Shepard, 1982

Psychological Review

VOLUME 89 NUMBER 4 JULY 1 9 8 2

Geometrical Approximations to the Structure of Musical Pitch

Roger N. Shepard Stanford University

' Rectilinear scales of pitch can account for the similarity of tones close together in frequency but not for the heightened relations at special intervals, such as the octave or perfect fifth, that arise when the tones are interpreted musically. In- creasingly adequate a c c o u n t s of musical pitch are provided by increasingly generalized, geometrically regular helical structures: a simple helix, a double helix, and a double helix wound around a torus in four dimensions or around a higher order helical cylinder in five dimensions. A two-dimensional "melodic map" o f these double-helical structures provides for optimally compact representations of musical scales and melodies. A two-dimensional "harmonic map," obtained by an affine transformation of the melodic map, provides for optimally compact representations of chords and harmonic relations; moreover, it is isomorphic to the toroidal structure that Krumhansl and Kessler (1982) show to represent the • psychological relations among musical keys.

A piece of music, just as any other acous- the musical experience. Because the ear is tic stimulus, can be physically described in responsive to frequencies up to 20 kHz or terms of two time-varying pressure waves, more, at a sampling rate of two pressure one incident at each ear. This level of anal- values per cycle per ear, the physical spec- ysis has, however, little correspondence to ification of a half-hour symphony requires well in excess of a hundred million numbers. I first described the double helical representation of Clearly, our response to the music is based pitch and its toroidal extensions i n 1978 (Shepard, Note on a much smaller set of psychological at- 1; also see Shepard, 1978b, p. 183, 1981a, 1981c, p. tributes abstracted from this physical stim- 320). The present, more complete report, originally ulus. In this respect the perception of music drafted before I went on sabbatical leave in 1979, has been slightly revised to take account of a related, elegant is like the perception of other stimuli such development by my colleagues Krumhansl and Kessler as colors or speech sounds, where the vast (1982). number of physical values needed to specify The work reported here was supported by National the complete power spectrum of a stimulus Science Foundation Grant BNS-75-02806. It owes much to the innovative researches of Gerald Balzano is reduced to a small number of psycholog- and Carol Krumhansl, with whom I have been f o r t u n a t e ical values, corresponding, say, to locations enough to share the excitement of various attempts to on red-green, blue-yellow, and black-white bridge the gap between cognitive psychology and the dimensions for homogeneous colors (Hurv- perception of music. Less directly, the work has been ich & Jameson, 1957) or on high-low and influenced by the writings of Fred Attneave and Jay 'Dowling. Finally, I am indebted to Shelley Hurwitz for front-back dimensions for steady-state vow- assistance in the collection and analysis of the data and els (Peterson & Barney, 1952; Shepard, to Michael Kubovy for his many helpful suggestions on 1972). But what, exactly, are the basic per- the manuscript. ceptual attributes of music? Requests for reprints should be sent to Roger N. Shepard, Department of Psychology, Jordan Hall, Just as continuous signals of speech are Building 420, Stanford University, Stanford, California perceptually mapped into discrete internal 94305. representations of phonemes or syllables,

305 306 ROGER N. SHEPARD continuous signals of music are perceptually pitch (Balzano, 1980; Krumhansl & Shep- mapped into discrete internal representa- ard, 1979; Shepard, 1982) and time (Jones, tions of tones and chords; just as each speech 1976; Pressing, in press), the dimensions sound can be characterized by a small num- within which higher order musical units such ber of distinctive features, each musical tone as melodies and chords are capable of struc- can be characterized by a small number of ture-preserving transformations. perceptual dimensions of pitch, loudness, In this paper I confine myself to the case timbre, vibrato, tremolo, attack, decay, du- of pitch and to the question of how a single ration, and spatial location. In addition, psychological attribute corresponding, i n the much as the internal representations of case of pure sinusoidal tones, to a simple one- speech sounds are organized into higher level dimensional physical continuum of fre- internal representations of meaningful words, quency affords the structural richness req- phrases, and sentences, the internal repre- uisite for tonal music. One can perhaps sentations of musical tones and chords are readily conceive how structural complexity organized into higher level internal repre- is achievable in the dimension of time, sentations of meaningful melodies, progres- through overlapping patterns of rhythm and sions, and cadences. stress, but the structural properties of pitch seem to be manifested even in purely melodic The Fundamental Roles of Pitch and sequences of purely sinusoidal tones (Krum- Time in Music hansl & Shepard, 1979). In the absence of physical overlap of upper harmonics of the The perceptually salient attributes of the sort considered by Helmholtz (1862/1954) tones making up the musically significant and by Plomp & Levelt (1965), wherein does chords and melodies are not equally impor- this structure reside? tant for the determination of those higher order units. In fact, for the music of all hu- Cognitive Versus Psychoacoustic man cultures, it is the relations specifically Approaches to Pitch of pitch and time that appear to be crucial Until recently, attempts to bring scientific for the recognition of a familiar piece of methods to bear on the perception of musical music. Other attributes, for example, loud- stimuli have mostly adopted a psychoacous- ness, timbre, vibrato, attack, decay, and ap- tic approach. The goal has been to determine parent spatial location, although contribut- the dependence of psychological attributes, ing to audibility, comfort, and aesthetic such as pitch, loudness, and perceived du- quality, can be varied widely without dis- ration, on physical variables of frequency, rupting recognition or even musical appre- amplitude, and physical duration (Stevens, ciation. 1955; Stevens & Volkmann, 1940) or on The reasons for the primacy of pitch and more complex combinations of physical vari- time in music are both musical and extra- ables (de Boer, 1976; J. Goldstein, 1973; musical. From the extramusical standpoint Plomp, 1976; Terhardt, 1974; Wightman, there are compelling arguments, recently 1973). advanced by Kubovy (1981), that pitch and By contrast, the cognitive psychological time alone are the attributes that are "in- approach looks for structural relations within dispensable" for the perceptual segregation a set of perceived pitches independently of of the auditory ensemble into discrete tones. the correspondence that these structural re- From the musical standpoint a case can be lations may bear to physical variables. This made that the richness and power of music approach is particularly appropriate when depends on the listener's interpretation of such structural relations reside not in the the tones in terms of a discrete structure that stimulus but in the perceiver—a circum- is endowed with particular group-theoretic stance that is well known to students of mu- properties (Balzano, 1980, in press, Note 2). sic theory, who recognize that an interval In the case of the human auditory system, defined by a given physical difference in log moreover, the requisite properties appear to frequency may be heard very differently in be fully available only in the dimensions of different musical contexts. To elaborate on STRUCTURE OF PITCH 307 an example mentioned by Risset (1978, p. cally significant relations of pitch to emerge 526), in a C-major context the interval B- as invariant in these scales seems to be a F is heard as having a strong tendency to direct consequence of the assiduous avoid- resolve by contraction into the smaller in- ance, by the psychoacoustic investigators, of terval C-E, whereas in an F#-major context any musical context or tonal framework the physically identical interval (now called within which the listeners might interpret B-E#, however) is heard as having a simi- the stimuli musically. Attneave and Olson larly strong tendency to resolve by expansion (1971) showed that when familiar melodies, into the larger interval A^-F*. Moreover, as opposed to arbitrarily selected nonmusical Krumhansl (1979) has now provided system- tones, were to be transposed in pitch, listen- atic, quantitative evidence that the perceived ers required that the separations between the relations between the various tones within tones be preserved on the musically relevant an octave do indeed depend on the context- scale of log frequency—not on a nonlinearly induced musical key with respect to which related scale such as the mel scale. That the those tones are interpreted. How, then, are scale underlying j u d g m e n t s of musical pitch we to represent the relations of pitch be- must be logarithmic with frequency is in fact tween tones as those relations are perceived now supported by several kinds of empirical by a listener who is interpreting the tones evidence (Dowling, 1978b; Null, 1974; Ward, musically? 1954, 1970). Second, because of the unidimensionality Previous Representations of Pitch of scales of pitch such as the mel scale, perceived similarity must decrease monotoni- Rectilinear Representations cally with increasing separation between tones on the scale. There is, therefore, no The simplest representations that have provision for the possibility that tones sep- been proposed for pitch have been unidi- arated by a particularly significant musical mensional scales based on judgments made interval, such as the octave, may be per- in nonmusical contexts. Examples are the ceived as having m o re in common than tones "mel" scale that Stevens, Volkmann, and separated by a somewhat smaller but mu- Newman (1937) and Stevens and Volkmann sically less significant interval, such as the (1940) based on a method of fractionation major seventh. Indeed, even the musically or the similar scale that Beck and Shav/ more relevant log-frequency scale, also being (1961) later based on a method of magnitude unidimensional, is subject to this same lim- estimation. Pitches are represented in such itation. Yet, in the case of the octave, which scales by locations along a one-dimensional corresponds to an approximately two-to-one line. In these scales, moreover, the location ratio of frequencies, the phenomenon of aug- of each pure tone is related to the logarithm mented perceptual similarity at that partic- of its frequency in a nonlinear manner dic- ular interval (a) has long been anticipated tated by the psychoacoustic fact that pains (Boring, 1942, pp. 376, 380; Licklider, 1951, of low-frequency tones are less discriminabk pp. 1003-1004; Ruckmick, 1929), (b) was than are pairs of high-frequency tones sep- in fact empirically observed at about the arated equally in log frequency. Such scales time that the unidimensional mel scale was thus deviate from musical scales (or from being perfected (Blackwell & Schlosberg, the spacing of keys on a piano keyboard) in 1943; Humphreys, 1939), (c) has since been which position is essentially logarithmic with much more securely established (Allen, 1967; frequency. From a musical standpoint such Bachem, 1954; Balzano, 1977; Dowling, scales are in fact anomalous in two respects. 1978a; Dowling & Hollombe, 1977; Idson First, because of the resulting nonlinear & Massaro, 1978; Kallman & Massaro, relation between pitch and log frequency, the 1979; Krumhansl & Shepard, 1979; Thur- distance between tones separated by the low & Erchul, 1977), and (d) probably un- same musical interval, such as an octave or derlies the remarkable precision and cross- a fifth, is not invariant under transposition cultural consistency with which listeners are up or down the scale. The failure of musi- able to adjust a variable tone so t h a t it stands 308 ROGER N. SHEPARD

b between the tones C and C, an octave apart, in Figure 1.) Moreover, this is true whether the curve is embedded in a cylinder, as proposed by Drobisch, a flat plane, as suggested by Donkin, or a bell-shaped surface of revolution, as advocated by Ruckmick (1929). Despite their differences, the representations proposed by these authors were alike in having adjacent turns more closely spaced toward the low-frequency end, where given differences in log frequency are less discriminable. (See the figures reproduced in Pikler, 1966; Ruckmick, 1929.) These representations were analogous, in this respect, to the unidimensional psychophysical representations of Stevens and his colleagues (Stevens et al, 1937; Stevens & Volkman, 1940). In 1954, before learning of these early proposals, I had attempted to accommodate a heightened similarity at the octave by means of a tonal helix (Note 3) that differed, however, from the ones proposed by Drob- Figure 1. A simple regular helix to account for the in- isch, Donkin, and Ruckmick in being geo- creased similarity between tones separated by an octave. metrically uniform or regular. (See Figure (From "Approximation to Uniform Gradients of Gen- eralization by Monotone Transformations of Scale" by 1, which, except for relabeling, is reproduced Roger N. Shepard. In D. I. Mostofsky (Ed.), Stimulus from Shepard, Note 3—as it later appeared Generalization, Stanford, Calif.: Stanford University in Shepard, 1965). Because it is geometri- Press, 1965, p. 105. Copyright 1965 by Stanford Uni- cally regular, this is the helical analog of the versity Press. Reprinted by permission.) unidimensional scale having the musically more relevant logarithmic structure advo- in an octave relation to a given fixed tone cated by Attneave and Olson (1971); in it (Burns, 1974; Dowling, 1978b; Sundberg the distance corresponding to any particular & Lindqvist, 1973; and, originally, Ward, musical interval is invariant throughout the 1954).1 representation. The uniform helix also possesses an additional advantage over curves Simple Helical Representations embedded in variously shaped surfaces of revolution, such as Ruckmick's (1929) "tonal We can obtain geometrical representa- bell." Only when the helix is regular, and tions that are consistent with an increased hence embedable in a cylindrical surface, similarity at the octave by deforming a rec- will tones standing in the octave relation, in tilinear representation of pitch into a higher addition to coming into closer proximity with dimensional embedding space to form a he- each other, fall on a common straight line lix, as proposed for this purpose by Drobisch parallel to the axis of the helix. Such lines as early as 1846, or a spiral, as proposed by Donkin in 1874 (seePikler, 1966; Ruckmick, 1 Incidentally, t h e s e studies agree in showing that the 1929; and for a recently proposed planar ratio of physical frequencies of pure tones that subjects spiral, Hahn & Jones, 1981). For, unlike a set in the octave relationship is about 2.02:1 and not straight line, a helix or spiral that completes exactly 2:1. This small "stretched-octave" effect (Bal- one turn per octave achieves the desired in- zanol 1977; Burns, 1974; Dowling, 1973a; Sundberg & Lindqvist, 1973; Ward, 1954) still leads to an ap- crease in spatial proximity between points proximately logarithmic relation between pitch and fre- an octave apart—at least if the slope of the quency and does not vitiate any of the conclusions to curve is not too steep. (Compare Paths a and be drawn here. STRUCTURE Of PITCH 309

