An experimental reexamination of the theory of the neural quantum in the sensory discrimination of pitch and loudness

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AH EOTRMEHTAL RS*mJtIHATI6N OF THE THEORY OF THE KSURAL QUANTUM IN THE SENSORY DISCRIMINATION OF FITCH AHD LOUDNESS

by John Flermonte Corso

A dissertation submitted In partial fulfillment of the requirements for the degree of Doctor of Philosophy, in the Department of Psychology in th© Graduate College of the State University of Iowa June 1950

ProQuest Number: 10598587

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ProQuest 10598587 Published by ProQuest LLC (2017). Copyright o f the Dissertation is held by th e Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C o d e Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346

the writer wishes to express his sine^re&t appreciation to Professor Bon Lewis for suggesting the problem and offering critical guidance through­ out the course of the study, .Further aeknowledgssent Is mad© to Hr, H* H, Burfthardt for the design of the electronic circuits used in the ©xperimental apparc.tus,

11

TABLE OF COMTira

Gimp tor

1

page Theoretical background

%

Introduction* . . . . . * The classical theory of discrimination . ....... . . The theory of the mural quantum* * XI XII

1 7

Statement of the problem..............

33

Experimental procedure...............

34

♦ . Subjects* . , . . ......... . Apparatus . . . . . . . . . . . . . Loudness discrimination. . . . Pitch discrimination . . . . . Additional devices . . . . . . Procedure » ......... Pitch discrimination . . . . . Loudness discrimination* . . . Levels of certainty* .. . . . IV

1

Results

. . . . . . . . . . . . . . . .

3^ 35

35 46 4? 49

50 52 52 54 55 60 67

Pitch discrimination....... Loudness discrimination. . . . Levels of certainty* . . . . . V

Discussion of results. . . . . . . . . .

69

71

Summary and conclusions. . . . . . . . .

74

Appendix A . . . . . . . . . . . . . .

*

77

Appendix B . . . . . . . . . . . . . .

.

76

............

Appendix 0 . . . Appendix D ♦ . . .

............

Appendix E . . . . . . . . Bibliography . ill

80

86

.......

91

......

93

TABLE OF FI0UBE3 Figur©

pag©

1. Th© psychometric function - two categories of Judgment .........

*

3

2* Boring’s chart of th© psychometric function for the two-point threshold . . . . . . . . .

6

3. A schematic representation of th© basic notions involved in the theory Of the neural quantum ........... . . . . . . . . . . . .

14

4* Th© distribution of surpluses and th© probability function .............

IS

5# The fora of the psychometric function in the sensory discrimination of pitch and loudness according to the theory of the neural quantum

24

6. Block diagram of the apparatus required to provide tonal stimulation in the Investigation ....... of loudness discrimination . • . 7* Circuit-diagram of the band-pass filters used to eliminate transient frequencies in pitch and loudness discrimination . . . . . . . . .

36 3$

8. Frequency selectivity curves for the filters used in loudness discrimination

.........

40

9. Block diagram of the apparatus required to ^ provide tonal stimulation in the investigation of pitch discrimination ♦ . , . . ......... 10. Frequency selectivity curves for the filters used In pitch discrimination . . . . . . . .

43 48

11. Representative non-linear psychometric functions obtained in pitch discrimination

»

59

12. Representative linear psychometric functions obtained in pitch discrimination

•

/6l

-IV

TABLE OF FIGURES (Oontlnued) Figure

page

13* Representative non-linear psychometric? functions obtained in loudnessdiscrimination'

64

14, The two linear psychometric functions obtained In loudnessdiscrimination * * , „ *

65

15* Psychometric functions obtained in pitch discrimination for two trained observers under a low and under a high certainty criterion * * .......... ....

68

TABLE OF TABLES Table X

page Chi*square values obtained 1b pitch discrimination

56

Ratio values for the linear functions obtained In pitch discrimination * . ♦ „ . * #

57

Ghl*square value® obtained In loudness discrimination .........

62

Ratio values for the linear functions obtained In loudness discrimination « . « * .

66

V

Total scores- on Seashore tests • * * » « . . *

77

VI

Observer R* B» * * * . * ............. .. „ .

80

Observer R* R* . • • . . . • ♦

- . ..........

80

Observer £»K» « • * « * • * * * « « * » * • »

81

Observer F* ?.........

81

Observer

82

Observer W«

82

It III IV

VII VIII IX X XI XII XIII XIV XV

Observer R* S* . * * . *. • • • » • •

* < *. •

83

*

83

*

84

Observer K. K. * Observer J* B« * , • . . «

. * .. * * • * *

Observer S. S......... * » * ,

*

84

XVI

Observer K* H* B*

85

XVII

Observer Jo F* C,

85

XVIII XXX

Observer G* V.

♦ ...........

Observer f* S.............. vi

86

86

TABLE OF TABLES (Continued) Table

Pag®

XX. Observer A. 3. XXI XXII XXIII XXIV XXV

•

ObserverM. R, . . ............. ObserverJ. C. . . . . . . ObserverIf. w. . . . . .

ObserverL. S, . . . . .

87

................ . ...........

ObserverJ. S. . . . . . . . . . . . . . . .

§7

88 88

.

..........

89 89

XXVI

observerH. 0. . . . . . . . . . . . . . . .

.

90

XXVII

ObserverK. 8. . . . . . . . . . . . . . . .

.

90

Til

1

Chapter I THEORETICAL BAOECrROUOT Introduction the discriminatory ability of th© human organism in audition may b© considered from two points of view: first, the ability to perceive the presence, or absence, of a given auditory stimulus under specified conditions and, second, the ability to detect a change in a given auditory stimulus whenever one of its attributes is Increased or de­ creased In physical magnitude. The basic concepts Involved In these two aspects of discrimination are the absolute threshold and the differential threshold, respectively (22), The present study utilises the concept of the differential threshold as an approach to the theoretical formulations which have been advanced to account for the behavior ob­ served under conditions of pitch and loudness discrimina­ tion, The first of these formulations is called the classi­ cal theory of discrimination; the second, and more recent formulation, the theory of the neural quantum, The Classical Theory of Biscrimination The classical view of sensory discrimination may be introduced through the determination of the differential threshold, or difference Ilmen. This threshold is defined as the smallest stimulus difference which can b© perceived

an observer a specified proportion of th© times pres­ ented, Since the method of right and wrong oases Is gen­ erally regarded as the most satisfactory psychophysical procedure for determining threshold values (5)* this method will now be presented in detail. At the beginning of a di aer1mlna 11on the several values of the variable stimulus to be are carefully selected, These stimuli., some greater and some less than th© standard in physical magnitude, are presented to- the subject a large number of times in an irregular order, or in a prearranged order uhimown to th© subject. On each trial the variable is preceded, accomp­ anied, or followed by the standard.* The subject makes a judgment of ' rtgreater** or nlessn with reference either to the standard or to the variable, or In accordance with -some pre­ arranged plan. In the calculation of the differential thresh­ old values, the proportion of each of th© two types of judgments is determined for each value of the variable stimulus- used, The results can be presented graphically in the form of a psychometric function {21) as shown in Fig, 1, Hi© upper differential threshold Is conventionally defined as the distance between th© standard stimulus {Rs} and the variable stimulus (By.) for which the psychometric curve has an ordinate of p = 0,75* This is th© increment which must b© added to th© standard stimulus in order for

3

1.00

Smaller

50

PROPORTION

OF JUDGMENTS

75

VARIABLE STIMULUS

y-

Fig. 1. The psychometric function - two categories of Judgment*

4 the subject to make the correct discrimt,tmtion in threefourth® of his attempt®-* The earn© line of reasoning may he followed in the determination of the lower differential threshold value* It must he pointed out, however, that actual expert* mentation doe© not give the curves shorn in Fig,# 1* but only a series of empirical points through or among which these curves can be drawn* The curves can b# drawn either by free­ hand approximation or by assuming ma,theme,tioal equations for the obtained relationships* The best-fitting curves, in the latter case., can be determined by any one of several curve-fltt ing procedures, such as the method of least squares. The choice of a mathematical formula to represent a psychometric function is essentially a statement -of an hypothesis about the psychometric- function. The classical theory of sensory discrimination maintains that discrim­ ination data ©an- be most adequately represented by the phi function of gamma. This hypothesis states that th© psycho­ metric curve may be described by the integral of the normal probability function. This is equivalent to saying that the threshold of the fepman organism varies somehow, perhaps due to •physiological and/or psychological cause®, in accordance with the normal law of error. Guilford (9) summarizes Boring*s (3) argument in the following maimers

