. Its parameters are: profile ¼ 2, minimal length ¼ 2, and the unit of the onset grid in the graphical representation of the weight is 1/8. The mixed weight is also an option of this RUBETTE®. Figure 3.5 shows the mixture of the parts: clarinet (part 11), bassoon (part 16), horn (part 14), and violin (part 27). The parameters are same as above; the weights are scaled by the factors 3, 2.5, 1, and 0.1 for the respective parts. For a more in-depth application of this RUBETTE® in musicology, we refer to Fleischer et al. (2000). This paper is an excellent example of an unexpected musicological application of a RUBETTE® which was not intended to be interesting on its own. The metrical analysis turns out to be quite sophisticated, although the concept of a local meter is a very elementary one. The surprising effect of this setup is that the combination of the simplistic concept reveals unprecedented insights into the time structure of classical works. This is a good hint to the musicologists, teaching them that interesting insight can result from a complex

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.4 The weight graphics of the above composition with profile ¼ 2, minimal length ¼ 2. Horizontal axis: onset, vertical axis: (relative) weight. The unit of the onset grid in the graphical representation of the weight is 1/8

Fig. 3.5 The mixed weight graphics of four parts clarinet (part 11), bassoon (part 16), horn (part 14), and violin (part 27) with scaling factors 3, 2.5, 1, and 0.1 for the respective parts

aggregation of simple ingredients. In this case, the simple elements are provided by the maximal local meters, whereas their complex aggregation is conceived by the nerve of the covering they define. A second very interesting application of this RUBETTE® has been presented in Mazzola et al. (1995a). It was recognized that the longest possible minimal lengths of local meters for the left-hand part of Schumann’s “Tra¨umerei” yields a 3 + 5 quarters periodicity over two bars, whereas the same analysis of the right hand yields the expected periodicity of four quarters with stress on the bar lines. The sonification of this fact can be heard on the audio example; see the CD-ROM in Mazzola et al. (2002, p. xxx). In this RUBATO® version, however, the output is a weight which is by no means a denotator. This output is a final data and must be used in its special format. This will be the case for operators of the PerformanceRUBETTE®. For the Java-based distributed RUBATO® (Fig. 3.2), such a restrictive usage would be forbidden.

3.2 The RUBETTE® Family

3.2.2

33

MeloRUBETTE®

In general, the author does not believe in the possibility or even desirability of enforcing strict musical definitions. Rudolph Reti on the concept of a motif (Reti 1978)

Summary The MeloRUBETTE® is an excellent example of the tension between abstract concepts and operational implementation. We expose the routines for motivic analysis and the interface concept, and we discuss the performance problem, including proposals for performance improvement and their theoretical limits. The MeloRUBETTE® refers to the theory of motivic topologies in Mazzola et al. (2002, 22) and, in particular, to Mazzola et al. (2002, 22.9) about motivic weights. The score is loaded as for the MetroRUBETTE®; its projection to the onset-pitch space is then analyzed and yields a numeric weight for each onset-pitch event. One such weight is visualized in Fig. 3.6. It corresponds to part 30 (celli) of the above composition “Go¨tterda¨mmerung.” The weight values are encoded in gray levels of the disks which represent the events in onset and pitch. The calculation relies on these parameters which relate to melodic topology: • Symmetry Group. This is the paradigmatic group of the shape type. In each group, we include the translation group in pitch and onset. The choice is then between the translation group (encoded by “trivial”), the one generated by the translations plus the retrograde, or the one generated by the translations plus inversion, and the full counterpoint group, i.e., generated by the inversion and retrograde over the translations. • Gestalt Paradigm. This is one of three possible shape types: diastematic, elastic, and rigid. The first candidate is in fact the type which we called “diastematic index shape type” in Mazzola et al. (2002, 22). Observe that we have no topology for the diastematic type but may nevertheless define neighborhoods! (Fig. 3.7).

Fig. 3.6 The weight graphics for the celli part 30 in the above score denotator deduced from Richard Wagner’s “Go¨tterda¨mmerung”

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.7 The main window of the MeloRUBETTE®

• Neighborhoods. This is the neighborhood radius E which was used in the expression DμE (M ) in Mazzola et al. (2002, 22.9). • Span. This is the maximal admitted onset difference between motive events. • Cardinality. This is the maximal admitted cardinality of motives. Together with the Span, this condition defines the selection μ of motives which is addressed in the approach from Mazzola et al. (2002, 22.9). The presence and content functions are defined as follows: 1. Presence. For a motif M, the presence value prμ,E(M ) is the sum of all these numbers: for each N 2 DμE (M ), we count the number m of times where M has a submotif M0 of N that at a distance less than E from M. We also look for the difference of cardinalities d ¼ card(N)  card(M ). This gives a contribution p (N ) ¼ m  2 d, and we add all these numbers. 2. Content. Similarly, for each motif N 2 μ such that M 2 DμE (N), we take p(N ) ¼ m  2 d with d ¼ card(M )  card(N ) and m the number of times, where N has a submotif N0 of M that at a distance less than E from N. We add all these numbers p(N ) and obtain the content ctμ,E(M ). 3. Weight. Given a motif M 2 μ, this is the product

3.2 The RUBETTE® Family

35

nW E ðMÞ ¼ pr μ, E ðMÞ  ctμ, E ðMÞ; i.e., taking the function ω(x,y) ¼ xy from Mazzola et al. (2002, 22.9). We have already given musicological comments on this construction in Mazzola et al. (2002). It is however remarkable to see the overwhelming amount of calculations which arise in this routine. For example, we have calculated the number C of comparisons of motives (for distance measurements) Schumann’s “Tra¨umerei” (“Kinderszene” number 7) which comprises 463 notes. If we take Span ¼ 1/2 bar and Cardinality ¼ 4, we obtain 25,745 motives and C ¼ 1,023,490,904  1.023  109. This is beyond any explicit human calculation power. It demonstrates that the task of finding a dominant motif is a very hard one, and that this one, if it is recognized by a human listener, can at most be present in a very hidden layer of consciousness. This becomes even more dramatic for larger pieces, such as an entire sonata, say! Here, the combinatorial extent of motivic units exceeds any calculation power of humans and machines, as is easily verified. This means that a huge composition bears a motivic complexity that will escape to (human or machine-made) classification forever. Nonetheless, the usage of statistical methods, of simplified approaches to motivic topology, or of topological invariants that are more easily perceived could help find a rough orientation in the virtually infinite motivic variety of music. This implementation also makes evident the tension between fuzzy concepts in musicology and implementation of a precise model. Although the concept of a motif is rather elementary, it entails a very sophisticated motivic analysis which could eventually converge to the intensions hidden2 behind those fuzzy motive theories.

3.2.3

HarmoRUBETTE®

Eine Theorie aber, die gerade dort versagt, wo auch das Pha¨nomen, das sie erkla¨ren soll, ins Vage und Unbestimmte gera¨t, darf als ada¨quat gelten. Carl Dahlhaus on Hugo Riemann’s harmony (Dahlhaus 1966)

Summary The HarmoRUBETTE® makes clear that a vague theoretical approach does not reflect a vague phenomenon but an extremely complex one. The implementation of this RUBETTE® reveals several deep deficiencies of traditional “messy” analysis in harmony. We account for this on the level of preferences that have to be defined in order to get off ground with the analysis. The chapter concludes with a discussion of combinatorial problems due to the global complexity of harmony and to the local character of tonal paradigms.

2

We are not sure whether they are really hidden and not only faked. . .

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.8 The main window of the HarmoRUBETTE®

The HarmoRUBETTE® is probably the most interesting RUBETTE®, since it is situated on a turning point of several critical issues in harmony. To begin with, the context problem in harmony is a multilayered and ramified one which is (we said it repeatedly) not clarified by music theory. This is manifest in the preliminary question of what is a chord in a given score. Should one only look at groups of notes as a common onset, should one also consider onset groups which are not manifest, but can be deduced from plausible rules, or is the selection of the relevant set of chords within a score also a function of the harmonic statements which could result thereof? We have implemented two variants. The first one takes as the sequence (ai)i all maximal zero-addressed local compositions ai ¼ {ai,j|j ¼ 1, . . . ti} of pitches of note events with identical onsets. The second one is less naive. Within the given score, we take all local compositions ai ¼ {ai,j|j ¼ 1, . . . ti} of pitches with this property: There is at least one onset which is the offset time of an event, and the chord ai is the nonempty set of pitches of all note events which either start or still last at this offset time. This option is chosen by the button “use duration” on the RUBETTE®’s main window; see Fig. 3.8.

3.2 The RUBETTE® Family

37

This second variant encodes all changes of chord configurations, not only the onset commonalities. Using either of these methods, the generated chord sequence is the basis for the following analysis which at the end will yield a harmonic weight for each note event and for which we refer to the harmonic tension theory presented in Mazzola et al. (2002, 27.2.2). According to that approach, each chord ai of the chord sequence (ai)i must be given a Riemann matrix (TFf,t(ai) ¼ ϕf,t(ai) ^)f,t, from which we deduce the weights ω( f, t, ai) ¼ ln(ϕf,t(ai)) and also call this data the Riemann matrix of ai. According to that discussion, we may also downsize weights below a threshold ϕmin to 1. This is what the user sets when defining the “global threshold” in the main window. The local threshold is just the same for relative weights within a given Riemann matrix. The percentages in the main window mean that we downsize values below a defined percentage relative to the global or local (only within a fixed Riemann matrix) value range. Following the rules for the value 1, we may neglect any path through a chord which has this value. So this singular value means that a chord is “inharmonic” insofar as it cannot contribute to a positive harmonic evaluation. This is a mathematical rephrasing of the classical but fuzzy concept of inharmonic chords. Here it just means that the harmonic weight of a chord is too small to be considered as a contribution to the global harmonic path and that the minimal size of allowed weights is set without further theoretical justification. It is a regulatory limit for the sensitivity of the path maximization with respect to the involved weights. We should stress that this matrix (ω( f, t, ai))f,t defines Riemann weights without any contextual considerations. This is information that comes along from the isolated calculation on the chord ai. There are different calculation methods for this matrix, one by the author, and using chains of thirds, as discussed in Mazzola et al. (2002, 25.3.3), and one by Thomas Noll, using self-addressed chords—the user may choose his preferred method by a pull-down button on the RUBETTE®’s main window. The Riemann matrix (ω( f, t, ai))f,t is visualized on the “ChordInspector” window for each chord (see Fig. 3.9). In this window, the non-logarithmic values ϕ(ai) are shown. The window also shows the chord’s pitch classes as well as all its minimal third chains. Once the Riemann matrices (ω( f, t, ai))f,t are calculated, the optimal path in the Riemann quiver is calculated. This one follows the method discussed in Mazzola et al. (2002, 27.2.2). To this end, we need preferences for the matrices T VALtype , T VALmode , TTON. The matrices T VALmode , TTON are defined in the windows shown in Fig. 3.10. To the left, we have TTON; however as a distance value according to the amount of fourth between pairs of tonalities, to the right, we have T VALmode . The matrix T VALmode is shown in the left lower corner of the window in Fig. 3.11. In the middle lower part of that window, we have check buttons for every Riemann locus (i.e., the position in the Riemann matrix), meaning that if the check is disabled, no path is possible through such a locus. The large upper matrix encodes the third weights needed for chord weight calculations according to the formulas (Mazzola et al. 2002, [25.8]) and (Mazzola et al. 2002, [25.9]).

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.9 The ChordInspector of the HarmoRUBETTE® shows each chord of the chosen chord sequence with its pitch classes, the third chains, and the Riemann matrix according to the chosen calculation method. The gray level of the values is proportional to their relative size

Fig. 3.10 This preference window (for the author’s third chain method) shows the tonality distance matrix T VALmode (left) and the mode matrix TTON (right)

With these settings, the best path is calculated. This is however a tedious task, to say the least. In fact, if we are given 200 chords (a very small example), we may choose from a number of 12200 ~ 6.8588169039290515  10215 paths. This number exceeds any calculation power of present computers. This exuberant number is due to different factors. First of all, no larger paths are taken into account, i.e., we have

3.2 The RUBETTE® Family

39

Fig. 3.11 The upper matrix encodes the weights of thirds (relative to a fixed reference tonic, the lower left shows the matrix T VALmode ). The lower middle matrix encodes the a priori allowed Riemann matrix locus position

not implemented cadences as preferred paths nor have we implemented modulatory constraints. More precisely, we do not give preference to maximal subpaths within a fixed tonality. We only take into account tonality changes a posteriori, i.e., via the weights of paths of length 1, when they show a tonality change. So we have to calculate the entire path and then hope that the negative points for tonality changes rule such paths out. It is also not clear whether human harmonic logic really can take into account such global path comparisons. In other words, it is more likely that humans only consider local optimization of paths. This is what we have in fact implemented in the following sense. In each index i of a chord ai, we consider only a local part of the entire chord sequence. Such a part is defined by two nonnegative entire variables CD ¼ Causal Depth, FD ¼ Final Depth. This means that we look at the subsequence of chords from index i  CD to i + FD (inclusive) and therein select an optimal path pi,CD,FD. Within this path, chord ai is positioned at a determined Riemann locus ( fi), ti)). The path which we finally select is the path p through all triples ( fi), ti), ai). The causal part is a tribute to the influence of preceding chords down to index i  CD on the harmonic position of chord ai. The final part influences the harmonic position of ai relating to the future chords up to index i + FD. The result is visualized in the Riemann graph which is shown in Fig. 3.12. This is however not the end of the job. We do not have the weights of the single notes. To this end, we first need the globally calculated weight of each chord ai. In a provisional form, the weight of chord ai is defined by the weight ω( pi) of the chosen path p from the first chord to ai. This value is finally modified by a global slope σ preference such that the final and the first weight can be set to build a defined slope. Thereby, we meet the requirement of a control over the global tension which cannot be deduced from the locally (only 0 and 1 lengths of subpaths) defined weights. Let us denote by ωi the weight ωσ ( pi) corrected by the slope preference.

