..V . m . . . . ‘ , _ ‘ . . . ‘26.. . , , . ; it? .. , . V. D. . 7 . u... .. . . ., _ _ ‘ \ 0 .m. . V . . VMfl 72f :. _ «Wu iA W . .H _ . e U r mm..s, .. mmm ; “V! S , v ‘ 3 .un ; M f, N. R. A m . m C 5 . . m M . ., .m 5 n. V V.._..H....., a, _‘_.1_..u._w_“‘...§.__u .....‘...$._%§mx_. éfigfi fig? I: . LIBR AP RY Michigs 118 wt ; UnibVCL :y ‘49. 34. yaw:::m- ' This is to certify that the » thesis entitled Symmetry In Judgments of Musical Pitch presented by Cynthia H. Null has been.accepted towards fulfillment of the requirements for ph,D, deg?9h1 Psychology flaw/x M Major professor c: ABSTRACT SYMMETRY IN JUDGMENTS OF MUSICAL PITCH By Cynthia H. Null Five ear-trained musicians and five other persons served as subjects in three concurrent experiments designed to examine measurement structures for pitch. Symmetry of judgments of pitch change was the key property to which each experiment was directed. In the first experiment subjects made magnitude estimations of the distance between two consecutive sinusoidal tones. The subjective distances between pitches for ascending and descending intervals were the same and the data for eight of the subjects were repre- sented by the musical or log model for pitch. Two of the ear-trained subjects provided data suggestive of the helical model for pitch. The second experiment used the method of bisection to look at effects of order and to test the bisymmetry axiom for pitch. The bisymmetry axiom was supported for six of the ten tests for ear-trained subjects. The order of presentation_of the tones in the interval to be bisected had little affect on the bisection point. Reaction time was used in the third experiment to examine the process Cynthia H. Null by which a subject decides whether a two-tone sequence was ascending or descending. Presentation order did not affect reaction time. Overall, the musical or log model for pitch explains subjective judgments better than the mel or power models for pitch. SYMMETRY IN JUDGMENTS OF MUSICAL PITCH BY Cynthia H. Null A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Psychology 1974 To Tim ii ACKNOWLEDGMENTS I would like to express my appreciation to the members of my thesis committee. Dr. Lester M. Hyman taught Ine programming for the PDP—8/I computer and was a constant help while I programmed these experiments. I wish to thank Dr. John E. Hunter for his suggestions on analyzing the data and to Dr. Ralph L. Levine for becoming a committee member on such short notice. I am especially grateful to Dr. David L. Wessel, my committee chairman and advisor, for his inestimable assistance in all phases of this research. A special thanks to Mark Dionne and Bruce Marshall for their assistance while I learned computer hardware and designed and built the Digital-to-Analog conversion inter- face. Bruce also rebuilt the PDP-8/I's internal clock which made my reaction time experiment possible. I am also indebted to several of my friends for serving as subjects for 15 hours in these experiments without remuneration. This research would have been impossible without their dedication. A special thanks goes to my husband for long lasting patience, understanding and encouragement. iii LIST OF TABLES LIST OF FIGURE Chapter TABLE OF CONTENTS S I O O O O O O O O O I O O O O O O O I . INTRODUCTION 0 O O O O O I O O O O O O O C II. EXPERI Meth Background . . . . . . . . . . . . . Distance models . . . . . . . . . . . Some properties of two-tone sequences suggesting violations of symmetry . Some properties of three-tone sequences . . . . . . . . . . . . . Reaction time . . . . . . . . . . . . Models for two- and three-tone pitch perception . . . . . . . . . . . . Pitch drift--a processing model for asymmetry . . . . . . . . . . . . . MENT I O O O O O I O O O O O O O O 0 od . . . . . . . . . . . . . . . . . Stimuli I O O O O O O O O O O 0 O O 0 Subjects 0 O O O O O O O O O O O O 0 Procedure 0 O O O O O O O O O O O O 0 Results 0 O O O O O O O O O O 0 O O O O 0 III. EXPERI WNT II 0 O O O O O O O O I O O O O O Sllbjects O O O O O O O O O O O O O O Stimuli O I O O O I O O O O O O O I 0 Procedure . . . . . . . . . . . . . . ReSUltS C O O O O O O O O O O O O O O O 0 iv Page vi vii 13 18 20 24 26 26 26 29 29 31 41 41 41 42 44 Chapter IV. EXPERIMENT III 0 O O O O O O O 0 O 0 Subjects . . . . . . . . . . . Stimuli . . . . . . . . . . . . Procedure . . . . . . . . . . . Results . . . . . . . . . . . . . . V0 DISCUSSION 0 O O O O O O O O O O 0 0 Influence of ear training . . . Existence of asymmetries . . . PtiCh mOdels O O O O O O O O 0 VI 0 CONCLUSIONS 0 O O O O O I O O O O O 0 Appendix A. LIST OF THE STIMULI FOR EXPERIMENTS I AND III 0 O O O O O I O O O O O O O I B. LIST OF THE 10 MIDDLE TONES FOR EACH INTERVAL FOR THE BISECTION EXPERIMENT C. ANALYSIS OF VARIANCE TABLES FOR EACH SUBJECT FOR EXPERIMENT I . . . . . . D. ANALYSIS OF VARIANCE TABLES FOR EACH SUBJECT FOR EXPERIMENT III . . . . . REFERENCES . . . . . . . . . . . . . . . . . Page 53 53 53 53 54 58 59 60 62 66 67 68 69 74 79 LIST OF TABLES Table Page 1. Mean magnitude estimates for the distance between two tones . . . . . . . . . . . . . . . l3 2. Medians for each of the 10 subjects for the lowest and highest frequency at each of six interval lengths and two orders of presentation . . . . . . . . . . . . . . . . . . 36 3. A summary of the comparisons between the log and power models of pitch for magnitude estimation data 0 O O O O O O O O I I O O I I I 39 4. A summary of the results from the bisection experiment for the non—ear-trained subjects . . 45 5. A summary of the results from the bisection experiment for ear-trained subjects . . . . . . 46 6. A summary of the results from the test of bisymmetry . . . . . . . . . . . . . . . . . 48 7. A test of the log and power models of pitch for the bisection data . . . . . . . . . . 52 vi 10. LIST OF FIGURES These are examples of chord pairs for an experiment of finality effect . . . . . . . A graphical illustration of bisymmetry . . An information processing model for the identification of marked and unmarked concepts 0 O O O O I O O O O O O O O O O 0 Schematic diagram of the experimental set-up O O O O O O O O O O O I C O I O O O A graph of mean magnitude estimates as a function of interval size for non-ear- trained subjeCts O O O O O O O O O O I O O A graph of mean magnitude estimates as a function of interval size for ear—trained subjects I O O O O O O O O O O O I O O O O A graph of median magnitude estimates as a function of interval size for subject 8 . A graph of median magnitude estimates as a function of interval size for subject 10 Scatter diagrams of the subjective mid- points of the intervals for the bisection task as a function of the log estimates for these mid-points for non-ear-trained subjects . . . . . . . . . . . . . . . . . Scatter diagrams of the subjective mid- points of the intervals for the bisection task as a function of the log estimates for these mid-points for ear-trained subjects . . . . . . . . . . . . . . . . . vii Page 12 16 19 28 32 33 38 38 49 50 Figure 11. 12. Page A graph of reaction times for trained and non-trained subjects as a function of interval lengths . . . . . . . . . . . . . . 56 An illustration of differential decay of frequency components for ascending and descending chord sequences . . . . . . . . . 61 viii CHAPTER I INTRODUCTION When listening to a sequence of tones one often has the impression that the pitch changes are analogous to spatial changes. It often sounds as if the intervals between the tones are distances in some abstract geometrical space. These compelling impressions appear to afford the possibility of characterization by a measurement structure (Krantz, Luce, Suppes, & Tversky, 1971). And indeed musical scales have evolved with the notion of modeling subjective experience in the numerical domain. In this thesis selected properties that are essen- tial to the characterization of pitch changes as distances are explored. Primary attention is given to the symmetry assumption which implies a certain form of subjective equivalence between ascending and descending intervals of the same frequency difference. The existence of asymmetries requires a measurement structure, such as the bisection system, which does not require symmetry. The existence of asymmetries also demands a theory that will shed light on the information processing mechanisms responsible for such subjective differences. Although the focus is on symmetry, the existing schemes for pitch scales will also be examined. Background The standard equally tempered musical scale has the pitch of a tone logrithmically related to frequency.1 The subjective criterion underpinning this log scale is derived from the notion that equal frequency ratios give rise to equivalent subjective impressions of interval size regard- less of position on the frequency scale. On the standard equally tempered scale there are 12 equally spaced intervals within an octave. The frequency of each note in the scale 1/12 times the frequency of its lower neighbor. is exactly 2 Given the starting frequency, FZ' notes for a general musical scale can be specified by laying off intervals in octave units of size, s. The frequency of the next note in the scale, F is a constant multiple of the frequency of n+l' the lower note, Fn, as given by the formula F =F-2. (1) The log model rests upon the assumption that trans- position of a musical piece from one frequency range to 1The use of the term frequency for the physical quantity to be scaled is not intended to imply a position on the periodicity versus place controversy in pitch per- ception (Green, unpublished manuscript). In the experiments conducted in this thesis pure tones were used and the fre- quency and periodicity values of a given tone are equal. another does not change the character of the relations among the tones. There is considerable disagreement in musical circles as to the consequences of changing a piece from one key to another and many doubt that the character of the music is subjectively invariant with transposition. Special key characteristics and other register effects could be due to timbral properties of the musical instruments rather than to the inappropriateness of the musical scale for pitch. If, however, the results of S. S. Stevens and his associates (Stevens, 1957; Stevens & Volkmann,1940; Stevens & Galanter, 1957) on the mel scale apply to pitch sensations in musical passages then the psychological inapprOpriateness of the musical scale should be taken seriously, as suggested by Lindsay and Norman (1972). Stevens and Volkmann (1940) presented subjects with different tones and asked the subjects to adjust a variable tone until it was half the pitch of the standard. The scale derived from these and other similar tasks was not in agree- ment with the logrithmic relationship between pitch and frequency. Stevens and Volkmann presented an empirically derived function relating pitch and frequency known as the mel scale. This scale indicates that changing the key or register of a piece of music will change its character. Stevens and Volkmann write of their results, "these facts contradict the . . . notion that 'equal ratios of frequency give rise to equal intervals of pitch.