5 Let us assum® that th® stimulated organs, including the brain with its momentary states of equilibrium, are variously disposed toward a given impression. These disposition© are dependent upon a great many different factors (*coins ’), each of which can be favorable for a Judgment of a given category (1two1). These dispositions, being independent, are favorable and unfavorable in chance combinations. For a weaker, or smaller, stimulus more of them must be favorable than for a stronger, or larger, stimulus* To simplify matters,' let us assume six such factors operative in the two point limen experiment. All six factors must be favorable simultaneously in order that a separation, of 8mm., let us say , can arouse a report of *two1. At least five must be favorable for a report of 1two1 at a separation of 9 mm.% at least four at 10 ms*, and so on, as Fig. 2 shows-.- Assuming 64 trial# with each.stimulus, the theoretical probability of a Hwo* at 8 mm, is 1/64; at 9 mm. * 1/64 plus S/64, and so on. In terns of proport ions, these probabilities are .016, .109., *344, *656, ♦891,- and *984* respectively* {See Fig. 2.} One other feature of the classical theory of sensory discrimination remains to be mentioned* This Is the assumption of the existence of two continual (l) a physical continuum and (2) a psychological continuum* Hie physical continuum consists of a continuous gradation In th# magnitude of a stimulus attribute; the psychological continuum, a continuous gradation In the magnitude of the corresponding sensory response* This view holds that sensation can be regarded as a continuous function of the stimulus, Titchener (20) although accepting this notion, pointed out that a continuous gradation of stimuli would, under certain .conditions, give rise to a discrete sensory seal#-*

6

64

1.000 0.914 Cumulative frequency of all grades of chance disposition not less than a ^ certain amount ^

0.891

56

48

40

Phi-gamma function

32

0.500

Frequency of each grade of chance disposition

0.344 15,

16

8

0.109 0.016

0 5

6

4

3

2

1

0

Grades of Chance Disposition

7

Fig. 2

24

8

9 10 11 12 R (STIMULUS) VALUES IN MM.

13

14

Boring’s chart of the psychometric function for the two-point threshold.

FREQUENCIES

PROPORTIONS

0.656

T We obtain this series (Titehener says} by increas ing our stimulus very slowly, so that sensation follows it, not continuously, but in little Jerks. We owe the jerkiness to the fact of friction (a phenomenon resulting from the incoming disturbance)* Specifically stated, the general applicability of the phi function of gamma to psychophysical data appears to justify the following assumptions of the classical theory of sensory discriminations (1) the existence of a continuous series of psychological responses parallel to a correspond­ ing physical stimulus continuum ; (2) the dependence of the psychometric function upon a large number of unknown factors which produce moment i© moment threshold variations in accordance with the normal law of error; and (3 ) the mediation of a discriminatory response through the gross response of the organism to the total magnitude of the stimulus. The Theory of the Heural quantum Certain experimental evidence in audition has been presented recently which tends to indicate that the underlying assumptions of the phl~g.amma hypothesis may not, after all, be valid. It is necessary in the light of these findings to re-examine the notions which underlie the processes of auditory discrimination. The starting point for such a re-examination is the all-or-none principle which Boring (4) points out

a in a theory of iology is being forced to accept this principle as applying not only to motor fibers, but also to sensory fibers. As a consequence* the notion of 1 1sensory quanta8 has been introduced in the theoretical explanation of auditory dis­ crimination , The quanta! concept regards neural activity in terms of fixed units of influence. If this assumption Is Justified* it follows that appropriate experimental techniques should repeal discontinuities or “steps” in the sensory scales of pitch and loudness discrimination. Bekesy (2) obtained evidence indieating that such die continuities do exist in the audibility curve (minimum audible pressures) at extremely low frequencies* i.e. below 50 cycles. A themophone^ was used to generate the low frequencies beginning at about 2 cycles per second, This ton® was presented to the observer at a high Intensity and gradually reduced in intensity until nothing was heard. Th® frequency was then increased until the auditory sensation reappeared. The Intensity was again decreased and the entire process was repeated In this manner until a frequency of 50 cycles was reached. The threshold curve of

1*. The characteristics of this device are such that active* tion by two sinusoidal currents produces a sensusoldal ton® whose frequency Is the difference between the frequencies of the two applied currents. Thus* low frequency tones of large amplitude can be produced In a thermophone through the beating of pure alternating currents (I?)*

9 audibility was obtained by plotting pressure (db above 1 dyne/cms ) against frequency* Between 4 and 50 cycles this curve showed a series of wsteps* which occured at fairly regular intervals (4*5*

7*5* 1 1 # 14* 18, 22, 28, 32,

and 38 cycles)* fhese *steps* m y be taken as mi indication of the possible quental nature of the processes involved in the determination of the absolute threshold. In the ease of the differential threshold, the problem of obtaining significant quanta! data becomes con­ siderably more difficult,

This is due to the random

fluctuation in the sensitivity^ of th© human organism from moment to moment* Since this fluctuation is a continuous process, the effect of a stimulus increment (AX or A F) at a particular moment in time must be determined by an instan­ taneous presentation of the increment* It must be added and removed before the organism is able to change in sensitivity by more than a negligible amount. This dictates certain experimental conditions* (1) there must b# no time interval between the standard stimulus and the variable stimulus, and (2) the variable stimulus must be of a very short duration* If these conditions are not satisfied, the random fluctua­ tions in over-all sensitivity will result in the traditional psychometric function of the ogive type. 2* Absolut© sensitivity is expressed as th© reciprocal of the absolute difference Ilmenj I.e.* l/j\T for loudness discrimination and 1/aP for p Iten 4i scrimination.

10 Certain other conditions must also obtain in order for an experiment to repeal the individual quantum of sensory discrimination in pitch and loudness. (1 ) The first requisite is that the observer must be able to make Judg­ ments with relative ease* This means that the observer must be well-trained and that the experimental conditions must provide optimal conditions for focusing his attention and stabilising his criteria of Judgment* {£) The judgments must be mad® with sufficient rapidity to eliminate the necessity of averaging the results from widely separate ex­ perimental sessions* This precaution must be taken to minimize the effects of temporal variations* It Is pref­ erable to make all judgments In a single session* {3 } A ^warning* signal* such as a dim light* may aid some obser­ vers in focusing their attention at the proper moments In th® test trials. This reduces th® strain of sustained attention ard permits the observer to adjust himself to the pattern of stimulus presentations. (4) Although it is of crucial importance that the time interval between th# presentation of the standard stimulus and th® variable stimulus be negligible* the transition between stimuli can­ not be so abrupt as to introduce undesirable transient sounds. Such sounds* if present* will serve as extraneous cue® and tend to distort the resulting psychometric functions*

11

Be&eey (1 ) was the first to take these ©xperi* mental requirements into consideration and to investigate th© possible quant&l nature of loudness dlserimlnati on * lell-praotioed observers nrwe presented a 1000 cycle tone lasting 0*3 s©c. followed immediately by a second tone of the same frequency and duration but of a slightly different intensity. These tones, with all circuit noises removed, were presented monaurally through a high quality earphone, Th© observer# were asked to judge whether or not a differ­ ence in loudness between the two tones was perceived, A record was maintained of the percentage of times each observer heard the second tone as different In loudness from the first. Under these conditions, the obtained psycho* metric functions were reported to agree very closely with those predicted on the basis of the theory of the neural quantum* It was found, furthermore, that the observers usually required th© addition of two quanta in order to perform a discrirnination, Although no specific hypothesis was statistically tested, the result# of this study suggest that loudness discrimination may have quanta! character*' i© tlC S ,

The theory of the neural quantum in auditory discrimination as conceived by Steven©,. Morgan, and Yolkroann {18} Is derived from the assumption that the basic neural procaaaM which mediate pitch and loudness