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.12 The Riemann graph is the sequence of chords, together with their functional values as they result from the optimal path

Using the weight of a chord ai, we finally may define the weight3 ω(x) of a determined event x in ai. To this end, the weight ω( fi), ti), ai) is compared to the weight ω( fi), ti), ai  {x}) of the chord ai which contains x. With a positive preference quantity 0 < d  1, we consider the factor λð x Þ ¼

1 d þ ð1  dÞeωðf ðiÞ, tðiÞ, ai fxgÞωðf ðiÞ, tðiÞ, ai Þ

ð3:1Þ

which measures the weight differences. It evaluates to 1 for the difference zero and yields 1/d for the weight ω( fi), ti), ai  {x}) ¼  1. This means that the influence of x in the building of the chord’s weight is accounted for. If the weight decreases after omission of x, its influence is important and the factor increases λ(x). So we finally get the weight ωðxÞ ¼ λðxÞ  ωi The graphical representation of this weight is the same as for the MeloRUBETTE®, and we may omit this window of the HarmoRUBETTE®. Whereas the Riemann graph is conformal to the usual function-theoretic analysis (although it need not provide the common data in general), the weights of chords and events are far beyond the usual harmonic analysis and therefore cannot be compared without caution to established knowledge in harmony. It is however a common approach to harmony in its performance aspects to weight chords or notes in a more or less metaphoric way. Our present approach in the HarmoRUBETTE® is a concretization of these metaphors and also a point to be discussed with traditional performance theorists. 3

In this notation, we omit all the preferences.

3.2 The RUBETTE® Family

3.2.4

41

PerformanceRUBETTE®

All words, And no performance! Philip Massinger (1583–1640)

Summary The PerformanceRUBETTE® is a “macro” RUBETTE®: it manages the stemma generation, the weight input and recombination, the operator instantiation, and the production of output of performance data on the level of music technology. Originally, the PerformanceRUBETTE® was the very focus of RUBATO®. Its purpose was the implementation of a type of performance logic with arguments from an analytical output. Although the analytical Rubettes have earned a growing importance, one of the cornerstones of analysis is its success in the construction of a valid performance. In fact, playing a good performance is a way of demonstrating one’s understanding of music. Therefore the performance theory implementation is important beyond its autonomous interest. The PerformanceRUBETTE® implements the stemma theory of Mazzola et al. (2002, 38). The starting point is a selection of a score denotator. This will play the primary mother’s role, i.e., we are constructing a primary mother LPS in the sense of definition (Mazzola et al. 2002, 35.3). The score form is provided in the same format that we have known as input for the other Rubettes. For this RUBETTE®, the kernel is always given as a (zeroaddressed) local composition in the space form EHLDGC. This local composition is the kernel in the top space of a cellular hierarchy pertaining to the primary mother’s LPS. The primary mother’s LPS is instantiated according to “hard-coded” default parameters in the Objective C source code4. However, for each specific performance operator, the LPS data are adopted and define specific daughters. After setting the kernel (see Fig. 3.13) of the primary mother, the main window shows the stemmatic ramification with individual names for each LPS and arranged in a browser, stemmatic inheritance running from the left to the right. In the figure, the “Mother LPS” has a daughter named “PhysicalOperator,” and this one (all mothers are indicated on top of the daughters’ column) has two daughters, “SplitOperator 1” and “SplitOperator 2,” generated by the SplitOperator, etc. For each ramification, one or two daughters are generated according to a chosen operator. For example, in Fig. 3.13, the SplitOperator generates two daughters, whereas the TempoOperator always produces one single daughter. The operators need only be loaded at run-time as they are needed. So this RUBETTE® is nonterminal in the sense that it allows further ramifications via an arbitrary number of dynamically loadable performance operators. For every highlighted LPS in the stemma browser of the main window, we can visualize the top kernel of its hierarchy on the Kernel View window, as shown in Fig. 3.14, by means of the usual pianola graphics. Here, the gray level indicates the loudness. In order to apply 4 Objective C is a programming language for the NEXTSTEP-, OPENSTEP-, and Mac OS X-based RUBATO® projects.

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3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.13 The main window of the PerformanceRUBETTE® shows the stemmatic inheritance, descending from left to right. In the figure, the “Mother LPS” has a daughter named “PhysicalOperator,” and this one (all mothers are indicated on top of the daughters’ column) has two daughters “SplitOperator 1” and “SplitOperator 2,” generated by the SplitOperator, etc.

Fig. 3.14 The Kernel View window shows the top kernel of the hierarchy of a selected LPS (here the LPS named “PhysicalOperator” in the above stemma browser) in common pianola (piano roll) rectangles, loudness being codified by gray levels. The vertical bars are set to four bar intervals in the given score

an operator to a given LPS, one next needs a list of weights; this is conformal with the operator theory exposed in Mazzola et al. (2002, 44.7). The management of available weights as well as their concrete application is the business of the Weight Watcher system. Figure 3.15 shows a metrical weight in its splined interpolation shape. The weights to be used for a given operator can be loaded into the Weight Watcher; see top of Fig. 3.16. The loaded weights are then added or multiplied (according to the Boolean flag button “Combine as Product” to the right, below the weight list), and the resulting weight combination is applied to the given operator. For each weight, one can set the upper and lower limit of range (high norm, low norm), the nonlinear deformation quantity (deformation), the

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43

Fig. 3.15 In the PerformanceRUBETTE®, weights are used in their splined interpolation shape. Here, we see a metrical weight issued from the MetroRUBETTE®’s analysis

inversion/non-inversion flag (inverted weight button to the lower left corner), the influence in a combination of several weights (influence), and the slope of decrease to weight value 1 as the arguments tend to infinity (tolerance). The moral of this Weight Watcher system is a gastronomic one: weights may be mixed and dosed at will in order to experience their influence on a given operator. This is not merely a lack of theory; it is above all an experimental environment for effective performance research. In fact, since virtually nothing is known about the influence of weights on performance, we have provided the user with a great number of possibilities in order to realize an optimal testbed for future theory. Now, given a weight watcher combination of weights, an operator is fed by this combined weight and acts on the mother LPS to yield a new daughter LPS (or two in the case of the SplitOperator, where however no weight is needed). The detailed operation of a specific operator has already been described in Mazzola et al. (2002, 44.7); we need not repeat these details here. Figure 3.17 shows the inspector of the SymbolicOperator. The weight acts on selected parameters which are defined by Boolean buttons. The same procedure is performed for the PhysicalOperator, whose inspector is shown in Fig. 3.18. The action of an operator on the symbolic score is shown in the performed kernel in the same pianola representation as for the symbolic kernel. The action of a physical operator is shown in Fig. 3.19. A funny application of this operator to Schumann’s “Tra¨umerei” can be heard on the CD-ROM under the title Alptraeumerei; see Mazzola et al. (2002, p. xxx). This piece is the dead-pan version of the score with the melodic weight being applied to pitch via the physical operator and everything being played with Schumann’s original tempo indication. The inspector windows of the tempo-sensitive operators, the TempoOperator and the ScalarOperator, are shown in Figs. 3.20 and 3.21, respectively. The TempoOperator implements basically two methods: “Approximate” and “Real.” The former is a direct integration method, whereas the latter uses Runge–Kutta–

44

3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.16 The Weight Watcher window shows the loaded weights (top), the upper and lower limits of their range, the nonlinear deformation, and the Boolean flag of inverting/ non-inverting the weight (button in the lower left corner)

Fehlberg numerical ODE routines, including different parameters for numerical precision. The ScalarOperator uses exclusively Runge–Kutta–Fehlberg numerical ODE routines since it is an operator that acts on two or more parameters, where the naive approximation method cannot work. For the visualization of performance fields, the window shown in Fig. 3.22 is available. Finally, the parameters for the SplitOperator are determined on the window shown in Fig. 3.23. Here, one may define those lower and upper parameter limits of the total six-dimensional frame, where the subframe of the split daughter is cast.

3.2 The RUBETTE® Family

45

Fig. 3.17 The SymbolicOperator Inspector allows us to select a number of symbolic input parameters where the weight changes the values

3.2.5

PrimavistaRUBETTE®

It is hard if I cannot start some game on these lone heaths. William Hazlitt (1778–1830)

Summary Several musical predicates from score notation are paratextually loaded. The PrimavistaRUBETTE® takes care of the paratextual signification for the most important predicates regarding dynamics, agogics, and articulation. The PrimavistaRUBETTE® serves a different task insofar as it is neither analytic nor performance oriented. It deals with paratextual information as it is provided by verbal indications for dynamics, tempo, and articulation. It basically does this: it transforms verbal information into weights which may then be used to shape the symbolic data and the tempo before performance in the proper sense is shaped.

46

3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.18 The PhysicalOperator inspector allows us to select a number of physical output parameters where the weight changes the values

Fig. 3.19 The effect of an operator, here a PysicalOperator, is shown in the Kernel View of the performed kernel. This figure shows the performed symbolic kernel as shown above in Fig. 3.14

3.2 The RUBETTE® Family

47

Fig. 3.20 The TempoOperator inspector allows us to select different integration methods. The “Real” method uses Runge– Kutta–Fehlberg routines, whereas the “Approximate” method uses simple numerical integration

The input of this RUBETTE® is a local composition whose elements are events with verbal specification such as absolute dynamics (Fig. 3.24 right preference window), relative dynamics (Fig. 3.24 left middle preference window), articulation (Fig. 3.24 left upper preference window), and relative tempo (Fig. 3.24 left lower preference window). The functionality of this RUBETTE® is to transform these data into weights; this is performed on the window for primavista operations as shown in Fig. 3.25 and according to the numerical data that are defined the above preference windows. These methods have been discussed in detail in Mazzola et al. (2002, 39.2).

48

3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.21 The ScalarOperator inspector allows us to select different options as defined in the scalar operator theory, but Runge–Kutta–Fehlberg ODE integration is mandatory in this situation

3.2 The RUBETTE® Family

49

Fig. 3.22 The performance field of a selected LPS can be visualized. The user may select two parameters whereon the six-dimensional field is projected

Fig. 3.23 On this window, the user may define those lower and upper parameter limits of the total six-dimensional frame, where the subframe of the split daughter is cast for the SplitOperator

50

3 Introduction to RUBATO: The Music Software for Statistical Analysis

Fig. 3.24 A number for preference windows for dynamics, articulation, and tempo allow us to define numerical values of paratextual predicates

Fig. 3.25 The main window of the PrimavistaRUBETTE® manages the transformation of verbal (paratextual) predicates into weights

References

51

References ¨ ber den Begriff der tonalen Funktion, in Beitr€ C. Dahlhaus, U age zur Musiktheorie des 19. Jahrhunderts, ed. by M. Vogel (Bosse, Regensburg, 1966) A. Fleischer, G. Mazzola, T. Noll, Zur Konzeption der Software RUBATO fu¨r musikalische Analyse und Performance. Musiktheorie 4, 314–325 (2000) R.T. Kouzes et al., Collaboratories: Doing science on the internet. Computer, August 1996 G. Mazzola, RUBATO at SMAC KTH, Stockholm (1993) G. Mazzola, O. Zahorka, Geometry and logic of musical performance I, II, III. SNSF Research Reports, Universita¨t Zu¨rich, Zu¨rich (1993–1995), 469pp G. Mazzola, O. Zahorka, The RUBATO performance workstation on NeXTSTEP, in Proceedings of the ICMC 94, San Francisco, 1994, ed. by ICMA G. Mazzola et al., Analysis and performance of a dream, in Proceedings of the 1995 Symposium on Musical Performance, ed. by J. Sundberg (KTH, Stockholm, 1995a) G. Mazzola et al., The RUBATO platform, in Computing in Musicology, ed. by W.B. Hewlett, E. Selfridge-Field, vol. 10 (CCARH, Menlo Park, 1995b) G. Mazzola et al., The RUBATO homepage, since (1996). http://www.rubato.org. Univ. Zu¨rich G. Mazzola et al., The Topos of Music (Birkha¨user, Basel, 2002) R. Reti, The Thematic Process in Music, 2nd edn. (Greenwood, Westport, 1978) H. Riemann, System der musikalischen Rhythmik und Metrik (Breitkopf und Ha¨rtel, Leipzig, 1903) U.-J. Ru¨etschi, Denotative geographical modelling – an attempt at modelling geographical information with the denotator system. Diploma thesis, University of Zu¨rich, 2001 J. Sloboda, The Musical Mind: An Introduction to the Cognitive Psychology of Music (Calderon, Oxford, 1985) O. Zahorka, PrediBase – controlling semantics of symbolic structures in music, in Proceedings of the ICMC 95, San Francisco, 1995, ed. by ICMA O. Zahorka, RUBATO – Deep Blue in der Musik? Animato 97/3, 9–10, Zu¨rich (1997a) O. Zahorka, From sign to sound – analysis and performance of musical scores on RUBATO, in Symposionsband Klangart ’95, ed. by B. Enders (Schott, Mainz, 1997b)