can be thought of as projecting all tones with plitudes of the component frequencies de- the same musical name but differing by oc- termined by a fixed bell-shaped spectral en- taves (e.g., the tones C, C, C", etc.) down velope that was at its maximum near the into a single corresponding point in a middle of the standard musical range and "chroma circle" on a plane orthogonal to the that gradually fell away in both directions axis of the helix (see Figure 1). Moreover, to below-threshold levels for very low and this projective property, unlike the property very high frequencies. The different sounds of augmented proximity, holds regardless of generated in this way remain fully distinct the slope of the helix. in chroma but are all equivalent in height. Thus, in shifting through chroma, from C # Physical Realization of the Chroma Circle to C* to D to D and so on to B, the next step (though still heard as a step up in pitch), It is, in fact, the projective property of the instead of leaving one an octave higher at regular helix that subsequently led me to a C, leaves one back at the original starting method of physically separating the two un- tone C (Shepard, 1964b).2 Indeed, applica- derlying components of pitch implicit in the tion of multidimensional scaling to measures helical representation, namely, the rectilin- of similarity derived from judgments of rel- ear component called pitch height, corre- ative pitch between tones varied in this man- sponding to the axis of the helix (or of the ner (Shepard, 1964b) yielded the almost cylinder in which it lies), and the circular perfectly circular solution displayed in Fig- component variously called tone quality ure 2 (a) (see Shepard, 1978a). (A similarly (Revesz, 1954) or tone chroma (Bachem, circular representation for tones generated 1950, 1954), corresponding to the circum- in this way has also been reported by Char- ference of that cylinder (see Shepard, 1964b). bonneau and Risset, 1973.) What was required was the specification of Although this circular component, chroma, two physical operations corresponding to the emerges most compellingly when the recti- geometrical projection of the entire helix linear component, height, is artificially sup- onto the central axis, in the one case, and pressed, as with these special, computer-gen- onto an orthogonal plane, in the other. The erated tones, the claim is that this circular auditory realization of the required opera- component is necessarily present in all mu- tions was greatly facilitated by the devel- sical tones for which tones separated by an opment, at just this time, of computer tech- octave are perceived as more closely related niques for the additive synthesis of arbitrarily than tones separated by a somewhat smaller specified tones (Mathews, 1963). interval. Circular multidimensional scaling For the first operation I proposed a broad- solutions have in fact been obtained for or- ening of the band of energy around the cen- dinary musical tones. Figure 2 (b), for ex- ter frequency of each tone until the resulting ample, reproduces a similarly circular pat- narrow-band noise encompassed about an tern subsequently o b t a i n e d by Balzano (1977, octave. Because the different sounds gener- 1982) from a multidimensional scaling anal- ated in this way have different center fre- ysis of his own discriminative reaction time quencies, they still differ over the whole data for melodic intervals.3 range of pitch height. But, because they have all been spread alike around the chroma cir- 2 The illusion of circular or "endlessly ascending" cle, they can no longer be discriminated with tones can be beard on a commercial record ("Shepard's respect to chroma. This operation thus cor- Tones," 1970) or on a short 16-mm film (Shepard & Zajac, 1965), which we believe to be the first film in responds to collapsing the helix onto its cen- which both the sound and the animation were generated tral axis. by computer. I demonstrated the independent variation For the second operation I proposed, in- of linear pitch height and circular tone chroma at the stead, a harmonic elaboration of each tone 1978 meeting of the Western Psychological Association until it included, alike, all multiples and sub- (Shepard, Note 1). • 3 One should, however, exercise caution in basing the multiples of the original frequency (i.e., all inference of circularity solely on a multidimensional tones standing in octave and multiple-octave scaling solution. The curvature evident in some obtained relations to that original tone), with the am- solutions (e.g., the one reported by Levelt, Van de Geer, 310 ROGER N. SHEPARD a

Mai 7th

M»j 6th (8

Figure 2. Chroma circles recovered by multidimensional scaling (a) for 10 computer-generated tones especially designed to eliminate differences in pitch height and (b) for twelve ordinary musical tones. (Panel a is from "The Circumplex and Related Topological Manifolds in the Study of Perception" by R. N. Shepard. In S. Shye (Ed.), Theory Construction and Data Analysis in the Behavioral Sciences. San Francisco, Calif.: Jossey-Bass, 1978. Copyright 1978 by Jossey-Bass, Inc. Reprinted b y permission. Panel b is from Chronometric Studies of the Musical Interval Sense by G. J. Balzano. Unpublished doctoral dissertation, S t a n f o r d University, 1977. Reprinted by p e r m i s s i o n . )

By now the possibility of analyzing per- between points in the representational struc- ceived pitch into the rectilinear and circular ture, quite apart from whether that structure components of height and chroma has been is basically rectilinear or helical in overall accepted by a number of researchers (e.g., shape, should be adjusted to reflect how the Bachem, 1950,1954; Balzano, 1977; Deutsch, operating characteristics of the sensory 1972, 1973; Jones, 1976; Kallman & Mas- transducers shift as we move from low to saro, 1979; Pikler, 1955; Revesz, 1954; Ris- high input frequencies. Someone preoccu- set, 1978). Even among advocates of a helical pied with such sensory considerations might representation, though, opinions may still dif- even see some significance in the resem- fer concerning the relative merits of a geo- blance of a distorted spiral or helix to the metrically regular structure such as I pro- anatomical conformation of the cochlea. posed versus a more or less distorted variant By contrast, a more cognitively and mu- such as Ruckmick (1929) advocated. Here, sically oriented approach to pitch is likely again, one's predilection may depend on to regard such considerations of automa- whether one takes a more psychoacoustic or tic peripheral transduction (and anatomy) a more cognitive and musical point of view. as largely irrelevant. Adopting something like Chomsky's (1965) competence-perfor- Argument for a Geometrically Regular mance distinction, I suggest that if it is Structure musical pitch that interests us, the repre- From the psychoacoustic standpoint it sentation should reflect the deeper structure seems natural to suggest that the spacing that underlies a listener's competence to im- pose a musical interpretation on a stream of acoustic inputs under favorable conditions. & Plomp, 1966) may simply reflect an artifact that almost always arises when basically one-dimensional data Such an interpretive structure continues to are fit in a higher dimensional space (Shepard, 1974, exist whether or not the acoustic stimuli in pp. 386-388). a particular stream fall within the range of STRUCTURE OF PITCH 311 frequencies, amplitudes, or durations that best thought of not as a stimulus but as a can be adequately transduced by a particular "medium" in which auditory patterns (chords ear or whether or not a preceding context or tunes) can move about while retaining has been provided that is sufficient to acti- their structural identity, just as visual space vate and to orient or tune the internal struc- is a medium in which luminous patterns can ture required for a musical interpretation. move about while retaining their structural From this cognitive standpoint the already identity. We are not very good at recognizing cited evidence for invariance under trans- the position of an isolated point of light in position (Attneave & Olson, 1971; Dowling, otherwise empty space, but we can say with 1978b) would require not only t h a t the struc- great precision whether one such point is ture be helical (rather than spiral, say) but above or below another or whether three also that the helix be geometrically regular points form a straight line or a right triangle. and, so, be carried into itself by a rigid Likewise, many of us are poor at identifying screwlike motion under the musical trans- the pitch of an isolated tone but quite good position that maps any tone into any other. at saying whether one tone is higher or lower Only then will the geometrical structure pre- than another or whether three tones stand serve the property, essential to music, that in octave relations or form a major triad. In all pairs of tones separated by a given in- either the visual or the auditory case, any terval, such as an octave or a fifth, have the such pattern moved to a different location same musical relation regardless of their or pitch continues to be recognized as the overall pitch height. Indeed, from this stand- same pattern. point the reason that this structure must take I go beyond the formulations o f Attneave a helical form (just as much as the reason and Olson (1971) and of Dowling (1978b), that the DNA molecule must take a helical however, in saying that in the case of pitch, form) is a consequence of the fundamental such a motion must .be helical. For example, geometrical fact that the most general rigid suppose we alternate one well-defined pat- motion of space into itself is a combination tern, say a major triad, with exactly the same of a rotation and a translation along the axis pattern but each time with the second pat- of the rotation, that is, a screw displacement tern displaced farther away from the first in (Coxeter, 1961, pp. 101, 321; H. Goldstein, pitch height. As the two patterns are moved 1950, p. 124; Greenwood, 1965, p. 318; apart from their initially complete coinci- Hilbert & Cohn-Vossen, 1932/1952, pp. dence, each will retain its own structural 82, 285). identity. But the perceived degree of relation One should not suppose, here, that the between the two patterns will first decrease octave-related tones that project onto the and then increase again as all three com- same point on the chroma circle share some ponents of one come into the octave relation absolutely identifiable quality of chroma any with corresponding components of the other. more than the projection of any tone onto (The perceived relation may also increase the central axis of the helix corresponds to somewhat at certain intermediate positions an absolutely identifiable quality of pitch as the two triads pass through other special height. Only those rare individuals possess- tonal relations, just as the visual similarity ing "absolute pitch" can recognize a chroma to the original of a rotating copy of a triangle absolutely (Bachem, 1954; Siegel, 1972; will increase at certain intermediate orien- Ward, 1963a, 1963b). For most individuals tations before coming again into complete (including most musicians), in the case of coincidence at 360°. See Shepard, 1982.) pitch just as in the case of many other per- Such a phenomenon is not explicable solely ceptual dimensions (loudness, brightness, in terms of increasing separation within a size, distance, duration, etc.), it is the rela- purely rectilinear medium; it implies a me- tions between presented values that have dium with a circular component. Motions in well-defined internal representations, not the such a medium are thus auditory analogs of values themselves (Shepard, 1978c, 1981b). the operations of mental rotation and rota- In fact, as Attneave and Olson (1971) tional apparent motion in the visual domain have so persuasively argued, pitch is perhaps (see Shepard & Cooper, 1982). 312 ROGER N. SHEPARD