12 &1 scrimImti 03a of an &ll~or-none character, These processes are assumed to imrolT© neural structures which are diTlded Into functionally distinct units or quanta* At a particular foment a stimulus of a given magnitude will excite a cer­ tain number of qu&ni&l units* The sensory dl3orimim11on of one stimulus from another is assumed to depend upon the excitation of '-at least -one additional quanta* On the basis of th© assumption of th© existence of neural quanta, discrimination data would be expected to nftal a step*like function* This would theoretically be accomplished in the following manner* Suppose that the magnitude of a stimulus has been determined such that the stimulus completely excites a certain number of quanta and that no unused energy remains after this excitation,. Let increments be added to the predetermined stimulus under conditions of pitch or loudness discrim ination. if the neural units are im changing, then the increments will not excite the additional quantum needed for discrimination until the magnitude of these increments reaches a certain size, Thereafter, ©aeh time, that an increment is added to the original stimulus, a noticeable difference will result in pitch or loudness* If percent response were plotted against incremental magnitude, zero response would be ob­ tained up to a certain point; 100 per cent response would b# obtained for all increments above this point* These

theoretical predictions, however, are not evidenced in auditory discrimination data. Consequently, a second major assumption is made* This assumption holds that the sensitivity of th© human organism does not remain at a constant level* Lifsehitz (14) has shown, by introspective methods that the absolute threshold of hearing {and hence sensitivity) fluctuates in a "random* fashion* The data of Montgomery (15) also Indicate that the sensitivity of th© ear approximates a normal distribution as it varies with time* Myers and Harris {16) report that the lower auditory threshold varies within ±1 db from the mean threshold value for short periods of time. It follows, therefore, that the amount of energy required to stimulate a fixed number of neural units will vary with fluctuations in sensitivity* Conversely, a stim­ ulus of a given magnitude may excite a varying number of neural units* Since the variation in the number of units excited Is dir-continuous, or quanta!, all the available energy of a given stimulus will not necessarily be utilized. This means that, at a particular moment, the given stimulus will excite completely a certain number of quanta and leave a small amount of energy {?tsurplus") which is insufficient to excite th© next unit. Fig* 3 Is a schematic representation of the basic notions Involved In the theory of th® neural quantum up to

14 STIMULUS CONTINUUM

T T k i / // // // // // //

NEURAL UNITS

AS

ASq

1

X

T p

NEURAL UNITS COMPLETELY EXCITED BY S

Fig* 3* A schematic representation of the basic notions involved in the theory of the neural quantum.

15

this point. 3 1m the stimulus; a $q, th© six© of th© stimulus Increment v?hich will always excite an additional quantum; p, the amount of nsurplusM or “partial” excitation resulting from 3$ and AS, the amount of energy required to activate the “partially excited** neural unit* ■Two features of the diagram in Fig. 3 should he specifically noted t first* that the six© of the neural quantum is measured in terms of ASq and, second* that a “partially excited” unit can he stimulated by adding to 3 an increment (AB) smaller than the amount (ASq} required for stimulation when no “surplus” (p) exists. Fluctuations in sensitivity can, therefore, bring about dlscriminaiory responses to an Increaent smaller than one of neural unit sis©. This would account for th© absence of th© ©tep~ll&© function predicted on the assumption of the existence of neural units* As already Indicated, an Increment smaller than one of neural unit siss© will excite an additional quantum only when a surplus exists which is sufficiently augmented by th© Increment to provide a total supply of energy equivalent to that required for the activation of a neural unit* It follows that the stimulus Increment required Is small., when the surplus is large, and large when the surplus is small. Consequently, at any instant, the sis© of th© increment necessary to add another quantum to the

16

total number excited by the constant stimulus depends upon th© amount of surplus excitation* This statement can b© written In the following mathematical form: A S = ASq - p,

(1)

where A S is the stimulus increment required to activate an additional quantum 5 a Sq is the sis© of an increment which will always excite one quantum; and p is the amount of surplus or partial excitation:. Equation (1) indicates that a given A S will com­ pletely stimulate the additional quantum needed for a dis­ crimination whenever AS - ASq - p. As the size of the in­ crement becomes greater, an 'increase in th® number of dis­ criminations must be expected. This holds since large in­ crements will augment either large or small surpluses to neural unit size, whereas small increments will augment only large surpluses. Th© precis© manner in which th© num­ ber of discriminations increases as a function of increment sis© depends upon th© relative frequency with which the different surplus values occur. Assume that the overall fluctuation in th© sen­ sitivity of the organism is large compared to th© size of a neural quantum. This fluctuation will produce a varia­ tion in the number of neural units excited by a constant stimulus. Sine© a surplus cannot exceed neural unit size,

1? it must always partially stimulate the first unit beyond the last stimulated by the constant stimulus. The surpluses will b© spread out* therefore, over the same range of neu­ ral units which are stimulated during the course of fluc­ tuations in sensitivity. The relative frequency with which th© surpluses are distributed over this rang© is dependent upon the time distribution of the organism*s fluctuations. Assuming that the organism fluctuates in sensitivity be­ cause of a large number of unknown independent factors, the distribution of surpluses over th© range indicated will approximate a normal curve. The *chance” distribution of surpluses is shown in Part A of Fig* 4. The abscissa has been divided into six equal neural units* These units represent the range over which the organism fluctuates in sensitivity. Each neural unit has been divided further into ten equal surplus values. Thus, a surplus of a given magnitude may be found to occur in each of the six neural units. The ordinate of the distribution represents the theoretical relative fre­ quency of occurrence of the surplus values. For example, let the surplus value equal 0.3 of a neural unit. The vertical lines drawn on the graph Indicate the relative frequency of occurrence of this surplus value within each of the six neural units, Notice that this relative frequency is not th© same from unit to unit.

18

function

03 o>^ ,| H > CD +» 3 (0 ® H

p.

of surpluses and the probability

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the discrimination trials* Procedure The general procedure for both pitch and loudness discrimination was to present a reference tone, one of whose parameters, either frequency or intensity, was mo­ mentarily. increased at 3 sec* intervals* Each frequency change lasted 0*30 second; each Intensity change, 0*15 second* These increments were presented in a random order in blocks of 25 consecutive trials each, with no time in­ terval elapsing between trials. The same-sized increment appeared in all the trials for one block. The observer was instructed to press the response key (telegraph-key) each time that he perceived a momentary change, either in the pitch or in the loudness, of the reference tone* He was further instructed to refrain from pressing when no change was detected. Prior to each block of trials, the observer, seated in a sound-proof room, was given the preparation signal *ready51 through the Inter­ communication system* The observer, when prepared to listen, turned on the stimulus tone by depressing the starterbutton. At the end of each block of trials, the experimenter turned off the tone and instructed the observer to *relax”* The number of responses made by the observer for that series of trials was recorded by the experimenter.

m Fitch .d.isorImlnation* Af ter the ten subjects were selected on the baa is of the pitch teat scores and thresh­ old measurements, fire subjects were placed in each of two groups* Each group was then trained and tested in accord­ ance with the following plan* At the first practice hour* each of the fire observers in Oroup 1 was presented, monaurally, with a ref­ erence tone of 1000 cycles per second at a sensation level of 50 db {threshold values having been determined in the initial selection; of subjects)* Each observer mad© 25 judgments at each of several frequency increments*. Tha sis© of the incremental 51stepsn was adjusted to the sensitivity of the individual observer, this first hour served to locate the frequency increment value which were always perceived or never perceived. At the second practice hour, each, observer mad© approximately 500 judgments covering the rang© of in­ crements established in the first practice hour* For th© test trials, each observer in 0-roup 1 served in a series of four two-hour sessions* A fifteenminute rest period was provided midway through each session and th© observer was permitted to leave the sound-proof room.* Uo more than two hours were served by an observer in one day* Each observer made judgments at four sensation levels in the following orders 60, 40, 80, and 20 db. All

measurements for one sensation level were made in a single two-hour period. At least six frequency increments were presented at each sensation level, with approximately 200 judgments being made at each point. The five observers In Group 2 received their training and test trials under a different set of con­ ditions* At the first practice hour, each observer mad© judgments at each of three frequencies: 3-00, 1000, and 3000 cycles per second. Each frequency was presented at a sensation level of 60 db. As in the previous case, the first practice hour served to locate the increments at each fre­ quency which were always or never perceived. The sire of the Incremental *steps* and the number of trials presented at each increment value were determined by the sensitivity of the individual observer* At the second practice hour, each observer made approximately 175 judgments at each of the three fre­ quencies. These judgments covered the range of increments from those always perceived to those never perceived. For the test trials, each observer in Group 2 served in a series of three- two-hour sessions. As before, a fifteen minute rest period was provided midway through each session and the observer was permitted to leave the room. Mo more than, two hours were served by an observer In one day. Each observer made judgments at the three fre-

