Perception & Psychophysics 1987, 41 (6), 489-504 Tonal schemas: Evidence obtained by probing distorted musical scales DANIEL S. JORDAN Xerox Palo Alto Research Center, Palo Alto, California and ROGER N. SHEPARD Stanford University, Stanford, California Listeners were presented with a normal major diatonic scale or with a distortion of that scale obtained either by stretching or by equalizing its intervals. The scale was followed by a test tone, selected from a series of quarter tones spanning the range of the scale, and listeners rated how well the test tone fit in with the scale. The pattern of ratings indicated that listeners interpreted the successive tones of each scale by means of a template-like tonal schema tuned to the usual spacing of the major diatonic scale. For the scale obtained by stretching the major scale to span a semitone more than an octave, the schema evidently shifted monotonically in pitch to accom­ modate each successive tone. For the scale obtained by making all seven intervals in the scale equal, the schema tended to impose its tonal structure despite the absence of the unequal spac­ ing expected by the schema, and led to illusory impressions that the (usually smaller) third and seventh intervals, though now equal, were larger than the other intervals. The characterization of musical tones solely in terms octave, namely, the pattern 2,2, 1,2,2,2,1, where each oftheir physical attributes, such as frequency, amplitude, successive integer indicates the number ofsemitone steps and duration, is not sufficient to explain the experience between adjacent tones of the scale. (A semitone step is, of those tones in a musical context. A musical context on a physical scale of log frequency, 1/12 of an octave tends to activate, within a listener, a particular cognitive and on a piano keyboard corresponds to the interval be­ schema or mental framework with respect to which en­ tween two adjacent keys-for example, between the white suing tones are perceptually interpreted. Different con­ key C and the next black key C=.) A fundamental property texts, inducing different tonal schemas, may lead to quite that the major diatonic scale shares with most musical different interpretations of the same physical tone. scales is that the pattern of its successive intervals is asym­ The musical scales on which the prevailing music of metric in the sense that as we rigidly shift the whole pat­ a particular culture is based provide perhaps the most tern up or down in pitch, it comes into congruence with powerful such schemas. In the case of Western listeners, itself only after a shift of an integral number of octaves, the preeminent such scale is the familiar major diatonic which, in the case of the Western diatonic system, is a scale, whose successive tones are named do, re, mi, fa, shift of some multiple of 12 semitone steps. sol, la, ti, do, where the eighth tone, though given the By virtue of this asymmetry, each tone in the scale is same name as the first (do), is an octave above it. (Physi­ distinguished by a unique set of musical intervals to the cally, an octave corresponds to a doubling of the fun­ other tones in the scale. Hence, even though the scale damental frequency of the tone. The term octave, repeats indefinitely over successive octaves (so that an ar­ however, is not based on a physical property but on the bitrarily selected portion of successive intervals might ex­ musical schema. Within the diatonic musical scale, it is hibit, for example, the pattern ... 2, 1,2,2, 1,2,2,2, the eighth tone that corresponds to the doubling of fre­ 1,2,2, 1,2, ., .), we can always determine where a given quency.) tone is within the scale from the pattern of intervals that The major diatonic scale is characterized by the pat­ separates it from neighboring tones in the scale. tern of intervals between its successive tones within each For example, the tone do, called the tonic tone of the scale and regarded as the most important or stable tone This research was supported by the first author's National Institute of Mental Health Traineeship (MH-15157-06) at Stanford University (on which pieces of music traditionally end), is unique and by the second author's National Science Foundation Research Grants in that the 3 succeeding tones above it move up by a whole BNS 80-05517 and BNS 85-11685, also at Stanford University. We tone, a whole tone, and a semitone (i.e., in the pattern thank Eddie Kessler for assistance in generating the tone sequences. Cor­ 2,2, 1), whereas the tone immediately preceding it in the respondence concerning this paper should be sent to Daniel S. Jordan, Intelligent Systems Laboratory, Xerox Palo Alto Research Center, 3333 scale is 1 semitone below. Similarly, the tone sol, called Coyote Hill Road, Palo Alto, CA 94304, or to Roger N. Shepard, Depart­ the dominant of