‘ They also discourage what appears still to be a prevalent belief; namely, that the musical scale is a subjective scale." Stevens later championed the power law as the ubiquitous psychophysical relationship. The general form of this power law is wn = kFfi (2) where wn is the subjective pitch of tone n, Fn is the fre- quency of tone n, and k is a scale factor. Though the empirically derived mel scale has been on some occasions considered a power function, a distinction between these two notions will be maintained throughout this report. A third alternative is a two dimensional model pro- posed by Revesz (1954) and explored in a unique experiment by Shepard (1964). Revesz's model is graphically repre- sented by a helix. The two aspects of the model are tone height and cyclical chroma. That is, tone height increases continuously with frequency while the chroma is repeated every octave. The frequencies of tones on a vertical slice of the helix are related by powers of two. The tone below a particular tone, t, is half the frequency of t and the tone above t is double the frequency of t. An important feature of subjective data captured by the helical model is the strong subjective similarity and confusability of tones an octave apart. Shepard (1964) provided some data in support of the helical model for pitch with a clever experiment using a special set of computer generated tones. The basis for his tone generation procedure is analogous to collapsing the helix into a circle by forming complex tones made exclusively of a large set of octave components. Ten such complex tones were formed which divided the octave into equal intervals using a log frequency scale. When these tones are played in order they complete exactly one revolution about the circle. When played they sound as though they are continuously increasing or decreasing depending on the direction around the circle. By asking the subjects to make judgments as to the direction of a pair of these complex tones, Shepard demonstrated consistent circularity in the judgment of pitch. Arguments over the relative merits of each of these three schemes as subjective measurement structures for pitch have not been carried on with the same stimulus contexts in mind. The arguments for the musical scale are clouded by the cultural weight of Western musical tradition. The argu- ments for the mel scale and power functions are based on judgments of single tones and fractionation experiments and support for the helical model relies upon unusual complex tones capitalizing on octave generalization. Each of these pitch measurement schemes can be examined using a simple task of some musical significance. Subjects can make magnitude estimations of the pitch distance covered by a sequence of two tones, that is, the subjective length of an interval can be estimated. Before discussing the specifics of the two-tone sequence experi- ments, it is important to make some general observations on models incorporating the notion of subjective distance. Distance models Similarity or subjective distance between stimuli, such as, color, forms, or tones is widely used for gaining information on the ways these stimuli are perceived and coded. There are a number of ways these judgments can be made so as to learn about the subjective structure of the stimuli. They can be obtained directly by asking the subject to assign numbers as to the degree of similarity’ between the pairs of stimuli. More indirectly, these mea- sures can be derived from measures of confusability between pairs of stimuli. Many more possibilities are outlined by Wish (1972). One way of analyzing similarity data is based on a geometrical model. The stimuli are considered points in some multidimensional metric space with the similarity indices reflecting the distances between the points. The most similar objects will be closest in the solution Space and the most dissimilar objects will be the farthest apart. The scaling techniques, such as, Guttmann (1968), Shepard (1962) and Kruskal (1964), Torgerson (1959), and Carroll and Chang (1970), recover a metric configuration based on assumed distance data. What properties should these dissimilarity measures possess in order to be properly treated as distances? If we assume the stimuli can be described in terms of a finite number of underlying dimensions then it seems reasonable that if points x and y "approach" one another the dissim- ilarity of x to some 2 approaches the dissimilarity of y to z, for all points 2. We will write the dissimilarity of x and y as 6(x,y), therefore,the first assumption can be stated x + y implies 6(x,z) + 6(y,z), (4) for all 2. That is to say, the space is continuous. It would also seem reasonable to expect that the dissimilarity between two distinct points x and y cannot be less than the dissimilarity between x and itself or y and itself. That is 6(x,y) :max [6(X.x). 6(yry)]. (5) Because we are dealing with stimuli with a finite number of dimensions for some psychological distance model, the assumption of symmetry seems realistic. The order of the stimuli should not change the dissimilarity as indicated by d(x,y) = 6(y,x), for all x and y. (6) These conditions define a space that is semimetric (Carroll and Wish, 1973). By adding the triangle inequality d(x,z);gd(x,y)-+d(y,z), for all x, y, and z, (7) where d(x,y) indicates distance between x and y, the space is made metric. For a finite set of points the dissimilarities can be transformed into distances (satisfying the triangle inequality) by adding a large enough constant. The smallest constant that will work is c = max (6(x,z)-5(x,y)-6(y,z)) for all x, y, z. (8) The assumptions of this model are straightforward, but it is not immediately apparent that they are met by auditory judg- ments. In particular, the distance between a pair of tones is judged by listening to them successively. Order of presentation could effect the judgments which would violate the assumption of symmetry. Somegproperties of two-tonetgqugnces suggesting violations of symmetry Experimenters have long been aware of the possibility of order effects in perceptual tasks and have named them time-errors (Woodworth and Schlosberg, 1954). Hysteresis is another term used to describe such order effects. Postman (1946) investigated time-error for pitch and loudness. The subjects listened to pairs of stimuli and judged the second to be higher or lower in pitch (or to be louder or softer for the loudness experiment) than the first. The second tone was a variable length of time from the first. He looked for a bias in the number of higher or lower judg- ments as a function of inter-stimulus-interval (181). The judgments for pitch were not symmetrical but there was no consistent pattern to the deviations. Significant time-error was found for loudness, with the number of louder judgments increasing for 181's of more than 2 sec. Interestingly, in musical literature there is considerable discussion of perceptual effects. The idea that the pitch of the lower note of a two note interval is dominant and that the natural orientation of a tonal pattern is upward is a part of music lore (Farnsworth, 1958). Farnsworth (1938) supported this notion with musically naive subjects and male singers, especially basses. The subjects listened to simultaneously sounding intervals and indicated the apparent pitch of the interval. He found the upper note dominated these judgments for most musically trained subjects, while all other subjects were influenced by the lower note. These results suggest that the subjective size of an interval may depend on whether 10 the interval is ascending or descending, that is, whether the dominant note is first or second in time. To investigate these order effects, a preliminary study was conducted by a Michigan State University under- graduate, Anna Olivarez. She had 4 musically trained and 4 musically untrained subjects rate the distance between two notes with intervals ranging from 1 to 6 half-steps on an equal tempered scale. One musically trained subject gave larger values to the ascending intervals and one non-trained gave larger values to the descending intervals, but on the whole the data was symmetric. Spohn (1965) has observed that the learning of equal temperament names for intervals is much easier in an ascend- ing direction. It seems that the character of the interval is more apparent when the tones are ascending. This is sup- portive of Farnsworth's contention that the natural pattern is from below upward. Another musical phenomenon is the finality effect, that is in Western Music one note or chord seems to be an appropriate stopping place. Played in one order, two notes or chords may seem to come to rest, while presented in the other order, the sequence subjectively needs to be continued. When considering this sound quality, two notes or chords may not be subjectively symmetric. If one plays an alternating sequence of C's and F's on the piano it seems that the most 11 natural stOpping place will be on the F. The explanation of this effect is based on the special significance of the interval of a fifth. In the key of F, C is the perfect fifth above F. That is, C is the second most important note in the scale. But in the key of C,'F is the perfect fourth above C and this relationship is not as musically important. Hence, the familiarity with scales and their structure leads to the assumption that the sequence is in the key of F and F is the tonic and a "logical" final tone (Farnsworth, 1958). Another Michigan State University undergraduate, Kelly Van Vliet, looked at similarity judgments made on pairs of major chords to investigate the musical phenomenon of finality effect. The stimuli were designed so a pair of chords, such as C and F, appeared four times. For two of the pairs, C was followed by F and for the other two, F was followed by C. By raising C up an octave for one of the pairs where C was followed by F and for one where F was followed by C, the order C-F had both an ascending and a descending pair. Figure 1 shows these four pairs. Pairs one and three are a fourth apart and pairs two and four are a fifth apart. In this experiment four different musical keys were used. Three basic interval lengths were used in this study, the fifth, the third, and the second. Because of changing the octave so as to have both ascending and descending sequences and both orders of the chords, that is, 12 C-F and F-C, the intervals fourth, sixth,and seventh were also presented. The lengths fifth and fourth; third and sixth; and second and seventh contain the same chords but with octave displacement of some of the chords. \ O Q— 8 o O o o o o n o 8 o a 0 X75 0 R o s— %57 1 2 3 4 C F C F F‘ C F' C Figure 1. These are examples of chord pairs for an experiment on finality effect. From Farnsworth's arguments about finality those pairs ending in the tonic, F in the example given above, would come to rest. That is, the ascending fourth and the descending fifth, chords 1 and 2‘in Figure 1, would possess finality and would be judged in a similar manner. The results are presented in Table 1. For all subjects including two music students, a finality effect based on a supposed tonic chord will not explain the results. 