Chapter 4

Modeling the Structure of Raga Bhimpalashree: A Statistical Approach

4.1

Introduction

Music, according to Swami Vivekananda, is the highest form of art and also the highest form of worship (provided you understand it!). Understanding music, both esthetically and scientifically, becomes important. This is especially true for classical music, be it Indian or Western, since each is a discipline in its own right. While the former stresses on the emotional richness of the raga as expressed through melody and rhythm, the latter is technically stronger as, in addition to melody and rhythm, the focus is also on harmony and counterpoint. A raga is a melodic structure with fixed notes and a set of rules characterizing a particular mood conveyed by performance. The present chapter gives a statistical structure analysis of raga Bhimpalashree. Our analysis attempts to answer the following question: Can we find a working statistical model that can capture the essence of the Bhimpalashree raga structure? This question is important because the true model is both complex and unknown and that one of the strengths of statistics lies in modeling. Although statistical models are subjective and biased, we can at least make the data objective as far as possible. Also, behind these models stand some beautiful mathematical theorems, and they are unbiased (Klemens 2008). Moreover, the true model may contain multiple parameters related to music theory, the training and background of the artist, the genre of music, and even the place of performance and the audience, and we do not have an explicit idea as to how exactly (in what functional form) these parameters enter the model. Statistical models are, in contrast, approximate models that use fewer parameters to capture the phenomenon generated by these complex unknown true models. Although approximate, it is possible to verify the goodness of fit of these models as well as control the errors in them. In the light of these arguments, modeling a musical structure or a musical performance has been a coveted research area in computational musicology. © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_4

53

54

4 Modeling the Structure of Raga Bhimpalashree: A Statistical Approach

There are three fundamental steps in statistical modeling: deciding which model to fit, estimating the parameters of the chosen model, and verifying the goodness of fit of this model. We all know that statistics can be broadly divided into two categories: descriptive and inferential. In statistical modeling, both are involved—as we first describe a pattern (through modeling) and then infer about its validity. Two types of models are used in statistics: probability models and stochastic models. Through a probability model, we can tell the probability of a note or a note combination, but cannot predict the next note. Through a stochastic model, we can predict (make an intelligent guess of) the next note, given the previous. In the present chapter, a simple exponential smoothing is used to capture the note progression depicting the structure of the raga Bhimpalashree. This is a stochastic model used in time series analysis. As a final comment, there is no unique way of analyzing music, be it structure or performance, and hence statistics and probability are likely to play important roles (Beran and Mazzola 1999). For a good bibliography of statistical applications in musicology, see Nettheim (1997).

4.2

Methodology

Simple exponential smoothing is used to statistically model time series data for smoothing purpose or for prediction. Although it was Holt (1957) who proposed it first, it is Brown’s simple exponential smoothing that is commonly used nowadays (Brown 1963). Simple exponential smoothing is achieved by the model Ft + 1 ¼ αYt + (1  α)Ft, 0 < α < 1, to the data (t, Yt) where Ft is the predicted against Yt and initially Fo ¼ Yo. Here α is the smoothing factor. This is the only parameter in the model that needs to be determined from the data. The smoothed statistic Ft + 1 is a simple weighted average of the previous observation Yt and the previous smoothed statistic Ft. The term smoothing factor applied to α here is something of a misnomer, as larger values of α actually reduce the level of smoothing, and in the limiting case with α ¼ 1 the output series is just the same as the original series (with lag of one time unit). Simple exponential smoothing is easily applied, and it produces a smoothed statistic as soon as two observations are available. Values of α close to one have less of a smoothing effect and give greater weight to recent changes in the data, while values of α closer to zero have a greater smoothing effect and are less responsive to recent changes. There is no formally correct procedure for choosing α. Sometimes the statistician’s judgment is used to choose an appropriate factor. Alternatively, a statistical technique may be used to optimize the value of α. For example, the method of least squares might be used to determine the value of α for which the sum of the quantities (Ft  Yt)2 is minimized (see Wikipedia for further literature).

4.3 Statistical Analysis

4.2.1

55

Getting the Musical Data for Structure Analysis

Time series is a series of observations in chronological order. Musical data can also be taken as a time series in which a musical note characterized by pitch Yt is the entry corresponding to the argument time t which may mean time of clock in actual performance or just the instance at which the note is realized. In our case, since we are modeling only the structure of the raga, the arguments will be simply the instances 1, 2, 3. The desired note sequence is given in Table 4.2 in Appendix (Dutta 2006) from which we have taken the first 90 notes only for modeling. Western art music readers should refer to Table 4.1 where corresponding western notations are provided. The tonic (Sa in Indian music) is taken at natural C. Analyzing the structure of a musical piece helps in giving an approximate model that captures the note progression in general without bringing the style of a particular artist into play. On the other hand, performance analysis gives additional features like note duration and the pitch movements between notes (Chakraborty et al. 2009). In Table 4.1, pitches of notes in three octaves are represented by corresponding integers, C of the middle octave being assigned the value 0. We are motivated by the works of Adiloglu et al. (2006). The letters S, R, G, M, P, D, and N stand for Sa, Sudh Re, Sudh Ga, Sudh Ma, Pa, Sudh Dha, and Sudh Ni, respectively. The letters r, g, m, d, and n represent Komal Re, Komal Ga, Tibra Ma, Komal Dha, and Komal Ni, respectively. Normal type indicates the note belongs to middle octave; italics implies that the note belongs to the octave just lower than the middle octave, while a bold type indicates it belongs to the octave just higher than the middle octave. The terms “Sudh,” “Komal,” and “Tibra” imply, respectively, natural, flat, and sharp.

4.3

Statistical Analysis

Simple exponential smoothing fitted to Bhimpalashree note sequence The fit is found to be explaining the note progression well enough with smoothing factor 0.793645. Should such a model work well for the other ragas also, it is of interest to see how the smoothing factor varies. Here is a summary of our results obtained using Minitab Statistical package version 16: Single exponential smoothing plot for C1 is given in Fig. 4.1. Residual plots are given in Fig. 4.2. Table 4.1 Numbers representing pitches of musical notes in three octaves C S 12 S 0 S 12

Db R 11 R 1 R 13

D R 10 R 2 R 14

Eb G 9 G 3 G 15

E G 8 G 4 G 16

F M 7 M 5 M 17

F# m 6 m 6 m 18

G P 5 P 7 P 19

Ab d 4 d 8 d 20

A D 3 D 9 D 21

Bb n 2 n 10 n 22

B N 1 N 11 N 23

Western notation Notes (lower octave) Numbers for pitch Notes (middle octave) Numbers for pitch Notes (higher octave) Numbers for pitch

4 Modeling the Structure of Raga Bhimpalashree: A Statistical Approach

56

Simple Exponential Smoothing for Raga Bhimpalashree Single Exponential Method 15

Variable Actual Fits

10

Smoothing Constant Alpha 0.793645 Accuracy MAPE MAD MSD

C1

5

Measures 52.6132 2.0754 5.7965

0

-5

-10 1

9

18

27

36

45

54

63

72

81

90

Index

Fig. 4.1 Simple exponential smoothing captures the Bhimpalashree note sequence

Residual Plots for C1 Normal Probability Plot

Versus Fits

99.9

5.0

Residual

Percent

99 90 50 10

2.5 0.0 -2.5

1

-5.0

0.1

-8

-4

0

4

8

-5

0

Histogram

10

15

Versus Order

24

5.0

Residual

Frequency

5 Fitted Value

Residual

18 12

2.5 0.0 -2.5

6

-5.0

0 -4

-2

0 Residual

2

4

1

10

20

30

40

50

60

70

80

90

Observation Order

Fig. 4.2 Residual plots

4.4

Discussion

Interpretations from Figs. 4.1 and 4.2: The random pattern of the residuals (Fig. 4.2) together with the closeness of smoothed data with the observed one (Fig. 4.1) justifies the simple exponential smoothing. A detailed discussion of the findings is given next.

4.4 Discussion

57

Mean absolute percent error (MAPE)—measures the accuracy of fitted time series values. It expresses accuracy as a percentage. Mean absolute deviation (MAD)—measures the accuracy of fitted time series values. It expresses accuracy in the same units as the data, which helps conceptualize the amount of error. Mean squared deviation (MSD)—measures the accuracy of fitted time series values. MSD is always computed using the same denominator (the number of forecasts) regardless of the model, so one can compare MSD values across models and hence compare the accuracy of two different models. For all three measures, smaller values generally indicate a better fitting model. In case we fit other models to the same data, it is of interest to compare the corresponding MAPE, MAD, and MSD values. This is reserved as a rewarding future work. The normal probability graph plots the residuals versus their expected values when the distribution is normal. The residuals from the analysis should be normally distributed. In practice, for data with a large number of observations, moderate departures from normality do not seriously affect the results. The normal probability plot of the residuals should roughly follow a straight line. One can use this plot to look for the following: This pattern

Indicates

Not a straight line Curve in the tails A point far away from the line Changing slope

Nonnormality Skewness An outlier An unidentified variable

As is clear from Fig. 4.2, the plot roughly follows a straight line. The next graph plots the residuals versus the fitted values. The residuals should be scattered randomly about zero. One can use this plot to look for the following: This pattern

Indicates

Fanning or uneven spreading of residuals across fitted values Curvilinear A point far away from zero

Non-constant variance A missing higher-order term An outlier

As is clear from Fig. 4.2, the residuals are quite evenly spread on both the positive side and the negative side. A histogram of the residuals shows the distribution of the residuals for all observations. One can use the histogram as an exploratory tool to learn about the following characteristics of the data: • Typical values, spread or variation, and shape • Unusual values in the data

4 Modeling the Structure of Raga Bhimpalashree: A Statistical Approach

58

The histogram of the residuals should be bell shaped. One can use this plot to look for the following: This pattern

Indicates

Long tails A bar far away from the other bars

Skewness An outlier

Because the appearance of the histogram can change depending on the number of intervals used to group the data, one should use the normal probability plot and goodness-of-fit tests to assess whether the residuals are normal. We have already given the normal probability plot for residuals. The graph “residuals versus order” plots the residuals in the order of the corresponding observations. The plot is useful when the order of the observations may influence the results, which can occur when data are collected in a time sequence (as in our case) or in some other sequence, such as geographic area. This plot can be particularly helpful in a designed experiment in which the runs are not randomized. The residuals in the plot should fluctuate in a random pattern around the center line as in Fig. 4.2. One can examine the plot to see if any correlation exists between error terms that are near each other. Correlation among residuals may be signified by: • An ascending or descending trend in the residuals • Rapid changes in signs of adjacent residuals Remark: 1. Simple exponential smoothing is useful when (1) there is no trend, (2) there is no seasonal variation, (3) there is no missing value, and (4) we want short-term forecast. Conclusion A simple exponential smoothing model successfully captures the note progression depicting the structure of the raga Bhimpalashree. The authors have shown that such a smoothing model also fits the Bhairav raga structure well, whose performance is analyzed in Chap. 8 [see Tewari and Chakraborty (2013)]. Interestingly, the smoothing factor for Bhairav turns out to be 0.762839. Note that Bhimpalashree, belonging to Kafi thaat, is an afternoon raga, while Bhairav, of Bhairav thaat, is the first morning raga. The timings are such that a twilight is indicated in both! Both the ragas use seven notes, and both are of restful nature. We do not know as yet whether the closeness in smoothing factor has anything to do with these and possibly other similarities, if any, between these ragas. Chakraborty et al. (2010) have provided a comparative study between Bageshree, also of Kafi thaat, and Bhimpalashree. An important point of distinction is the note sequence {M D n S} in Bageshree in contrast to {M P n S} in Bhimpalashree.

Appendix

59

Appendix Table 4.2 Notes of raga Bhimpalashree Sl. No.

Note

Pitch

Sl. No.

Note

Pitch

Sl. No.

Note

Pitch

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

n S M g M P M P M g M g R n S M g P M P g M P n D P M P g M P n S R n S S n D

−2 0 5 3 5 7 5 7 5 3 5 3 2 −2 0 5 3 7 5 7 3 5 7 10 9 7 5 7 3 5 7 10 12 14 10 12 12 10 9

40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78.

P M P M g R S S n S M g M P M g M M g R S R n S M g M M g P M g M g R S S n D

7 5 7 5 3 2 0 0 −2 0 5 3 5 7 5 3 5 5 3 2 0 2 −2 0 5 3 5 5 3 7 5 3 5 3 2 0 0 −2 −3

79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115.