52

quencies in the foliowing order: 1000, 500, and 5000 cycles per second# All measurements for on® frequency were made in a single two-hour period. At least six frequency increments were presented at each frequency, with approximately 200 Judgments being mad© at each point# houteees dl aorlmlration. The training period and the test trials for th© observers in loudness discrimination followed the plan outlined for pitch discrimination, except that in this case loudness judgments were made* Group lt five observers, served at a frequency of 1000 cycles per second at four sensation levels: 20, 40, 60, and SO db* Group 2, five observers, served at a 60 db sensation level at three frequencies: 300, 1000, and 30O0 cycles per second* bevels of certainty, In th© phase of this study designed to determine the effects of altering the level of judgmental certainty on the form of the psychometric fmiction, th© two trained observers were submitted to the procedure outlined for pitch discrimination. As the observers had listened to the tonal stimuli for a considerable number of hours during the setting-up of the experimental apparatus, no practice trials were needed. For the test trials, each observer listened to a 10O0 cycle tone at a 60 db sensation level. At the first hour of test trials, the subjects were instructed to respond whenever a change in pitch was detected, even though th© level of certainty that an increment was

present was very low* The subjects were further instructed that In ease of doubt, as to whether or not an increment was perceived, th© response-key should be pressed. At th© second hour of test trials, the subjects were instructed to respond only when they were highly confident that an increment was present. In case of doubt no response was to be made.

54 Chapter It RESULTS In the analysis of the data, th© results obtained under each of the conditions for pitch and loudness dis­ crimination were treated In a similar manner. For each observer, th© proportion of perceived judgments was plotted against the sis© of the stimulus increment. Since there appeared to be no change in the sensitivity of the observer or in the variability of responses from th® first to the second hour of the test trials, the data obtained in these two hours were combined to provide approximately 200 Judg­ ments at each point investigated. These points provided an adequate basis for fitting a straight-line function to th© data obtained for each observer under each of th© several test conditions. In the curve-fitting process, the method of least squares was used, omitting all empirical propor­ tions greater than 0*97 or less than 0.03, A linear function was fitted in order to test the hypothesis that the data obtained In the sensory discrimination of pitch and loudness could be represented by a straight line as predicted by the quantum theory* The chi-square test was used to test th© goodness of fit of the obtained functions. In the calculation of the ehi~square values, all theoretical proportions greater than

55 0*97 and less than 0*03 were combined with a&jaeent propor­ tions 9 as were th© corresponding empirical proportions. In this way, no proportions representing theoretical frequen­ cies of less than five were used In determining the chisquare values. In addition9 no empirical proportions were used which did not enter into the solution for the param­ eters of the linear functions. Pitch dIacrimination.« The chi-square values ob­ tained for pitch discrimination are presented in Table I. The table is divided into two main parts? (1) the chisquare values obtained for Group 1 which listened to a 1000-cycle tone at four sensation levels (20, 40, 60, and 80 db), and (2) the chi-square values obtained for Group 2 which listened to three frequencies (300, 1000, and 3000 c.p*s.) at a 60 db sensation level. Of th® thirty-five chisquare values computed, only seven had P values greater than 0*05. This means that, in all but seven oases, the null hypothesis must be rejected; 1. e.. a real difference does exist between hypothesis and fact and a linear function cannot be used to represent the data obtained in pitch dis­ crimination. The P values for the chi-squares which did not permit the rejection of the linearity hypothesis were 0 .06,

0.07, 0.17, 0.22, 0.36, and 0.50. These values represent a range of fits from poor to excellent. Table II presents the ratio value, computed by

56

Table I Chi-Square Values Obtained In Fitoh Discrimination Group Is 1000 Gyeles Per Second.

Sensation Levels: Observers 1. 2. 3. 4*

H. B. R, R. D, H.

?. r. 5. J. F.

20 db V' cf 9.10® 17.69 9.52 15.44 3,08*

4 4 2 2 2

40 db v

28.68 7.12* 19.02 35,56 17,60

60 db ^ DP

m

2 3 4 3 3

29*64 3.60*

80 db x." BP

29.02 32*70

2 2

47.36 13*34

3

24*40

3

3.70* 13.30

3 13.22

2 2 3 1 3

Croup 2 s 60 Bb Sensation Level

Frequencies? Observers

300 e.j).S. BF V'

1* W. 2* r. 3 . K. 4* J*

71.70 32.58 88.26 23.64 45.20

M.

a* K* B*

5. s. a*

,

4 2 2 2

1000 Tf

C.D.S.

37.26 12.24 14.42 36.36 18.18

3000 e.p.s.

BP

>-

DF

2 3 1 3 2

16,48 26.52 3.72* 40.90 5.24*

2 4 4 3 6

* Ohl-Square has a P value greater than 0,05

ST

T bM M

II

Ratio Values for the Linear Functions Obtained in Pltoh Mserlmlnatlon

(AF) Cbserrer 1* 2* * *x >* 4. 5. 6. 7.

R* R* R. P. J* K. 5*

B* R# (40 8b) R* (60 db) P. P. L\4 s.

100$ 10.82 4.19 3.09 5.01 6.04 17.93 18.87

(AF) 0%

Ratio Value

4.52 1.46 1.21 1.33 .94 10.65 9.78

2*39 2*6? 2*53 III

1,69 1.93

58

dividing the size of frequency increment heard XOO per cent of th© time by th® size of the increment heard 0 per cant of the time, for each of the cases in which the hypothesis of linearity was retained* The incremental values used in determining these ratios were Calculated by solving the equations of the fitted functions for the abscissa values corresponding to the ordinate values of 100 per cent and 0 per cent, respectively. According to the theory of the neural quantum, th® obtained ratio value for each of the linear functions should equal two. An inspection of the data in Table II shows that only one of the obtained ratios (1*93) approximates this predicted value* Fig* 11 contains four representative psychometric functions obtained in pitch discrimination which do not conform to th© linear hypothesis* The straight line in each case Is the line fitted to the data by th© method of least squares* Each experimental point is based upon 200 judgment©! those points indicated by open circles were not used in computing the parameters of the fitted functions* Plot 1 is for observer R*.3*.» 1000 cycles per second at 80 db sensation level! plot 2, observer F.F., 1000 cycles, 20 db; plot 3, observer R.S., 300 cycles, SO db; and plot 4, observer ¥*M., 3000 cycles, 60 db* Th© ehi-square value for each linear function may be seen in Table I* Four of the seven rectilinear psychometric

59

PROPORTION

HEARD

1.00

Plot 1. R.B 1000 c.p.s. 80 db S.L.

.80

I

80

.60

60

.40 .20 .00

00

.5 3.0 af

HEARD

.00

PROPORTION

Plot 2. F.F 1000 o.p.s. 20 db S.L.

1.00

3.5

4.0

in c .p .s

4.5

5.0

1

.

4

2

A F IN C.P.S.

_Plot 3. R.S. 300 c.p.s. 60 db S.L.

1.00

.80

.Plot 4. W.M. 3000 c.p.s. 60 db S.L.

80

.&)

60

.40

40

.20

20

.00

.00

.5

1.0

1.5

A F IN C.P.S.

2.0

2.5

4

6 af

8

10

in c .p .s

12 .

Pig. 11. Representative non-linear psychometric functions obtained in pitch discrimination.

14

factions obtained in pitch discrimination are presented in Fig. 12* Plot 1 ia for observer S.Sy, 3000 cycles per second at 60 db sensation level; plot 2* observer E.IC.,

3000 cycles, 60 db; plot 3* observer J*F*, 1000 cycles, 20 db; and plot 4, observer P*?.*, 1000 cycles, 80 db. In each plot, the straight line Is the line fitted to th# data by the method of least squares* Each experimental point is based upon 150 to 200 Judgments, the points indicated by open circles were not used in the least squares solution, fable 1 contains the computed chi- square values for th# tests of goodness of fit* fh© ratio values, (a F)xqo1>/ ^ ^ o % 9 for each of the four linear functions are Included In -Table XI* A complete summary of the data obtained for each observer, under each of the conditions Investigated in pitch discrimination Is presented, in tabular form in Appendix 0. Each table Includes the size of the frequency increments, the number of trials per increment, and the proportion of times each increment u m perceived. Loudness discrimination* The chi-square values for testing th© goodness of fit of the linear functions for loudness discrimination are given in Table XIX. As In the ease of pitch discrimination,, the table is divided into two parts. The chi-square values presented under uroup 1 are the values obtained for those observers who listened to

61

Plot 1. 3.3 3000 c.p.s. 60 db S.L.