Limitations of the Simple Helix modulations of key most often occur (Forte, 1979; Helmholtz, 1862/1954; Schenker, The structure of the simple regular helix 1906/1954). (f) Finally, according to Bal- pictured in Figure 1 was dictated by two zano's (1980, Note 2) group-theoretic anal- considerations: invariance under transposi- ysis, the preeminence of the fifth in tonal tion and increased similarity at the octave. music has a basis in abstract structural con- In such a regular helix, moreover, the special straints independent of the psychoacoustic octave relation is represented by unique col- facts noted under (a) and (b). linearity or projectability as well as by aug- Despite these diverse indications of the mented spatial proximity. As it stands, how- importance of the perfect fifth, the fifth has ever, the helical structure does not provide largely failed to reveal its unique status in either augmented proximity or collinearity psychoacoustic investigations for the same for tones separated by any other special reason, I believe, that the octave often re- musical interval. Yet, beginning with Eb- vealed its unique status only weakly, if at binghaus, Drobisch's proposal of a helical all. In the absence of a muscial context, representation has been criticized for its fail- tones—particularly the pure sinusoidal tones ure to account for the special status of the favored by psychoacousticians—tend to be interval of a perfect fifth (see Ruckmick, interpreted primarily with respect to the sin- 1929). gle, rectilinear dimension of pitch height. There are, indeed, a number of converging Without a musical context there is insuffi- reasons for supposing that the fifth should, cient support for the internal representation like the octave, have a unique status, (a) As of more complex components of pitch—com- has been known at least since Pythagoras, ponents that may underlie the recognition after the 2-to-l ratio in the lengths of a vi- of special musical intervals and that (like the brating string that corresponds to the octave chroma circle) are necessarily circular be- (which as we now know determines a 1-to- cause, again, the musical requirement of in- 2 ratio in the resulting frequency of vibra- variance under transposition entails that tion), the fifth corresponds to the next sim- each such component repeat cyclically plest, 3-to-2, ratio (Helmholtz, 1862/1954). through successive octaves. (b) In the case of musical and, therefore, harmonically rich tones, those separated by Recent Evidence for a Hierarchy of Tonal a fifth also have, within the octave, the few- Relations est upper harmonics that deviate from coincidence by an amount expected to produce Motivated by the considerations just given, noticeable beats (Helmholtz, 1862/1954; Carol Krumhansl and I initiated a new s e r i e s Plomp & Levelt, 1965). (c) Moreover, such of experiments on the perception of musical beats can be subjectively experienced even intervals within an explicitly presented mu- when the harmonics that most contribute to sical context. In this way we were in fact them are not physically present (Mathews, able to obtain clear and consistent evidence Note 4; see also Mathews & Sims, 1981). that the perfect fifth is at the top of a whole (d) Correspondingly, simultaneously sounded hierarchy of special relations within each tones differing by a fifth—even pure, sinu- octave. In these experiments we established soidal tones—tend to be heard as particu- the necessary musical context, just prior to larly smooth, harmonious, or consonant, and presenting the to-be-judged test tone or the fifth, together with the similarly har- tones, simply by playing, for example, the monious major third, completes the uniquely sequence of tones of a major diatonic scale stable and tonally centered chord, the major (the tones named do, re, mi, fa, sol, la, and triad (Meyer, 1956; Piston, 1941; Ratner, ti and corresponding, in the key of C major, 1962; Schenker, 1906/1954, p. 252). (e) In to the white keys of the piano keyboard). addition, the interval of the fifth plays a piv- Ratings of the ensuing test tones yielded otal role in tonal music, being .the interval highly consistent orderings of the musical that separates musical keys that share the intervals across listeners having equivalent greatest number of tones and between which musical backgrounds. This was true both in STRUCTURE OF PITCH 313 the initial experiments in which listeners New Representations for Musical Pitch rated, in effect, the extent to which each individual test tone out of the 13 within one The Diatonic Scale as an Interpretive complete octave was substitutable for the Schema tonic tone (do) that would normally have In their characteristic eschewal of musical completed the major scale presented as con- context, psychoacoustically oriented inves- text (Krumhansl & Shepard, 1979; also see tigators missed the essential musical aspect Krumhansl & Kessler, 1982) and in further of pitch. By failing to elicit, within the lis- experiments in which listeners rated the sim- teners, the discrete tonal schema or "hier- ilarities between the two test tones in all archy of tonal functions" (Meyer, 1956, pp. possible pairs selected from one complete 214-215; Piston, 1941; Ratner, 1962) as- octave (Krumhansl, 1979). sociated with a particular musical key, these Only for listeners with little musical back- investigators left the listeners with no unique ground did the obtained orderings of the cognitive framework within which to inter- musical intervals agree with previous psy- pret the test tones. Even musically sophis- choacoustic results in which similarity was ticated listeners therefore had little choice determined primarily by proximity in pitch but to make their judgments on the basis of height between the two tones making up the the simplest attribute of tones differing in interval, that is, in which the ranking of the frequency—pitch height. intervals with respect to similarity or mutual Cognitively oriented researchers are now substitutability of their two component tones recognizing that interpretive schemata play was, from greatest to least, unison, minor an essential role in the perception of musical second, major second, minor third, major pitch, just as they do in perception generally. third, and so on. For the more musically In the case of pitch, the primary schema oriented listeners, the results tended, instead, seems to be the musical scale—usually, in toward the entirely different ranking: unison the case of Western listeners, the familiar and octave (nearly equivalent to each other), major diatonic scale (do, re, mi, etc.). As followed by the fifth and sometimes the ma- noted by Dowling (1978b, in press), even jor third, followed by the other tones of the though the most commonly used musical diatonic scale, followed by the remaining, scales differ somewhat from culture to cul- nondiatonic tones (those corresponding in ture, they all share certain basic properties. the key of C major to the sharps and flats Regardless of the total number of tones per- or black keys of the piano). mitted by each scale, most are organized In short, data collected from listeners who around five to seven "focal pitches" per oc- invest the test tones with a musical inter- tave. Moreover, the steps between such pretation consistently reveal a whole hier- pitches rather than being constant in log fre- archy of tonal relations that cannot be ac- quency are almost always arranged accord- commodated within previously proposed ing to a particular asymmetric pattern that geometrical representations of pitch whether repeats exactly within every octave. rectilinear, helical, or spiral. Accordingly, it The cyclic repetition of the pattern from now appears justified to present some alter- octave to octave can be explained in terms native, generalized helical structures to- of the perceived equivalence of tones differ- gether with the steps that led to their con- ing by an approximately 2-to-l ratio of fre- struction and some evidence that such quencies, which led to the proposed simple generalized structures are indeed capable of helix for pitch. The other structural univer- accommodating the musically primary tonal sals of musical scales have been attributed relations. The following is intended, there- to pervasive cognitive constraints on the fore, as the first full account of these new number of absolutely identifiable categories representations of pitch—first briefly de- per perceptual dimension (7 ± 2, as enun- scribed in 1978 (Shepard, Note 1; also see ciated by Miller, 1956; cf. Dowling, 1978b) Shepard, 198la, or, for a description con- and to the requirement that the scale have current with that presented here but follow- a structure that affords reference points or ing a different derivation, Shepard, 1982). tonal centers to which a melody can move 314 ROGER N. SHEPARD or come to rest (Balzano, 1980; Zucker- terpretation of continuously variable tones kandl, 1956, 1972). In this view, evenly with respect to this discrete internalized spaced scales, such as the whole-tone scale schema appears to be a kind of "categorical or the chromatic (half-tone or twelve-tone) perception" (Burns & Ward, 1978; Krum- scale, have not been widely used because, in hansl & Shepard, 1979; Locke & Kellar, the absence of a reference point, music lacks 1973; Siegel & Siegel, 1977a, 1977b; Za- the tension, motion, and resolution that en- torre & Halpern, 1979; Blechner, Note 5) gages the suitably tuned mind with such dy- much like that originally reported for the namic force. perception of speech sounds (Liberman, Balzano's (1980, in press, Note 2) group- Cooper, Shankweiler, & Studdert-Kennedy, theoretic analysis has carried this line of 1967).4 Indeed, during a child's development argument to a new level of formal elegance the process whereby the abstract and seem- and detail. He showed that the requirements ingly universal structure of the underlying that the organization of the scale be invari- tonal schema becomes tuned to the partic- ant under transposition and that each tone ular musical scale entrenched in that child's have a unique functional role within the culture may be like the process whereby, scale jointly constrain both the number of according to Chomsky (1968), the innate scale tones within each octave and the ap- schematism that underlies all human lan- proximate spacing between those tones. In guages becomes tuned to the particular lan- fact, he showed that among possible scales guage of that culture. presupposing a division of the octave into Laboratory studies substantiate the role less than 20 steps (a restriction I take to be of the diatonic scale for Western listeners. desirable, if not necessary, to avoid over- Cohen (1975, Note 6) showed that after loading the human cognitive system), the hearing a short excerpt from a piece of mu- diatonic scale is the only one satisfying these sic, listeners can generate the associated dia- requirements. Thus, it may not be accidental tonic scale with considerable accuracy. Fur- that this scale has become the basis of West- thermore, listeners recognize or reproduce ern music, in which the structural complex- a series of tones more accurately when the ities of harmony and counterpoint (though tones conform to a diatonic scale than when not of other aspects such as, particularly, they do not (e.g., Attneave & Olson, 1971; rhythm; see Pressing, in press) have reached Cohen, 1975; Dewar, 1974; Dewar, Cuddy, their greatest development. Nor is it a co- & Mewhort, 1977; Frances, 1958, Experi- incidence that the diatonic scale has arisen, ments 3 and 4; Krumhansl, 1979). More- with but slight variations, "independently in over, as I have already noted, Krumhansl different ages and geographical locations" (1979) established that the perception of (Pikler, 1955, p. 442) and, according to re- musical intervals depends on the key or cent archeological evidence, can be traced back over 3,000 years to the earliest deci- pherable records (Kilmer, Crocker, & B r o w n , 4 From the cognitive musical standpoint taken here, 1976). the important issues concern the structural relations In any case, as Dowling (1978b, in press) between tones as they are represented in a discrete internal structure, such as the diatonic schema. To the notes, there are many reasons to believe that extent that the perception of musical tones is categor- every culture has some such discrete tonal ical, the question of whether the physical stimuli that schema, which through early (and possibly are thus categorically associated with nodes in this struc- irreversible) tuning provides a framework ture came from a physical scale of equal temperament with respect to which listeners interpret all (in which all chromatic steps are uniform with respect to log frequency) or a scale of just intonation (in which music they hear. Music of another culture the frequency ratios of the musical intervals take the may therefore be misinterpreted by assimi- simplest integer forms), though affecting the timbral lation to the listener's own, somewhat dif- quality of simultaneously sounded harmonically rich ferent tonal schema (Frances, 1958, p. 49). tones (Helmholtz, 1962/1954) to a small extent (Ma- thews & Sims, 1981; Mathews, Note 4), is largely ir- Within a culture, children evidently inter- relevant. The same consideration leads me to disregard nalize the prevailing schema by 8 years of the distinction between a sharp of one note and the flat age (Imberty, 1969; Zenatti, 1969). The in- of the note just above (e.g., C* versus D'). STRUCTURE OF PITCH 315

tonality of the diatonic scale with respect to physically equal, the steps that should have which the listeners interpret the test tones. been half as large if the scale had been dia- Finally, in the informal experiment men- tonic (viz., the steps between Tones 3 and tioned earlier in which a major triad was 4 and between Tones 7 and 8) sounded too alternated with the same major triad dis- large. Moreover, in j u s t completed, more for- placed in pitch height, I noticed that between mal experiments, a student and I have now the unison and the octave displacement, the obtained strong quantitative confirmation greatest perceived relation was at the dis- of this phenomenon (Jordan & Shepard, placement of a perfect fourth and a perfect Note 7). fifth. But these intervals, which are adjacent Our tendency to hear the successive in- to the unison around the circle of fifths, do tervals of the diatonic scale as uniform, not have the greatest number of component which I take to underlie this auditory illu- tones in the octave or unison relation; rather, sion, probably depends on perceptual set. for these displacements alone, all tones in That tendency may be weakened in musi- either triad are in the diatonic scale deter- cians such as singers, trombonists, and string mined by the alternately presented triad. players who, u n l i k e passive listeners or those The rectilinear and the simple helical and who primarily play keyboard or other wind spiral representations of pitch bear little re- instruments, must learn to make vocal or lationship to the diatonic or related musical motor adjustments essentially proportional scales found in human societies. It is not sur- to actual differences in log frequency. (Such prising, therefore, that these previously pro- differences in set may in part account for the posed geometrical representations fail to departure from equality of successive scale provide an account of the various phenom- steps implied by the results of Frances, 1958, ena of culture-specific, context-dependent, Experiment 2, or Krumhansl, 1979). Nev- and apparently categorical perception. ertheless, our own results (Jordan & Shep- My own approach to the representation ard, Note 7) indicate that there is a tendency of pitch grew out of an informal observation to hear scale steps as more nearly e q u i v a l e n t concerning the diatonic scale: In listening to than they physically are. The work of Bal- the eight successive tones of the major scale zano (1980, 1982; Balzano & Liesch, in (do, re, mi, . . . , do), I tended to hear the press; Balzano, Note 2) has also provided successive steps as equivalent, even though support for the notion that in addition to I knew that with respect to log frequency, pitch height and tone chroma, the discrete some of the intervals (viz., the interval mi steps or "degrees" of the musical scale are to f a and the interval ti back to do an octave psychologically real. higher) are only half as large as the others. Suppose, then, that listeners interpret suc- This apparent equivalence of successive steps cessive tones of the major scale (e.g., C, D, of the diatonic scale could not be dismissed E, F, G, A, B, C, in the key of C major) by as an inability to discriminate between major assimilating each tone to a node in an in- and minor seconds. ternalized representation of the diatonic When I then used a computer to generate scale. If the steps in this internal represen- a series of eight tones that divided the octave tation are functionally equivalent (as steps), into seven equal steps in log frequency (a then the perception of uniformity would fol- series that does not correspond to any stan- low, despite the fact that the physical dif- dard musical scale), the successive steps ferences are only half as great for the steps sounded oddly nonequivalent. Apparently, E to F and B to C' as for the other steps. just as we cannot voluntarily override the visual system's tendency to interpret paral- Derivation of a Double Helix lelograms projected on the two-dimensional retina as rectangles in three-dimensional I propose to represent musical tones by space (Shepard, 1981c, p. 298), I could not points in space and, as in the case of the wholly override my auditory system's ten- simple regular helix, to represent the under- dency to interpret tones in terms of the dia- lying relations between their pitches by geo- tonic scale. Thus, after they had been made metrical relations of distance and collinear- 316 ROGER N. SHEPARD a imposed by this third, perhaps arguable, requirement.) 1. Invariance under transposition: Tones separated by a given musical interval must be separated by the same distance in the underlying structure regardless of the absolute location (height) of those two tones. 2. Octave equivalence: All the tones standing in octave relationships to any given tone, and only those tones, must fall on a unique (chroma) line passing through the given tone. 3. Uniformity of scale steps: The steps between tones that are adjacent within the major diatonic scale in any particular key should be represented as equal distances in the underlying geometrical structure. According to Requirement 1 the equivalence between half- and whole-tone steps im- Figure 3. Stages in the construction of a double helix posed within any one key by Requirement of musical pitch: ( a ) the flat strip of equilateral t r i a n g l e s , 3 must hold for all keys. Consequently, all (b) the strip of triangles given one complete twist per octave, and (c) the resulting double helix shown as major and minor seconds must be repre- wound around a cylinder. (Panel c is from "Structural sented by equal distances in the underlying Representations of Musical Pitch" by R. N. Shepard. invariant structure. Thus, the geometrical In D. Deutsch (Ed.), Psychology of Music. New York: implication of Requirements 1 and 3 to- Academic Press, 1982. Copyright 1982 by Academic gether is t h a t the points corresponding to any Press, Inc. Reprinted by permission.) three successive tones of the chromatic scale ity between those points. Thus, I propose must form an equilateral triangle, and these that the points corresponding to tones of the triangles must be connected together in an same chroma name (e.g., C, C, C", etc.) but endless series as is shown for one octave (and located in different octaves should fall on the in flattened form) in Figure 3 (a). same line within the geometrical structure For illustration, the tones included in one and, hence, should project down into the particular key, namely, the key of C, are same point on an orthogonal plane. indicated by open circles in the figure, Now, however, I propose that the geo- whereas the tones not included in that key metrical structure must also reflect certain (i.e., the tones corresponding to the black subjective properties of the diatonic scale. keys on the piano) are indicated by filled Because steps between adjacent tones of the circles. The pattern for any other key would diatonic scale are of only two physical be the same except for a translation and, in sizes—half-tone steps (minor seconds) and the case of half the keys, a reflection about whole-tone steps (major seconds)—I start by the central axis of the strip. Two things considering the geometrical constraints that should be noted about this pattern. First, are entailed by perceived distances between steps between adjacent open circles are of tones separated by half- and whole-tone uniform size, in accordance with Require- steps. My initial attempt to erect a new geo- ment 3. Second, the pattern of open circles metrical structure is then based on the fol- as a whole possesses the asymmetry neces- lowing three requirements, the first two of sary to confer on each tone within any one which are the same as for the earlier simple octave of the scale the unique structural role helix but the third of which is new, having or tonal function required by music theory. its origin in the just-mentioned informal ob- For example, the most stable tone within the servation about subjective uniformity of suc- scale, the tonic, is always the lowest tone in cessive steps of the diatonic scale. (Later, I the set of three adjacent scale tones along allow departures from the strict uniformity one edge of the strip, whereas the second STRUCTURE OF PITCH 317