P M P S n n S R n S M g M P M g M P n D P M g M P M g M g R S n S M g M P

−5 −7 −5 0 −2 −2 0 2 −2 0 5 3 5 7 5 3 5 7 10 9 7 5 3 5 7 5 3 5 3 2 0 −2 0 5 3 5 7

Italics, normal type and bold letters are used to represent notes in the lower octave, middle octave and higher octave respectively

60

4 Modeling the Structure of Raga Bhimpalashree: A Statistical Approach

References K. Adiloglu, T. Noll, K. Obermayer, A paradigmatic approach to extract the melodic structure of a musical piece. J. New Music Res. 35(3), 221–236 (2006) J. Beran, G. Mazzola, Analyzing musical structure and performance – a statistical approach. Stat. Sci. 14(1), 47–79 (1999) R.G. Brown, Smoothing Forecasting and Prediction of Discrete Time Series (Prentice-Hall, Englewood Cliffs, NJ, 1963) S. Chakraborty, R. Ranganayakulu, S. Chauhan, S.S. Solanki, K. Mahto, A statistical analysis of Raga Ahir Bhairav. J. Music Meaning 8, sec. 4, Winter (2009), http://www.musicandmeaning. net/issues/showArticle.php?artID¼8.4 S. Chakraborty, L.K. Krishnapryia, S. Chauhan, S.S. Solanki, K. Mahto, Melody revisited: tips from Indian music theory. Int. J. Comput. Cognit. 8(3), 26–32 (2010) D. Dutta, Sangeet Tattwa (Pratham Khanda), 5th edn. (Brati Prakashani, 2006) (in Bengali) (Kolkata, India) C.C. Holt, Forecasting trends and seasonal by exponentially weighted averages. Office of Naval Research Memorandum 52 (1957). Reprinted in C.C. Holt, Forecasting trends and seasonal by exponentially weighted averages. Int. J. Forecast. 20(1), 5–10 (2004). doi:10.1016/j.ijforecast. 2003.09.015 http://en.wikipedia.org/wiki/Exponential_smoothing. Accessed 28 Jan 2013 B. Klemens, Modeling with data: tools and techniques for scientific computing (Princeton University Press, Princeton, NJ, 2008) N. Nettheim, A bibliography of statistical applications in musicology. Musicol. Aust. 20, 94–106 (1997) S. Tewari, S. Chakraborty, A statistical analysis of Bhairav: the first morning raga. Int. J. Comput. Technol. 4(2), 371–386 (2013)

Chapter 5

Analysis of Lengths and Similarity of Melodies in Raga Bhimpalashree

5.1

Introduction

In monophonic melody analysis, significance of a melody is measured by multiplying its length with the number of times it occurs. An analysis of melody lengths begins a new line of investigation (for a more comprehensive account, see Chakraborty et al. 2011–2012; this is possibly the first work on melody lengths). The strength of our results stems from the fact that a structure analysis discovers several interesting facts of the raga without restricting to any specific artist. The sequence of the raga notes, taken from a standard text, is given in Appendix of Chap. 4. A raga is a melodic structure with fixed notes and a set of rules characterizing a certain mood conveyed by performance. Two melodies, of equal length, are similar if the correlation coefficient of their shapes is significant. These concepts are detailed in the next section.

5.2

Statistical Analysis of Melody Groups

Melody may be mathematically defined as a sequence of notes “complete” in some sense as determined by music theory, taken from a musical piece. A melody need not be a complete musical sentence. It suffices if it is a complete musical phrase. A segment is a sequence of notes which is a subset of melody but is itself incomplete. For example, {Sa, Sa, Re, Re, Ga, Ga, Ma, Ma, Pa} is a melody and {Sa, Sa, Re Re} is its segment in raga Kafi. Length of a melody or its segment refers to the number of notes in it. Significance of a melody or its segment (in monophonic music, where there is a single melody line, such as Indian classical music) is defined as the product of the length of the melody and the number of times it occurs in the musical piece. Thus, both frequency and length are important factor to assess the significance of a melody or its segment. By shape of a melody is meant the difference of © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_5

61

62

5 Analysis of Lengths and Similarity of Melodies in Raga Bhimpalashree

Table 5.1 Bhimpalashree melodies (see Appendix in Chap. 4) and their lengths

Note sr. no.

Group no.

Length

1–6 7–15 16–17 18–26 27 28–36 37–42 43–46 47–53 54–60 61–69 70–75 76–81 82–88 89–99 100–109 110–115

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 G13 G14 G15 G16 G17

6 9 2 (segment) 9 1 (segment) 9 6 4 7 7 9 6 6 7 11 10 6

Mean melody length ¼ 6.882353 and standard deviation sd ¼ 2.609538 cv ¼ 0.3791636 (coefficient of variation or cv is calculated as sd/mean; cv measures consistency, i.e., less cv implies more consistency)

successive pitches of the notes in it represented by the numbers given in Table 1.2 of chapter 1, e.g., the shape of the Kafi melody {Sa, Sa, Re, Re, Ga, Ga, Ma, Ma P}, i.e., {0, 0, 2, 2, 4, 4, 5, 5, 7}, is {0, 2, 0, 2, 0, 1, 0, 2). Table 5.1 gives the formation of melody groups (obtained by rendering each melody group in a harmonium and checking for completeness). Although intuitive, it should be understood that behind the intuition it is Indian music theory that works. See Chakraborty et al. (2010). These authors are motivated by the works of Adiloglu et al. (2006) who have provided quite a different formula for measuring the melody significance in polyphonic music where we have multiple melody lines. Our results on the significance of melody groups are summarized in Table 5.2. As regards similarity, only melody groups 13 and 17 were found to be close (Table 5.3). Remark: Since correlation coefficient is a number between
1 and +1, fractional values approaching
1 or +1 or even zero can be interpreted likewise based on the abovementioned definitions. Significance of a correlation coefficient can be tested using t-test. If r is the value of correlation coefficient and n be the number of pairs of observations (here successive differences), we calculate the statistic t ¼ r√(n
2)/√(1
r2). If the absolute value of t, i.e., |t| exceeds the table value of t at 5 % level of significance (say) and (n
2) degrees of freedom, then the value of r is significant at 5 % level otherwise insignificant. Here it is assumed that the n pairs are coming from a bivariate normal distribution. The formula for r is computed as r ¼ covariance (x, y)/√{sd(x) sd( y)} where sd ¼ standard deviation.

5.2 Statistical Analysis of Melody Groups Table 5.2 Melody groups and their significance

63

Melody group

Significance

G1, G17 G8 G9 G7, G12, G13 G15 G16 G2, G4, G6, G11 G10, G14 G3 and G5

24 16 14 12 11 10 9 7 Segmentsa

a Although G3 and G5 are segments and hence not considered in the analysis in Table 5.2, it must be kept in mind that segments are themselves sequential subsets of melodies and their significance should be compared among themselves. This argument gets a musical support also in our case as both the segments G3 and G5 contain the note M which is the Vadi swar (most important note) in Bhimpalashree and hence cannot be thrown away! In fact, a cursory glance at Appendix in Chap. 4 convinces us that M, corresponding to the segment G5, occurs 33 times with significance 1  33 ¼ 33, and {M, g}, corresponding to the segment G3, occurs times 18 with significance 2  18 ¼ 36

Table 5.3 Similarity study of melody groups (refer to [remark] for the formulae for r and t) Group

Correlation coefficient

|t|

Table t

Result

Musical meaning

13, 17 Others

+1 Details omitted

– Details omitted

– Details omitted

– Insignificant

Similar Different

Covariance (x, y) can be computed easily as {Sum(xy)}/n}
{mean (x)  mean ( y)}. sd(x) ¼ √[{Sum(x  x)}/n
{mean (x)  mean (x)}] sd( y) ¼ √[{Sum(y  y)}/n
{mean ( y)  mean ( y)}] Mean (x) ¼ Sum(x)/n and Mean ( y) ¼ Sum ( y)/n. Two melodies are in translation if the correlation coefficient r of their shapes equals +1. Two melodies are in inversion if the correlation coefficient r of their shapes equals
1. Two melodies are called different if the correlation coefficient of their shapes approaches 0. Thus correlation coefficient here is a measure of similarity between melodies. Remark: Squaring the standard deviation, one gets variance ¼ 6.8096885. This is close to the mean value. We know in Poisson distribution, mean and variance are equal. As there has to be at least three notes in a melody and at least one note in a segment, further considering the smallest single-note segment in G5 in our case (Table 5.1), one may fit a zero-truncated Poisson distribution and test for the

5 Analysis of Lengths and Similarity of Melodies in Raga Bhimpalashree

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goodness of fit (see Chakraborty et al. 2011–2012 where a 0–2-truncated Poisson model has been successfully fitted on a late night raga Malkauns). Conclusion Melody groups 13 and 17 are found to be similar. Melodies G1 and G17 have the highest significance. In fact, these two melodies are identical, namely, {n, S, M, g, M, P}.

References K. Adiloglu, T. Noll, K. Obermayer, A paradigmatic approach to extract the melodic structure of a musical piece. J. New Music Res. 35(3), 221–236 (2006) S. Chakraborty, L.K. Krishnapryia, S. Chauhan, S.S. Solanki, K. Mahto, Melody revisited: tips from Indian music theory. Int J. Comput. Cognit. 8(3), 26–32 (2010) S. Chakraborty, S. Tewari, S. Sundareisan, S. Kashyap, M. Pal, A Statistical Analysis of Melody Lengths in a North Indian Raga, Studii si Cercetari de Istoria Artei Teatru, Muzica, Cinemtographie, T. 5–6 (49–50), Bucuresti, 2011–2012, pp. 129–136

Chapter 6

Raga Analysis Using Entropy

6.1

Introduction

If P(E) is the probability of an event, the information content of the event E is defined as I(E) ¼ log2(P(E)). Events with lower probability will signal higher information content when they occur. The probability of a raga note, and hence its information content, depends on the raga concerned. The important raga notes will obviously be having higher probabilities. On the other hand, a weak note in a raga cannot be thrown away either for it would be carrying more surprise! The strength of entropy analysis lies here (entropy is the mean information content of a random variable). We have I(X ¼ 0) ¼ I(X ¼ 1) ¼ log2(1/2) ¼ 1. The units of information content are bits. So, we gain one bit of information when we choose between two equally likely alternatives. Let X be a discrete random variable which takes values x1, x2, x3,. . ., xn with corresponding probabilities p1, p2, p3,. . ., pn. Since X is a random variable, the information content of X is also random which we denote by I(X) (what value I(X) will take depends on what value X takes). When X ¼ xj which is an event with probability pj, then I(X) ¼ log2( pj). Accordingly, it makes sense to talk about the mean value of I(X) called its entropy, denoted by H(X), so that we have X   H ðX Þ ¼  pj log2 pj where the summation is over j ¼ 1 to n. It should be emphasized here that entropy is measuring surprise (not meaning!). The present paper gives our views on raga analysis using entropy with illustrative examples. Although only raga structure is analyzed, the ideas are applicable to performance as well. We consider the raga here, namely, Bhimpalashree, for analyzing the entropy. A raga is a melodic structure with fixed notes and a set of rules characterizing a © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_6

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certain mood conveyed by performance (Chakraborty et al. 2009a). The database consists of sequences of 90 notes, taken from a standard text (Dutta 2006). Before moving to the analysis, we make some additional comments. (a) For an impossible event E, P(E) ¼ 0, I(E) ¼ 1. As negative information is ruled out, it indicates the non-feasibility of ever obtaining information about an impossible event. The case of a possible event having a zero probability is discussed in Sect. 6.2. (b) We shall define plog( p) ¼ 0 when p ¼ 0. The range of plog( p) is thus [0, 1). (c) The use of entropy in music analysis has been successfully tried in Western music. We are motivated by the work of Snyder (1990). For further literature on entropy, see Applebaum (1996) and the references cited therein. Table 6.1 is formed taking the first 90 notes of Bhimpalashree (Appendix, Chap. 4). Mean entropy calculation X   H ðX Þ ¼  pj log2 pj Without octave ¼ 0:4308 þ 0:2866 þ 0:4308 þ 0:5085 þ 0:4176 þ 0:1636 þ 0:3876 ¼ 2:6255 With octave ¼ 0:3876 þ 0:1636 þ 0:2605 þ 0:0721 þ 0:4308 þ 0:0721 þ 0:5030 þ 0:1220 þ 0:3876 þ 0:0721 þ 0:1220 þ 0:3104 þ 0:1996 ¼ 3:1034

6.1.1

Mean entropy of raga Bhimpalashree

Ignoring octave

Considering octave

H(X) ¼ 2.6255

H(X) ¼ 3.1034

6.2

Discussion: Information on a Possible Event E with P(E) ¼ 0

Information theory falls flat for those possible events whose probability is zero. For example, in a raga, a vivadi swar is a note that is not allowable in the raga. However there have been occasions where a celebrated artist has used a vivadi swar deliberately at a “peak time” creating an electric feeling that did beautify the melody and drew immediate applauds from the enthusiastic listeners. Let us not forget that “art is good when beautiful and bad when not” (Brandt 2009), and the artist is within his/her right to seek such beautification, at times even breaking

6.2 Discussion: Information on a Possible Event E with P(E) ¼ 0

67

Table 6.1 Raga Bhimpalashree: computation of information content and entropy Without octave

Information content

With octave

Information content

S ¼ 15

I ðSÞ

Lower octave M ¼ 1

log2 (1/90) ¼ 6.4919

R¼7

I ðRÞ

Lower octave P ¼ 2

log2 (2/90) ¼ 5.4919

g ¼ 15

I ðgÞ

Lower octave D ¼ 1

log2 (1/90) ¼ 6.4919

M ¼ 24

I ðMÞ

Lower octave n ¼ 8

log2 (8/90) ¼ 3.4919

P ¼ 14

I ðPÞ

Middle octave S ¼ 12

D¼3

I ðDÞ

log2 (12/90) ¼ 2.9069 log2 (6/90) ¼ 3.9069

n ¼ 12

I ðPÞ

¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼

 log2ð15=90Þ 2:5850  log2ð7=90Þ 3:6845  log2ð15=90Þ 2:5850  log2ð24=90Þ 1:9069  log2ð14=90Þ 2:6845  log2ð3=90Þ 4:9069  log2ð12=90Þ 2:9069