13 Table 1. Mean magnitude estimates for the distance between two tones j Relationship between the two tones in musical notation F_—_l F_—__l fifth fourth third sixth second seventh Ascending 9.14 7.11* 7.62 13.71* 7.98 14.91* Descending 7.57* 6.82 . 6.69* 11.73 7.19* 11.09 *Indicates those chords hypothesized to possess the finality effect. The major effect was the direction of the interval with the descending intervals being judged consistently smaller than the ascending. The length of the interval was also subjec- tively important. If symmetry is violated, then a measurement structure not requiring the symmetry assumption must be explored. The bisection system proposed by Pfanzagl (1968) can handle such order affects and provide a subjective scale for pitch. Tests of the key properties of the bisection task will now be discussed. Some properties of three-tone sequences The bisection system developed by Pfanzagl (1968) requires a psychological bisection operation. In the bi- section task the subject may be asked to set a tone so that it "bisects" the pitch of a pair of tones or he could be asked to judge whether a variable tone is "above" or "below" 14 the halfway point for the interval. The point where the judgments are 50% above and 50% below is taken to be the subjective middle. For a weakly ordered set of objects, there is a unique B(x,y), for any x and y members of the set, that is interpreted as the bisection point between x and y. In addition there are four axioms for this bisection system. The bisection point of coinciding end points coincides with the end points, that is B(x,x) = x. (9) The second axiom asserts that if x has a higher pitch than y for example, then the midpoint between x and z is higher than the midpoint for y and z, for any 2 in the set of objects, as indicated by x 2 y implies B(x,z) Z B(y,z), (10) where 2 indicates weakly ordered. Thirdly, B is continuous in both of its arguments. The main assumption of the bisec- tion model is the bisymmetry axiom. This axiom states that for points a, b, c, and d, the bisection of the results of the bisection of a and b and of c and d is equivalent to bisecting the results of the bisection of a and c and of b and d. This is formally stated B(B(a,c), B(brd)) = B(B(a,b), B(de)). (ll) 15 A graphical illustration of this axiom appears in Figure 2. (See Krantz, Luce, Suppes, and Tversky, 1971 for a further discussion.) An important part of this model is that the bisection operation need not be symmetric. Pfanzagl has shown that the first four axioms guarantee that one can construct a numerical scale such that the order of the stimuli is preserved by their scale values and the scale value for the bisection point is a weighted average of the scale values of the end points. That is, there exists a function f that preserves the order of the stimuli as represented by f(x) :f(y) if and only if xzy (12) and assigns a scale value to the midpoint as given by f[B(x,y)] = pf(x) +qf(y), where p+q=1; PIQZO. (13) The p and q weights reflect the direction of the bias. When the bisection point is closer to the first end point the value of p is greater than 1/2. Pfanzagl has also shown that f is an interval scale and that the representation given above is unique. Thus, this bisection system can generate a unique subjective scale even when there are order effects. 16 B(B (arc) :B(brd)) L. _. ._,_\ l g i f Figure 2. l I i T I I l l 1 >1 1 /l I'é * .>| ‘l 111 I ! l '1' III [11: a b B(a,c) B(b,d) c d B(a,b) B(B(a,b),B(c,d)) B(c.d) A graphical illustration of bisymmetry. 17 Some of the work on pitch has used the method of bisection. Studies of pitch bisection have yielded ambiguous results concerning hysteresis effects and the bisymmetry axiom has not been tested. Stevens (1957) found hysteresis for some subjects. The bisection point was higher for the ascending intervals. The two subjects with reported "absolute pitch" gave the same bisection point in both directions. Also, two of his students found hysteresis in equisections (Stevens, 1957). In this case the subjects divided the interval 400-7000 Hz. into four equal intervals. Cohen, Hansel and Sylvester (1954), in a bisection experi— ment where the middle tone was adjusted by the subject, did not find evidence for hysteresis. For the interval 1000— 3000 Hz. the ascending middle was 1874 and the descending was 1838 Hz. For the interval 2000-4000 Hz. the ascending and descending middles were 2693 and 2808, respectively. The standard deviation (sd) was 303 Hz., overall. The differences in both cases were well under 1/2 sd. Also note, the directional difference was the opposite for the two intervals. Cross (1965) empirically supported the bisymmetry axiom in a bisection task with tones varying only in inten- sity but at the same time found hysteresis. The bisection points were nearer the softer tone (as the tones increased in intensity) and closer to the louder as intensity decreased between the two tones. 18 Reaction time Reaction time tasks have been successfully used by many researchers to study possible mechanisms for information processing (Sternberg, 1971; Clark, 1969; Clark & Chase, 1972; and others). If there are asymmetries in judgments of two or three tone sequences measurements of reaction time could be used to explore the possible processes involved. It was previously mentioned that musicians have talked about a natural order for pitches. The natural orientation indicated by musical literature is upward (Farnsworth, 1958). The task of learning to associate the appropriate equal temperament interval size name with a pair of tones seems to be easier and learned faster when the intervals are ascending (Spohn, 1965). This idea of natural order has a striking similarity to the concept of semantic marking (Clark, 1969). In this theory, a general term, such as tall, is classified as an unmarked word or concept. The contention is made that a large number of things can be described as "so tall" but only a small set can be described as "so short." The marked term is the unmarked plus a marker which changes the value to the opposite of the unmarked term. Clark assumes that the marked term is more complex than the unmarked. He concludes that in order to process information with a marked term a transformation must be made, "This is the opposite of the unmarked term." 19 Thus processing the marked terms involves an extra stage and takes more time. A simplified version of the model appears in Figure 3. Encode ‘ Is it the L yes _ Execute stimulus unmarked? A response ‘lno Change value Figure 3. An information processing model for the identification of marked and unmarked concepts. Olson and Laar (1973) have shown more time is needed to process the term "left" than the term "right" for right- handed subjects. They have accounted for this increased time by the idea of spatially marked and unmarked terms. That is, the term "left" must be transformed to a function of the term "right." When recognizing descending intervals, those intervals considered more unnatural and more difficult to learn, a transformation may be needed. The first ques- tion asked when determining the order of tones becomes, "Are they ascending?" If the answer is "no" then check for the mark that transforms ascending to descending. Ascending intervals, if ascending is really a more general concept, will be recognized more quickly than descending intervals. 20 Models for two- and three-tone pitch perception In the beginning of this chapter three models for pitch perception of single tones were presented: log, power, and helical. If the subject listens to two tones and makes a decision as to the distance between the two tones, what do these pitch models predict? The pitch of the upper tone and the lower tone of a given interval will be written Wu and w£, respectively. The log scale values for wu and WK are 6 II a-+k °1og Fu (l4) and WK a-tk.°log F£ (15) The usual model for the difference between two unidimensional stimuli says that the difference is the difference of their respective scale values. So for the log model the distance between a pair of tones is wu-Z = wu - WK = k ° (log Fu - log F£)' (16) The size of an interval in octave units, 5, is given by (log F -log F ) . s = u 1' (17) log 2 21 Thus the pitch difference is linearly related to the interval size in octaves, wu-Z = k '8 'log 2. (18) The power scale values for Wu and WK are Wu = koFE (19) and _ . P . respectively. The difference between the upper and lower tones is hypothesized to be = - = o p- p ¢u_£ Wu WK k (Fu F£)°, (21) When the interval size, s, is fixed, the frequency Fu is a constant multiple, ZS, of the frequency EZ' For clarity, the constant 25 will be written CS, so the relationship between Fu and FK can be written F = F -c . (22) Equation 21 can be simplified by substituting P = p.. P Fu Ez cS .