PROPORTION

Plot 2. K.K 3000 c.p.s. 60 db S.L.

1.00

HEARD

1.00

80 60

60

20

20 .00 11 af

17

19

in c .p .s .

AF

HEARD

Plot 3. J.F 1000 c.p^s. 20 db S.L.

PROPORTION

11

13

15

17

19

IN C.P.S.

Plot 4. F.F 1000 c.p.s. 80 db S.L.

1.00

o K < W X X o

80

.60

.20 00

.00

4 af

in c .p .s .

5

6

1

2

4 AF

6

IN C.P.S.

Fig. 12. Representative linear psychometric functions obtain­ ed in pitch discrimination.

Table III Chi-Square Values Obtained in Loudness Discrimination Croup 1: 1000 Cycles Per Second

Sensation travels t Observers 1. 2. 3* 4. ■5.

G. V. A, M. *?.

20 db per : DF

V, S. S* R. 0*

10*24 l4*10 Be 26 24*26 XXX•02

3 3 3 3

5

4o at> DF 86.68 32*14 22*26 26.74 34.19

60 db 7^ DF 40.88 30.97 59*53 37.54 32.3^

3 3 2 2 3

80 db m

2 5 5 3 1

25*82 43.44 18.16 14.76 2,48*

2 2 2 1 2

Group 2s 60 Db Sensation Level

N. J* L. H. K.

w. S. S. 0. B.

*& * DF

r

25*33

5 .27* 65.38 71.86

3 2

83*82

87.8? 51.62

4 5

34,67 66*62 1X5*52

4

t),s. DF

e Q

X. 2. 3. 4. 5*

300 C.i

O O O H

Frequencies t Observers

3 2 5 2 5

3000 c«P*» -p'

DF

95*83

3

31*42 74,08 80*88

2 2 2

129.18

* Chi-aouare has a F value greater than 0.05.

1

63

a. 1000-cycle tone at four sensation levels (20, 40, 60, and 80 db); "the Taluks imdtr G-roup 2 are for those observers who listened to three frequencies (300, 1000, and 3000 c.p.s*) at a sensation level of 60 db. Of the thirty-five chi- square values contained In the table, only two have P valties great­ er that 0.05. This indicates that for all but two eases a linear function is not a good fit for th© data obtained in loudness discrimination. Fig., 13 shows four representative non-linear psychometric functions obtained in loudness discrimination... Plot 1 Is for observer G.V., 1000 cycles per second at 20 db sensation level, plot 2, observer A.S. , 1000 cycles, 40 db; plot 3* observer J.S., 300 cycles, 60 db; and plot 4, observer K.B*, 3000 cycles, 60 db. The straight line In each plot is the line fitted to the data by the method of least square®. Each experimental point used In the curvefitting process was based upon 1?5 to 200 judgments« Th© points designated by open, circles, based upon 100 to 200 judgments, were not used in fitting the linear functions. The chi-square values for the tests of goodness of fit of each function may b© found In Table III. The two linear psychometric function obtained in loudness discrimination are shorn in Fig. 14. Plot 1 is for observer J.D*, 1000 cycles per second at BO db sensation level; plot 2, for observer W.W., 300 cycles at 60 db sen-

64

PROPORTION HEARD

1.00

Plot 1. G.V - 1000 c.p.s. 20 db S.L.

Plot 2. A.a. “ 1000 c.p.s. 40 db S.L.

1.00

80

00

00

1.0

1.4

1.8

2.2

0

2.6

.3

.6

AI IN DB

PROPORTION HEARD

1.2

1.5

1.8

1.0

1.2

AI IN DB

Plot 3. J.s 300 c.p.s. 60 db S.L.

1.00

.9

3000 c.p.s 60 db S.L. 1.00

80

80 60 40

20

20

00

00

2

5

8

1.1

AI IN DB

1.4

1

.2

.4

.6

8

AI IN DB

Fig. 13. Representative non-linear psychometric functions obtained in loudness discrimination.

65 1000 c.p.s. 80 db S.L. Q X < W X X o M w o pL. 0 X 01

80 60 40 20 .00 .0

2

.4

6

8

AI IN DB

Plot 2. N.W 300 c.p.s. 60 db S.L.

1.00 § $ x x o o §

80 60 40 20 00 .0

.3

.6

.9

1.2

1.5

A l IN DB Fig. 14. The two linear psychometric functions obtained in loudness discrimination.

sabion lev1®!. All of the experimental points represented were based upon 150 to 200 Judgments* The straight line in ©aoh plot is th© function obtained by the least squares solution,, The P value for the function in plot 1 is 0*30* indicating a good fit to the experimental data; for plot 2, the P value la 0+15# indicating a medium fit. The ratio values for the linear functions shown in Fig* 14 are given in fable IV, neither of these values satisfies the two-to-one ratio predicted by the theory of the neural quantum* fable IV Ratio Values for the Linear Functions Obtained In Loudness Discrimination

Observer

(AX)ioo^

0%

Ratio Values

1* J. C*

*61 1.21

.13 *36

3*18

2. B. W*

4.69

A complete summary of the data, obtained for each observer under each of th© conditions investigated in loud­ ness discrimination is presented in tabular form in Appen­ dix D, Each table includes the size of the intensity In­ crements , th© number of trials per increment, and the

67 proportion of times each Increment was perceived. Lovel-p o£ certainty*. Fig. 15 contains the psycho­ metric functions obtained for the two trained observers under each of two levels of certainty. Plot 1 is for ob­ server B.B.B,* low certainty-level; plot 2, for the same observer* high certainty-Xevel* Plot 3 Is for observer 3 .F.CLf low certainty-lev el; plot 4, for-the same observer* high certainty-level. In each case* the reference ton® presented in pitch discrimination was 1000 cycles per second at a 60 db -sensation level... laoh mep&rSm-mtaX point is based upon 100 judgments. Mo fmotions were fitted to the data as the number of judgments per point precluded a satisfactory test of goodness of fit. However, the general trend of the data is apparent*. Under the criterion of low certainty* both psychometric functions appear to be somewhat linear in; form, When the criterion Is raised to a high certainty-level* the points depart more markedly from linearity and the psycho­ metric function appears to be ogival in form, excluding the single inversion in plot 2*

68

Plot 1. R.H.B. Low Certainty-Crlterion

PROPORTION

HEARD

1.00

1.00

.80

.80

.60

.60

. 40

-

.20

-

.40

.20

.00 0

1

2

_L

_L

3

4

1 2 AF

Plot 3* J.F.C. Low CertaintyCrlterlon

. 80

80

. 60

.60

. 40

. 40

.20

20

HEARD

1.00

1 1

2

3

4

IN C.P.S.

High CertnlrtyCrlterlon

1.00

.00

±

.00

A F IN C.P.S.

PROPORTION

Plot 2. R.H.E. High Certalnty- Criterion

1

_L

3

4

A F IN C.P.S.

.00

4 A F IN C.P.S,

Fig. 15. Psychometric functions obtained in pitch discrimination for two trained observers under a low and under a high certainty criterion.