most stable tone, the dominant, a fifth above, cause the resulting structure then curves is always the next-to-lowest tone in the set back into itself, merging points correspond- of four adjacent scale tones along the other ing to distinct tones and eliminating the rec- edge of the strip. tilinear component of pitch height. The case Thus, by going to a more complex rep- in which alternate folds are made in opposite resentation than a simple unidimensional directions does not lead to these undesirable scale of pitch height, we avoid an objection consequences, however. As might be ex- to the subjective equality of the intervals of pected from the fact that the most general the diatonic scale, namely, that such unifor- rigid motion of three-dimensional space into mity would "deny a major source of melodic itself is a combination of a rotation and a variety" (Dowling, 1978b, p. 350). The translation along the axis of rotation, folding structural uniqueness of the diatonic scale in alternating directions produces an endless that underlies the desired variation of mel- helical structure with the amount of twist ody and modulation of key can be embodied per octave determined by the uniform angle in a qualitative asymmetry rather than in a of folding. Because there are two edges to purely quantitative one. No such structural the originally flat strip, each corresponding uniqueness is possible within scales that are to one of the two distinct whole-tone scales, both quantitatively and qualitatively sym- such folding leads to a double helix. metric, such as the whole-tone scale, rep- In order to achieve the collinearity of tones resented by just one edge of the strip of tri- that are equivalent except for height, in ac- angles, or the chromatic scale, represented cordance with Requirement 2, there must be by the symmetrical zigzag path that alter- an integer number of full twists of the struc- nates between the two edges of the strip. ture per octave. The flat version displayed Still, although the strip of triangles shown in Figure 3 (a) corresponds to the trivial 0° in Figure 3 (a) is consistent with Require- twist and, as I noted, must be excluded be- ments 1 and 3, it is not consistent with Re- cause it does not segregate the chroma lines: quirement 2, according to which any two They collapse into the two whole-tone scales. tones standing in an octave relation (such At the other extreme, two full twists per as C-C, C'-C", etc.) must fall on their own octave lead to a different kind of degeneracy unique chroma line. For, in this flattened in which all the triangles collapse into a sin- form, the line passing through C and C also gle triangle. This different kind of flat con- passes through D, E, F1, G*, and A*, which figuration must also be excluded because in are not octave or chroma equivalents to C it not only octaves but also minor thirds map and C. Likewise, the line passing through onto each other and, again, we lose the com- C* and C* also passes through D#, F, G, A, ponent of pitch height. The single remaining and B. However, there is no requirement that case of just one full twist per octave is the this structure remain flat. In fact, any fold- unique solution we seek. It is the nondegen- ing of the strip of triangles along the sides . erate double-helical structure of which one of the triangles will preserve the equilater- octave is portrayed in Figure 3 (b). In three- ality of the triangles imposed by Require- dimensional space it alone satisfies Require- ments 1 and 3. But in order to ensure the ments 1-3. full satisfaction of Requirement 1, the folding must be done in a uniform manner Emergent Properties of the Double Helix throughout the strip. Only then will the transformation of the strip into itself induced Musically significant properties emerge by transposition into a different key consist from the double-helical representation that only of rigid translations, rotations, and re- were not explicitly used in its derivation. flections of the structure as a whole and, First, successive tones of the chromatic scale thus, preserve all distances within it. Spe- project onto the axis of the helix in equal cifically, all folds must be made at the same steps of pitch height. More remarkably, as angle. is illustrated in Figure 3 (c), the same tones The case in which all folds are also made project down onto the plane orthogonal to in the same direction can be ruled out be- that axis to form a circle, the "cycle of 318 ROGER N. SHEPARD

helix partitions all tones into two disjoint sets: a set containing all the tones included in a particular diatonic key (two of which, in each octave, fall on the plane) and the complement of that set, containing all the tones not included in that key. The second consequence is that partitionings corresponding to the major diatonic keys can be obtained by rotating the plane about the central axis, with modulations between more closely related keys obtained by smaller angles of rotation. In Figure 3 the tones of the diatonic scale in C major are indicated by open circles, whereas the corresponding nondiatonic tones are represented by filled circles. In the view portrayed in Figure 3 (b) and, slightly tilted, in Figure 3 (c), the three-dimensional structure has been positioned so that the plane partitioning the tones into those included in the key of C major and those not included is seen almost edge on, with the tones belonging to C major falling to the right of the Figure 4. The two-dimensional melodic map of the double helix obtained by cutting and unwrapping the cyl- diametral plane. As can be seen from the 3- inder in Figure 3 (c). (From Shepard, 1981 a. Copyright 2-3-2- . . . grouping of the black circles on 1981 by Music Educators National Conference. Re- the left, the arrangement on the piano key- printed by permission.) board of the black keys, which correspond in the key of C to the sharps and flats or fifths" fundamental in music theory. In this "accidentals," is not itself an accident. projection tones separated by an octave map onto each other, and tones separated by a The Two-Dimensional Melodic Map of perfect fifth are closest neighbors around the the Double Helix circle. Because the tones in the double helix fall As I noted, it was the possibility of pro- on the surface of a right circular cylinder jecting the geometrically regular version of (Figure 3 [c]), we can make a vertical cut the single helix (Figure 1) down into the in this surface and spread it on a plane along chroma circle (Figure 2) that originally led with the embedded double helix. The re- me to devise a method of generating tones sulting two-dimensional "map" of the cyl- that in going endlessly around the chroma inder, illustrated in Figure 4, facilitates vi- circle, give the illusion of ascending endlessly sualization of some of the musically in pitch (Shepard, 1964b). Similarly, here, significant properties of the double helix. the emergence of the cycle of fifths as a pro- The rectangle bounded by the heavy dashed jection of the double helix led me, more re- line is from the region of the surface between cently, to devise a method of generating one C and the C an octave above. The hor- tones that glide continuously around the cir- izontal and vertical axes of this rectangular cle of fifths, that is, that pass continuously map correspond to the circle of fifths and to from C to G to D, and so forth, without pitch height, respectively. (The order in passing through intermediate chromas along which the tones project onto the axis cor- the way.5 responding to the circle of fifths is reversed The emergence of the cycle of fifths also has two important consequences (cf. Bal- 5 These tones, too, were demonstrated at the 1978 zano, 1980). The first is that a diametral meeting of the Western Psychological Association plane passing through the central axis of the (Shepard, Note 1). STRUCTURE OF PITCH 319

because the unwrapped surface of the cyl- to the diatonic scale, differing only in which inder is viewed in Figure 4 as if from the tone is taken as the principal (beginning, inside of the cylinder.) In order to represent final, or tonic) tone in the scale. Moreover, the unbounded character of the cylindrical the most common pentatonic scale is given surface, the rectangle can be endlessly re- by the complement of the diatonic scale peated in the plane as indicated in the figure. (e.g., by C#, D#, F#, G#, and A* in the figure The chromatic scale is represented in this or by the black keys on the piano). Notice two-dimensional map by the sequence of that apart from the always permissible key- notes on any of the straight lines directed changing translation, such a pentatonic scale upward and to the right. The two whole-tone is equivalent to a diatonic scale in which the scales are represented by the two distinct two most outlying tones within the vertical sequences of notes on the somewhat steeper band have simply been deleted (e.g., B and straight lines directed upward and to the left. F in the diatonic key of C). The resulting The diatonic scale, designated by the stip- 2-3-2-3-. . . pattern preserves many of the pled band, exhibits a 3-4-3-4- . . . zigzag desired structural properties of the 3-4-3-4- pattern in which strings of whole-tone steps . . . diatonic scale and, again, changes of are asymmetrically broken by single half- key correspond to horizontal shifts of the tone steps. (now narrower) vertical band. Corresponding to the division of the tones The adjacency of tones differing by half- into those that are and those that are not in and whole-tone steps in the embedded two- a particular key by a plane pivoted about the dimensional lattice preserves proximity in axis of the helix, the tones belonging to a pitch height. Thus, this two-dimensional particular key fall, in the flattened represen- structure is particularly suited for the rep- tation, within a particular vertical band de- resentation of melodies as well as scales. For, marcated in the figure for the key of C by as might be expected on the basis of Gestalt the lighter dashed lines. Modulation to an- principles of good continuation and grouping other key corresponds, here, to a horizontal by proximity (e.g., see Deutsch & Feroe, shift of the vertical band with, again, more 1981), transitions in pitch between succes- closely related keys obtainable by smaller sive tones of a melody are most commonly shifts. Alternatively, transpositions from any a single step in the diatonic scale (Dowling, major key to any other can be thought of as 1978b; Fucks, 1962; Merriam, 1964; Philip- translations of the stippled zigzag pattern of pot, 1970, p. 86; Piston, 1941, p. 23). For the diatonic scale from one location to an- example, in an analysis of nearly 3,000 me- other within this two-dimensional plane. lodic intervals in 80 English folk songs, That the two keys most closely related to a Dowling (1978b, p. 352) found, even after given key (e.g., C) are the two obtained by omitting the unison, that 68% of the tran- a shift of a fifth up (to G) or down (to F) sitions were no larger than one step on the is reflected in the geometrical fact that this diatonic scale, and 91% no larger than two zigzag pattern overlaps most with itself when steps.6 On the basis of these considerations, the straight group of three adjacent points is superimposed on the straight group of 6 There may be more than one reason for this striking four, slipped into either of the two alterna- predilection for small melodic steps. As Dowling (1978b) tive positions within that group. notes, it probably stems, in part, from basic limitations Other commonly used scales take similar of human memory capacity. It seems to be related to zigzag patterns within this space. In one of Deutsch's (1978) finding that accuracy of recognition its versions, the relative minor scale is iden- of a repeated tone falls off inversely with the average size of the intervals i n an interpolated sequence of tones. tical to its associated major scale except that As I have suggested, however, its close connection to the sequence is started and ended on a dif- Gestalt principles of visual perception (Koffka, 193S; ferent tone in the sequence (e.g., on A rather KOhler, 1947), to phenomena of "melodic fission" or than on C in the example illustrated). In- "auditory stream segregation" (Bregman, 1978; Breg- man & Campbell, 1971; Dowling, 1973b; McAdams deed, the seven so-called authentic church & Bregman, 1979; van Noorden, 1975), and to the modes (which derive rather directly from the closely allied phenomena of the "trill threshold" (Miller earlier Greek modes) all correspond exactly & Heise, 1950) and apparent movement in pitch (Shep- 320 ROGER N. SHEPARD