(23) and becomes ((1., = k . (FED - (c2— 1)). (24) 22 By taking logs of both sides, we find that the log of pitch is a linear function of the log of the frequency of the lower tone plus an additive constant based on the interval length, log wu_£ = log meg-1) +p - log FZ. (25) Predictions from the helical model are not straight- forward. For the magnitude estimation task, judgments of distance for a two octave interval should resemble those of a one octave interval more than other lengths. One might also predict that distances between intervals of over and under one-half octave would be less than the judgment for one-half octave, since the model is a circle and the farth- est point is half-way around. One other prediction is possible. Intervals of length 5 and of 5 plus an octave might also have similar distance judgments. What do the log and power scale models look like for the bisection task? The subjective midpoint, wm, is given by (3) +11) ,9 =_u___’€h.. (26) m 2 By substituting in the above equation the log scale values for each w we get the equation a-tk '1og Fu-ka-tk °1og Ff a+kolog F = . (27) m 2 23 This equation simplifies to log Fu-tlog PK log Fm = 2 . (28) For the power scale we again substitute into the general formula for the midpoint and get P P k(Fu+F£) , P _ k Fm - 2 . (29) By taking logs we get the equation p °1og meklog k = log k-tlog (FE-+F$)-log 2. (30) This can be simplified by making the substitution as before for F , u log in-log (CE-+l)-log 2 log F = . (31) m P The power scale again yields the subjective value, in this case the log of the subjective midpoint, as a function of the lower frequency and an additive constant based on the interval size. The helical pitch model makes no clear cut predic- tions for the bisection task. 24 Pitch drift--a processing model for asymmetry One way to explain asymmetry in the perception of tone sequences might be that the first tones processed gradually change in memory while later tones are being presented. For most musical instruments including the human voice, keeping a tone up to pitch requires tension. Making a tone sharp requires even more tension while relax- ing causes a tone to 90 flat. A choir singing a cappella will often gradually go flat. A logical direction for a change in memory for a tone would be descending. If memory for pitch resembles some counting system, thinking about the system loosing a few numbers may be more reasonable than considering the system gaining values. When listening to a descending interval the first tone would be approaching the second, thus the distance between them would become shorter. While listening to an ascending interval the first tone would be moving away from the second. Thus, the prediction could be made that ascend- ing intervals are longer than descending intervals. For the bisection task this model would predict the subjective middle of an interval will be lower than an esti- mate of the middle from any scale for both ascending and descending intervals. In the descending case,the first note approaches the second narrowing the interval and forcing the middle closer to the second. For an ascending interval, the 25 first tone moves away from the second expanding the interval in a descending direction and again the middle will be lower. Thus, the first bisymmetry axiom is violated by the pitch drift notion. The bisection point of coinciding end points would be lower than the end points, the first tone played would decrease in pitch and be below the second, therefore, B(x,x) # x. (32) In the following three chapters, three studies examining musical scales and perceptual asymmetries are discussed. In Chapter II order effects in auditory judg- ments are dealt with using the technique of magnitude estimation. The three pitch models are discussed in light of judgments made on two-tone sequences. Chapter III looks at an empirical test of the bisection measurement model, a model which is capable of dealing with asymmetries. Again the data is analyzed with respect to the pitch models. A reaction time experiment is discussed in Chapter IV to examine the process of deciding whether a two-tone sequence is "ascending" or "descending." The last chapters in this thesis contain a discussion of the results and some conclu- sions in terms of consistency across experiments and with respect to the models of pitch perception. CHAPTER II EXPERIMENT I This experiment dealt directly with possible order effects in auditory judgments using the technique of magni- tude estimation. The subject listened to a pair of tones, either ascending or descending, and estimated directly the distance between the two tones. Ascending and descending intervals were chosen to be of various sizes and were selected from the frequency range 100-2500 Hz. Method Stimuli The stimuli were sine tones generated by an Interstate Electronics Corporation series F-34 voltage controlled function generator. The intensity of a 1000 Hz. tone was measured to be 73 db. An Advent Frequency Balance Control was set to equalize the loudness of all other fre- quencies to that of the 1000 Hz. tone. The amplitude envelopes were rectangular. The characteristic electronic click at the onset and offset of the tones was partially' eliminated by filtering frequency components above 2500 Hz. with the frequency balance control set as a low pass filter. 26 27 A Digital Equipment Corporation PDP-8/I determined the tones to be played for each trial, initiated the trials, started the tones at the subject's command, and recorded subjects' responses. The range of tones presented was set on the oscillator as well as the tone duration. The oscillator was triggered externally with a pulse from the PDP-8/I. Figure 4 contains a diagram of the system. The digital to analog (D-A) converter was manufactured by DATEL System, Inc. The tones were played through a Scott model 299-f stereo amplifier into Koss Electronics, Inc. Pro-4 head phones. Pairs of sine tones were presented in ascending and descending order with six interval sizes: 2/10, 6/10, 9/10, 10/10, 16/10, 20/10 of an octave. The octave equivalent was based on the musical scale for pitch. The upper frequency, Fu' for an interval of size, s, in octave units with a lower frequency, F2’ is computed using F =F£-2. (33) The interval sizes for this experiment were chosen so as to be different from the standard tunings for Western Music. This was done to minimize the possible influence of over- learned reSponses to the familiar 12 tone equal temperament or just intonation scales. Four different pairs for each interval length were used, thus forming a 2 x 6 x 4 factorial 28 PDP-8/I tr [lZ-bit D-A Tri er 99 Function 'L——————4> Generator Response Teletype Box (TTY) Equalizer Head [ Amplifier] Phones ~ ' Monitor Foot Pedal Speaker Figure 4. Schematic diagram of the experimental set-up. 29 design. Each tone had a 250 msec. duration and the interval between the offset of the first tone and the onset of the second was also 250 msec. A list of the stimuli can be found in Appendix A. Subjects Five musicians having completed a two-year sequence of courses for ear-training and five persons not having this training served as subjects. The five ear-trained subjects and one of the non-trained subjects received $2.20 per hour remuneration for their participation. Subjects numbered 1 through 5 are the non-trained and 6 through 10 are the ear-trained. Procedure Before each subject began this experiment, he and the experimenter discussed the study. It was emphasized to the subject that the experiment was not an attempt to assess his musical ability, but rather that it was a basic study on auditory information processing. This experiment was run concurrently with Experi- ments II and III. The subject participated in sessions lasting approximately one hour. Within each session blocks of trials from each of the experiments were presented insa random order. 30 The subject was instructed to listen to a pair of tones and react to the distance between them by indicating a number on the teletype (TTY). A range of 0-20 was sug- gested with "0" indicating the two tones did not differ in pitch and "20" indicating that they were very far apart. The subjects could use numbers to 99 as responses if they desired. The subjects were told there were no correct answers and that accuracy in estimating the actual length was not to be sought, but that a subjective impression of the distance or separation between the pitches was to be given. The computer indicated that it was ready to present a trial by typing a "?" at the TTY. When the subject was ready to listen he typed "Y" and 1/2 sec. later the tones were played. The subject had four sec. to make a response. If the sub- ject did not respond to a trial the pair was put back in the pool of unused stimulus pairs and a new trial was initiated. The stimuli were presented in a different random order each time through the forty-eight pairs constituting a complete set (two orders, six 1engths,and four frequency levels). The DECUS random number generator, statistical subroutine number 5-25, was used for computing these random orders. The complete set of stimuli was rated twenty-five times with the first five used as practice. 31 Results The analysis of the magnitude estimation data begins by looking for possible asymmetries in the judgments of ascending and descending intervals. Figures 5 and 6 contain graphs of mean magnitude estimates for ascending and descend- ing intervals as a function of interval size for each of the 10 subjects. With the exception of subjects 8 and 10, the mean magnitude estimates increase monotonically as the inter- val length increases. Subjects 8 and 10 (Figure 6) show a dip of considerable magnitude at an interval length of 1.6 octaves. (This effect will be examined further when the type of scale the subject used is considered.) No obvious systematic relationship between ascending and descending intervals was revealed. For subjects 4, 5, and 9 (Figures 5 and 6) the ascending intervals have larger subjective length than the descending intervals. Subjects 2 and 3 (Figure 5) show differences in subjective length in the opposite direction. The differences in judgments are not consistent for subjects 1, 6, and 8 (Figures 5 and 6) and subjects 7 and 10 (Figure 6) show no apparent differ- ences with respect to the order of the pitches. To analyze more fully these differences and to look at the influence of interval length and frequency range, an analysis of variance was run for each subject with a 2 x 6 x 4 design (two orders x six interval lengths x four pitch 32 .muowflndm conflmnuuumolcoc uom omen Hm>umuca mo cowuocsw m we mmumfiwumw upsuflcmse sews wo gamma ¢ .m unavwh cues: e>euoo 5 oucuuouuwo oozesgum o.~ 04 0..” m. o. m. o.~ min 04 m. o. N. I 1 1 1 1 1 1 1 d a u 1 N I N I v L e W .. o 1 o w 8118083 . O 6 u I m 1 m N. medians .. ‘ m e L oa .. ca u .4 7.. 1 NH 1 «H m m m 80.33 .. Z l 3 0H muses mesuoo ea oosouowuao ousosvoum l I o. o. m. o. C v 8033. o.~ #4 m. . w 323 «$300 :w oucououuao 00:250.“..— o.m 04 04 m. m. a. o.~ wé 04 a. o. N. ... - ~ )1~ u « 4 T a q 1 q I. N l N v i v M 0 l 0 w u m. m L m a a 8 S .. on m NH .. NH a 3 A z n 8033 N 8333 a 3386 OH ma. ON eamase epnuubu use" 33 .muownnsm Umcfimuuuumw How mafia Hm>uquw uo :OMHUCSN m an mwumaflumm wuauwnmnfi Gums no ammum 1 .m whamwh mafia: 3:300 cw mucuuwuuwu 8:259; o.~ o.H o.H m. m. N. o.N w.H o.H m. m. N. < < 4 1 4 4 1 1 4 1 4 1 N l v l wcflucwomwu I O 1 o l mega—won» I C .. m l .. 3 I .. NH 1 va I 3 303.54 a uuofinsm mud; 95.50 :H wonmuwuuwu wocgwwuh o.~ m: o4 m. o. N. ed w; o4 m. m. N. o.~ a; 04 m. m. m. . 1 . . J < 1 . . . J . ll] . . . . ‘ m l u a l v I v I v a w J o l w .m L m -m 53 L 3 13 12 . 