Middle octave R ¼ 6 Middle octave g ¼ 15 Middle octave M ¼ 23 Middle octave P ¼ 12 Middle octave D ¼ 2 Middle octave n ¼ 4 Higher octave S ¼ 3 Higher octave R ¼ 1

log2 (15/90) ¼ 2.5850 log2 (23/90) ¼ 1.9683 log2 (12/90) ¼ 2.9069 log2 (2/90) ¼ 5.4919 log2 (2/90) ¼ 4.4919 log2 (3/90) ¼ 4.9069 log2 (1/90) ¼ 6.4919

a certain music theory. This is what is debated rather hotly as “romanticism vs. classicism” in Indian music, a conflict between unrestricted beautification and disciplined beautification (Chakraborty 2010). The probability of a vivadi swar is theoretically zero in a raga. It is still possible that it comes practically. To make a sensible analysis, with an eye on performance, let us agree to write P(vivadi) ¼ ε where ε is a very small positive number. Since P(vivadi) ¼ 0 theoretically, it means we are practically replacing the zero probability as ε where ε tends to 0+. Next assume there is a large positive number M such that 2M ¼ ε. Then we have I(vivadi) ¼ log2(ε) ¼ log2(2M) ¼ M, indicating that the occurrence of the vivadi does carry a lot of information and hence a lot of surprise (of course this must be done by an expert!). A more comprehensive link between raga and probability can be found in Tewari and Chakraborty (2011). For more on entropy analysis in raga, see Chakraborty et al. (2011a, b). Probability theory, however, does not explain the decision process of the artist as music is always planned and not random (Chakraborty et al. 2009b). Conclusion The mean entropy of Bhimpalashree is found to be higher when octave is considered. Musically this means the performer has to move from one octave (continued)

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to another fairly rapidly rather than sticking to one octave for a long period in order to maintain listener’s interest. We leave it as an exercise to the reader to compare this raga with others that use the same notes (e.g., Bageshree). The note probabilities would be different in the two ragas and so would be the information content and the entropy. The paper by Chakraborty et al. (2011a) deserves a mention here in which Bhupali and Deshkaar, two ragas that use the same notes but belong to different thaats (Kalyan and Bilaval, respectively), have been compared. An interesting finding based on entropy is that the Pa-Ga (P-G) meend (glide) in Bhupali is more important than the Sa-Dha (S-D) meend. Musically, the Pa-Ga meend is crucial because the correct usage of this meend demands touching the Teevra madhyam (m) as a grace note on the way, and this is one of the reasons, according to some musicians, why Bhupali is placed in the Kalyan thaat!

References D. Applebaum, Probability and Information: An Integrated Approach (Cambridge University Press, Cambridge, 1996). Chapter 6 P.A. Brandt, Music and the abstract mind. JMM J. Music Meaning 7, Winter (2009). http://www. musicandmeaning.net/issues/showArticle.php?artID¼7.3 S. Chakraborty, Review of the book Computer Music Modeling and Retrieval: Genesis of Meaning in Sound and Music, 5th International Symposium, CMMR 2008, Copenhagen, Denmark, May 2008. Revised papers by S. Ystad, R. Kronland-Martinet, K. Jensen, Springer, New York, 2009 published in Comput. Rev., 7 May 2010, www.reviews.com S. Chakraborty, R. Ranganayakulu, S. Chauhan, S.S. Solanki, K. Mahto, A statistical analysis of Raga Ahir Bhairav. J. Music Meaning 8, sec. 4, Winter (2009a), http://www.musicandmeaning.net/ issues/showArticle.php?artID¼8.4 S. Chakraborty, M. Kumari, S.S. Solanki, S. Chatterjee, On what probability can and cannot do: a case study in Raga Malkauns. J. Acoust. Soc. India 36(4), 176–180 (2009b) S. Chakraborty, S. Tewari, G. Akhoury, S.K. Jain, J. Kanaujia, Raga analysis using entropy. J. Acoust. Soc. India 38(4), 168–172 (2011a) S. Chakraborty, S. Tewari, G. Akhoury, How do ragas of the same thaat that evoke contrasting emotions like joy and pathos differ in entropy? Philomusica Online 10, 1–10 (2011b) D. Dutta, Sangeet Tattwa (Pratham Khanda), 5th edn. (Brati Prakashani, 2006) (Bengali) (Kolkata, India) J.L. Snyder, Entropy as a measure of musical style: the influence of a priori. Assumptions Music Theory Spectr. 12, 121–160 (1990) S. Tewari, S. Chakraborty, Linking raga with probability. Ninad J. ITC Sangeet Res. Acad. 25, 25–30 (2011)

Chapter 7

Modeling Musical Performance Data with Statistics

7.1

Statistics and Music

Today, the relationship between music and mathematics is a common factor. In the last two or three decades, the advances in mathematics, computer science, psychology, semiotics, and related fields, together with technological progress (in particular computer technology), lead to a revival of quantitative thinking in music [see, e.g., Archibald (1972), Babbitt (1961), Balzano (1980), Lewin (1987), Lendvai (1993), Forte (1964, 1973), Morris (1987, 1995), Johnson and Wichern (2002), Leyton (2001), Andreatta (1997), Solomon (1973), Beran and Mazzola (1999), Beran (2004), Meyer (1989), Morehen (1981)]. Musical events can be expressed as a specific ordered temporal sequence, and time series analysis is the observations indexed by an ordered variable (usually time). It is therefore not surprising that time series analysis is important for analyzing musical data as it is always be the function of time. Music is an organized sound. But the equation of these sounds does not produce the formula of how and why sounds are connected. Statistics is a subject which can connect theoretical concept with observable phenomenon and statistical tools that can used to find and analyzing the structure to build a model. But applications of statistical methods in Indian musicology and performance research are very rare. There were some researches that had been done on Western musicology and mostly consist of simple applications of standard statistical tools. Due to the complex nature of music, statistics is likely to play an important role where the random variables are the musical notes which are function of time.

© Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_7

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7.2 7.2.1

7 Modeling Musical Performance Data with Statistics

Time Series Analysis and Music Time Series Data

A time series is a set of statistics, usually collected at regular intervals. Arrangement of statistical data in chronological order, i.e., in accordance with occurrence of time, is known as “time series.” Mathematically U t ¼ f ðt Þ where Ut ¼ value of the phenomenon under consideration at time t. Time series models have been the basis for any study of a behavior of process or metrics over a period of time. In decisions that involve factor of uncertainty of the future, time series models have been found one of the most effective methods of forecasting. Most often, future course of actions and decisions for such processes will depend on what would be an anticipated result. The need for these anticipated results has encouraged organizations to develop forecasting techniques to be better prepared to face the seemingly uncertain future. Also, these models can be combined with other data mining techniques to help understand the behavior of the data and to be able to predict future trends and patterns in the data behavior. Time series analysis accounts for the fact that data points taken over time may have an internal structure (such as autocorrelation, trend, or seasonal variation) that should be accounted for Desikan and Srivastava (www-users.cs.umn.edu/~desikan/ publications/TimeSeries.doc).

7.2.2

Goal of Time Series Analysis

There are two main goals of time series analysis: (a) identifying the nature of the phenomenon represented by the sequence of observations and (b) forecasting (predicting future values of the time series variable). Both of these goals require that the pattern of observed time series data is identified and more or less formally described. Once the pattern is established, we can interpret and integrate it with other data (i.e., use it in our theory of the investigated phenomenon, e.g., seasonal commodity prices). Regardless of the depth of our understanding and the validity of our interpretation (theory) of the phenomenon, we can extrapolate the identified pattern to predict future values.

7.3 Autoregressive Integrated Moving Average

7.3

71

Autoregressive Integrated Moving Average

Identifying patterns in time series data involved knowledge about the mathematical model of the process. However, in real-life research and practice, patterns of the data are unclear, individual observations involve considerable error, and we still need not only to uncover the hidden patterns in the data but also generate forecasts. The ARIMA methodology developed by Box and Jenkins (1976) allows us to do just that; it has gained enormous popularity in many areas and research practice because of its power and flexibility. The following sections will introduce the basic ideas of this methodology.

7.3.1

Autoregressive Process

Most time series consist of elements that are serially dependent in the sense that you can estimate a coefficient or a set of coefficients that describe consecutive elements of the series from specific, time-lagged (previous) elements. This can be summarized in the Eq. (7.1): Xt ¼ ξ þ Ф1  Xðt1Þ þ Ф1  Xðt2Þ þ Ф1  Xðt3Þ þ    þ є

ð7:1Þ

where ξ is a constant (intercept) and Ф1, Ф2, Ф3 are the autoregressive model parameters. Each observation is made up of a random error component (random shock) є and a linear combination of prior observations. An autoregressive process will only be stable if the parameters are within a certain range; for example, if there is only one autoregressive parameter then is must fall within the interval of 1 < ϕ < 1. Otherwise, past effects would accumulate, and the values of successive xt’s would move towards infinity, that is, the series would not be stationary. If there is more than one autoregressive parameter, similar (general) restrictions on the parameter values can be defined.

7.3.2

Moving Average Process

Independent from the autoregressive process, each element in the series can also be affected by the past error (or random shock) that cannot be accounted for by the autoregressive component; Eq. (7.2) describes the moving average process: Xt ¼ μ þ єt  Ф1  єðt1Þ  Ф1  єðt2Þ  Ф1  єðt3Þ

ð7:2Þ

where μ is a constant and Ф1, Ф2, Ф3 are the moving average model parameters.

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Put into words, each observation is made up of a random error component (random shock, є) and a linear combination of prior random shocks.

7.3.3

ARIMA Methodology

When modeling time series statistically, one may use one of the following approaches: (a) parametric modeling, (b) nonparametric modeling, and (c) semiparametric modeling. In parametric modeling, the probability distribution of the time series is completely specified a priori. One typical parametric model is integrated ARIMA process, ARIMA ( p, d, q). The general model introduced by Box and Jenkins (1976) includes autoregressive as well as moving average parameters and explicitly includes differencing in the formulation of the model. Specifically, the three types of parameters in the model are: the autoregressive parameters ( p), the number of differencing passes (d), and moving average parameters (q). In the notation introduced by Box and Jenkins, models are summarized as ARIMA ( p, d, q); so, for example, a model described as (0, 1, 2) means that it contains 0 (zero) autoregressive ( p) parameters and 2 moving average (q) parameters which were computed for the series after it was differenced once.

7.3.4

Modeling Musical Data

A great deal of attention has been paid in the recent past in modeling musical structure and performance. Musical events can be expressed as a specific ordered temporal sequence and time series analysis which is basically the observations indexed by an ordered variable (usually time). It is therefore not surprising that time series analysis is important for analyzing musical data as it is always be the function of time. As musical data will always be the function of time and the stochastic processes such as the time series, ARIMA models can be used as predictive models in music. We use ARIMA models for analyzing higher-level structures in musical performance. ARIMA models can be used as statistical models in music. It can analyze and identify the nature of the phenomenon represented by the sequence of pitch for ragas performed on different period of the day separately. Musical notes are basically an ordered pitch within an octave, which forms the basis of a raga. So a raga is a framework comprising of a set of rules prescribed for the melody, which allows for endless variations within the set of notes. A raga is a particular arrangement of notes and melodic movements. A raga, in Indian classical music (both Hindustani and Carnatic), may be defined as a melodic structure with fixed notes and a set of rules that characterize a particular mood conveyed by performance. In this book we are interested to modeling two ragas with different

7.4 Identification of ARIMA ( p, d, q) Models

73

time of rendition taken from different thaat. Here in the next section, we analyze the musical features of raga Bihag.

7.3.5

Analysis of Bihag

Raga: Bihag

Musical Features (Dutta 2006) Thaat: Bihag is usually assigned to the thaat Bilaval, but if Teevra Madhyam is given more importance, Bihag seems to be more akin to Kalyan Thaat. Aroh (ascent): N S G m P N S Awaroh (descent): S N (D) P M P G m G R S Vadi Swar (most important note): G Samvadi Swar (second most important note): N Prakriti (nature): restful and romantic Pakad (catch): P N D P M P G m G R S Time of rendition: night

Abbreviations The letters S, R, G, M, P, D, and N stand for Sa (always shudh), Shudh Re, Shudh Ga, Shudh Ma, Pa (always Shudh), Shudh Dha, and Shudh Ni, respectively. The letter m represents. A note in normal typeface indicates that it belongs to middle octave.