Thus, we need a still more complex structure that somehow incorporates the essential features of both the double and the single helix. Such structures can be constructed, but only at the cost of increased dimensionality. We have to resort to an embedding space of four dimensions—or even five, if we wish to retain the linear component of pitch height. However, such higher dimensional structures are to some extent accessible to three-dimensional visualization. Instead of regarding the chroma circle as obtained by projection of the simple helix, -CHROM&- we could regard it as obtained by first cutting Figure 5. The double helix wound around a torus. (From a one-octave segment out of either a recti- "Structural Representations of Musical Pitch" by linear or a simple helical representation of R. N. Shepard. In D. Deutsch (Ed.), Psychology of Music. New York: Academic Press, 1982. Copyright pitch and then bending that segment until 1982 by Academic Press, Inc. Reprinted by permission.) its two ends coincide to form a complete circle. Likewise, we could cut a one-octave seg- 1 propose that the unwrapped version of the ment out of the double helix portrayed in double helix presented in Figure 4 be called Figure 3 (c) and, by bending the cylindrical the melodic map. tube in which it is embedded until its two ends coincide, obtain a double helix wound The Double Helix Wound Around a around a torus as illustrated in Figure 5. Torus The two strands of the double helix (orig- The double helix has of course retained inally the two edges of the flattened strip of the rectilinear component of the earlier sim- triangles) have become two interlocking ple helix, namely, the component called rings embedded in the surface of the torus, pitch height. Moreover, the circle of fifths, and the surface of the original strip of tri- which has replaced the earlier chroma circle, angles bounded by these two edges has be- is still closely related to the chroma circle, come a twisted band. Unlike the only half- being obtainable from it simply by inter- twisted band of MObius, however, the two changing every other point with its dia- ends differed by a full 360° twist before they metrically opposite point around the circle. were joined, and hence, the band retains its This is why the single helix becomes a double two-sided or "orientable" topology (cf. helix following such a substitution. Hence Shepard, 1981c, p. 316). (a) chroma-equivalent tones still project into Only in a four-dimensional embedding the same point on any plane orthogonal to space does the torus possess the full degree the central axis and (b) chroma-adjacent of symmetry implicit in the fact that it is the tones still project into adjacent points on the "direct," Cartesian, or Euclidean product of central axis. Even so, these vestiges of the two circles (Blackett, 1967)—in this case the chroma circle in the double helix do not seem chroma circle and the circle of fifths. (In a to reflect adequately the robustness of that sense, to be illustrated in a later section, the circle as it has emerged from studies using embedded double helix can be thought of as ordinary musical tones (see Figure 2 [b]), a kind of Euclidean sum of those same two or, certainly, the computer-synthesized cir- circles.) As a consequence there exists a rigid cular tones (Shepard, 1964b; also see Figure rotation of the whole structure such that the 2 [a]). points corresponding to the 12 tones of the octave project onto the plane defined by one pair of orthogonal axes as a chroma circle ard, 1981c, p. 319) suggest a somewhat broader, par- tially perceptual basis. Possibly, too, melodies consisting and onto the plane defined by the remaining of small steps provide a quicker and more effective in- pair of orthogonal axes as a circle of fifths. dication of the underlying tonality or key of the piece. In other words, the two circular components STRUCTURE OF PITCH 321 enter into the structure in completely analogous ways, and in four-dimensional space, where rotations are about planes rather than about lines, we can rigidly rotate the whole structure about either of the two mentioned planes in such a way that the circle projected onto the other plane is carried into itself through any desired angle. The complete symmetry between the two circular components of the toroidal version of the double helix is revealed more clearly in the unwrapped version already displayed in Figure 4. Such a planar map represents the toroidal surface just as it did the straight cylindrical surface (Blackett, 1967). The horizontal axis of the map still corresponds to the circle of fifths, but the vertical axis now corresponds to the chroma circle rather than to the rectilinear dimension of pitch height. Also, all four corners of the rectangular map now correspond to the same tone Figure 6. The double helix wound around a helical cyl- and, hence, the same point (C) in the torus. inder. (From "Structural Representations of Musical Pitch" by R. N. Shepard. In D. Deutsch (Ed.), Psy- The lattice pattern of the melodic map is chology of Music. New York: Academic Press, 1982. now strictly repeating vertically as well as Copyright 1982 by Academic Press, Inc. Reprinted by horizontally, and the two types of rotations permission.) just described correspond, respectively, to horizontal and vertical translations in the nitely in pitch height, as shown in Figure 6. two-dimensional plane. As before, we are distorting the true metric structure of this geometrical object by The Double Helix Wound Around a visualizing it in only three dimensions. It Helical Cylinder achieves its full inherent symmetry only in a space of five dimensions, where it exists as Actually, because tones can be synthe- the Euclidean "sum" of the two-dimensional sized such that those differing by an octave circle of fifths, the two-dimensional chroma realize any specified degree of perceptual circle, and the one-dimensional continuum similarity or "octave equivalence," we need of pitch height. In this way the structure some way of continuously varying the struc- continues to obey the already-stated prin- ture representing those tones between the ciple that the most general rigid motion of double helix with the rectilinear axis (Figure space into itself is the product of rigid ro- 3 [c]) and the variant with the completely tations and rectilinear translations. It is in circular axis (Figure 5). What modification this sense, too, that the structure crudely of the latter, toroidal structure will bring depicted as if three-dimensional in Figure back a separation of chroma-equivalent tones 6 can rightly be -regarded as a higher di- on a dimension of pitch height? Again the mensional generalization of the helix. analogy with the original simple helix is A final point to be made about this gen- helpful. Just as a cut and vertical displace- eralized helical structure will be relevant ment of the earlier chroma circle can yield when, in a following section, I compare it one loop of the simple helix, a cut and ver- against empirical data. Whereas the double tical displacement of the torus portrayed in helix as it was first derived (Figure 3 [b]) Figure 5 yields one loop of a higher order was regarded as rigid (in order to preserve tubular helix. By attaching identical copies the equilateral character of its constituent of such a loop, end to end, we can extend triangles), the more general structure illus- this higher order helical structure indefi- trated in Figure 6 can be regarded as ad- 322 ROGER N. SHEPARD justable through variation of three parame- yield robust, orderly, and informative data ters: a weight for the circular component for concerning the effects of musical contexts on fifths, a weight for the circular component the interpretive schema induced within the for chroma, and a weight for the rectilinear listener (Krumhansl & Kessler, 1982; Jor- component for height. Thus, we can accom- dan & Shepard, Note 7). In this technique modate the relations between musical pitches the context (e.g., a musical scale, a melody, as they are perceived by different listeners a chord, a sequence of chords, or some richer who may vary, for example, in the extent to musical passage) is immediately followed by which they represent the cognitive structural a probe tone (selected, for example, from the component of the circle of fifths versus the 13 chromatic tones inclusively spanning a purely psychoacoustic component of pitch one-octave range), and the listener is asked height. to rate (on a 7-point scale) how well the From this standpoint the original double probe tone "fits in" with the preceding con- helix (Figure 3) was perhaps too rigid. In text. present terms it can be seen to be the Eu- For any context, the average ratings from clidean sum of just the circle of fifths and trials using different probe tones form a pro- the dimension of pitch height. But we now file o v e r the octave that in the case of musical know that listeners vary widely in their re- listeners, reveals the hierarchy of tonal func- sponsiveness to these two attributes (Krum- tions induced by that context—with the rat- hansl & Shepard, 1979; Shepard, 198la). ings highest for a tone interpreted as the So, in fitting the helix to data, we should tonic, next highest for a tone interpreted as perhaps make what might be regarded as a the dominant, and so on (see Krumhansl concession to performance, as opposed to & Shepard, 1979; and, especially, Krum- competence, and allow a differential stretch- hansl & Kessler, 1982). Indeed, Krumhansl ing or shrinking of the vertical extent of an and Kessler demonstrate that the circularly octave of the helix relative to its diameter. shifted position (or phase) of the profile per- This implies, of course, a departure from the mits one to infer which of the 24 possible constraint imposed by my third requirement major or minor diatonic keys is instantiated (uniformity of scale steps), but I noted at as the listener's momentary interpretive the time that such a requirement may be framework. appropriate only under a certain "perceptual The rating of how well a given probe tone set." If listeners differ in their judgments of fits in with the preceding context can be in- the relations between tones, they are not all terpreted as a measure of the spatial prox- operating under identical perceptual sets. imity of that probe to the ideal tonal center In terms of the two-dimensional map of or tonality implied by that context. When the double helix, differences in relative sa- applied to a suitably complete set of such lience of pitch height and the circle of fifths proximity measures, techniques of multidi- would be accommodated by a certain class mensional scaling (see Shepard, 1980) should of linear transformations of the rectangle, therefore enable one to reconstruct the un- namely, those restricted to relative stretch- derlying spatial structure. ing or shrinking of the rectangle along its In the original experiment by Krumhansl vertical and horizontal axes only. In the lim- and Shepard (1979), however, only the dia- iting cases in which there is a degenerate tonic scale of a single key (C major) was collapse of the rectangle in the horizontal or presented as context. As a result, the ob- vertical direction, we obtain a rectilinear tained rating profiles directly provided in- (and logarithmic) dimension of pitch height formation about the spatial proximities of or a simple circle of fifths, respectively. the 13 probe tones to a single tonal center, C. However, there is every reason to believe Recovery of the Geometrical that under transposition into any other key, Representation From Empirical Data the profile would be essentially invariant except for random fluctuations in the data— The probe technique introduced by Krum- and this assumption already has some em- hansl and Shepard (1979) is continuing to pirical support (Krumhansl & Kessler, 1982; STRUCTURE OF PITCH 323

Jordan & Shepard, Note 7; also see Krum- quired sort was obtained separately from the hansl, Bharucha, & Kessler, 1982). Accord- average rating profile obtained from each of ingly, it seemed reasonable to approximate the 23 subjects in the experiment by Krum- the complete matrix of proximity measures hansl and Shepard (1979). Application of needed for the application of multidimen- individual-difference multidimensional scal- sional scaling by simply duplicating the p r o - ing (INDSCAL; Carroll & Chang, 1970) t o file of ratings in each row of a square matrix, the entire resulting set of 23 individual ma- after circularly shifting each succeeding row trices then yielded the four-dimensional s o - by one cell so that the highest number (cor- lution presented in Figure 7. responding to the functional identity of the Panel a shows the projection of the solu- tonic tone and the ideal tonal center) fell on tion onto the plane of Dimensions 1 and 2, the principal diagonal of the matrix. Then, whereas Panel b shows its projection onto as is customary in multidimensional scaling, the plane of Dimensions 3 and 4. As sug- each entry in the matrix was averaged with gested by the circular dashed line, the first its diagonally opposite counterpart to yield projection (a) is essentially the chroma cir- a symmetric matrix of proximity measures. cle, going clockwise from C (through C#, D, Because large individual differences, which D*, etc.) around to C' an octave above. The are related to extents of musical b a c k g r o u n d, configuration departs from the chroma cir- characteristically emerge in these experi- cle, however, in that the spacing is wider ments (Krumhansl & Shepard, 1979; Shep- near C and C' and, p a r t i c u l a r l y , in that C ard, 198la), a symmetric matrix of the re- and C' do not coincide as they should if oc-

(£) --'-

*-®+ -

DIMENSION 1 DIMENSION 3

C INDIVIDUAL LISTENERS d INDIVIDUAL LISTENERS *"" • Group 1 (most musical) ^ ^f • G'°»P ' ,, „ • Group 2 (intermediate) ^Cj^* • Group 2 ^V"*1 4 Group 3 (least musical) / A Group 3 ^ «

) V 1,4 /i« • >r3 V Var-J1 "

/"«•?;•"' / -* HEIGHT / /' A /' /' k.'