2 .S m uommndm Irv." h downs—am A 3” 0 0003.5 Iva mumps; apnuubmu uvaw 3321111133 apnnxubem ueau 34 levels). Due to an undetected error in the stimuli, the length 1.6 octaves did not have a second pitch level, that is, the pair of tones beginning about 400 Hz. and 1.6 octaves in length did not appear in the stimulus set. This pair of stimuli was missing from both the ascending and descending condition, that is, there were only forty-six conditions in the analysis instead of forty-eight. For all subjects the length of the interval was significant at less than .0001 and accounted for consider- able variance (n2 values were between .42 and .81). The order variable was significant at less than .01 for four subjects (3, 4, 5, and 9) but in each case under 5% of the total variance is explained by order while interval length eXplained between 42 and 67% of the total variance for these subjects. The means are slightly larger for the ascending intervals in each case. All other main effects and all interactions were either non-significant or account for little of the total variance (n2 values under .10). Com- plete details are in Appendix C. Although many of these other conditions were significant, they are probably not meaningful. First, the effect of interval length was over- whelming and all other effects are quite small in comparison. Second, with 874 degrees of freedom in the error term the likelihood of detecting small effects is larger than with fewer trials and conditions. 35 Due to extreme values in the magnitude estimates the median may be a better representative value than the mean which has been considered to this point. The medians were analyzed using a sign test to look for possible asymmetries not previously found. For each subject the twenty-three pairs of stimuli (six interval lengths and four frequency ranges minus one) were compared. Only subject 5 showed a significant difference between ascending and descending intervals with the ascending ones being judged longer than the descending. Sign tests were also run comparing the medians for the lowest pitched interval with the medians for the highest pitched interval at each interval length. These values are contained in Table 2. Sign tests were run both across sub- jects by interval length and within subjects across interval length. With respect to this data, intervals with pitches of equal frequency ratios seem to give rise to equal sub- jective intervals regardless of the interval's position in the frequency range or the order of presentation. To further investigate the pitch scale used by each subject, the medians of the magnitude estimates were deter- mined for each interval length across order and range. For two of the ear-trained subjects (8 and 10) no distinction was made between an octave and a two octave jump. Also, no distinction was made between the length 6/10 octaves and 36 .Ho. an ucmoHuacmHm. nomHaH and m\m m\N m\v m\N m\H .NH\H HH\0 cm\o 0H\m c0H\0H umnu 0H «0 :oHuMOQOum VH\m N h N 5 OH OH mH 0H 0H mH h m 0H mH vH MH mH m.mH m.mH m.mH H: h N h N 0N 0H NH mH mH mH m m.h mH mH m.HH 0H m.mH vH NN NN 0H o.N MH\m v v m o m N mH mH 0H mH m m.m m.h m.mH m.HH MH m.NH VH mH NH H: m.v v m m N N mH mH a m.@ v m.m 0H m m m m.NH 0H HN NH 0H o.H NH\o N N v m 0H 0H 0H 0H 0H m.m m N o m.NH m o m.m m.m m m OH H: N N m m 0H OH OH OH m m m.m m 0H m.NH m w m m.HH 0H mH m.NH oH o.H vH\m o m o o m.m m m m m m m N m.h m m m.0H N m m.w m.NH H: o o v v m m m m m m n v m h m m.v m.m 0 HH m.MH 0H 0.0 HH\v v v v v n N o o o o v m.m v m.m m m.o m.m N N m H: v v n v N N o o o o m n m v m m N m.v m VH 0H 0.0 mH\m H H m H N N N N N N n v n n N N N N n v H: N N N N v m.N m m N m.N m.N N N H N N m N m v 0H N.o uwan: mum o ¢ o 4 o 4 o a o 4 o < o 4 o a o < o a mm>wuoo uacu 0H u0 :H GOHuuomoum 0H m m h o m m H nuvcwq muowflnsm :oHumucmmoum mo muocuo 03» can mzumcoH Hd>uwucH me mo sumo um >ocoskuu ummzmHn can umoon and How muoonnsm 0H ecu mo some now ncdeox .N oHnma 37 16/10 octaves. Graphs of the medians against interval length for these subjects are in Figures 7 and 8. For all other subjects the subjective distance increases with physical length. The median magnitude estimates for all subjects except 8 and 10 were tested against the log and power pitch models as stated in equations 16 and 25 in Chapter 1. Using least squares linear regression k was estimated for the log model. The root mean square error was determined between the model estimates, wu-Z' and the magnitude estimates, e, using 23 2 (WW, - e) 2 1:1 . (34) 23 For the power scale the term log k (C:-l) will be a constant, a, for a particular interval length s. Least squares linear regression was used separately for each interval length for each subject using log wu-K = a-+p -log Pk. (35) The average power, p, value and sd of the p values for each subject are in Table 3. Across subjects the p values ranged from -.38 to .55. For each subject there is a large range of p values, also. The model is based on having one Median for magnitude estimates Figure 7. Median for magnitude estimates Figure 8. H O g... 0 (I) 0‘ .p. N 38 . L l 1 1 1 .2 .6 .9 1.0 1.6 2.0 Frequence differences in octave units A graph of median magnitude estimates as a function of interval size for Subject 8. l l 1_1 1 .2 .6 .9 1.0 1.6 2.0 Frequence differences in octave units b A graph of median magnitude estimates as a function of interval size for Subject 10. 39 Table 3. A summary of the comparisons between the log and power models of pitch for magnitude estimation data Root mean squared Power values for error power scale Subjects Log Power Mean S.d. l .59 1.29 -.17 .15 2 .44 .87 .Ol .37 3 .40 .77 .24 .37 4 .49 1.06 —.01 .54 5 .48 .43 .26 .37 6 .47 1.13 .16 .49 7 .48 1.09 —.04 .20 9 .42 .37 -.02 .43 p value throughout the frequency range. To compare the log and power models, the mean p value for each subject was used to determine the best intercept value, a, using least squares linear regression. Using one estimate for p and one for a, for each subject,the power model estimate wu-L' was calcu- lated and compared to the magnitude estimates, e, and the root mean square error was determined using 23 Z (10 log w _ -e)2 i=1 u 1' . '(36) 23 The root mean squared error values by subject for both models are available in Table 3. 40 Except for subjects 5 and 9 the log scale is definitely superior to the power scale for the magnitude estimation task. The power model has definitely been violated by the large range of exponent values within a subject across interval size. In summary, for magnitude estimates the length of the interval is the main determiner of the subjective length with order and frequency range having little influence. There appear to be two instances of octave generalization with ear-trained subjects which is sup- portive of the helical model for pitch. The log model is clearly superior to the power model for explaining the data of the remaining subjects. CHAPTER III EXPERIMENT II For this experiment the subject bisected four intervals, each in ascending and descending order, using the method of constant stimuli to determine if there is an order or hysteresis effect in pitch bisection. The bisymmetry axiom was tested using the results of the initial bisections. Subjects The same subjects as in Experiment I participated in Experiment II. Stimuli Four intervals, 400-500 Hz., 1000-1600 Hz., 400- 1000 Hz., and 500-1600 Hz. with lengths 1/3, 2/3, 1 1/3 and 1 2/3 octaves, respectively, were used in this exper- iment. For intervals under one octave the middle tones were 1/40 octave apart and for intervals over one octave the middle tones were 1/20 octave apart based on the log scale for pitch (an equal tempered scale) using equations 37 and 38, respectively. P = F ~21/‘° n+1 n (37) _ . 1/20 Fn+1 — Fn 2 (38) 41 42 The ten tones were centered around the log scale estimate for the center of the interval. The log estimate of the midpoint, Fm, was determined using F=F° -‘l, (39) where FL is the lower end point and Fu is the upper end point of the interval bisected. Appendix B has a list of tones used. For the bisymmetry test (see Figure l in Chapter I for a geometrical illustration of bisymmetry) the subjective nddpoint for the above intervals were deter- mined and used as the end points with the ten middle tones determined as above. The intervals bisected are listed in the first column of Table 6. Each tone had a 250 msec duration and the interval between the offset of one tone and the onset of the next tone was also 250 msec. Procedure Before each subject began this experiment, he and the experimenter discussed the study. It was emphasized to the subject that the experiment was not an attempt to assess his musical ability, but rather it was a basic study on auditory information processing. This experiment was run concurrently with Experi- ments I and III. The subject participated in sessions lasting approximately one hour. Within each session, 43 sets of trials from each of the experiments were presented in a random order. Using the method of constant stimuli the subject judged whether the middle of a set of three tones was above or below the bisection point of the interval defined by the first and third tones. The subject responded in a two- alternative forced choice situation whether the middle tone was "above" or "below" the perceived center. This was more easily conceptualized when the question was stated, "Is the middle tone closer to the higher tone (above) or the lower tone (below)?" A light flashed on the response box when a trial was ready. The subject initiated a trial by pressing a foot pedal. The tones began one-half sec. later. The subject indicated his choice by pressing the appropriately labeled button on the response box. The trial was put back in the pool of remaining trials if the subject had not responded in four sec. and the light flashed to indicate a new trial. As before each set of eighty (four intervals with ten different middle tones in two directions) trials were presented in a different random order. The complete set of stimulus triples were presented twenty-five times with the first five considered practice. The right button was randomly assigned above and below with the constraint that for the last twenty blocks of trials the right was labeled above for 10 blocks and labeled below for 10 blocks. 