7 INTERVALS SCALES AND TUNING* EDWARD M. BURNS Department of Speech and Hearing Sciences University of Washington Seattle, Washington !. INTRODUCTION In the vast majority of musical cultures, collections of discrete pitch relation- shipsmmusical scales--are used as a framework for composition and improvisa- tion. In this chapter, the possible origins and bases of scales are discussed, includ- ing those aspects of scales that may be universal across musical cultures. The perception of the basic unit of melodies and scales, the musical interval, is also addressed. The topic of tuningmthe exact relationships of the frequencies of the tones composing the intervals and/or scales~is inherent in both discussions. In addition, musical interval perception is examined as to its compliance with some general "laws" of perception and in light of its relationship to the second most important aspect of auditory perception, speech perception. !!. WHY ARE SCALES NECESSARY? The two right-most columns of Table I give the frequency ratios, and their val- ues in the logarithmic "cent" metric, for the musical intervals that are contained in the scale that constitutes the standard tonal material on which virtually all Western music is based: the 12-tone chromatic scale of equal temperament. A number of assumptions are inherent in the structure of this scale. The first is that of octave equivalence, or pitch class. The scale is defined only over a region of one octave; tones separated by an octave are assumed to be in some respects musically equiva- lent and are given the same letter notation (Table I, column 3). The second is that pitch is scaled as a logarithmic function of frequency. The octave is divided into *This chapter is dedicated to the memory of W. Dixon Ward. Copyright 1999 by Academic Press. The Psychology of Music, Second Edition 215 All rights of reproduction in any form reserved. 0 g.o I~ r O0 O0 ~ 0'1 ~ ,...-4 O~ ~0 O0 O0 O0 O0 O0 ~~ ~ "~ ~ ~~~ ~~ ~~ ~ ~%~~~ .. Z o ~ I~ (~ ~ ~0 oo ,-~ ~ ~ ~ ~ ~0 ~ ~. ~ ,-~ oo % ~o o .. ~- 4. ~. ~" ~. ~ ~. 4. ~" .. ~ 0 .,-~ .,.~ 0 .s ei O~ 0 0 ILl J m ~.- .~ .~ -~o ~ ~ o .~.. .~ I- 7. INTERVALS, SCALES, AND TUNING 217 12 logarithmically equal steps (semitones), the presumption being that all inter- vals that contain the same number of semitones are perceptually equivalent, re- gardless of the absolute frequencies of the tones composing the intervals. The names of the 13 intervals (including unison) that are contained in the 12-tone chro- matic scale are given in column 1 of Table I. Given that present Westem music uses a relatively small set of discrete pitch relationships, an obvious question occurs: Is this use of discrete scale steps univer- sal? That is, are there musical cultures that use continuously variable pitches? The evidence from ethnomusicological studies indicates that the use of discrete pitch relationships is essentially universal. The only exceptions appear to be certain primitive musical styles, for example, "tumbling strains" (Sachs & Kunst, 1962), or "indeterminate-pitch chants" (Maim, 1967), which are found in a few tribal cultures. Of course, pitch glides--glissandos, portamentos, trills, etc.mare used as embellishment and ornamentation in most musical cultures. However, these embellishments are distinct from the basic scale structure of these musics. The concept of octave equivalence, although far from universal in early and primitive music (e.g., Nettl, 1956; Sachs & Kunst, 1962), also seems to be common to more advanced musical systems. A related question follows: does the 12-tone chromatic scale represent a norm or a limit to the number of usable pitch relationships per octave? A number of Westem composers have proposed quarter-tone (i.e., 24 approximately equal in- tervals per octave) and other microtonal scales, but these scales have not gained wide acceptance. Numerous cultures have scales that contain fewer than 12 notes per octave. There are, however, apparently only two musical cultures that, in theory at least, use more than 12 intervals per octave: the Indian and the Arab- Persian. Both musical systems of India (Hindustani and Karnatic) are, according to tradition, based on 22 possible intervals per octave. They are not, however, equal, or even approximately equal intervals. The basic structure of the inclusive scale is essentially the same as that of the Western 12-interval chromatic scale, and the microtones (shrutis) are (theoretical) slight variations of certain intervals, the exact values of which are dependent on the individual melodic framework (raga) being played. There is evidence that in actual musical practice these microtonal variations of intervals are not played as discrete intervals but are denoted by a purposefully induced variability in