WEIGHTS ON DIMENSION t WEIGHTS ON DIMENSION 3 • Figure 7. A four-dimensional solution obtained by application of INDSCAL to the data collected by Krumhansl & Shepard (1979). Panels a and b show the projections of the obtained configuration on the plane of Dimensions 1 and 2 and the plane of Dimensions 3 and 4, respectively; Panels c and d show the weights that these same dimensions have for listeners differing in musical background. (From Shep- ard, 1981a. Copyright 1981 by Music Educators National Conference. Reprinted by permission.) 324 ROGER N. SHEPARD tave equivalence had been complete for all Figure 5 at C and springing it slightly apart listeners. Dimension 1 t h u s seems to combine (with respect to chroma) in that same four- one dimension of the chroma circle with the dimensional space rather than, as in Figure dimension of pitch height. The second pro- 6, in an orthogonal fifth dimension. Presum- jection (b), however, is an almost perfect ably, if similar data were collected and an- circle of fifths, with the points representing alyzed for tones spanning two or three oc- C and C nearly superimposed, indicating taves, the data could no longer be fit by a complete octave equivalence. Apart from the small distortion of this sort in the four-di- stretching and separation between the points mensional space and, thus, the truer, five- representing C and C on Dimension 1, the dimensional structure would emerge. four-dimensional configuration is, in fact, Further support for these conclusions the double helix on the torus depicted in comes from a linear regression used to assess Figure 5, which corresponds to the Euclid- the importance of the various proposed geo- ean product of the chroma circle and the metrical components of pitch, including—in circle of fifths. addition to the one-dimensional component Panels c and d display the INDSCAL for height and the two circular components weights for each of the listeners on each of for chroma and for perfect fifths already the four dimensions. Panel c shows that the discussed here—a two-dimensional compo- listeners with the least extensive musical nent for major thirds. The use of linear backgrounds (represented by the triangles) regression for this purpose is made possible had the heaviest weights on Dimension 1, by the principle of "Euclidean composition," which separated the tones with respect to according to which the squared distance be- pitch height, whereas Panel d shows that the tween any two points in the final, higher listeners with the most extensive musical dimensional configuration is a weighted sum backgrounds (represented by the circles) had of the squared distances between the cor- the heaviest weights on Dimensions 3 and responding points in each of the component 4, which contained the circle of fifths and configurations. What the regression analysis implied complete equivalence between oc- thus yields is the set of weights, one for each taves. Moreover, the fact that the points for component configuration, that provides the all listeners fall on a 45° line in Panel d best fit to the data. (See Shepard, 1982, for means that the circle of fifths emerges, to a fuller explanation of the analysis and the whatever extent that it does for any one lis- results.) The results indicated that the circles tener, as an integrated whole and never one of chroma and of fifths did indeed account dimension at a time. That the points for for a significant portion of the variance but Group 1 listeners also fall close to the (bro- that the factors of height and of major thirds ken) 45° line in Panel c indicates that under were also significant for the least and the the complete octave-equivalence character- most musical listeners, respectively. More istic of the most musical listeners, the chroma specifically, the principal factors (with frac- circle, too, comes and goes as an integrated tions of variance accounted for in the sym- unit. metrized data) were for the most musical I interpret the obtained four-dimensional listeners, fifths (.43), chroma (.21), and structure in Figure 7 as a one-octave piece thirds (.19); for the intermediate listeners, of the endless five-dimensional theoretical chroma (.36), fifths (.21), and thirds (.16); structure portrayed in Figure 6. But because and for the least musical listeners, chroma it includes only one octave, the gap between (.39) and height (.31) only (Shepard, 1982, the two ends, which should have been rep- Table 1). resented by a displacement in a separate Clearly, then, pitch is multidimensional, fifth dimension, has (with a small distortion) and the different dimensions differ in sa- been accommodated in the four dimensions lience for different listeners (as well as in of the embedding space of the torus. In other different musical contexts). Moreover, be- words, the separation in pitch height because the final configuration (whether fitted tween C and C an octave above has been by multidimensional scaling or Euclidean achieved by cutting through the torus in composition) contained circular compo- STRUCTURE OF PITCH 325

nents, the obtained structures provide fur- multaneously (that is, as harmonic inter- ther support for the claim that some of the vals). Harmony, which governs the selection dimensions of pitch are circular. The struc- of tones to be sounded simultaneously in ture as a whole thus appears to be consistent chords, is therefore based on the consonant with the theoretical expectation of a helical intervals of the major and minor third, along or, under complete octave equivalence, to- with the consonant perfect fifth, which toroidal character. gether make up the particularly harmonious, stable, and tonality-defining major triad The Problem of Harmonic Relations (e.g., C-E-G, in the key of C). There are two directions in which the geo- The generalizations of the double helix for metrical models proposed here might be gen- pitch presented in the two preceding sections eralized in order to accommodate harmonic depended on an implicit weakening of Re- relations. One possibility, already suggested quirement 3 underlying the original deriva- at the end of the preceding section, is to add tion of that double helix, namely, the re- further components to the structural repre- quirement that steps within a diatonic scale sentation. In addition to the one-dimensional correspond to equal distances within the component of pitch height, the two-dimen- model. Only by accepting a weakening of sional component of chroma, and the two- this strong constraint can we provide for the dimensional component of the circle of fifths, quantitative variations in the relative weights we could include another two-dimensional of underlying components (height, chroma, component of major thirds. As I noted, such fifths, etc.) necessary to fit the data of dif- an extended model will indeed permit a ferent listeners. In terms of the two-dimen- somewhat better fit to the data (for the most sional melodic map of the manifold in which musical listeners, an increase from 63% to the double helix is wound, changes in the 83% of the variance explained; see Shepard, relative weights correspond to linear stretch- 1982, Table 1). However, the prospect of ings or shrinkings of the rectangular melodic increasing the number of dimensions of the map along either or both of its two principal embedding space from five to seven is not axes. One of these two axes corresponds to terribly attractive. the circle of fifths, whereas the other cor-, A second possibility is to remove the re- responds to pitch height, chroma, or a mix- striction that the linear expansions or ture of these in the case of the straight cy- compressions of the melodic map are to be lindrical model (Figure 3 [c]), the toroidal permitted only along the two orthogonal model (Figure 5), or the cylindrical helical axes of that map. If we allow elongations model (Figure 6), respectively. and contractions along arbitrarily oriented The coefficients of this linear transfor- directions in the plane of that map, we can mation determine the relative importance of bring tones separated by other musical in- certain musical intervals, namely, the per- tervals into closer proximity without increas- fect fifth, the octave, and (through emphasis ing dimensionality. Linear transformations of either pitch height or chroma) the minor of this more general type are called affine second or chromatic step. Within this re- transformations (Coxeter, 1961); they pre- stricted class of linear transformations of the serve straight lines and parallelism, which melodic map there is, however, no way to is desirable if we are to continue to use col- emphasize the intervals of the major or mi- linearity and parallel projection to represent nor third. Yet these two intervals, though of musical relations. Even so, we are not free limited importance fpr melodic structure, to choose any affine transformation of the are fundamental to harmonic structure plane but must confine our choice to one that (Piston, 1941, p. 10). Thus, whereas the is compatible with the toroidal interpretation most common intervals between successively of the plane. Expansions or contractions sounded tones (melodic intervals) are, as we along the orthogonal axes of the rectangular noted, the minor or major seconds, which are map can be of any magnitudes, correspond- adjacent in the melodic map (Figure 4), such ing to changes in the relative sizes (or intervals are dissonant when sounded si- weights) of the two circular components 326 ROGER N. SHEPARD

(chroma and fifths) that generate the torus. (b). However, because the transformation But expansions or contractions along other is on the torus, the lower triangular half of directions can take on only certain discrete the resulting parallelogram wraps around to values, corresponding to operations of cut- fill in the vacated upper triangular half of ting through the torus in a plane parallel to the rectangular map (as shown in b). The one of the two generating circles, giving one second 360° twist operates in the same way free end of the resulting cylindrical tube an (to take b into c). Finally, a relative expan- integer number of 360° twists relative to the sion on the horizontal axis and a circular other end, and then reattaching the two ends. shift of the entire pattern around the torus Twists that are not multiples of 360° would to bring the tone C into the center of the fail to rejoin the two ends of each line on the map yields the final, transformed map (d) plane (or helical path on the torus) and, shown at the bottom of Figure 8. hence, would disrupt continuity required for Because the shearing transformation in- an affine transformation and collinearity de- duced by the double twist was confined to sired for our musical interpretation. the vertical axis corresponding to the chroma circle, the horizontal axis was unaffected and The Harmonic Map and Its Affinity to the still corresponds to the circle of fifths. As a Melodic Map consequence the tones falling within any particular key continue to form a vertical band In the present case we seek an affine trans- (illustrated for the key of C major in the formation (of the admissible sort) that will figure), and modulations between keys still bring tones separated by the harmonically correspond to horizontal shifts of this band important major and minor thirds and per- and, hence, to rotations around the torus. fect fifth into close mutual proximity. In However, owing to the twist in the torus Figure 4 we see that for any given tone, say entailed by the affine transformation, the C at the lower left corner of the rectangle, harmonic map is not best described as a dou- the major third (E), the minor third (D*), ble helix wrapped around a torus. In it the and the perfect fifth (G), upward and to the series of perfect fifths now forms a single right, form the vertexes of an elongated par- helix wrapped three times around the torus; allelogram in which the lower triangular half the three series of major thirds form a triple corresponds to the major triad built on C helix wound once around the torus crosswise (viz., C-E-G) and the upper, complemen- to the series of fifths; and the four series of tary triangular half corresponds to the minor minor thirds form four separate circular triad built on C (viz., C-D*-G). Because no rings around the torus crosswise to both of other tones fall within such parallelograms, the first two types of series. (See the hori- an appropriate affine transformation should zontal and diagonal rows of letter names for bring all such sets of four tones into the de- the tones in Figure 8 [d].) sired, more compact form. Fortunately, this The important result is that tones related can be achieved in a way that is compatible to any tone by major and minor thirds, as with the toroidal interpretation. well as by perfect fifths, have now been First, we cut through the original torus brought into spatial proximity to that tone, (Figure 5) at some point on the circle of whereas the formerly proximal tones related fifths. Then we give the chroma circle at one by major and minor seconds have been dis- end of the resulting cylindrical tube two full placed to greater distances. This is illus- 360° twists relative to the chroma circle at trated for the tone C in Figure 8 (d), where, the other end. Finally, we reattach the two as can be seen, all the tones forming major ends, forming a new torus. These operations or minor triads with C constitute a compact are most adequately visualized in terms of hexagonal cluster around C. Moreover, the the two-dimensional map of the torus, as tones traditionally considered to be conso- shown in Figure 8. The first twist takes the nant when sounded simultaneously with C rectangle of the original melodic map (a, now fall in a compact circular region around which is a simplified version of the earlier C. These same relations also hold for any Figure 4) into the sheared parallelogram other tone chosen as a reference point. Ac- STRUCTURE OF PITCH 327

- c C-D*-F*-A-C fc—0—E —C

\ & D B Q A* NO" E G \ ^^_ A* F D \ F C* A \ — E -— (

Melodic XD B Q Map * \ Fff D 1st 2nd 360° \ 360° Twist Twist

> r \ . >

d Harmonic Map

(circularly shifted around torus to bring C into center of map)

In key of C majpr i Not in C major

«_ Fifths Figure S. The harmonic map (d) as obtained from the melodic map (a) by an affine transformation consisting of two 360° twists of the torus (b and c). cordingly, I propose that this transformed tones, including augmented and diminished map be called the harmonic map. Whereas chords, though more complex, also have the melodic map provided for compact rep- compact representations in this map.7 resentations of musical scales and, hence, the most common melodies, the harmonic map 7 As an amusing instance of "converging evidence," provides for compact representation of con- in the version of this space that Attneave presented at the symposium (Note 8), which was reflected with re- sonant intervals and, hence, the most com- spect to the version shown here, the pattern for the sev- mon chords. Although not explicitly illus- enth chord took, as he observed, the unmistakable shape trated, chords consisting of more than three of the numeral 7. 328 ROGER N. SHEPARD