44 For the seven subjects, whose judged middle point fell within a "reasonable" range (see Results for further explanation), a test of bisymmetry was run. In a final session lasting about two hours these subjects bisected, as before, intervals based on their previous results. All presentations and hand positions were random as before, but unlike other sessions this was the only experiment run. Results For each subject the bisection data was analyzed using least squares regression to plot a normal ogive for each interval length in both directions. (See Woodworth and Schlosberg, 1954, for details on this technique.) The estimated subjective bisection points are summarized in Table 4 for the non-ear-trained subjects and in Table 5 for the ear-trained subjects along with log estimates of the midpoints, as determined by equation 39 in the method sec- tion. For three of the non-ear-trained subjects (2, 4, and 5) some nddpoints were not estimable, that is, regression analysis put the midpoints outside the interval being bisected. In these cases the direction of the estimate is indicated. Where possible a 2 test was run to test for a significant hysteresis effect (see Woodworth and Schlos— berg, 1954). These values are also available in Tables 4 and 5. 45 Table 4. A summary of the results from the bisection experiment for the non-ear-trained subjects Estimates of midpointsa Bisected Length Log Subjective .Test of interval in hysteresis Subject in Hz. octaves Ascending Descending Z 1 400-500 0.33 447 447 450 -0.18 1000-1600 0.67 1265 1232 1186 +0.26 400-1000 1.33 632 629 623 +0.13 500-1600 1.67 894 873 880 -0.14 2 400-500 0.33 447 433 469 -2.34 1000-1600 0.67 1265 - 1322 400-1000 1.33 632 - + 500-1600 1.67 894 799 + 3 400-500 0.33 447 424 460 -4.82* 1000-1600 0.67 1265 1251 1389 -4.71* 400—1000 1.33 632 653 685 -0.55 500-1600 1.67 894 1003 1146 -l.70 4 400-500 0.33 447 481 - 1000-1600 0.67 1265 + - 400-100 1.33 632 815 - 500-1600 1.67 894 993 759 +4.29* 5 400-500 0.33 447 418 + 1000-1600 0.67 1265 1294 + 400-1000 1.33 632 789 701 +1.04 500-1600 1.67 894 895 765 +1.11 a I I 0 c I O "+" indicates an extreme overestimation of the midp01nt; ”—" indicates an extreme underestimation of the midpoint. *Significant at p‘<.Ol. 46 Table 5. A summary of the results from the bisection experiment for ear-trained subjects Estimates of midpoints Bisected Length Log Subjective Test of interval in hysteresis Subject in Hz. octaves Ascending Descending Z 6 400-500 0.33 447 450 449 +0.16 1000-1600 0.67 1265 1267 1246 +1.13 400-1000 1.33 632 562 615 -2.11 500-1600 1.67 894 946 978 -0.95 7 400-500 0.33 447 437 446 -2.17 1000-1600 0.67 1265 1248 1243 +0.24 400—1000 1.33 632 628 613 +0.83 500-1600 1.67 894 902 886 +0.60 8 400-500 0.33 447 445 449 -0.54 1000-1600 0.67 1265 1275 1274 +0.04 400-1000 1.33 632 618 687 -l.7l 500-1600 1.67 894 839 1053 -2.82* 9 400-500 0.33 447 443 436 +1.11 1000-1600 0.67 1265 1061 1267 -l.44 400-1000 1.33 632 519 896 -7.41* 500-1600 1.67 894 849 1053 -2.87* 10 400-500 0.33 447 436 443 -0.98 1000-1600 0.67 1265 1276 1259 +1.09 400-1000 1.33 632 647 638 +0.21 500-1600 1.67 894 879 836 +0.34 *Significant at p < .01. 47 Subject 3 showed a consistent asymmetry for all interval lengths. He always judged the middle on the descending intervals higher than those on the ascending intervals. In two cases the effect is significant. This effect appears in subject 2 but because of missing estimates the results are less straightforward. Subject 4 shows a consistent but opposite effect and again there are missing values. Subject 9 judged the midpoint to be higher in the descending condition for three of the interval lengths and for two of those the effect was significant. All other sub- jects have inconsistent patterns of differences between the ascending and descending subjective midpoints. Subjects 4 and 8 each have one significant hysteresis effect, but no coherent pattern emerges. The data for the test of bisymmetry was analyzed using the least squares regression methods as before to determine the subjective midpoints. The intervals that each subject bisected, their length in octaves, the log estimate of the midpoints and the subjective midpoints are in Table 6. Again 2 tests were run for a test of bisymmetry. These values are also available in Table 6. Subjects 7 and 8 gave empirical support for the bisymmetry axiom. Subjects 3 and 8 successfully illustrated bisymmetry on ascending intervals and subjects 6, 9, and 10 did the same for descending intervals only. Figures 9 and 10 contain graphs of the subjective midpoints against log estimates of the midpoints for all 48 Table 6. A summary of the results from the test of bisymmetry Estimates Bisected Length of midpoints Test of interval in Interval Bisymmetry Subject in Hz. octaves direction Log Subjective Z 1 629-873 0.47 ascending 741 726 _4 97* 447-1232 1.46 742 946 ‘ 623-880 0.50 descending 740 750 +3 09* 450-1186 1.40 731 612 ' 3 653-1003 0.62 ascending 809 783 _2 26 424—1251 1.21 728 864 ' 685-1146 0.71 descending 886 940 +3 92* 460-1389 1.29 799 837 ' 6 562-946 0.75 ascending 729 712 -8 00* 450-1267 1.50 755 838 ' 615-978 0.67 descending 748 770 +1 10 449-1246 1.47 776 750 ' 7 628-902 0.53 ascending 753 745 +0 27 437-1248 1.51 738 741 ' 613-886 0.53 descending 737 768 +2 10 446-1243 1.48 745 739 ° 8 618-839 0.44 ascending 720 713 _0 52 445-1275 1.52 753 722 ' 687-1053 0.62 descending 851 858 +2 23 449-1274 1.50 756 804 ° 9 519-849 0.71 ascending 664 648 _2 68* 443-1061 1.26 686 707 ’ 896-1053 0.23 descending 971 985 +12 33* 436-1267 1.54 743 711 ‘ 10 647-879 0.44 ascending 754 770 +2 90* 436-1276 1.55 746 561 ' 836-638 0.39 descending 730 721 _1 27 443-1259 1.51 747 755 *Significant at p < .01. 49 .nuounnsa vosHeuuluoelsoa How 3.30.3.1. 0005 Han nouns—Hume moH 05 mo soHuocsu a an xnou coHuooaHn on» you 32,232.: on» no eucHomvHE gHuOOnnan on» NO Bushman. noun—60m .a 0.26.: 3509qu 05 no nous-Hun. 90H 05 an 85.5. 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D. . .- OOOH l OOOH IOOOH My . 82 ._ 8.2 182 n 1 8m." J 003 ngmd m 80.38 p 8038 w 30an anus-(qu- 10; mum 601 u‘; Iotauonbazg 51 bisected intervals to examine the log model for pitch bisection. Root mean square error values were determined using "bad (10 log Fm- f) 2 1 , (40) n where n is the number of intervals successfully bisected, log Fm is the particular model's estimate of the middle of the interval and f is the subjective midpoint. Equation 28, in Chapter I, was used in determining the estimate for the log model. The equation for the power model can be found in Chapter I, equation 30. These values can be found in Table 7. Because each subject bisected an interval length only once in each direction the exponent, p, values for the power scale cannot be determined without averaging over interval length. The p values from Experiment I were used as estimates for the p value for this experiment, except for subjects 8 and 10. The mean power value, .05, across the eight subjects was used in these cases. Root mean square error values for the power model were determined using equation 40 with log Fm being the power model estimate for the midpoint. The two models are nearly indistinguishable. As the exponent, p, approaches zero the frequency estimates of the power model for the subjective midpoints approach those of the log model. At zero, however, the power law changes to log Wu-£ = log F£° (41) 52 Table 7. A test of the log and power models of pitch for the bisection data Root mean squared error in frequency units Subjects Log Power 1 21.09 21.36 2 28.55 28.75 3 28.63 24.00 4 62.51 62.56 5 36.38 38.57 6 12.50 10.97 7 4.13 4.03 8 15.58 15.13 9 32.62 32.70 10 16.41 17.10 That is, the subjective pitch of an interval is based on the lower frequency alone and not on the interval length. In summary, there is evidence for bisymmetry for four of the ear-trained subjects. Three of the subjects demonstrate a consistent hysteresis effect which is clouded by out of range bisection values. The log and power models give essentially the same estimates for the bisection points because the power values are close to zero. CHAPTER IV EXPERIMENT III This study dealt with possible differences in information processes involved in deciding whether an inter- val was "ascending" or "descending" by measuring reaction times for the decision on the direction of the interval. Interval size and approximate frequency range were again considered. Subjects The same subjects as in Experiment I participated in this experiment. Stimuli The same set of forty-eight pairs of tones as in Experiment I was used in this experiment. Procedure Before each subject began this experiment, he and the experimenter discussed the study. It was emphasized to the subject that the experiment was not an attempt to assess his musical ability, but rather it was a basic study on auditory information processing. 53 54 This experiment was run concurrently with Experiments I and II. The subject participated in sessions lasting approximately one hour. Within each session, sets of trials from each of the experiments were presented in a random order. The subject listened to a pair of tones and indi- cated as quickly and as accurately as possible whether the interval was ascending or descending. For any wrong or missing responses the pair of tones was put back into the pool of remaining trials. A light flashed on the response box to indicate the next trial was ready. When the subject was ready for the tones he pressed a foot pedal and the pair of tones were played one-half sec. later. As in ExperimentIi the trials were presented in a random order within a block of forty-eight. The blocks of stimulus pairs were presented twenty-five times with the first five considered practice. The right button was randomly assigned ascending and de- scending with the constraint that for the last twenty blocks of trials the right was labeled ascending for 10 blocks and the left was labeled ascending for 10 blocks. Results To begin looking at how order may affect reaction. time (RT), means were calculated for each interval length keeping order and training separate. A graph of these means 55 appears in Figure 11. The means for the non-ear-trained subjects are longer than those for the ear-trained. The RT's for subject 4 were on the average 200 msec longer than those of any other subject and 400 msec longer than the group average. This accounts for the differences between trained and non—trained. The differences between ascending and descending intervals is unclear from this representation. Also note, the length of the interval does not seem to influence RT. For each subject a 2 x 6 x 4 analysis of variance was run on the RT data to examine the influence of order on RT and look at possible influences of interval length and frequency range. As in Experiment 1 only forty-six condi- tions were used in the analysis due to an undetected error in the stimuli. The pair of tones with length 1.6 octaves and lower frequency of about 400 Hz. did not appear in the stimulus set in either presentation order. Although across the subjects many main effects and interactions were significant, the n2 values were very low (less than .07) and a minimum of 80% of the variance was in the error term. Complete details are in Appendix D. Subject 7 had the largest n2 for order. His RT's were longer for the descending trials. 