7.4

Identification of ARIMA ( p, d, q) Models

The autoregressive, integrated, moving average (ARIMA) model of a time series is define by three terms ( p, d, q). The meaning of these terms will be explained. Identification of a time series is the process of finding integer, usually very small (e.g., 0, 1 or 2), values of p, d, and q that model the patterns in the data. When the value is 0, the element is not needed in the model. The middle element, d, is investigated before p and q. The goal is to determine if the process is stationary and, if not, to make it stationary before determining the values of p and q. Recall that a stationary process has a constant mean and variance over the time period of the study. In the simplest time series, an observation at a time period simply reflects the random shock at that time period, at, that is:

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7 Modeling Musical Performance Data with Statistics

Y t ¼ at The random shocks are independent with constant mean and variance and so are the observations. If there is trend in data, however, the score also reflects that trend as represented by slope of the process. In this slightly more complex model, the observation at the current time, Yt, depends on the value of the previous observation, the slope, and the random shock at the current time period: Y t ¼ θ0 ðY t1 Þ þ at

7.4.1

Autoregressive Components

The autoregressive components represent the memory of the process for preceding observations. The value of p is the number of autoregressive components in an ARIMA ( p, d, q) model. The value of p is 0 if there is no relation between adjacent observations. When the value p is 1, there is a relationship between the observations at lag 1, and correlation coefficient ϕ1 is the magnitude of the relationship. When the value of p is 2, there is a relationship between the observations at lag 2, and the correlation coefficient ϕ2 is the magnitude of the relationship. Thus, p indicates the lag in the dependence of observations which is needed to build the relationship. For example, a model with p ¼ 2, ARIMA (2, 0, 0), is Y t ¼ ϕ1 Y t1 þ ϕ2 Y t2 þ at

7.4.2

Moving Average Components

The moving average components represent the memory for preceding random shocks. The value q indicates the number of moving average components in an ARIMA ( p, d, q). When q is zero, there are no moving average components. When q is 1, there is a relationship between the current score and the random shock at lag 1, and correlation coefficient θ1 represents the magnitude of the relationship. When q is 2, there is a relationship between the current score and the random shock at lag 2, and the correlation coefficient θ2 represents the magnitude of the relationship. Thus, an ARIMA (0, 0, 2) model is Y t ¼ at  θ1 at1  θ2 at2

7.4.3

Mixed Models

A series has both autoregressive and moving average components so that both types of correlations are required to model the patterns. If both elements are present only at lag 1, the equation is

7.5 ACFs and PACFs

75

Y t ¼ ϕ1 Y t1 þ ϕ2 Y t2 þ at In the present work, this is the model that we have found to be a suitable one.

7.5

ACFs and PACFs

Models are identified through patterns in their autocorrelation functions (ACFs) and partial autocorrelation functions (PACFs). Both autocorrelations and partial autocorrelations are computed for sequential lags in the series. The first lag has an autocorrelation between Yt  1 and Yt, the second lag has both an autocorrelation and partial autocorrelation between Yt  2 and Yt, and so on. ACFs and PACFs are the functions across all the lags. The equation for autocorrelation is that the mean of the y-series is subtracted from each Yt and from each Yt  k, and the denominator is the variance of the whole series (more correctly, it is the mean sum of squares of y-series). 1 NK

rk ¼

N k  X

  Y t  Y Y tk  Y

t¼1 1 N1

N  X

Yt  Y

2

t¼1

where N is the number of observations in the whole series and k is the lag. Y is the mean of the whole series. The standard error of an autocorrelation is based on the squared autocorrelations from all previous lags. At lag 1, there are no previous autocorrelations, so r20 is set to 0.

SErk ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u k1 u X u1 þ 2k  r 2l u t l¼0 N

If an autocorrelation at some lag is significantly different from zero, the correlation is included in the ARIMA model. Similarly, if a partial autocorrelation at some lag is significantly different from zero, it too, is included in the ARIMA model. The significance of full and partial autocorrelations is assessed using their standard errors.

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7 Modeling Musical Performance Data with Statistics

7.6

Estimating Model Parameters

Estimating the values of parameters in models consists of estimating the parameter (s) from an autoregressive model or the parameters from a moving average model as indicated by McDowall et al. (1980); the following rules apply: 1. Parameters must differ significantly from zero, and all significant parameters must be included in the model. 2. Because they are correlations, all autoregressive parameters must be between 1 and 1. If there are two such parameters ( p ¼ 2), they must also meet the following requirements: ϕ1 þ ϕ2 < 1 ϕ1  ϕ2 < 1 These are called the bounds of stationarity for the autoregressive parameter(s). 3. Because they are also correlations, all moving average parameters must also be between 1 and 1. If there are two such parameters (q ¼ 2), they must also meet the following requirements: θ1 þ θ2 < 1 θ1  θ2 < 1 These are called the bounds of invertibility for the moving average parameter (s). Complex and iterative likelihood procedures are used to estimate these parameters. The equation for θ1 is: θ1 ¼

7.7

covðat at1 Þ σ 2N

Model Diagnostics

How well does the model fit the data? Are the values of observations predicted from the model close to actual ones? If the model is good, the residuals (differences between actual and predicted values) of the model are a series of random errors. These residuals form a set of observations that are examined the same way as any time series.

7.9 Results and Discussions

7.7.1

77

Ljung–Box (Q) Statistic for Diagnostic Checking

The Ljung–Box Q statistic or Q(r) statistic can be employed to check independence instead of visual inspection of the sample autocorrelations. A test of the hypothesis can be done for the model adequacy by choosing a level of significance and then comparing the value of calculated x2 with the x2-table at (k  m) degree of freedom, k be the maximum considered lag and m is the no. of parameters in the model. The Q(r) statistic is calculated by the following equation (Ljung and Box 1978): Q ð r Þ ¼ nð n þ 2Þ

m X

ðn  kÞ1 r 2k

k¼1

where n is the no. of observation in the data series.

7.8

Modeling: Finding Fitted Model

The present analysis is based on a vocal performance of Hindustani Musician Manilal Nag’s raga Bhairav (Sa set to natural C) and recorded by a laptop at 44.100 KHz, 16 bit mono, 86 kb/s mode. Solo Explorer 1.0 software, a WAV to MIDI converter, and an automatic music transcriber for solo performances (hence the name) were used to generate the onsets and the fundamental frequencies of the notes. The software generates the notes in Western notation, but we generated the same in Indian notation; cross-checking with MATLAB. The original series was found non-stationary so a different transformation is required. We try to fit a model which gives minimum error. To do that, we check the residuals from the model, if residuals are important over the various lags for all the models. To do that, we did another Ljung–Box test which indicates the residuals. A model is good if the residuals were independent over the various lags for all the models. The original series was found non-stationary so a different transformation is required. We try to fit a model which gives minimum error (by error we mean the difference between the predicted value).

7.9

Results and Discussions

In Bihag which is a night raga where the original series was found non-stationary, a different transformation is required. We try to fit a model which gives minimum error (by error, we mean the difference between the predicted value from the model and the actual ones). We test the data for various standard error checking methods like R-squared, root mean square error (RMSE), and mean absolute percentage error (MAPE) and find which method gives minimum error between predicted

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Table 7.1 Fitted ARIMA model Model parameters

Note

Model

Fit statistics Rsquared RMSE

Night raga

ARIMA (1, 1, 1)

0.091

75.041

43.349

ARIMA (0, 1, 1) ARIMA (1, 1, 0)

0.422 0.091

2.992 2.983

50.548 50.585

MAPE

Lag

Model chosen

AR(1) 0.172 MA(1) 0.998 MA(1) 0.910 AR (1) 0.452

ARIMA (1, 1, 0)

Table 7.2 Results of L–B test Note

Model

L–B Test Statistics

Degrees of freedom

Sig

Night raga

ARIMA (1, 1, 1) ARIMA (0, 1, 1) ARIMA (1, 1, 0)

16.402 14.207 58.736

16 17 17

0.425 0.652 0.001

value and actual data value. We also checked the residuals from the model. To do that, we did Ljung–Box (L–B test) test which indicates that residuals. A model is good if the residuals were independent over the various lags for all the models. The results were obtained using SPSS statistical package version 17.0. In this test, from Tables 7.1 and 7.2, we find the fitted ARIMA model which is chosen on the basis of minimum error. In case of Bihag, we choose ARIMA (1, 1, 0) which results in minimum error. The statistics is 58.736 which is larger than the value for the statistical test for raga Malkauns (Chakraborty and Shukla 2009, Chakraborty et al. 2009) which is 19.800.

7.9.1

Results for Night Raga Bihag

Conclusion Using this ARIMA (1, 1, 0) model, we obtain Fig. 7.1 which shows the graphical plot for the actual pitch series versus the fitted pitch series, and from the visual inspection of the plot, it is quite evident that the chosen model is good enough as the fitted series is very close to the observed series.

References

79

Fig. 7.1 Observed versus fitted ARIMA model (1, 1, 0) for Night Raga

References M. Andreatta, Group-theoretical methods applied to music. Ph.D. thesis, University of Sussex, 1997 B. Archibald, Some thoughts on symmetry in early Webern. Perspect. New Music 10, 159–163 (1972) M. Babbitt, Set structure as a compositional determinant. JMT 5(2), 72–94 (1961) G.J. Balzano, The group theoretic description of 12-fold and microtonal pitch systems. Comput. Music J. 4(4), 66–84 (1980) J. Beran, G. Mazzola, Analyzing musical structure and performance: a statistical approach. Stat. Sci. 14(1), 47–79 (1999) J. Beran, Statistics in Musicology (Chapman & Hall, New York, 2004) G.E.P. Box, G.M. Jenkins, Time Series Analysis, Forecasting and Control (Holden Day, San Francisco, 1976) S. Chakraborty, R. Shukla, Raga Malkauns revisited with special emphasis on modeling. Ninad J. ITC Sangeet Res. Acad. 23, 13–21 (2009) S. Chakraborty, M. Kumari, S.S. Solanki, S. Chatterjee, On what probability can and cannot do: A case study on raga Malkauns, in National Symposium on Acoustics 2009, Hyderabad, India, 26–28 Nov 2009 P. Desikan, J. Srivastava, Time series analysis and forecasting methods for temporal mining of interlinked documents. Department of Computer Science, University of Minnesota, wwwusers.cs.umn.edu/~desikan/publications/TimeSeries.doc, accessed on January 26, 2014 D. Dutta, Sangeet Tattwa (Pratham Khanda), 5th edn. (Brati Prakashani, 2006) (Bengali) (Kolkata, India) A. Forte, A theory of set-complexes for music. JMT 8(2), 136–183 (1964) A. Forte, Structure of Atonal Music (Yale University Press, New Haven, CT, 1973) R.A. Johnson, D.W. Wichern, Applied Multivariate Statistical Analysis (Prentice Hall, Englewood Cliffs, NJ, 2002) E. Lendvai, Symmetries of Music (Kodaly Institute, Kecskemet, 1993)

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7 Modeling Musical Performance Data with Statistics

D. Lewin, Generalized Musical Intervals and Transformations (Yale University Press, New Haven, CT, 1987) M. Leyton, A Generative Theory of Shape (Springer, New York, 2001) G.M. Ljung, G.E.P. Box, On a measure of lack of fit in time series models. Biometrika 65, 297–303 (1978) D. McDowall, R. McCleary, E.E. Meidinger, R. Hay Jr., Interrupted Time Series Analysis (Sage Publications, Thousand Oaks, CA, 1980) L.B. Meyer, Style and Music: Theory, History, and Ideology (University of Pennsylvania Press, Philadelphia, PA, 1989). Re statistics: 57–65 J. Morehen, Statistics in the analysis of musical style, in Proceedings of the Second International Symposium on Computers and Musicology, Orsay (CNRS, Paris, 1981), pp. 169–183 R.D. Morris, Composition with Pitch-Classes (Yale University Press, New Haven, CT, 1987) R.D. Morris, Compositional spaces and other territories. PNM 33, 328–358 (1995) L.J. Solomon, Symmetry as a determinant of musical composition. Ph.D. thesis, University of West Virginia, 1973

Chapter 8

A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

The present chapter gives an interesting statistical comparison of two ragas, namely, Bhairav (morning raga) and Bihag (night raga). Metric and melodic properties of notes are analyzed using RUBATO along with distinct transitory and similar looking non-transitory pitch movements (but possibly embedding distinct emotions!) between the notes for both the ragas. According to Strawn (1985), “a transition includes the ending part of the decay or release one note, the beginning and possibly all of the attack of the next note and whatever connects the two notes.” Hence in addition to the study on modeling a performance, a count for distinct transitory and similar looking non-transitory frequency movements (but possibly embedding distinct emotions!) between the notes was also taken. In the characterization of ragas in Indian music, not only the notes and note sequences but how they are rendered are important. There is a concept of alankar in Indian music meaning ornament (of course in a musical sense!). The shastras have categorized alankars into Varnalankar and Shabdalankar (http://www.itcsra.org/alankar/ alankar.html). The varnas include sthayi (stay on a note), arohi (ascent or upward movement), awarohi (descent or downward movement) and sanchari (mixture of upward and downward movement). This classification of alankars relate not only to the structural aspect of the raga but also to the raga performance. A non-transitory movement would depict stay on a note for short or long duration. In the graph it looks close to a horizontal line with some tremor (the tremor is because pitch is never steady). Our analysis is summarized in Table 8.1. In raga Bhairav recording rising transitions are more than falling transitions in the given recording. This phenomenon is purely an artist-dependent characteristic in which the recording supports an arohi or ascending tendency. Although rising transitions exceed the falling ones and the difference 190–157 is noteworthy, this raga cannot be classified as of arohi or awarohi varna. For another artist, the picture could be quite different. In contrast, in Bihag, the rising transitions are nearer to the falling ones (417–408) which means the artist has shown about equal inclination to

© Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_8

81

Bhairav

408

Bhairav

190

417

Bihag

25

52

113

39

74

295

19

17

121

23

15

379

Convex Concave Linear Convex Concave Linear Convex Concave Linear Convex Concave Linear

157

Falling transition

Bihag

Rising transition

Table 8.1 Transitory and non-transitory pitch movement in Bhairav and Bihag

202

588

Bihag

Hats

Valley

38

93

75

139

71

131

Bhairav Bihag Bhairav Bihag Bhairav Bihag

Mixed transition No transition Bhairav

82 8 A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

8.1 IOI Graph

83

ascend and to descend. This raga also cannot be classified as of arohi or awarohi varna. Hats and Valleys are approximately equal in both the recordings. A hat may be interpreted as a rise followed by immediate fall, a valley as a fall followed by immediate rise.