the scale and generally considered to be ment of Psychology, Building 420, Stanford, CA 94305. the next most stable tone, has the different unique struc- 489 Copyright 1987 Psychonomic Society, Inc. 490 JORDAN AND SHEPARD tural property that although the 3 succeeding tones above rated highest, other relatively stable tones such as the it again move up in the pattern 2, 2, 1, the tone immedi­ dominant and mediant were rated next highest, less sta­ ately preceding it in the scale is 2 semitones (i.e., a whole ble tones of the scale (including the leading tone) were tone) below. rated lower, and tones not included in the scale (the chro­ Every other tone in the scale has a unique such pattern matic tones between the 7 scale tones) were rated lowest of relations with its neighbors, including tones that are of all. The hierarchy of tonal functions as thus quantified heard as quite unstable and as having a tendency to resolve by the probe-tone method is quite robust, having now been to a neighboring more stable tone. For example, the tone replicated in many different studies reported, particularly ti, which is called the leading tone because it is heard as by Krumhansl, Shepard, and their collaborators (e.g., having a strong tendency to resolve to the neighboring Kessler, Hansen, & Shepard, 1984; Krumhansl, 1985; tone (do), is the only tone that is preceded in the scale Krumhansl, 1987; Krumhansl & Kessler, 1982; Krum­ by 3 successive whole-tone steps (2, 2, 2). The unique hansl & Shepard, 1979). structural relations that characterize each tone of a scale In the series of experiments reported here, we further are invariant under changes of key. Regardless of whether explored the idea that the perception of musical tones is the scale begins on, say, C, on F, or on Ep(as when we mediated by a tonal schema with the properties of a tem­ are in the major keys of C, F, or Ep, respectively), the plate that can be rigidly shifted in pitch to achieve a fit leading tone, ti (which would then be B, E, or D, respec­ with the incoming tones. Ordinarily, we are not aware tively), will always have the unique property of being of the role of the hypothesized schema. When music preceded by 3 whole-tone steps in the scale of that key. matches the schema-as the preponderance of popular, We have previously proposed that the principal cogni­ classical, and folk music in the Western world will-the tive schema by means of which Western listeners inter­ schema operates so automatically that it is effectively pret tonal music is a kind of pitch template (Shepard & "transparent" (cf. Shepard, 1981, for visual analogs). Jordan, 1984). The template, being tuned to the pattern The operation of the schema is likely to become evident of intervals of the major diatonic scale (2, 2, 1, 2, 2, 2, only when the incoming tones depart from the pattern to 1), can be shifted up and down in pitch until it comes into which the template is tuned. In this case, the effect of the register with the pitches of incoming tones. (Often, a schema will be to induce some systematic distortions in default interpretation of the first-presentedtone as the tonic the relations among the physically presented tones, as they will be correct.) When a position is found such that the are perceived or remembered. We have explored two dis­ incoming tones sufficiently match the pitches to which the tortions of the familiar major diatonic scale. shifted template is tuned, the tones are categorically as­ One distortion that we investigated is a uniform stretch­ signed their unique tonal functions (tonic, dominant, lead­ ing of the major scale in which the relational pattern (2, ing tone, etc.) in accordance with their positions in the 2, 1, 2, 2, 2, 1) of the semitone intervals remains un­ template. The categorical nature of this interpretive changed but the absolute sizes of the intervals are linearly process is, we suggest, why listeners are often quite in­ expanded (on the log-frequency scale) by the ratio 13/12. sensitive to small mistunings (e.g., Bums & Ward, 1978; Hence, the 8th tone of this stretched diatonic scale reaches Siegel & Siegel, 1977; Zatorre & Halpern, 1979). not just to an octave above its first tone but to a minor In our graphical depiction of the template, we 9th above its first tone (i.e., to a semitone beyond the oc­ represented the hierarchy of tonal functions by attaching, tave). If our notion of the tonal schema as a template is to each position in the template, a bar whose length is correct, we might expect a slight, and perhaps unnoticed, proportional to the importance or stability of that tone (as upward adjustment of the template with each successive, specified, qualitatively, by music theory or, quantitatively, slightly sharper tone, in order to maintain registration of by experimental measurements about to be described).