56 600 . 500 r- 400.. 300,_ 200.. ascending descending trained A A non-trained O o 100,_ 1 J l 1 L L .2 .6 .9 1.0 1.6 2.0 Frequence difference in octave units Figure 11- A graph of reaction times for trained and non-trained subjects as a function of interval length. 57 Length was significant for subjects 1, 4, and 6 with n2 values of 6 to 8% with the direction of longer intervals being identified faster. For three of the ear-trained sub- jects (8, 9, and 10) there were no significant differences in any condition. It should be noted that most subjects had extreme values in many of the conditions which seem to be anticipa- tion on the one end and inattention or distraction at the other. These anticipatory responses occur when the first tone of the pair is either very low or very high in fre- quency and the subject has a good chance of guessing the direction of the interval before the second tone is played. Due to these extreme responses the medians were analyzed using a sign test to assess processing differences in the terms "ascending" and "descending" not found using analysis of variance. Subjects 1, 3, and 7 show significant differences between ascending and descending decisions, at the .01 level. For subjects 1 and 3 the ascending intervals took longer to recognize than the descending, the Opposite holds true for subject 7, as noted above. Overall, processing times for ascending and descend- ing trials are the same. However, two non-ear-trained subjects took longer to identify the ascending intervals. and one ear-trained subject took longer on the descending intervals. CHAPTER V DISCUSSION Overall, no systematic differences were found in perception of ascending and descending intervals. This lack of asymmetry indicates that dissimilarity judgments for simple tones are at least semi-metric and, as stated before, can be trivially transformed into metric or dis- tance measures for multidimensional scaling analysis. These results suggest that experiments that vary on other dimensions like timbre might be carried out without too much concern for time-error or hysteresis effects. Only subject 3 shows any constant order effect across tasks. In the magnitude estimation task the descending intervals are perceived as being larger. One explanation of this would be that the pitch of a tone increases in memory. The first tone will be sharper when compared to the second. For a descending sequence the distance between the tones enlarges and for an ascending sequence the dis- tance is shortened. It follows from this lengthening and shortening that reaction time for the ascending intervals will be longer because the pitches are closer together in 58 59 the ascending case and harder to recognize. Subject 3 takes consistently, a longer time to recognize ascending intervals. With the intervals being changed in an upward direction, the bisection points should be above the actual. This is true for subject 3 but only for ascending intervals. These results compliment those of Farnsworth (1938), who indicated for musically naive subjects the natural orientation would be upward. One other non-ear—trained subject (2) shows an upward orientation but only for the magnitude estimation task. However, for two other non-ear-trained subjects (4 and 5) and one ear—trained subject (9) a model with the tones decreasing over time could explain their magnitude estimates. Influence of ear training Are there differences between the non-ear-trained subjects and the ear-trained subjects? There are no dif- ferences between trained and non-trained for the magnitude estimation of interval length task. For the reaction time task two non-trained subjects took longer to identify the ascending intervals and one trained subject took longer to identify the descending, but on the whole the data was sym- metric. There are differences between subjects on the bisection task. There is strong evidence for bisymmetry- for four of the ear-trained subjects, yet the non-trained show little support of the bisymmetry axiom. Three of the 60 non-ear-trained subjects did not provide bisection points for all of the intervals and the test of bisymmetry could not be made. The bisection task seems to be a difficult one. The difference between trained and non-trained may show up here because the ear—training gives the subject more practice with critical listening. The bisection task breaks down into the comparison of two intervals. This task may be more familiar to those subjects with ear- training and therefore, they do better. Because of the results of the magnitude estimation study one should not conclude that non-trained subjects hear intervals differ— ently. If the differences are a factor of the practice with intervals then the non-trained subjects should show a decrease in response variability with increased experience. It would be interesting to continue running the non-trained subjects in this experiment to see if bisection values could actually be determined after the subjects were given enough listening. Existence of asymmetries Asymmetries were found by Van Vielt, as discussed in Chapter I, in judgments of distance between major chords presented in ascending and descending order. No systematic asymmetries were found in the three experiments described in this dissertation. One explanation of the asymmetries 61 found in the chord study involves memory for the tones. From the data presented here the pitch drift model (see Chapter I) for single tones is not supported. Other changes besides loss of frequency are possible. For example, when presenting a chord the frequency components could decay at different rates. If the higher frequency components decay faster than the lower components, the higher components of the first chord would be degraded by the time the second chord was presented. In Figure 12, the dotted circles represent those notes which have decayed. If the chords were ascending, the first chord would not overlap the fre- quency range of the second. However, if the chords were descending the lowest note of the first chord would overlap the frequency range of the second and lower chord. If having notes that overlap in frequency creates similarity then ascending intervals would be more distant than de- scending intervals. Differential decay of frequencies would not influence the distance judgments made on single tones. Figure 12. An illustration of differential decay of frequency components for ascending and descending chord sequences. 62 The question arises as to why there are no asymmetries in the reaction times for the identification of ascending and descending intervals. Olson and Laar (1973) and Chase and Clark (1971) found that asymmetries disappeared when the words in the stimuli were replaced by pictures or directional arrows. These authors conclude that the processing time asymmetries occur only when the subject must mentally represent the terms to carry out the comparison. When the task is changed in these studies from words and pictures to only pictures the subject only had to compare two displays for simple pictorial matches. It could be that the decision "ascending?" or "descending?" can be dealt with without an internal representation of the stimuli. If this is the case maybe a harder task like deciding whether two sets of intervals are in the same direction would show the possible asymmetries. In this case the stimuli would have to be mentally represented before they could be compared. Pitch models These experiments point to the log model for the perception of pitch. The log scale fits the magnitude estimates more accurately than the power model. The large range of power values within a subject further raises doubts to the appropriateness of the power scale. For the range of p values estimated by the magnitude estimation experiment 63 the power model is approximately equal to the log model when applied to the bisection task. It is curious to note that there are no systematic differences across frequency range for any of the experi- ments. Stumpf (1883) argued that upper octaves are per- ceptually longer than lower octaves. Stevens and Volkmann (1940) showed that the subjective length of equal frequency ratios enlarges as the frequency increases until approxi- mately 4000 Hz. The lack of range differences is another indication that neither the power nor mel scale is appropriate for explaining this data. Two of the ear-trained subjects in the magnitude estimation experiment generated identical values for one octave and two octaves and for 6/10 octave and 1 6/10 octaves. Neither the log, mel, or power scales for pitch will account for this data, but they are compatible with Revesz's (1954) two component model. There is no chroma difference between octaves, only height. For these two subjects chroma is the dominating factor in determining distance. Nevertheless, another explanation of this data is possible. The instructions given to the subject were purposely vague. However, this vagueness could have led the subject to believe the expected response was based on the simple interval (those under one octave) and would respond the same to 6/10 octave as to 1 6/10 octaves since they are based on the same simple interval, 6/10 octave. 64 It is interesting to note that judgments for single tones result in the mel or power scale for pitch in numerous instances. The two-and three-tone sequences in these experiments seem to be represented by the log scale. Shepard (1964) using complex tones demonstrated the helical model for pitch. Much of the determining of a scale for pitch seems to be based on the method used. In fact, even Stevens and Gallanter (1957) did not support the mel scale when subjects generated pitch information using ratio scaling. The models for both the magnitude estimation and the bisection of intervals tasks were based on the idea that subjective values are determined by generating sub- jective values for the end points and then either sub- tracting or averaging the two values, depending on the task. In the case of magnitude estimation, it is assumed that the subject subtracts the two end values and gives the absolute difference as the subjective distance. In dealing with "ratio scaling" judgments, Krantz (1972) suggests that judgments made to a pair of stimuli are not based on prop- erties of the single stimulus. That is, judgments are mediated by perceived relations between pairs of stimuli. This idea is referred to as "relation theory.