69 Chapter V DISCUSSION OF RESULTS The theory of the neural quantum, as outlined in a preceding section, predicts that under quanta1 experi­ mental conditions 5 (1) the form of the psychometric func­ tions obtained in pitch and loudness discrimination will be linear, and (2) the si&e of the smallest increment always heard will be twice that of the largest increment never heard. Both of these predictions must hold If the theory in its present form is to be accepted as a hypothetical expla­ nation of the processes which mediate sensory discrimination. It is essential, therefore, that the results of the present study be examined to determine whether or not the experi­ mental findings tend to confirm the theoretical predictions. It must be realised, first of all, that the use of chi-square as a test of goodness of fit makes It im­ possible to evaluate critically the entire form of the psychometric function. This is due to certain fundamental assumptions underlying the use of chi-square which require that the theoretical frequencies b© reasonably large, I.e., 10 or greater, preferably, and in no case less than 5- In order to meet this assumption, extremely small and extremely large theoretical frequencies, and hence proportions, must be eliminated. This was accomplished in the present case by

?0 fitting th@ linear function®; only bo those empirical propor­ tions falling between 0*03 and 0*97* With 200 judgments for a given point, this procedure establishes an arbitrary criterion of 6 as the minimum expected frequency value which can enter Into the calculation of chi-square. Consequently, a chi-square test of goodness of fit may indicate that a linear function is acceptable, whereas* had the extreme proportions not been eliminated, the hypothesis might have been rejected* Despite this weakness of the experimental data, only nijie of the seventy psychometric functions obtained in this study were accepted as conforming to the hypothesis of linearity. Seven of these nine functions were in pitch dis­ crimination, while two were in loudness discrimination. Since only about 8 per cent of all the obtained psychometric functions were linear in form, it is reasonable to conclude that the experimental findings fall to confirm the first prediction of the quantum theory, namely, r©c bilinearity of the psychometric function, this is in agreement with Koeater and Sohoenfeld (12) who also failed to obtain rectilinear psychometric functions under quantal conditions of pitch discrimination, Flynn (8) experienced similar difficulties In pitch discrimination, obtaining only ten rectilinear function® out of a total of thirty. The ratio values presented in Table II and l¥ .chow that the values computed for the nine rectilinear

psychometric functions range from 1.69 to 6*42* Only a single value (1*93 in pitch discrimination) approaches the two- to-on© ratio predicted by the theory of the neural quantum* The inability to satisfy this predicted ratiocriterion has been reported in several other studies in which rectilinear psychometric functions were obtained. Among these studies are those of Flynn (8} In pitch dis­ crimination* J©rom#- (11) in olfactory discrimination, and BeOlllls (6} in tactual discrimination* It appears, there­ fore, that although rectilinear psychometric functions are ooeassionally obtained, the predicted two-to-one ratio is sifarly always absent* In view of the data cited, it may he stated in general, that the experimental findings In sensory discrim­ ination fail to satisfy the predictions ©f the theory of the neural quantum* Nevertheless, the existence of recti­ linear psychometric functions has been experimmitally dem­ onstrated In several Instances and an attempt should be made to explain these occurrences* The first explanation has already been suggested. It may be that the inability to test critically the form of the psychometric function from G to 100 per cent re­ sponse leads to erroneous conclusions* This is suggested by plots 3 and 4 of Fig. 12 arid plots 1 and 2 of Fig. 14. In each of these cases, the obtained P value for chl-aquare Is

greater than 0.05* Indicating the testability of a linear hypothesis. Despite this, an examination of the form of the psychometric functions, as revealed by the plotted experimental points, shows that the functions may be ogival In accordance with the phi-gamma hypothesis. As in the case of the linear hypothesis, a critical evaluation of the latter hypothesis cannot be mad© due to the inherent weak­ ness of the experimental data. A second possible explanation for the occasional existence of a rectilinear psychometric function concerns the level of certainty under which the diecriminal responses are made, Although the same instructions are given to each observer prior to experimentation, each observer, depending on his Individual tendencies, will establish an arbritrary certainty-criterion which must foe met before a response will occur. Consequently, the psychometric functions obtained in a given study may foe based upon several different levels of judgmental certainty. It is conceivable that the form of the psychometric function obtained under a low certainty-erlterlon represents the summation of two separate functions, one function for responses made to perceived increments and the other for responses made when no increments are perceived* The latter class of responses could arise from a large number of uncontrolled factors, such as extra-loud heart beats, tne

73

anticipation of an increment at fixed internals of time, or the reinforcing properties of a discriminal response. Sine© the experimenter has no -ray of distinguishing the two types of responses, the total number of responses made in a given block of trials Is taken as the number of times an increment Is perceived* This results in a distorted representation of the psychometric function. Under a high certainty*criterion, the proportion of the total responses due to uncontrolled factors would be minimissed and the total number of responses would more accurately represent the ‘’true** psychometric function. This hypothesis is supported by the psychometric functions presented in Fig. 15, where it is shown that the form of the psychometric function is directly related to the eert&tety-criterion under which the dis criminal judgments are made.

74 Chapter VI SOKMAEY AND CONCLUSIONS 1* The main purposes of the present study were, .first, to test the hypothesis that the data obtained In the sensory discrimination of pitch and loudness satisfied the conditions predicted by the theory of the neural quantum, and, second, to determine the effects of both frequency and intensity upon the size of the quanta of pitch and loudness dlscrimination. A minor purpose was to Investigate the man­ ner in which the form of the psychometric function varied with the level of certainty with which an observer;1s judg­ ments were made. 2. The general procedure for both pitch and loud­ ness discrimination was to present an observer with a pure tone, one of whose parameters, either frequency or intensity, was momentarily increased at 3 see* Intervals. Each frequen­ cy change lasted 0.30 second; each Intensity change, 0,15 second. The Increments were presented in a random order In blocks of 25 consecutive trials each, with no time interval elapsing between trials. The observer, listening monaurally, responded by depressing a telegraph-key each time a change In the tone was detected. Approximately 200 judgments were made at each point. 3. After the selected obsez^vers had been given a prescribed amount of training, psychometric functions were

75

obtained for two groups* in pitch discrimination and two groups in loudness discrimination. Each of these groups was composed of five observers . In pitch discrimination , Group 1 was tested at a frequency of 1000 cycles per second at four sensation levels; 20, 40, 60, and BO db» Group 2 was tested at a 60 db sensation level at three frequencies t 300, 1000, and 3000 cycles per second. The same plan was followed for the group in loudness discrimination, except that in this case loudness judgments were made. In all cases, only one sensation level, or on© frequency, was tested at each laboratory session which lasted for two hours. 4.

In the analysis of the data, a straight line

was fitted by the method of least squares to each of the psychometric functions obtained in pitch and loudness dis­ crimination. The chi-square test was used to test the goodness of fit of each of the linear functions. Of the seventy chi-square values computed, only nine had a P value ^0.05. Seven of these were in pitch discrimination and two, in loudness discrimination. Of the nine psychometric func­ tions In which the hypothesis of linearity was retained, only one (In pitch discrimination) had a ratio value which approached the predicted two-to-one criterion. 5* On the basis of the data obtained, it was con­ cluded that the experimental findings In pitch and loudness discrimination failed to satisfy the predictions of the

theory of the neural quanto,* I.e.»

rectilinear!ty

of the

psychometric function and the two-to-one ratio* 6* ’ Two explanations were offered which might account for the occasional rectilinear psychometric functions obtained in auditory discriminations (1) the inability to test critically the entire form of the psychometric function by means of the chi-square test of goodness of fit, and (2) the dependence of the form of the psychometric function on the certainty-criterion under which the diecriminal respon­ ses are made* 7* Although one of the primary purposes of this study was to determine the effect of both frequency and intensity upon the ®1&@ of the quanta of pitch and loudness discrimination, these relationships could not be established since the existence of sensory quanta was not experimentally demonstrated.

71

Appendix A

Table 7 Total Scores, on Seashore Tests

Pi tch Observers

D. w. S. X. p. J* R. 8. d 9. R. 10. H. 11. J. 12. 0. 13. M. 14. S. 15. w. 16. L, 17a P. 18. R. 19. B. 20. B. 1. ■a. 3. 4. 3, 6. 7.

*

H. K. S.

r. F. F. S. B.

B. H. H. R. H. 0. B.

L. E. W.

I. E.

Loudnes s Scores

8? 86 86 85 83 82 82 82 80 79 75 75 75 73 70 70 6? 6? 65 6l

Observers

1. 2. 3* 4. 5.

8.

7. 8. 9* 10. n. 12.

J. w. 0. K. H.

0. S. V.

B.

0.

M. R.

A. s.* L. S. J. s. ^ . w* E. s.

* p. *!"X* P. B. 14. ¥. 0. S* 15. S 16. E. i. 17. L. A. 18. S. S. 19. H. R. 20. S. 1. 21. 1. E, 22. L. B. 23 a E. K. 24. D« 0.