Representations that more or less resem- to a logarithmic scale of frequency and in ble the harmonic map portrayed in Figure which each of two or more circular com- 8 have been independently proposed by a ponents completes one turn within a certain number of music theorists (e.g., Balzano, musical interval: the octave, the perfect fifth, 1980; Hall, 1974; Lakner, 1960; Longuet- or, possibly, the major third. Higgins, 1962, 1979; O'Connell, 1962; Att- I have illustrated how techniques of mul- neave, Note 8; Balzano, Note 2). What I tidimensional scaling can reconstruct such have added here is the demonstration that a structure from data collected from musical this two-dimensional space, which is so well listeners within a musical context. I have also suited to the representation of harmonic re- noted that unless certain constraints are im- lations, is related to the map of the double posed, the results of multidimensional scal- helix, which was not in any way based on ing may not be strictly invariant under trans- harmonic considerations, by a simple math- position. For example, if tones spanning only ematical transformation—specifically, an one octave are scaled, pitch height may be affinity. approximately accommodated by a distor- Two recent developments provide addition of the chroma circle. This disortion tional converging evidence for the impor- could be eliminated by scaling tones span- tance of this harmonic structure. One is Bal- ning several octaves or by means of a linear zano's erection of essentially this structure regression analysis (Shepard, 1982). It could from group-theoretic considerations, which also be eliminated by using the circular tones make no use of acoustic notions of conso- that I devised to achieve complete octave nance (Bajzano, 1980, Note 2). The other equivalence (Shepard, 1964b). In this latter is Krumhansl and Kessler's (1982) striking case, however, we forgo the component of demonstration that this same toroidal struc- pitch height. ture can be recovered by the multidimen- These considerations suggest two com- sional scaling of correlations between pro- ments concerning the multidimensional scal- files o b t a i n e d by applying t h e probe technique ing representations for musical pitch ob- in musically rich contexts. tained by Krumhansl (1979) and by Krum- By presenting contexts in both major and hansl and Kessler (1982). minor keys, Krumhansl and Kessler have Krumhansl's (1979) analysis of judged shown that thig toroidal manifold can rep- similarities between tones presented in the resent—in addition to the harmonic relations context of a diatonic key yielded a conically between individual tones considered here— shaped representation of the tones within an the relations between musical keys, with the octave. The disposition of the tones on the 12 minor tonalities falling in regular diag- conical surface corresponded to the hierar- onal rows between the diagonal rows shown chy of tonal functions in this way: First, the in Figure 8, which under their interpretation tonic of the contextually established key and would represent the 12 major tonalities. Spe- the other two tones making up the major cifically, each major tonality (e.g., C major) triad based on that tonic (i.e., the major is flanked, in their representation, by its cor- third and the perfect fifth above it) were responding relative and parallel minors (e.g., clustered together near the apex of the cone. A minor and C minor). Second, this central cluster was surrounded by an intermediate band of the remaining Discussion and Conclusions four tones that belong to the contextually I have argued that musical pitch, if not established key. Third, the remaining five psychophysical pitch height, should be rep- tones not belonging to that key were widely resented by a geometrically regular structure spaced around the perimeter of the cone. that is capable of rigidly mapping into itself I believe that Krumhansl's multidimen- under transposition from one musical key sional solution, like my own illustrated in into another. I have further argued that such Figure 7 (a), here, is somewhat distorted a structure is a generalized helix—probably because tones from only one octave were in- a double helix—in which the rectilinear axis cluded. The conical representation cannot be of the helix, called pitch height, corresponds extended into higher and lower octaves in STRUCTURE OF PITCH 329 such a way that musical transposition cor- their case because pitch height is not rele- responds to rigid motions of the structure vant to tonality. To say that a piece is in F into itself. However, if we cut off the apex major is not to say whether it is to be played of the cone and then cut through the re- by a piccolo or a tuba. Krumhansl and Kes- sulting closed band, we obtain a strip that sler's work elegantly uses the tone-probe with a slight twist, can be continued upward technique to show how a listener's internally and downward as a helical structure (much represented tonal center shifts in this closed as the torus of Figure 5 was cut through, toroidal surface under the influence of an twisted, and continued to form the higher evolving musical context. order helix of Figure 6). A further consid- As I mentioned earlier, there may be fun- eration is raised by Tversky's (1977) demon- damental parallels between the cognitive stration that perceptually more salient stim- structural constraints governing visual-spa- uli are judged to be both more similar to tial and auditory-musical representation. each other when the judgments are of sim- Pitch is the "morphophoric" medium that ilarity and more different from each other is most analogous to physical space in the when the judgments are of dissimilarity. It case of vision (Attneave, 1972; Attneave is possible that the tones that are perceived & Olson, 1971; Kubovy, 1981; Shepard, as closely related to the tonal center are 1981c, 1982). Musically meaningful ob- rated as more similar when similarity is jects, such as melodies and chords, are the judged because they are more salient and closest auditory analogs of visual shapes be- not because they are closer in the underlying cause such objects preserve their structure representational space. If so, the underlying under rigid transformations within their re- structure might be more similar, though ev- spective morphophoric media. Moreover, idently not isomorphic, to one of the gen- because the medium for the rigid transfor- eralized helical structures proposed here. mation of melodies and chords has circular Alternatively, it may be that the departures components of chroma and fifths, these rigid from helical regularity in Krumhansl's con- transformations include rotations as well as ical solution reflect real differences in the translations. Just as in the case of visual cog- relative stabilities of the tones within the nition, the time needed to carry out such diatonic context. (These possibilities are transformations mentally is expected to de- further considered in Shepard, 1982, pp. pend on the angular extent of these spatial 383-384.) transformations (Shepard, 1978a, 1981c; The toroidal manifold of possible tonali- Shepard & Cooper, 1982). Likewise, mod- ties obtained more recently by Krumhansl ulations from one key to another correspond and Kessler (1982) is, apart from its inclu- to rotations (Balzano, 1980; Shepard, 1982; sion of the minor tonalities, isomorphic to Balzano, Note 2; Shepard, Note 1). The to- the toroidal manifold whose two-dimen- roidal space in which these latter rotations sional harmonic map is shown in Figure 8. occur has now been fully elaborated by However, whereas the map in Figure 8 al- Krumhansl and Kessler (1982), who have lows the inclusion of pitch height and, hence, given all major and minor keys angular co- the extension of the structure into higher and ordinates within this same two-dimensional lower octaves in the manner illustrated in manifold. Figure 6, Krumhansl and Kessler's solution Finally, although I have in this paper fo- is a closed torus with complete octave equiv- cused exclusively on just one of the two in- alence and, hence, no component of pitch dispensable musical attributes, namely, pitch, height. there are reasons to believe t h a t the structure This closure of the torus was ensured in described here may carry over to the other Krumhansl and Kessler's experiment by gen- indispensable musical attribute, namely, erating the tones presented to the listeners time. Relations of beat and rhythm have the according to a scheme that achieves com- same structural properties of augmented plete equivalence of octaves and suppression correspondence at the simplest ratios of of pitch height (after Shepard, 1964b). tempo, such as the 2-to-l or 3-to-2 ratios, Moreover, such a closure is appropriate in which in the case of frequency of tones cor- 330 ROGER N. SHEPARD respond to the octave and perfect fifth, re- References spectively. Accordingly, illusions of contin- Allen, D. Octave discriminability of musical and non- ually accelerating tempo can be constructed musical subjects. Psychonomic Science, 1967, 7,421- that are analogous to the illusion of contin- 422. ually ascending pitch (Risset, Note 9) and, Attneave, F. The representation of physical space. In A. W. Melton & E. Martin (Eds.), Coding processes hence, that demonstrate the existence of a in human memory. Washington, D.C.: V . H. Win- temporal analog of the circle of chroma. ston, 1972, pp. 283-306. Further, recent ethnomusicological studies Attneave, F., & Olson, R. K. Pitch as a medium: A new of the highly developed rhythms of Indian approach to psychophysical scaling. American Jour- and West African cultures have even re- nal of Psychology, 1971,84, 147-166. Bachem, A. Tone height and tone chroma as two dif- vealed temporal analogs of the diatonic scale ferent pitch qualities. Acta Psychologica, 1950, 7,80- (Pressing, in press). By the same group-the- 88. oretic arguments already given for pitch by Bachem, A. Time factors in relative and absolute pitch Balzano (1980, in press), this finding implies determination. Journal of the Acoustical Society of America, 1954, 26, 751-753. (as Pressing notes) the existence of a t e m - Balzano, G. J. Chronometric studies of the musical in- poral analog of the circle of fifths. Remem- terval sense (Doctoral dissertation, Stanford Univer- bering that we must add the rectilinear time sity, 1977). Dissertation Abstracts International, line to these two circular components, we 1977, 38, 2898B. (University Microfilms No. 77-25, arrive at the conclusion that a structure that 643). Balzano, G. J. The group-theoretic description of is isomorphic to the five-dimensional gen- twelvefold and microtonal pitch systems. Computer eralized helix portrayed in Figure 6 will be Music Journal, 1980, 4, 66-84. needed for the representation of rhythmic Balzano, G. J. Musical versus psychoacoustical vari- relations as well. ables and their influence on the perception of musical intervals. Bulletin of the Council for Research in Music Education, 1982, 70, 1-11. Reference Notes Balzano, G. J. The pitch set as a level of description for studying musical pitch perception. In M. Clynes 1. Shepard, R. N. The double helix of musical pitch. (Ed.), Music, mind, and brain. New York: Plenum In R. N. Shepard (Chair), Cognitive structure of Press, in press. musical pitch. Symposium presented at the meeting Balzano, G. J., & Liesch, B. W. The role of chroma and of the Western Psychological Association, San Fran- scalestep in the recognition of musical intervals in and cisco, April 1978. out of context. Psychomusicology, in press. 2. Balzano, G. J. The structural uniqueness of the dia- Beck, J., & Shaw, W. A. The scaling of pitch by the tonic order. In R. N. Shepard (Chair), Cognitive method of magnitude estimation. American Journal structure of musical pitch. Symposium presented at of Psychology, 1961, 74, 242-251. the meeting of the Western Psychological Associa- Blackett, D. W. Elementary topology. New York: Ac- tion, San Francisco, April 1978. ademic Press, 1 9 6 7 . 3. Shepard, R. N. Unpublished doctoral dissertation Blackwell, H. R., & Schlosberg, H. Octave generaliza- proposal, Yale University, 1954. tion, pitch discrimination, and loudness thresholds in 4. Mathews, M. V. Personal communication, January the white rat. Journal of Experimental Psychology, 7, 1982. 1943, 33, 407-419. 5. Blechner, M. J. Musical skill and the categorical Boring, E. G. Sensation and perception in the history perception of harmonic mode (Status Report on of experimental psychology. New York: Appleton- Speech Perception SR-5,1/52). New Haven, Conn.: Century-Crofts, 1942. Haskins Laboratories, 1977, pp. 139-174. Bregman, A. S. The formation of auditory streams. In 6. Cohen, A. J. Inferred sets of pitches in melodic p e r - J. Requin (Ed.), Attention and performance VII. ception. In R. N. Shepard (Chair), Cognitive struc- Hillsdale, N.J.: E r l b a u m , 1978. ture of musical pitch. Symposium presented at the Bregman, A. S., & Campbell, J. Primary auditory meeting of the Western Psychological Association, stream segregation and the perception of order in San Francisco, April 1978. rapid sequences of tones. Journal of Experimental 7. Jordan, D., & Shepard, R. N. Effects on musical Psychology, 1971, 89, 244-249. cognition of a distorted diatonic scale. Manuscript Burns, E. M. Octave adjustment by non-Western mu- in preparation, 1982. sicians. Journal of the Acoustical Society of Amer- 8. Attneave, F. Discussant's remarks. In R. N. Shepard ica, 1974, 56, 525. (Abstract) (Chair), Cognitive structure of musical pitch. Sym- Burns, E. M., & Ward, W. I. Categorical perception- posium presented at the meeting of the Western Psy- phenomenon or epiphenomenon: Evidence from ex- chological Association, San Francisco, April 1978. periments in the perception of melodic musical inter- 9. Risset, J.-C. Demonstration tape of auditory illu- vals. Journal of the Acoustical Society of America, sions. (Available from J.-C. Risset, UER de L,uming, 1978, 63, 456-468. 13008 Marseille, France.) Carroll, J. D., & Chang, J.-J. Analysis of individual STRUCTURE OF PITCH 331