8.1

IOI Graph

The notes are said to be in rhythm if the inter-onset times between successive notes is equal. This has been detected in our paper using an inter-onset-interval (IOI) graph where the term onset refers to the point of arrival time of a note (the idea also applies to beats in a percussion instrument). Figures 8.1 and 8.2 give the IOI graphs Fig. 8.1 IOI graph for Bhairav. Mean ¼ 0.946061, standard deviation ¼ 0.996182

Fig. 8.2 IOI graph for Bihag. Mean ¼ 0.75, standard deviation ¼ 0.678823

84

8 A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

for the two recordings. The mean IOI and standard deviation are approximately equal in Fig. 8.1 implying there is rhythm in the notes in this recording. Here are the IOI (inter-onset-interval) graphs for detecting rhythm: By seeing above IOI graphs, we can see that standard deviation is less in raga Bihag. When there is less deviation, rhythm is more. So raga Bihag recording is more in rhythm than raga Bhairav (Fig. 8.2).

8.2

Duration Graphs

The duration of the note is important as it is directly related to the conception of pitch stability proposed by Carol Krumhansl. This concept of pitch stability is psychological in nature reflecting a stay on the note. A note is important in a psychological sense if its average duration is high with low standard deviation. From above and the statistical analysis and Figs. 8.3 and 8.4, it is clear that the mean note duration and the standard deviation in the two ragas are not much different confirming that both the ragas are of similar restful nature.

8.3

RUBATO Analysis

RUBATO is probably the best software for analyzing musical structure and performance for statistical purpose. RUBATO studies the metric (rhythmic), melodic, and harmonic properties of scores. RUBATO demands a note in the score be described by four characteristics, namely, (1) its onset (point of time of arrival of the note in the score in seconds), (2) its pitch (which gives the shrillness or hoarseness of the note) (measured by frequency in Hertz), (3) its duration (the time difference between its departure and arrival in seconds), and (4) its loudness (measured by amplitude in volts). In the present approach to performance theory,

Fig. 8.3 Duration graph of Bhairav. Duration Bhairav mean ¼ 0.85034, SD ¼ 0.996419

8.4 RUBATO Analysis of Bhairav

85

Fig. 8.4 Duration graph of Bihag. Duration Bihag mean ¼ 0.9, SD ¼ 0.933381

through RUBATO, each note in the score, by virtue of the four characteristics, can be assigned a weight that measures its metric, melodic, or harmonic importance. In RUBATO, there are explicit rules derived from general music theory and practice for analyzing a score and transforming the results into numerical weights (Mazzola 1996, 2002). The metric analysis used by RUBATO essentially considers all periodically repeating metric structures called local meters. A note in the score is metrically important if it is a part of many local meters. The melodic analysis considers similarities between musical motifs (configurations) of the score. A note is melodically important if it is part of motifs that are similar to many other motifs that occur elsewhere in the score. A comprehensive account of the mechanism of RUBATO can be found in Chap. 3.

8.4

RUBATO Analysis of Bhairav

Metro 2

86

Metro 4

Metro 10

Metro 18

8 A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

8.4 RUBATO Analysis of Bhairav

Metro 24

Metro 32

Metro 36

87

88

8 A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

meloweight

melo_settings

8.5 RUBATO Analysis for Raga Bihag

8.5

RUBATO Analysis for Raga Bihag

Weights

W3

W5

89

90

W10

W15

W20

8 A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

8.5 RUBATO Analysis for Raga Bihag

W25

W30

W35

91

92

W40

W45

W50

8 A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

8.5 RUBATO Analysis for Raga Bihag

W54

Meloweight3

93

94

8 A Statistical Comparison of Bhairav (a Morning Raga) and Bihag (a Night Raga)

Screen Shot 2012-12-05 at 10.07.17 AM

Conclusion By comparing the RUBATO analysis of raga Bihag and raga Bhairav, we can see that Bihag is more in rhythm than raga Bhairav, a fact observed from IOI graphs as well. Regarding the melodic part, the black spots are heavy weights, and gray spots are light weights. Table 8.2 gives a comparative study of musical features of the two ragas.

References

95

Table 8.2 Comparison of musical features of Bhairav and Bihag Musical feature

Raga Bhairav

Raga Bihag

Thaat (a raga group based on scale)

Bhairav

Bilaval

Aroh (ascent)

S r G M, P d, N S

NSGMPNS

Awaroh (descent)

SNdPMGrS

SNDPmGMGRS

Vadi (most imp note)

d

G

Samvadi (second most imp note)

r

N

Pakad (catch)

G M P d, P d M P G M, r S

N S, G M P, G M G

Jati (no. of distinct notes used in ascent and descent)

Sampoorna-Sampoorna (7 distinct notes in ascent; 7 distinct notes in descent)

Audav-Sampurna (5 distinct notes in ascent; 7 distinct notes in descent)

Nyas swar (stay notes)

r, M, P, d

G P and N

Anga

Uttaranga-Pradhan meaning that the second half is more important

Purvanga Pradhan meaning that the first half is more important

Time of rendition

First raga of the morning (5 AM–8 AM)

Night (second quarter of night i.e. 9 PM–12 PM)

Nature

Restful and serious

Restful

Italics, normal type and bold letters are used to represent notes in the lower octave, middle octave and higher octave respectively

References G. Mazzola, RUBATO on the internet, 1996–2014 (1996), http://www.rubato.org G. Mazzola, The Topos of Music (Birkhauser, Basel, 2002) J. Strawn, Modeling musical transitions. Ph.D. dissertation, Stanford University, Stanford, CA, USA, 1985

Chapter 9

Seminatural Composition

9.1

Introduction

Music composition can be natural, artificial, or seminatural. The first category refers to a composition where a human being decides both what to play (or sing) as well as how to. The second refers to a composition where a mechanical device such as a computer is trained to accomplish both the what and the how part. The third category is of interest here in which the computer will decide the what part, while a human being will take up the how part. Consider a music composer interested in composing a raga-based song. He is looking for the starting line or a clue for the next line. Can computer help? We answer this question through seminatural composition giving an example in raga Bhimpalashree. Seminatural composition was introduced in Chakraborty et al. (2009) and is discussed more formally in Chakraborty et al. (2011) recently. However, for the sake of completeness, we are providing below the algorithm seminatural composition algorithm (SNCA) after which the illustrative example will follow. For an extensive literature on algorithmic composition, the reader is referred to Nierhaus (2008). We assume a transition probability matrix (TPM) is available. In our case we build it with a long sequence of raga notes (Dutta 2006). Raga notes are given in Table 9.4 in Appendix. In the Appendix, notes in middle octave are given in normal font; those in lower octave are given in italics, while bold type is used to depict notes in higher octave. To find P(x/y) which is the probability for the next note to be x given that the present note is y, the number of times y is followed by x is divided by the total number of times y occurs in the entire sequence (ignoring octave). If y is the last note in the sequence, there is no information of the next transition. Hence, in that special case, we simply subtract one from the denominator. A first-order Markov chain is used to build the TPM. This means we assume the note at instance j to be influenced by the note at instance j
1 only, and not the notes realized at instances j
2, j
3, etc. Nierhaus (2008) has clearly indicated that it may not be wise to indiscriminately increase the order in a Markov chain. There are several © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_9

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9 Seminatural Composition

Table 9.1 TPM of raga Bhimpalashree (first-order Markov chain used)

S (or C) R (or D) g (or Eb) M (or F) P (or G) D (or A) n (or Bb)

S

R

g

M

P

D

n

3/15 3/7 0 0 1/14 0 8/12

3/15 0 4/14 0 0 0 0

0 0 0 13/24 2/14 0 0

5/15 0 8/14 2/24 9/14 0 0

0 0 2/14 9/24 0 1 0

0 0 0 0 0 0 3/12

4/15 4/7 0 0 2/14 0 1/12

Corresponding Western notation is provided taking the tonic Sa at natural C Table 9.2 Class matrix of raga Bhimpalashree S

R

G

M

P

D

N

[0.7333, 0.7333] [0.4285, 0.4285] [0.8571, 1] [0.625, 1]

[1, 1]

[1, 1]

P

[0, 0.07143] [0, 0] [0, 0.6666]

[0.07143, 0.07143] [0, 0] [0.6666, 0.6666]

[0.4000, 0.7333] [0.4285, 0.4285] [0.2857, 0.8571] [0.5416, 0.625] [0.2143, 0.8571] [0, 0] [0.6666, 0.6666]

[1, 1]

[0, 0]

[0.4000, 0.4000] [0.4285, 0.4285] [0.2857, 0.2857] [0, 0.5416] [0.07143, 0.2143] [0, 0] [0.6666, 0.6666]

[0.7333, 0.7333] [0.4285, 0.4285] [1, 1]

M

[0.2000, 0.4000] [0.4285, 0.4285] [0, 0.2857] [0, 0]

[0.7333, 1]

g

[0, 0.2000] [0, 0.4285] [0, 0]

[0.8571, 0.8571] [0, 1] [0.6666, 0.6666]

[0.8571, 0.8571] [1, 1] [0.6666, 0.91666]

[0.8571, 1]

S R

D n

[0.4285, 1]

[1, 1] [0.91666, 1]

Table 9.3 Random numbers (independent U[0, 1] variates) 1 2 3 4 5 6 7 8 9 10

0.68748 0.931552 0.881453 0.360767 0.050769 0.14729 0.98519 0.00813 0.006948 0.447831

11 12 13 14 15 16 17 18 19 20

0.395649 0.105849 0.233853 0.29305 0.119797 0.48212 0.559246 0.357652 0.327365 0.286544

21 22 23 24 25 26 27 28 29 30

0.264452 0.650879 0.967518 0.467493 0.367386 0.92207 0.653119 0.13736 0.728659 0.785223

31 32 33 34 35 36 37 38 39 40

0.399981 0.867007 0.212576 0.236418 0.963381 0.547728 0.819231 0.074684 0.917081 0.024483

41 42 43 44 45 46 47 48 49 50

0.0251463 0.845766 0.585291 0.483267 0.423388 0.219986 0.543505 0.647764 0.505027 0.118593

techniques to determine the order. For example, if an appropriate time series prediction model suggests that knowledge of Y( j
1) is enough to predict Y( j), we automatically know that we can safely use a Markov chain of first order. We omit a rigorous discussion here and refer the reader to Chakraborty and Shukla (2009). Table 9.1 gives the TPM, and Table 9.2 gives the class matrix. Table 9.3 gives the independent U[0, 1] variates.

9.2 Experimental Results

99

Algorithm SNCA Step 1: Without any loss in generality, take the note at instance 1 to be the tonic Sa (S). Sa can always be the first note in any raga. Also it is musically logical to start with the tonic, the base note, or note of origin from which other notes are realized. Step 2: Using the transition probabilities at Sa, one simulates the next note. The strategy is to generate a continuous uniform variate X in the range [0, 1], and, depending on the class in which X falls (see row one of Table 9.2 giving the classes), the note at instance 2 is obtained as either S, R, g, M, P, D, or n with respective transition probabilities as given in row 1 of Table 9.1. The logic that defends the strategy is that since X is a continuous U[0, 1] variate, so P (a < X < b) ¼ b
a, obtained by integrating the probability density function of X, which is unity, between the limits a and b. Step 3: Using the note simulated at instance 2 and using the corresponding transition probability row of this note, the next note at instance 3 is simulated similarly using cumulative probabilities to form classes using another independent U[0, 1] variate. The process is repeated.

9.2

Experimental Results

Applying algorithm SNCA on the set of 50 independent U[0, 1] variates, using Sa as the starting note, we immediately have a sequence of 51 notes as our illustrative example of raga Bhimpalashree as given below (the octaves, note duration, pause between notes, and loudness are to be decided by the composer). SMPnSSSnSSMgRSRSMMgMgRnnSRnSSMPMPgRnSnS n S R n S M g R n S M g. Render it! Conclusion A simulated raga note sequence may not confirm to the raga correctly but may suffice for a line of a raga-based song where the raga need not be maintained correctly. An important contribution of a raga-based song is that it promotes Indian classical music among the general mass. We do not support artificial composition in Indian music as calculated artistry is not a substitute for emotion and devotion. In seminatural composition, some scope is there for imagination once the note sequence is obtained. In some cases, the composer may change a few notes here and there to make the line meaningful. As a final comment, a seminatural composition will not, of course, in general beat a natural composition, but we feel it is still a better strategy to exploit the computer’s power to simulate than “lifting” tunes indiscriminately. For better results, a large database of raga notes should be used to build the TPM. (continued)

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9 Seminatural Composition

Building a TPM for any song in general is always a challenging research problem. We conclude this chapter with some musical features of raga Bhimpalashree. Musical features of raga Bhimpalashree (or Bhimpalasi): Thaat (a method of grouping ragas according to scale): Kafi Aroh (ascent): n S g M, P, n, S Awaroh (descent): S n D P M g R S Jati: Aurabh-Sampoorna (five distinct notes allowed in ascent; seven in descent) Vadi Swar (most important note): M Samvadi Swar (second most important note): S Prakriti (nature): restful Pakad (catch): n S M, Mg, PM, g, M g R S Stay notes (nyas swars): g M P n Time of rendition: 1 PM–3 PM

Appendix Table 9.4 Notes of raga Bhimpalashree (Dutta 2006) Sl. No.