intonation such as a slow vibrato (Callow & Shepherd, 1972; Jhairazbhoy & Stone, 1963). The one system that may use true quarter tones (i.e., intervals that bisect the distance between the Western chromatic intervals) is the Arab-Persian system. In this system, there are various claims as to the number of possible intervals (rang- ing from 15 to 24) and some controversy as to whether they are true quarter tones or, as in the Indian system, merely microtonal variations of certain intervals (e.g., Zonis, 1973). The limited data on measured intonation in this system are ambigu- ous as to how accurately these quarter tones are actually produced (e.g., Caron & Safvate, 1966; Spector, 1966). It is clear, however, that neither the Indian nor 2 1 8 EDWARD M. BURNS Arab-Persian scales are chromatically microtonal. That is, the quarter tones (or microtones) are never played contiguously but only as alternative versions of cer- tain larger intervals. Thus, the evidence indicates that no musical cultures exist wherein the smallest usable interval is smaller than the semitone. Although this, of course, does not prove that scales containing intervals smaller than semitones, or composed of more than 12 intervals, are not possible, it is certainly suggestive. What then is the probable basis for the use of a discrete number of tones in scales and for the appar- ent limitation on the number of tones per octave? In pairwise discrimination tasks, normal listeners can discriminate literally hundreds of different frequencies over the range of an octave, and well over a thousand over the range of frequencies used in music, so the discriminability of individual tones is clearly not a limitation. In keeping with Miller's (1956) "magical number 7 + 2" limit for identification of stimuli along a unidimensional psychophysical continuum, most listeners are very poor at identifying individual tones varying only in frequency, that is, they can only place the tones over the entire frequency range of human heating into about five categories with perfect consistency (e.g., Pollack, 1952). However, this limi- tation is irrelevant for melodic information in music. As is overwhelmingly evi- dent from both everyday musical experience, from music theory, and as has been shown in formal experiments (e.g., Attneave & Olson, 1971; White, 1960), me- lodic information in music is mediated by the frequency ratio relationships among tones (i.e., the musical intervals) not by their absolute frequencies. Although it is true that the absolute frequencies of the tones composing the equitempered scale have been fixed (ranging from Co= 16.4 Hz to B8 = 7902.1 Hz, with A4, the desig- nated tuning standard, at 440 Hz), this has been done primarily for the purpose of standardizing the manufacture of fixed-tuning instruments and has virtually no relevance for the perception of music (with the exception of the relatively few individuals who possess absolute pitch; see Chapter 8, this volume). The most likely reason for the adoption of a relatively small number of discrete intervals as the tonal material for music is that discretization or categorization is a typical, if not universal, strategy used by animals in order to reduce information overload and facilitate processing when subjected to the high-information-content signals and/or high information rates from a highly developed sensory system (e.g., Estes, 1972; Terhardt, 1991). An obvious example is the processing of speech information by humans, wherein each language selects a relatively small portion of the available timbral differences that can be produced by the vocal tract as information-carrying units (phonemes) in the language. A related reason for discrete scales, and another obvious analogy with speech perception, lies in the social aspect of music. Music first developed as, and still largely remains, a social phenomenon associated with religious or other rituals that, like language, necessi- tated an easily remembered common framework. Given this information-transfer-based speculation on the origin of discrete- tone musical scales, we might further speculate that the apparent limit of 12 semi- 7. INTERVALS, SCALES, AND TUNING 219 tones per octave is, in fact, related to the "channel capacity" for information trans- mission as delineated by the 7 + 2 rule. According to this empirically based rule, subjects can place stimuli along a unidimensional psychophysical continuum into only five to nine categories with perfect consistency, which corresponds to about 2.3 bits (log2 7) of information. This phenomenon is remarkable for its robustness, having been observed for a wide variety on continuua, in all five sensory modali- ties (Miller, 1956). Although we shall see in Section III that musicians' channel capacity