25. B. B.

Scores

93 92 92 91 90 89 89 89

89 89 88 88 88

87 86 86 86 85 85 84 82 82 81 78 78

78

Appendix B Mathematical Derivation of the Formula Used in the Calculation of th© Loudness Increments By definition (B .1} where db is the number of decibels and the P 1s are pressure Talues. Th© number of decibels (dbg) which the steady-state voltage (Vg) is below a given reference voltage (E) may, therefore, be expressed as follows? db = 20 log

*

(B*2)

S In a similar manner, th© number of decibels (dbj} which the increment voltage (Vj) is below the same reference Jlbvel {£) may be written as dbj = 20 leg

*

(B.2)

When the increment voltage is added, In phase, to th© steady-state voltage, the number of decibels by which the steady-state voltage is increased (db^J is given by the following ©xpression: (B.4) In order to economically utilise the relationships shown in formula (B.4), it is necessary to represent the

T9 and

values In terms of decibels which may subsequently

be set on the attenuators in the steady-*state and increment channels*, The solution for

Is mad© directly from equation

(B*2). It can be shown that (B,5)

V$ = afttllog 20

A similar expression can also be obtained for Vj, Starting with equation (B.3); (B.6)

1= antilog

ST

By substituting the expressions in (B.5) and (B.6) for Vs and

In equation (B.4) and dividing the numerator

and the denominator by B, the following formula la obtained; antilog

a.bA = 20 log

!+___go

7) may be reduced to a more convenient form; dbA = 20 log

’dbc- dfe« 1+ antilog (ay i j

(B.8)

80

appendix G B0MRARY TA8LSS FOR PITCH SISCRIMISfATXei &roire 1 -3LQPQ Cycles Per Second at Four Sensation Levels Table VI Observer R. 8. mil mpiw- 1 20 db S.]L, B Ar P 5.0 6*0 T.O 8*0 9.0 10*0

175 175 175 200 175 200

*06 .25 •62 .62 *80

40 db S •L. AF 8 P

60 db S,■L. AF H P

5.0 3.5 4.0 4.5 5.0 5.5

3.5 4.0 4.5 5.0 5.5 6*0

200 200 200 200 200 200

.01 .0? « 49 .63 .88 .96

200 200 200 200 200 200

.00 .22 .60 .80 .90 .94

80 db S.]L. AF

B

P

2.5 3«c 3*5 4.0 4.5 5.0

200 200 200 200 200 200

.00 .08 .18 .71 .85 .90

Table VII Observer H, R*

20 db s.:L. AF

H

p

1.0 2.0 3.0 4*0 5.0

100 100

.10 *11 .19 .36 .41 *79 .97

6*0 8.0

100 100 100 100 100

40 db 3,,L. AF B P 1.0 1.5 2.0 2.5 3.0 3.5 4.0

200 200 200 200 200 200 200

*00 .04 .17 .32 .62 .76 .91

60 db SJ AF 1*0 1 it 2.0 2.5 3*0 3.5

*

B

P

200 200 200 200 200 200

*01 *13 .47 .66 .95 .98

80 db S,.L. AF B P 1.0 1.5 2.0 2.5 3.0 3*5

200 200 200 200 200 200

.00 .04 .40 .68 .88 .94

81

fable VIII Observer D, H.

20 db S,►■L* N A F P 2*0 .3*0 4>0 3*0 6.0

?.o

200 200 200 200 200 200

40 db B,tL * AF H P

2*5 3*0 3*3 4*0 *94 5-0 *99 5*5 6*0

*07 «29 *64 *BO

200 200 200 200 200 200 200

60 db S.]b « 1 AF P

*10 *19 -30

3*0 3*5 4.0 .60 5*0 *84 7.0 *90 8.0

200 200 200 200 200 200

.04 .10 *26 .60

.89

.94

80 db B.fit* AF H P 2.0 2.5 3*0 4.0 4.5 5*0

200 200 200 200 200 200

.09 *20 *46 *60 *89 *96

*96 ?abl© : IX

Observer F. F*

20 db 3 e>1** M AF P 1.0 2.0 3.0 4*0 5*0 6.0

200 200 200 200 200 200

*00 .04 .33 *73 *90 *98

40 db S. T H AF P t

3-0 3-5 4*0 4*5 5-0 5-5

200 200 200 200 200 200

0

60 db 3 . ,b AF N P

.04 4 *0 *32 5*0 .56 6*0 *75 7*0 .82 8.0 *91 9-0

.

200 200 200 200 200 200

*04 *36 *65 .78 .84 *94

80 db 8.L. N P

af

1.0 2.0 3*0 4*0 5*0 6.0

200 200 200 200 200 200

-01 *14 .48 .81 *93 98

.

82 Table x Observer J. P.

20 db S ,,L • AP

1.0 2.0 3.0 4*0 5.0

6*0

N

P

40 db S.I AF P

60 db S,,1. AF

200 *06 2.0 200 .15 1.0 200 .13 2.5 200 .32 2.0 200 .36 3.0 200 .48 3.0 200 .63 4*0 200 *87 4.0 200 .88 4.5 200 .88 5.0 200 .94 5.0 200 .96 6.0

W

200 200 200 200 200 200

P

80 db S.,L. AF

.02 .05 .10 1.0 .31 2.0

.62 .77

3.0 4.0 .86 5.0

6.0

N

P

200 200 200 200 200 200 200

.06 .14 .30

.62 .88 .94 .98

Group 2 Three Frequencies at a Sensation level of 60 Decibels Table XI Observer W. M.

•300 c *P.s« AF N P 1*5

2.0 2.5 3.0 3.5 4.0

200 .04 200 .10 200 .67 200 .77 200 .91 200 .96

1000 e.p.s.

3000 c.p. 8.

AF

H

P

AF

1.0 1.5 2.0 2.5 3.0

200 200 200 200 200 200

.02 .16

4.0

3.5

.58 .74 .94 .95

6.0 8.0 10.0 12.0 14.0

H

P

200 200 200 200 200 200

.00 .24 .52 .77 *91 .95

83

Table XII Observer R, S.

300 e.p 3* «

AF

N

P

.50 ,75 1.00 1.25 1.50 1.75 2.00

200 200 200 200 200 200 200

*05 .14 .42 .68 .84 .95 .97

1000 G.p • s* AF H P 2.0 2.5 3.0 3.5 4.0 4.5 5.0

200 200 200 200 200 200 200

.02 .07 .12 .34 .66 .80 .92

300C) c.p. s. AF H P 4.0 5.0 6.0 7.0 8.0 9.0 10.0

200 200 200 200 200 200 200

.04 .12 .42 .61 .80 .85 .96

Table XIII Observer I. it.

300 c.p.s. AF 1.00

1.25 1.50 1.75 2,00 2.50

1000 c.p.s. H p

H

p

af

200 200 200 200 200 200

.01 ,14 .30 ,82 .90 .96

1.0 1.5 2* 0 3-0 4.0

200 200 200 200 200

5*0

200

,00 .02 .14 .52 ,83 .94

3000 c.p.s. AF K p 11.0 12.0 13.0 14,0 15.0 16.0 17.0

200 200 200 200 175 200 200

,02 .16 .32 .52 *59 ,72 .87

84

Table

XIV

Observer J* B.

30c> e.p. s. AF

1*0 2.0 2.5

3*0

3.5 4.0

I

200 200 200 200 200 200

1000 c.p s» •

P

AF

ff

P

.01 .14 .40 .76 .80 • 94

1 tz

200 200 200 200 200 200

.04 .24 .62 .75 .92 .94

2.0 2.5 3.0 3*5 4.0

3000 e.p. s. AF 4.0

6.0 8.0

10.0 12.0 14.0

K

P

200 .05 200 .20 200 .62 200 .67 200 .82 200 .92

Table XV Observer S. 5

300 e.p. s. AF

2.00 2.25 2,50

3.00 3*50 4.00

H

P

1000 e.p. s, AF

N

P

3000 e.p. s. AF

p

200 .01 200 .16

2.0

2.0

200 .01 200 .20

11.0 12.0

150 .11 150 .29

200 200 200 200

3*0 3*5 4,0 4.5

200 200 200 200

13.0 l4.0 15*0

150 150 150

.55 .74 .88 .97

.52 175 ,86 *94

16.0 17*0

18.0

150 150 150

.36 .46 .53 .73 .80 .90

85 Levels of Certainty 1Q00 Cycles Per Second at a Sensation Level of 60 Decibel a Table XVI Observer R. H* B.

Low Ce Ftain ty-Criterion AF * P 0*5 1.0 1.5 2.0 2*5 3-0 3.5

ICO 100 100 100 100 100 100

High Gerta inty-Grl terion AF N p

.11 *24

1*5 2.0 2*5 3*0 3.5 4.0

*6? .75 *90 .97

100 100 100 100 100 100

.09 *20 *58 *74 *67 .92

Table XVII Observer .J* p. C.