Note

Sl. No.

Note

Sl. No.

Note

Sl. No.

Note

Sl. No.

Note

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

n S M g M P M P M g M g R n S M g P M

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

P g M P N D P M P g M P n S R n S S n

39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

D P M P M g R S S n S M g M P M g M M

58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76

g R S R n S M g M M g P M g M g R S S

77 78 79 80 81 82 83 84 85 86 87 88 89 90

n D P M P S n n S R n S M g

Italics, normal type and bold letters are used to represent notes in the lower octave, middle octave and higher octave respectively

References

101

References S. Chakraborty, R. Shukla, Raga Malkauns revisited with special emphasis on modeling. Ninad J. ITC Sangeet Res. Acad. 23 (2009) S. Chakraborty, M. Kumari, S.S. Solanki, S. Chatterjee, On what probability can and cannot do: a case study in Raga Malkauns. J. Acoust. Soc. India 36(4), 176–180 (2009) S. Chakraborty, M. Kalita, N. Kumari, Semi-natural composition. An experiment with North Indian ragas. Int. J. Comput. Cognit. 9(2), 51–55 (2011) D. Dutta, Sangeet Tattwa, 5th edn., vol. 1 (Brati Prakashani, 2006) (in Bengali) (Kolkata, India) G. Nierhaus, Algorithmic Composition: Paradigms of Automated Music Generation, 1st edn. (Springer, Heidelberg, 2008)

Chapter 10

Concluding Remarks

What is primarily common between music and statistics? I think it is the fact that a musician is imagining and creating musical patterns during a composition. Statistics, on the other hand, is the science of exploring and studying patterns in numerical data. Musical data are certainly numerical in character as they pertain to pitch, onset and departure of notes, loudness, timbral characteristics, etc. All these can be subjected to a careful statistical analysis. I have also worked, and am still working, with a team of doctors and another statistician studying the therapeutic impact of Hindustani ragas on patients with brain injury, and I can assure you that it is very difficult, if not impossible, to establish the aforesaid impact without a sound statistical analysis, even if we all know music can heal. In this issue, we open with an exciting new study that implements one of the most influential genres of music in India and measures its applied effects in two populations not often studied in music and medicine trials (referring to our work, see Singh et al. 2013). . .. Readers will broaden their scope of music by considering the impact of chant and the tradition of such healing music. —Ralph Spintge, MD and Joanne V. Loewy, DA, LCAT, MT-BC

Editorial comment titled Seeking the Essence of the Study within the Study, Music and Medicine 5(2), 2013, 65–66 (this specific comment appears on p. 66) These editors wrote on p. 65 of the aforesaid reference “. . . the securing of an appropriate design and statistical analysis of the variables in hand is most important” which sums up the story. Whether it is the analytic or therapeutic side of music, the statisticians have a lot to do. I hope my words will excite other professional statisticians to take up music analysis and music therapy seriously as a part of their professional career. Computational musicology and music therapy are definitely connected. If a particular piece or genre of music is found to be working well for a particular malady (e.g., the raga Pilu has antidepressant properties, and this is one of the ragas we used in our study on brain injury patients), it becomes important to know what are the features of the concerned raga (Pilu in our example), demanding both an aesthetic and scientific study of the raga characteristics. It is also important to do © Springer International Publishing Switzerland 2014 S. Chakraborty et al., Computational Musicology in Hindustani Music, Computational Music Science, DOI 10.1007/978-3-319-11472-9_10

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Concluding Remarks

brain imaging studies to explore what happens inside the brain when a patient hears this raga. The latter may require tools like functional magnetic resonance imaging (fMRI), positron emission tomography (PET), etc., for substantiation. In an attempt to promote both computational musicology and music therapy/ medicine in the context of Indian music, I submitted a very interesting proposal to the Director of SPIC MACAY, Dr. Kiran Seth (himself an applied probabilist at Indian Institute of Technology, Delhi). The proposal talked about presenting art of music and science of music on the same stage in which a scientific researcher and an artist (with other accompanists) would contribute jointly. The proposal comprises of three phases, namely, (1) the concert phase, (2) the talk phase, and (3) the interaction phase. In the concert phase, the artist gives his/her own scheduled concert. In the talk phase, the scientist talks about scientific research in music highlighting both analytical and therapeutic aspects of music research. The third and last phase is the most crucial phase—the interaction phase—where the scientist and the artist are present together. Lest the reader wonder what will happen here, I give a brief account of the third phase from my personal maiden adventures of the aforesaid scheme. The first such adventure was with the Carnatic flutist Jayaprada Ramamurthy held in The Institute of Mathematics and Applications in Bhubaneswar, arranged on my own request, and the second with the shehnai player Sanjeev Shankar (of Banaras Gharana) at the Taurian World School in my own home town Ranchi, arranged by SPIC MACAY, the Society for the Promotion of Indian Classical Music And Culture Amongst Youth. I heard Jayaprada for the first time in 2009 in Hyderabad during a conference arranged by Acoustical Society of India and was very impressed. In her flute playing, I discovered simultaneously the romanticism of Pt. Hariprasad Chaurasia, the classicism and tonal quality of the legendary Pannalal Ghosh and the intricate shruti ornamentations of Dr. N. Ramani (Jayaprada is a direct disciple of the last one). Sanjeev’s blowing is also very soft, and he builds the raga mood cautiously before moving to technical embellishments. After Jayaprada finished her own recitals, in ragas Hanswadhwani and Bhimpalasi (same as Bhimpalashree), she began a lecture-demonstration on flute and Carnatic music. Then, all of a sudden, she started talking about scientists who are working on music. At that point, I stood up from my chair (I was sitting in the front row) and, facing the audience, said: “Why do we listen to music? For pleasure? To relieve tension? Spiritual Sadhana? Carol Krumhansl, a Professor of Psychology at Cornell University, has discovered the answer to this intriguing question dwelling in the psychology of music. According to her, we listen to music to fulfill our expectations (Krumhansl 2002). To this, I would like to add that we also listen to music to fulfill our un-expectations! That is to say, un-expectations or surprises are also highly appreciated by an enthusiastic music listener. The use of entropy, as a measure of surprise, in music analysis is important therefore, and I refer to Snyder (1990) for more information. Musical expectations, as opposed to un-expectations or surprises, can be scientifically measured using probability. This is what Temperley (2007) has successfully accomplished using a classical Bayesian approach. Bayesian analysis rests on the concept of conditional expectation.” Having said that I told the audience “Well, we have got the answer to

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the question why we should listen to music. But why should we listen to classical music? That Jayaprada will explain!” Jayaprada picked up from where I left and started explaining to the audience the benefits of listening to classical music. An important point she mentioned is that if one can grasp classical music, one can understand any other form of music with little difficulty. I personally feel that in today’s context, with a sudden rise of crimes in various forms, as well as the sudden rise of diseases like hypertension, diabetes, and heart problems which have much to do with our fast and tensed lifestyle, it is important that music, especially classical music being a discipline, is promoted. Music must be made compulsory in every educational institute and that everybody should be encouraged to learn to play at least one musical instrument of his/her choice, if not to become a musician then certainly to become a better and healthy human being. In the final analysis, that should be our highest expectation! Jayaprada’s concert was in the afternoon (of a bright and sunny day in Sept, 2010), and her choice of Bhimpalasi, one of my favorite ragas, was also very fitting. Jayaprada’s effort is particularly commendable as she was defending Carnatic music and flute with a single percussionist, the Mridangam player Srikanth, and with neither the sound system nor the mikes working properly. In fact, only one mike was audible, the one I used for my experiment (there must be some sense in the nonsense!). In contrast, Sanjeev’s concert was in the morning, and his choice of raga Megh was befitting of the cloudy weather (it was August, 2012). When I stood up on stage, after my formal introduction to the audience as a statistician cum music researcher by Sri Rajiv Ranjan, the SPIC MACAY zonal coordinator, I requested Sanjeev to play the notes of Megh in straight transition and then again play them using meend or glide between the notes. Sanjeev did this nicely as requested, probably anticipating my plan. I asked the audience—“Which do you think creates the raga atmosphere better—the first rendition or the second?” When the audience said it was the second, I added “this is because Indian classical music moves in curves more than in straight lines like nature (pointing to the flowers in the garden which were also moving in curves!). It is because of this movement in curves that we require fractional dimension to capture the chalan or melodic movement of a raga.” (see also Chakraborty et al. 2010). From here, I moved onto music therapy and briefed the audience about the ongoing research work at RIMS hospital in Ranchi. A commonality in both the adventures is that we jumped to the third (interaction) phase from the first (concert) phase, bypassing the second phase (talk) as there was no overhead projector. I never got a chance to give a formal talk on my music research which I normally do in conferences. I do feel that the overhead projector should be made available to the scientist on the stage itself. The success of this program depends on proper coordination between the scientist and the artist. The organizer should make sure both get the proper facilities. It should be understood that such programs fulfill three objectives:

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1. Promote music appreciation, as knowledge and understanding of music is more complete when our aesthetic appreciation is coupled by quantified information as provided by scientific research in music. 2. Promote music therapy, which can be a success and an established discipline if and only if medical professionals, musicians, and music researchers work together, given the healing powers of music in relieving psychological diseases and negative emotions as concluded by music researchers. 3. Explain how scientific research in music can assist the music learners (e.g., in optimizing their practice routine, in making better use of their instruments and voice, in assessing versatility in performance and improving the same, etc.). It is important for the scientist to convince this to the music learner. In the words of Prof. Richard Parncutt (2007), professor of Systematic Musicology at the University of Graz, Austria, “Research on musical expression (structural, emotional, bodily) may raise students’ awareness of their expressive strategies and help them to plan and practise the expression that they bring to specific works, linking analysis to interpretation. Research on memory, sight-reading, improvisation and intonation may help students to enhance their skills in these specific areas. Research on performance anxiety may help students to turn anxiety to their benefit. Research on music medicine may help students to prevent and treat injuries.” It is unfortunate that since music performance is more intuitive than logical, these aspects are not included in our music curriculum. Another commonality is that both the artists were wind instrumentalists. This is also befitting, given that I myself played the Hindustani flute in the prime of my youth! (I currently play the harmonium, another wind instrument, as an amateur solo performer and accompanist in addition to my professional assignments as a statistician). This book, written with the sole objective of promoting computational musicology in Indian music, is primarily aimed at teaching how to do music analysis in Indian music. It assumes that the musical data is already available. Thus, this is not a book that teaches you how to acquire the musical data using signal processing. Consequently, several aspects of music analysis involving signal processing such as raga identification and tonic (Sa) detection had to be left out. The reason for this is that while there are good books available in musical signal processing (see, e.g., Roads et al. 1997; Klapuri and Davy 2006), there was not a single book in computational musicology in Indian music especially with a thrust in statistics. Most of the works are available as research papers, and in most cases they are not accessible without a personal or an institutional subscription for the concerned journal. Hopefully this book will meet some of the requirements of music enthusiasts and analysts (young and old!) interested in Indian music. However, we promise to write a follow-up manuscript on music information retrieval (MIR) in the near future where musical signal processing would be extensively dealt with in the context of Indian music. Most of the issues that could not be addressed here would be taken up then.

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References S. Chakraborty, S. Tewari, G. Akhoury, What do the fractals tell about a raga? A case study in raga Bhupali. Conscious. Lit. Arts 11(3), 1–9 (2010) A. Klapuri, M. Davy (eds.), Signal Processing Methods for Music Transcription (Springer, New York, 2006) C.L. Krumhansl, Music: a link between cognition and emotion. Curr. Dir. Psychol. Sci. 11(2), 45–50 (2002) R. Parncutt, Can researchers help artists? Music performance research for music students. Music Perform. Res. 1(1), 1–25 (2007) C. Roads, S.T. Pope, A. Piccialli, G.D. Poli (eds.), Musical Signal Processing (Studies on New Music Research) (Routledge, New York, 1997) (an imprint of Taylor and Francis) S.B. Singh, S. Chakraborty, K.M. Jha, S. Chandra, S. Prakash, Impact of Hindustani ragas on visual acuity, spatial orientation, and cognitive functions in patients with cerebrovascular accident and diffuse head injury. Music Med. 5(2), 67–75 (2013). doi:10.1177/ 1943862113485903 J.L. Snyder, Entropy as a measure of musical style: the influence of a priori. Assumptions Music Theory Spectr. 12, 121–160 (1990) D. Temperley, Music and Probability (MIT Press, Cambridge, 2007)