differences in multidimensional s c a l i n g via an N-way ed.). New York: Holt, Rinehart & Winston, 1979. generalization of Eckart-Young decomposition. Psy- Frances, R. La perception de la musique. Paris: Vrin, chometrika, 1970, 35, 283-319. 1958. Charbonneau, G., & Risset, J.-C. Circularite de juge- Fucks, W. Mathematical analysis of formal structure ments de hauteur sonore. Comptes Rendus Hebdo- of music. IRE Transactions on Information Theory, madaires des Seances de L'Academie des Sciences 1962, 8, 225-228. Paris, 1973, 277B, 623-626. Goldstein, H. Classical mechanics. Reading, Mass.: Chomsky, N. Aspects of the theory of syntax. Cam- Addison-Wesley, 1950. bridge, Mass.: M.I.T. Press, 1965. Goldstein, J. L. An optimum processor theory for the Chomsky, N. Language and mind. New York: Har- central formation of the pitch of complex tones. Jour- court, Brace & World, 1968. nal of the Acoustical Society of America, 1973, 54, Cohen, A. Perception of tone sequences from the West- 1496-1516. ern-European chromatic scale: Tonality, transposi- Greenwood, G. D. Principles of dynamics. Englewood tion and the pitch set. Unpublished doctoral disser- Cliffs, N.J.: Prentice-Hall, 1965. tation, Queen's University at Kingston, Ontario, Hahn, J., & Jones, M. R. Invariants in auditory fre- Canada, 1975. quency relations. Scandinavian Journal of Psychol- Coxeter, H. S. M. Introduction to geometry. New York: ogy, 1981, 22, 129-144. Wiley, 1961. Hall, D. E. Quantitative evaluation of musical scale tun- de Boer, E. On the "residue" and auditory pitch per- ings. American Journal of Psychics, 1974, 42, 543- ception. In W. D. Keidel & W. D. Neff (Eds.), Hand- 552. book of sensory physiology (Vol. 5, Pt. 3: Clinical Helmholtz, H. von. On the sensations of tone as a phys- and special topics). New York: Springer-Verlag, iological basis for the theory of music. New York: 1976, pp. 479-583. Dover, 1954. (Originally published, 1862.) Deutsch, D. Octave generalization and tune recognition. Hilbert, D., & Cohn-Vossen, S. Geometry and the Perception & Psychophysics, 1972, //, 411-412. imagination. New York: Chelsea, 1952. (Originally Deutsch, D. Octave generalization of specific interfer- published, 1932.) ence effects in memory for tonal pitch. Perception Humphreys, L. F. Generalization as a function of & Psychophysics, 1973, 13, 271-275. method of reinforcement. Journal of Experimental Deutsch, D. Delayed pitch comparison and the principle Psychology, 1939, 25, 361-372. of proximity. Perception & Psychophysics, 1978, 23, Hurvich, L. M., & Jameson, D. An opponent-process 227-230. theory of color vision. Psychological Review, 1957, Deutsch, D., & Feroe, J. The internal representation of 64, 384-404. pitch sequences in tonal music. Psychological Review, Idson, W. L., & Massaro, D. W. A bidimensional model 1981,55, 503-522. of pitch in the recognition of melodies. Perception Dewar, K. M. Context effects in recognition memory & Psychophysics, 1978, 24, 551-565. for tones. Unpublished d o c t o r a l dissertation, Queen's Imberty, M. L acquisition des structures tonales chez University at Kingston, Ontario, Canada, 1974. I'enfant. Paris: Klincksieck, 1969. Dewar, K. M., Cuddy, L. L., & Mewhort, D. J. K. Jones, M. R. Time, our lost dimension: Toward a new Recognition memory for single tones with and without theory of perception, attention, and memory. Psy- context. Journal of Experimental Psychology: Hu- chological Review, 1976, 83, 323-355. man Learning and Memory, 1977, 3, 60-67. Kallman, H. J., & Massaro, D. W. Tone chroma is Dowling, W. J. The 1215-cent octave: Convergence of functional in melody recognition. Perception & Psy- Western and Nonwestern data on pitch-scaling. Jour- chophysics, 1979, 26, 32-36. nal of the Acoustical Society of America, 1973, 53, Kilmer, A. D., Crocker, R. L., & Brown, R. R. Sounds 373A. (Abstract) (a) from silence: Recent discoveries in ancient Near Dowling, W. J. The perception of interleaved melodies. Eastern music. Berkeley, Calif.: Bit Enki Publica- Cognitive Psychology, 1973, 5, 322-337. (b) tions, 1976. Dowling, W. J. Listeners' successful search for melodies Koffka, K. Principles of Gestalt psychology. New York: scrambled into several octaves. Journal of the Acous- Harcourt Brace, 1935. tical Society of America, 1978, 64, S146. (Abstract) Kdhler, W. Gestalt psychology. New York: Liveright, (a) 1947. Dowling, W. J. Scale and contour: Two components of Krumhansl, C. L. The psychological representation of a theory of memory for melodies. Psychological Re- musical pitch in a tonal context. Cognitive Psychol- view, 1978,55, 341-354. (b) ogy, 1979, //, 346-374. Dowling, W. J. Musical scales and psychophysical Krumhansl, C. L., Bharucha, J. J., & Kessler, E. J. scales: Their psychological reality. In T. Rice & R. Perceived harmonic structure of chords in three re- Falck (Eds.), Cross-cultural approaches to music. lated musical keys. Journal of Experimental Psy- Toronto, Ontario, Canada: University of Toronto chology: Human Perception and Performance, 1982, Press, in press. 8, 24-36. Dowling, W. J., & Hollombe, A. W. The perception of Krumhansl, C. L., & Kessler, F. J. Tracing the dynamic melodies distorted by splitting into several octaves: changes in perceived tonal organization in a spatial Effects of increasing proximity and melodic contour. representation of musical keys. Psychological Re- Percepton & Psychophysics, 1977, 21, 60-64. view, 1982, 89, 334-368. Forte, A. Tonal harmony in concept and practice (3rd Krumhansl, C. L., & Shepard, R. N. Quantification of 332 ROGER N. SHEPARD

the hierarchy of tonal functions within a diatonic con- Ratner, L. G. Harmony: Structure and style. New text. Journal of Experimental Psychology: Human York: McGraw-Hill, 1962. Perception and Performance, 1979, 5, 579-594. Revesz, G. Introduction to the psychology of music. Kubovy, M. Concurrent pitch-segregation and the the- Norman: University of Oklahoma Press, 1954. ory of indispensable attributes. In M. Kubovy & Risset, J.-C. Musical acoustics. In E. C. Carterette & J. R. Pomerantz (Eds.), Perceptual organization. M. P. Friedman (Eds.), Handbook of perception (Vol. Hillsdale, N.J.: Erlbaum, 1981. 4). New York: Academic Press, 1978, pp. 521-564. Lakner, Y. A new method of representing tonal rela- Ruckmick, C. A. A new classification of tonal qualities. tions. Journal of Music Theory, 1960, 4, 194-209. Psychological Review, 1929, 36, 172-180. Levelt, W. J. M., Van de Geer, J. P., & Plomp, R. Schenker, H. Harmony (O. Jones, Ed. and E. M. Triadic comparisons of musical intervals. British Borgese, trans.). Cambridge, Mass.: M.I.T. Press, Journal of Mathematical and Statistical Psychology, 1954. (Originally published, 1906). 1966, 19, 163-179. Shepard, R. N. Attention and the metric structure of Liberman, A. M., Cooper, F. S., Shankweiler, D., & the stimulus space. Journal of Mathematical Psy- Studdert-Kennedy, M. Perception of the speech code. chology, 1964, /, 54-87. (a) Psychological Review, 1967, 74, 431-461. Shepard, R. N. Circularity in judgments of relative Licklider, J. C. R. Basic correlates of the auditory stim- pitch. Journal of the Acoustical Society of America, ulus. In S. S. Stevens (Ed.), Handbook of experi- 1964, 36, 2346-2353. (b) mental psychology. New York: Wiley, 1951. Shepard, R. N. Approximation to uniform gradients of Locke, S., & Kellar, L. Categorical perception in a non- generalization by monotone transformations o f scale. linguistic mode. Cortex, 1973, 9, 355-369. In D. I. Mostofsky (Ed.), Stimulus generalization. Longuet-Higgins, H. C. Letter to a musical friend. Stanford, Calif.: Stanford University P r e s s , 1965, pp. Music Review, 1962, 23, 244-248. 94-110. Longuet-Higgins, H. C. The perception of music (Re- Shepard, R. N. Psychological representation of speech view Lecture). Proceedings of the Royal Society, sounds. In E. E. David & P. B. Denes (Eds.), Human London, 1979, 205B, 307-332. communication: A unified view. New York: McGraw- Mathews, M. V. The digital computer as a musical in- Hill, 1972, pp. 67-113. strument. Science, 1963, 142, 553-557. Shepard, R. N. Representation of structure in s i m i l a r i t y Mathews, M. V., & Sims, G. Perceptual discrimination data: Problems and prospects. Psychometrika, 1974, of just and equal tempered tunings. Journal of the 39, 373-421. Acoustical Society of America, Suppl. 1, 1981, 69, Shepard, R. N. The circumplex and related topological 538 (Abstract) manifolds in the study of perception. In S. Shye (Ed.), McAdams, S., & Bregman, A. Hearing musical streams. Theory construction and data analysis in the behav- Computer Music Journal, 1979, 3, 26-43. ioral sciences. San Francisco: Jossey-Bass, 1978, 29- Merriam, A. P. The anthropology of music. Evanston, 80. (a) 111.: Northwestern University Press, 1964. Shepard, R. N. Externalization of mental images and Meyer, L. B. Emotion and meaning in music. Chicago: the act of creation. In B. S. Randhawa & W. E. University of Chicago Press, 1956. Coffman (Eds.), Visual learning, thinking, and com- Miller, G. A. The magic number seven, plus or minus munication. New York: Academic Press, 1978. (b) two. Psychological Review, 1956, 63, 81-97. Shepard, R. N. On the status of "direct" psychophysical Miller, G. A., & Heise, G. A. The trill threshold. Jour- measurement. In C. W. Savage (Ed.), Minnesota nal of the Acoustical Society of America, 1950, 64, studies in the philosophy of science (Vol. 9). Min- 637-638. neapolis: University of Minnesota Press, 1978. (c) Null, C. Symmetry in judgments of musical pitch. Shepard, R. N. Multidimensional scaling, tree-fitting, Unpublished doctoral dissertation, Michigan State and clustering. Science, 1980, 210, 390-398. University, 1974. Shepard, R. N. Individual differences in the perception O'Connell, W. Tone spaces. Die Reihe, 1962, 8, 34-67. of musical pitch. In Documentary report of the Ann Peterson, G. E., & Barney, H. L. Control methods used Arbor symposium: Applications of psychology to the in a study of the vowels. Journal of the Acoustical teaching and learning of music. Reston, Va.: Music Society of America, 1952, &, 175-184. Educators National Conference, 1981. (a) Philippot, M. L'Arc, Beethoven, No. 40, 1970. Shepard, R. N. Psychological relations and psycho- Pikler, A. G. The diatonic foundations of hearing. Acta physical scales: On the status of "direct" psycho- Psychologica, 1955, ;/, 432-445. physical measurement. Journal of Mathematical Pikler, A. G. Logarithmic frequency systems. Journal Psychology, 1981, 24, 21-57. (b) of the Acoustical Society of America, 1966,39,1102- Shepard, R. N. Psychophysical complementarity. In M. 1110. Kubovy & J. R. Pomerantz (Eds.), Perceptual or- Piston, W. Harmony. New York: Norton, 1941. ganization. Hillsdale, N.J.: Erlbaum, 1981. (c) Plomp, R. Aspects of tone sensation: A psychophysical Shepard, R. N. Structural representations of musical study. New York: Academic Press, 1976. pitch. In D. Deutsch (Ed.), Psychology of music. New Plomp, R., & Levelt, W. J. M. Tonal consonance and York: Academic Press, 1982. critical band width. Journal of the Acoustical Society Shepard, R. N., & Cooper, L. A. Mental images and of America, 1965, 38, 548-560. their transformations. Cambridge, Mass.: MIT Press/ Pressing, J. Cognitive isomorphisms in pitch and rhythm Bradford Books, 1982. in world musics: West African, the Balkans, Thailand Shepard, R. N., & Zajac, E. (Producers). A pair of and Western tonality. Ethnomusicology, in press. paradoxes. Murray Hill, N.J.: Bell Telephone Lab- STRUCTURE OF PITCH 333

oratories, Technical Information Library, 1965. (Film) Tversky, A. Features of similarity. Psychological Re- "Shepard's Tones." On M. V. Mathews, J.-C. Risset, view, 1977, 84, 327-352. et al., The voice of the computer (Decca Record DL van Noorden, L. P. A. S. Temporal coherence in the 710180). Universal City, Calif.: MCA Records, Inc., perception of tone sequences. Unpublished doctoral 1970. dissertation, Technishe Hogeschool, Eindhoven, The Siegel, J. A. The nature of absolute pitch. In E. Gordon Netherlands, 1975. (Ed.), Research in the psychology of music (Vol. 8). Ward, W. D. Subjective musical pitch. Journal of the Iowa City: University of Iowa Press, 1972. Acoustical Society of America, 1954, 26, 369-380. Siegel, J. A., & Siegel, W. Absolute identification of Ward, W. D. Absolute pitch. Pt. I. Sound, 1963, 2, 14- notes and intervals by musicians. Perception & Psy- 21. (a) chophysics, 1977, 21, 143-152. (a) Ward; W. D. Absolute pitch. Pt. II. Sound, 1963, 2, Siegel, J. A., & Siegel, W. Categorical perception of 33-41. (b) tonal intervals: Musicians can't tell sharp from flat. Ward, W. D. Musical perception. In J. V. Tobias & Perception & Psychophysics, 1977, 21, 399-407. (b) E. D. Hubert (Eds.), Foundations of modern auditory Stevens, S. S. The measurement of loudness. Journal theory (Vol. 1). New York: Academic Press, 1970, of the Acoustical Society of America, 1955, 27, 815- pp. 407-447. 829. Wightman, F. L. The pattern-transformation model of Stevens, S. S., & Volkmann, J. The relation of pitch to pitch. Journal of the Acoustical Society of America, frequency: A revised scale. American Journal of Psy- 1973,54, 407-416. chology, 1940, S3, 329-353. Zatorre, R. S., & Halpern, A. R. Identification, dis- Stevens, S. S., Volkmann, J., & Newman, E. B. A scale crimination, and selective adaptation of simultaneous for the measurement of the psychological magnitude musical intervals. Perception & Psychophysics, 1979, of pitch. Journal of the Acoustical Society of Amer- 26, 384-395. ica, 1937, S, 185-190. Zenatti, A. Le developpement genetique de la perception Sundberg, J. E. F., & Lindqvist, J . Musical octaves and musicale (Monographies Francaises de Psychologic, pitch. Journal of the Acoustical Society of America, No. 17). Paris: Centre National de la Recherche 1973, 54, 922-929. Scientifique, 1969. Terhardt, E. Pitch, consonance, and harmony. Journal Zuckerkandl, V. Sound and symbol. Princeton, N.J.: of the Acoustical Society of America, 1974,55,1061- Princeton University Press, 1956. 1069. Zuckerkandl, V. Man the musician. Princeton, N.J.: Thurlow, W. R., & Erchul, W. P. Judged similarity in Princeton University Press, 1972. pitch of octave multiples. Perception & Psychophys- ics, 1977, 22, 177-182. Received October 21, 1981 •