Low Certainty-Criterion AF H p 1.0 1.5 2.0 2*5 3.0 3.5 4.0

100 100 100 100 100 100 100

*02 *17 *22 .51 .71 .86 .96

High Certainty-Crlteri on AF H p 1.5 2*0 2*5 3*0 3*5 4.0

100 100 100 100 100 100

.05 .15 *25 .62 .74 *88

86

Appendix B SUMICAHT TABLES FOB L0OTHES8 DISGRIHIH&TIGN Group 1 1Q9.9. Q74L.fi For Second at Four Sensation Levela Table XVIII

Observer 0. V*

20 ab s.;

t Lt a

AI 1.35 1*67 2*05 2.2? 2.51 2.7?

N

P

200 200

.07 .23 ,5? .75 .90 .95

200 200 200 200

40 db B .L. H AX P

60 db S.;L. AX

P

200 .03 .45 200 .05 1.21 175 .13 .53 200 .29 1.09

1.28 1.35 1.50 1.67

175 175 175 200

.59 .66 .70 .78

.51 .77 .79 .91

200 200 200 200

.76 .77 *97 .99

80 db S B AX

.10 100 .16 200 .26 .36 .45 .56 .66

P

.00

.09 200 .25 200 .71 200 .85 200 .97 150 1.00

i Table XXX ,

Observer w. s

20 db S «L. N 4I P 5.08 5-30 5.54 5.78 6.00

125 125 125 125 125

.10 .29 .61 .74 .81

40 db S.♦ # AX H P 1.35 1.50 1.67 1.87 2.05 2.2?

200 200 200 200 200 200

.02 .18 .38 .71 .75 .91

60 db S.L. AX .92 1.21 1.35 1.50 1.76 1.87 2.05 2.16

R 200 200 200 175 175 175 200 200

P

80 db S,hit. B AX P

•02 .36 .21 .45 *29 .56 .50 .78 .73 .98 .75 1.15 .68 .89

200 200 200 200 200 200

.03 .23 .52 ,86 .89 ,98

87

Table XX Gb&esnrer a. S.

20 db SL a K AI P 2.16 2*39 2.51 2*77 2.91

3-06 3*37

175 150 125

125

125 125 150

40 db S.r L » AI P

60 db S.L,

j *

*20

200 175 175 175 *56 1.76 175 .69 2.05 175 .91 ■2..27 200

.92 *36 1.15 *34 1.35 *55 1*50

AI

.40 -59 .16 *78 *35 *98 .57 1.21 *90 *98 *05 22

*

80 db S Xj K AI P *

H

P

200 200 200 200 200

*03 *19 .49

*75 .95

.14 .24

150 200 *34 200 *45 200 *56 200 .66 200

«

.07 .15 ,4a .70

.89 .98

QTOVLp 2

at a Sensation Level of 60 Decibels

Three

fable XXIII Observer If. W*

300 e.p.s* AI K p *30 .50

*59 .70 .92

1*03 1.21

1.42

175 *01 175 *15 175 *25 175 -38 175 -70 175 -84 175 94 175 97

1000 c.p.s. AI If p *21

*40 *59 .66 *78 .98

1.21

1*42 1.58

175 175 175 178 175 175

.00 .05 *06 *22 .49 .65 .85 *91 98

3000 c.p.s. AI N p

.40 .50 .59 .66 .78 .98 1.21

175 175 175 175 175 175 175

*06 *48 .71 -87 *85 .97 .98

89

Table XXIT Observer j. s.

300 c.p.s*

1000 c.p.s* AI M p

AI

N

p

.50 *56

200 75 200 200 200 200 200

.00 .00

.56 .70

.65 .91 .94 .97

.92 1.09 1.28

.63 •TO .92 1.09 1.28

.8 3

200 200 200 200 200 200

.00 .10 .51 .75 .89 .95

3000 c.p.s. AI N p

.40 .45 .48 .50 .59 .70 ,92

75 200 75 200 200 200 200

.00 .00 .00 .51

.89 .95 .97

Table XXV Observer L. S.

300 c.p.s. A I M p .21 .40 .56 .70 1.03 1.28 1.58 1.87

200 200 175 175 175 175 200 200

.02 .05 .25 .44 .66 .81 .86 .97

1000 c.p.s, AX W p

.42 *53 .63

,74 .83

,92 1.15 1.35 1.58

175 175 175 125 125 125 175 175 175

.10 ,13 *13 .22 .59 .68 .85 .92 .97

A

3000 c.p.s. I H p

.21 .31 *40 .50 .59 *70 .83

200 200 200 200 200 200 200

.03 ,17 .33 .79 .06 .97 .99

90

'Sable xxvi Observer H* 0.

300 e.p*s. AI

.36

.50 *63

.74 .87 .98 1.09 1.21 1.35

H

P

75 .00 175 .11 175 .52 175 .81 175 .65 175 *79 175 *54 175 •94 175 .95

1000 e.p.S. H p

AI

*40 .45 .50 *59 .70 .83 .92 1*03 1.15

100 .00 175 175

•01 •10

175 .6 5 175 *77 175 .9 0 175 .94 175 *93 100 1.00

3000 Q.P. s.

AI

.19 .38 *59 .78 .98 1.15 1*25 1*58

1

P

150 150 150 150 150 150 125 125

.07 .09 .81 .89 .93 *97 *99 *99

T&bl© XXVII Observer K. B.

.63

*83 1.03 1.09 1.21 1.42 1.67 1.87 2.05

75 175 175 175 175 175 175 175 175

.00 .07 .27 .26 *57 .72 .82 *80 .94

A

1000 c.p.s. I H p

*53 .63

*74 .83

*92 1.03 1.21 1*42

175 175 175 175 175 175 175 175

•05 *30 .21

*66 *?8 .89 .94 .96

3000 c.p.s. AI

B

p

*30 *34 .40 •45 .50 .59 *70

125 100 200 175 175 175 175 200

* O 0

300 c.p.s. AI B p

.83

.00 *11 *18

.71 .89 .93 .99

91

A p p e n d i x 25

STUDY OH THE THEORY OP THE NEURAL QUANTUM Observer _______________________(II,F) Total score on Seashore

Date tests

Threshold settings for ear; 300 cycles } 1000 cycles

93

Hour

; 3000 cycles

DISCRIMINATION DATA Conditions for Frequency Sensation level

trials, experimental session $ cycles db Setting______

Inere-_ No. of 25 iTotal | ments trial blocks! trials| In !presented | f T T

No. of increments heard per block

rm

jTotalj % |heard: heard

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13. Lewis, Don, "Quantltatlve Methods in Psychology,” Ann

94 Arbors Edimrds Bros., 1948, pp. 254-257, l4« Lifschitg, S„, "Fluctuation of the Hearing 'Threshold," Journal of the Acoustical Society of America. ¥01. 11. 1939, PP. llBTiST -------15* Montgomery, H. C., "Influence of Experimental Technique on the Measurement of Differential Intensity Sensitivity of the Ear," Journal of the Acoustical Society of America* Vol. 7, 1935, pp7 39-45o ' ' ~ ' 16. Myers, c* Km and Harris, D* J., "The Inherent Stability of Auditory Threshold," U.S.K. Bureau of Medicine and Surgery Report, Project BM-003-021, April, 1949. 17. Stevens, 3* S. and Davis, H., "Hearing% Its Psychology and Physiology." John Wiley and Go., Hew York, II. t., 1938. 18. Stevens, S. S., Morgan, 0. T., and Volkraann, J., "Theory of the Heural Quantum in the Discrimination of Loudness and Pitch," American Journal of Psychology* Vol. 54, 1941, pp. T “" 19. Stevens, S, S. and Volkmann, J., "The Quantum of Sensory Discrimination," Science, Vol. 92, 1940, pp. 583-585. 20. Titchener, £. B., "Experimental Psychology*tf Hew forks Macmillan, 1905, II, p. xxxli. 21. Urban, ?. M., "The Application of Statistical Methods to the Problems of Psychophysics *" Phil.; Psychological Clinic Press, 1908, p . l O ? . r 22. Woodworth, R. S., "Experimental Psychology," Hew York; Holt, 1938, p.