“ For the magnitude estimates this might change the log model for distance from 65 wu_£ = k °(1og Fu-log Fi) (16) to something like wk, = k- (rug) (42) where ru£ is some relational value for the pair of stimuli u and Z. Due to the character of the power scale formulations it seems unlikely that the notion of relations for pairs of stimuli is compatible. A direct test of the log models for pitch could be accomplished by having subjects concurrently participate in experiments of magnitude estimation for single stimuli and for pairs of stimuli as well as ratio scaling of pairs. The results from the single stimuli judgments would be used to make predictions for the other studies. CHAPTER VI CONCLUSIONS 1. The presentation order of the tones in an interval did not affect the subjective estimates of the length of an interval. 2. The frequency range did not affect the subjective estimates of the length of an interval as hypothesized by Stumpf (1883). 3. The bisymmetry axiom was empirically supported for seven of twelve tests for ear-trained subjects. Non- ear-trained subjects had much difficulty with this task as indicated by the failure of three of these subjects to provide midpoints for all intervals. There was very little hysteresis effect in the bisection results. 4. For two- and three-tone sequences the musical or log scale seems most appropriate. Two of the ear-trained subjects provided data suggestive of the helical model for pitch. 5. There were no differences between the reaction times when deciding whether an interval was ascending or descending. 6. Symmetry of judgments for auditory stimuli was apparent throughout these experiments. 66 APPENDIX A LIST OF THE STIMULI FOR EXPERIMENTS I AND III APPENDIX A LIST OF THE STIMULI FOR EXPERIMENTS I AND III Length 2/10 Octave 1 2 3 4 200-230 460-529 980-1126 1310-1505 Length 9/10 Octave l 2 3 4 170-317 390-726 825-1536 1000-1862 Length 16/10 Octave wa 220-667 Missing 680-2063 758-2300 Length 6/10 Octave 1 2 3 4 180-273 350-531 610-925 1075-1630 Length 10110 Octave buts) Length book) 67 145-290 333-666 785-1570 850-1700 20/10 Octave 125-500 325-1300 450-1800 575-2300 APPENDIX B LIST OF THE 10 MIDDLE TONES FOR EACH INTERVAL FOR THE BISECTION EXPERIMENT (II) APPENDIX B LIST OF THE 10 MIDDLE TONES FOR EACH INTERVAL FOR THE BISECTION EXPERIMENT (II) Interval l Interval 2 Interval 3 Interval 4 68 400-500 1000-1600 400-1000 500-16000 l/3 Octaves 2/3 Octaves 1 1/3 Octaves 1 2/3 Octaves 1-413 1-1170 1-533 1-754 2-420 2-1191 2-553 2-781 3-428 3-1212 3-573 3-810 4-436 4-1234 4-593 4-849 5-444 5-1256 5-615 5-869 6-451 6-1279 6-637 6-900 7-460 7-1302 7-660 7-933 8-468 8-1325 8-683 8-966 9-476 9-1349 9-708 9-1001 10-485 10-1373 10-733 10-1037 APPENDIX C ANALYSIS FOR VARIANCE TABLES FOR EACH SUBJECT FOR EXPERIMENT I APPENDIX C ANALYSIS FOR VARIANCE TABLES FOR EACH SUBJECT FOR EXPERIMENT I Three-way analysis of variance of magnitude estimation for subject #1 E 2 Source df ms F P n Order 1 21.00 1.00 .3168 Length 5 5671.41 270.95 <.0001 .5683 Frequency Range 3 492.88 23.55 <.0001 .0296 Order x Length 5 77.66 3.71 .0025 .0078 Order x Range 3 87.74 4.19 .0059 .0053 Length x Range 14 42.75 2.04 .0129 .0120 Order x Length x Range 14 35.34 1.69 .0529 Error 874 20.93 Total 919 Three-way analysis of variance of magnitude estimation for subject #2 _ :— 2 Source df ms F P n Order 1 46.58 5.97 .0143; .0023 Length 5 2415.27 309.61 <.0001 .6058 Frequency Range 3 47.87 6.14 .0004 .0072 Order x Length 5 6.45 0.83 .5311 Order x Range 3 68.55 8.79 <.0001 .0103 Length x Range 14 33.92 4.35 <.0001 .0238 Order x Length x Range 14 9.68 1.24 .2399 Error 874 7.80 Total 919 70 Three-way analysis of variance of magnitude estimation for subject #3 2 Source df ms F P n Order 1 32.16 9.35 .0023 .0023 Length 5 1904.97 553.80 <.0001 .6681 Frequency Range 3 341.95 99.41 <.0001 .0720 Order x Length 5 3.09 0.90 .4815 Order x Range 3 38.62 11.23 <.0001 .0081 Length x Range 14 31.73 9.23 <.0001 .0312 Order x Length x Range 14 6.52 1.90 .0235 .0064 Error 874 3.44 Total 919 Three-way analysis of variance of magnitude estimation for subject #4 2 Source df ms F P n Order 1 473.48 37.50 <.0001 .0161 Length 5 2807.56 222.36 <.0001 .4782 Frequency Range 3 344.49 27.28 <.0001 .0352 Order x Length 5 63.10 5.00 .0002 .0011 Order x Range 3 228.46 18.09 <.0001 .0233 Length x Range 14 86.02 6.81 <.0001 .0410 Order x Length x Range 14 40.60 3.22 <.0001 .0193 Error 874 12.63 Total 919 Three-way analysis of variance of magnitude estimation for subject #5 Source df ms F P n Order 1 200.98 99.97 <.0001 .0444 Length 5 376.12 187.09 <.0001 .4158 Frequency Range 3 141.23 70.25 <.0001 .0937 Order x Length 5 13.89 6.91 <.0001 .0154 Order x Range 3 25.18 12.53 <.0001 .0167 Length x Range 14 5.32 2.65 .0009 .0165 Order x Length x Range 14 2.91 1.45 .1254 Error 874 2.01 Total 919 Three-way analysis of variance of magnitude estimation for subject #6 Source df ms F P n2 Order 1 6.44 1.03 .3106 Length 5 3304.41 527.92 <.0001 .6818 Frequency Range 3 274.84 43.91 <.0001 .0340 Order x Length 5 16.91 2.70 .0197 .0034- Order x Range 3 10.22 1.63 .1801 Length x Range 14 84.26 13.46 ' <.0001 .0487 Order x Length x Range 14 8.10 1.29 .2048 Error 874 6.26 Total 919 72 Three-way analysis of variance of magnitude estimation for subject #7 Source df ms F P n Order 1 0.16 0.28 .8667 Length 5 2875.32 517.99 <.0001 .7163 Frequency Range 3 51.61 9.30 <.0001 .0077 Order x Length 5 4.57 0.82 .5335 Order x Range 3 6.10 1.10 .3484 Length x Range 14 43.84 7.90 <.0001 .0306 Order x Length x Range 14 2.43 0.44 .9628 Error 874 5.55 Total 919 Three-way analysis of variance of magnitude estimation for subject #8 2 Source df ms F P n Order 1 10.87 3.53 .0607 Length 5 1357.76 440.83 <.0001 .6409 Frequency Range 3 58.38 18.95 <.0001 .0165 Order x Length 5 9.39 3.05 .0098 .0044 Order x Range 3 30.68 9.96 <.0001 .0087 Length x Range 14 45.37 14.73 <.0001 .0600 Order x Length x Range 14 10.80 3.51 <.0001 .0143 Error 874 3.08 Total 919 73 Three-way analysis of variance of magnitude estimation for subject #9 I Source df ms F P n2 Order 1 42.61 47.29 <.0001 .0141 Length 5 406.77 451.39 <.0001 .6732 Frequency Range 3 9.12 10.12 <.0001 .0091 Order x Length 5 2.58 2.86 .0143 .0043 Order x Range 3 8.39 9.31 <.0001 .0083 Length x Range 14 4.30 4.77 <.0001 .0199 Order x Length x Range 14 2.27 2.51 .0017 .0105 Error 874 0.90 Total 919 Three-way analysis of variance of magnitude estimation for subject #10 Source df ms F P n2 Order 1 0.11 0.13 .7224 Length 5 677.43 787.12 <.0001 .8015 Frequency Range 3 9.36 10.88 <.0001 .0066 Order x Length 5 0.74 0.86 .5058 Order x Range 3 1.50 1.74 .1573 Length x Range 14 2.53 2.94 .0003 .0084 Order x Length x Range 14 1.07 1.24 .2405 Error 874 0.86 Total 919 APPENDIX D ANALYSIS OF VARIANCE TABLES FOR EACH SUBJECT FOR EXPERIMENT III APPENDIX D ANALYSIS OF VARIANCE TABLES FOR EACH SUBJECT FOR EXPERIMENT III Three-way analysis of variance of reaction time for subject #1 1 #- - - Source df ms F P n Order 1 2019047.61 12.80 .0004 .0131 Length 5 1724103.76 10.93 <.0001 .0561 Frequency Range 3 154619.52' 0.98 .4815 Order x Length 5 196623.26 1.25 .2855 Order x Range 3 362178.31 2.30 .0764 Length X Range 14 116627.18 0.74 .7356 Order x Length x Range 14 61566.37 0.39 .9779 Error 874 157768.07 Total 919 Three-way analysis of variance of reaction time for subject #2 —W 2 Source df ms F P n Order 1 1014260.01 19.32 <.0001 .0193 Length 5 235966.02 4.49 .0005 .0224 Frequency Range 3 238961.03 4.55 .0036 .0136 Order x Length -5 99297.00 1.89 .0934 Order x Range 3 433416.50 8.26 <.0001 .0247 Length x Range 14 79439.19 1.51 .0998 Order x Length x Range 14 63623.65 1.21 .2605 Error 874 52497.28 Total 919 74 75 Three—way analysis of variance of reaction time for subject #3 Source df ms F P n Order 1 1018048.7l 25.27 <.0001 .0256 Length 5 109819.20 2.73 .0188 .0138 Frequency Range 3 302196.62 7.50 <.0001 .0228 Order x Length 5 128782.30 3.20 .0073 .0162 Order x Range 3 257372.49 6.39 .0003 .0194 Length x Range 14 31252.95 0.78 .6965 Order x Length x Range 14 16627.73 0.41 .9714 Error 874 40289.92 Totalv 919 Three-way analysis of variance of reaction time for subject #4 w 2 Source df ms F P n Order 1 102103.11 0.73 .3917 .0008 Length 5 1697878.34 12.21 <.0001 .0629 Frequency Range 3 140901.72 1.01 .3860 Order x Length 5 61887.31 0.45 .8170 Order x Range 3 452608.11 3.26 .0212 .0101 Length x Range 14 164529.91 1.18 .2823 Order x Length x Range 14 189857.25 1.37 .1634 Error 874 139019.11 Total 919 76 Three-way analysis of variance of reaction time for subject #5 m 2 Source df ms F P n Order 1 613.04 0.02 .8900 Length 5 88626.13 2.77 .0172 .0148 Frequency Range 3 35444.24 1.11 .3451 Order x Length 5 44952.19 1.40 .2201 Order x Range 3 194567.99 6.08 .0005 .0194 Length x Range 14 30725.78 0.96 .4928 Order x Length x Range 14 20162.59 0.63 .8414 Error 874 31997.30 Total 919 Three-way analysis of variance of reaction time for subject #6 m 2 Source df ms F P n Order 1 70367.52 6.65 .0101 .0062 Length 5 181733.29 17.18 <.0001 .0802 Frequency Range 3 60733.94 5.74 .0007 .0161 Order x Length 5 25292.79 2.39 .0363 .0112 Order x Range 3 116027.43 10.97 <.0001 .0307 Length x Range 14 7038.49 0.67 .8093 Order x Length x Range 14 24996.22 2.36 .0032 .0309 Error 874 10576.38 Total 919 77 Three-way analysis of variance of reaction time for subject #7 =5 2 Source df ms F P n Order 1 2661701.33 74.13 <.0001 .0699 Length 5 150924.80 4.20 .0009 .0198 Frequency Range 3 181676.39 5.06 .0018 .0143 Order x Length 5 166636.24 4.64 .0004 .0219 Order x Range 3 277454.92 7.73 <.0001 .0219 Length x Range 14 28560.37 0.80 .6750 Order x Length x Range 14 46580.22 1.30 .2024 Error 874 35906.48 Total 919 Three-way analysis of variance of reaction time for subject #8 Source df ms F P n Order 1 2800.03 0.05 .8319 Length 5 81663.25 1.32 .2551 Frequency Range 3 54772.40 0.88 .4498 Order x Length 5 53930.75 0.87 .5015 Order x Range 3 22483.51 0.36 .7804 Length x Range 14 66740.52 1.08 .3764 Order x Length x Range 14 89983.40 1.45 .1240 Error 874 62071.79 Total 919 Three-way analysis of variance of reaction time for subject 78 Source df ms F P n Order 1 173250.68 3.22 .0730 Length 5 10393.72 0.19 .9651 Frequency Range 3 149621.53 2.78 .0399 .0089 Order x Length 5 123333.91 2.30 .0437 .0123 Order x Range 3 36438.78 0.68 .5656 Length x Range 14 73059.52 1.36 .1666 Order x Length x Range 14 64762.18 1.21 .2657 Error 874 53739.92 Total 919 Three-way analysis of variance of reaction time for subject #10 Source df ms F P n Order 1 385073.57 3.12 .0775 Length 5 204156.32 1.66 .1426 Frequency Range 3 116481.81 0.95 .4182 Order x Length 5 86255.38 0.70 .6236 Order x Range 3 186627.63 1.51 .2093 Length x Range 14 93745.43 0.76 .7127 Order x Length x Range 14 87091.08 0.71 .7691 Error 874 123230.22 Total 919 REFERENCES REFERENCES Carroll, J. 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