PARTCH'S ERRORS

by

B. McLaren

Harry Partch stands as one of the most influential innovators in 20th century music. However, his book Genesis of A Music is riddled with psychoacoustic errors.

The gross pervasive errors of fact in his writings so flagrantly contradict so many documented facts about the human ear that they offer a minefield for prospective non-12 composers.

Without exception, his statements on psychoacoustics embody the exact opposite of the truth. In short, if Partch said it about the human ear, you can count on the reverse being factually correct.

Why was Partch so reliably and so completely wrong when it came to matters of musical perception?

For one thing, he listened exclusively to harmonic-series timbres when tuning sustained tones on his instruments. For another, Partch does not appear to have been familiar with any of the psychoacoustic literature published after 1945--the Stone Age of psychoacoustics.

Partch's information was woefully out of date. He penned "Genesis of a Music" between 1928 and 1947, an era when ignorance of the ear/brain system reigned. In the early 1930s no one had heard of the critical band, no one realized that the ear hears stretched octaves and stretched fifths and stretched thirds as "just," and "pure," while hearing purportedly "natural" small-integer-ratio fifths and thirds and octaves as "too narrow" and "impure" and "out of tune." Prior to 1945, no one imagined that the rules of harmony changed radically as soon as sustained timbres became inharmonic.

Thus Partch completely missed out on the enormous post-World War II explosion of knowledge in modern psychoacoustics.

The modern era of ear/brain research dates from the period 1954 onward--in particular, from Max Mathews' seminal paper in Bell Labs Technical Journal in 1959. Because computers can generate sounds with any desired spectrum and amplitude, and because the computer peerlessly analyzes acoustic sounds in precise detail, our knowledge of the ear/brain system increased explosively during the late 1960s and throughout the seventies.

Partch blithely ignored all of the advances in musical psychoacoustics from 1928 through 1974. And so his pervasively false claims about the behavior of the human ear serve as a warning to prospective just intonation composers: before you embarrass yourself by making provably untrue pronouncements about the human ear/brain system, it's best to do some research first and get your facts straight.

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"The number of cycles--per second--determines the pitch of the tone." Harry Partch, "Genesis of a Music," 2nd Ed., Da Capo Press: New York, 1974, pg. 76.

"For sine tones in particular, the pitch does not depend exclusively on the frequency; the intensity is also relevant." - Johan Sundberg, "The Science of Musical Sounds," Academic Press, Inc.: San Diego, 1991, pg. 45.

"The pitch of pure tones depends not only on frequency, but also on other parameters such as sound pressure level." H. Zwicker & . Fastl, "Psychoacoustics: Facts and Models," 1993, pg. 105.

"The pitch of harmonic complex tones depends on level. Figure 4.9 shows the pitch shift of a complex tone with a 200-Hz fundamental frequency as a function of its level. An increasing negative pitch shift shows up with increasing level of the complex tone." H. Zwicker & E. Fastl, "Psychoacoustics: Facts and Models," 1993, pg. 111.

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"After hearing a `major third' on the organ or piano or some other instrument with tempered intonation, this interval becomes fixed in the mind as a pretty poor consonance, at least by some comparisons." Harry Partch, "Genesis of a Music," 2nd. Ed., pg. 6.

"As will be seen and demonstrated in the recording, there is...reason to doubt the musical merit of a superparticular ratio as low as 4:5." --Fritz Kuttner & J. Murray Barbour, program notes, Musurgia Records' series on the History of Musical Theory. In "Theory and Practice of Just Intonation."

"A small difference can be seen for the fifth, the dyad version being close to the barbershop result of 705 cents. One can also see that these musicians preferred the Pythagorean major third to the just." - Johan Sundberg, "The Science of Musical Sounds," pg. 101.

"The average extent of each of the five intervals approximated its theoretical magnitude in Pythagorean intonation. The unquestionably crucial cases were those of the minor second, the minor third, and the major third, where Pythagorean intonation differed significiantly from natural and equally tempered intonation. Since the performers on the average apprixmated the Pythagorean and larger natural scale values of the major second, and did not significantly differentiate between the two theoretical varieties of this interval in the natural scale, it may be justifiable to assume that the criterion in actual performance here also was Pythagorean intonation." Greene, Paul C. "Violin intontion," Journal of the Acoustical Society of America, Vol, 9, 1937, pp. 43-44

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"This 2 to 1 relationship is a constant one...the fact is that nature does not offer one tone and its doubling (200 to 400) as a given quality of relationship, and the same quality of relationship in two tones which are not a ratio of doubling (200 to 600, for example)" Harry Partch, "Genesis of a Music," 2nd Ed., Da Capo Press: New York, 1974, pg. 77.

"...a soft sine tone at 300 Hz may sound as a pure octave of a loud sine tone at 168 Hz. The mathematically pure octave, however, has the frequency of 150 Hz. The tone that sounds as a pure octave is 12% too high. This means that mathematically it is a minor seventh [300/188 = 10.04 semitones]! This is a good argument for avoiding confusion of perceptual and physical entities." - Johan Sundberg, The Science of Musical Sounds," Academic Press, Inc.: San Diego, 1991, pg. 46.

"If a frequency of 8 kHz is chosen for f₁, subjects produce for the sensation of `half pitch' not a frequency of 4 kHz, but a frequency of about 1300 Hz." E. Zwicker and H. Fastl, Psychoacoustics: Facts and Models," 1993, pg. 103.

"Dowland has reported that measurements of Western and non-Western fixed pitch instruments support Ward's conclusion that the perceptual octave is some 15 cents larger than the physical or mathematical octave. Western musical practice supports these conclusions (play sharp in higher octave). Balinese gamelan tunings take advantage of this apparently widespread characteristic of pitch perception to create a multi-octave beating complex in their fixed pitch instruments." [Erickson, R., "Timbre and the Tuning of the Balinese Gamelan," Soundings, Vol. 15, pg. 100, 1984]

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"The ear informs us that tones which are in small-number proportion, say in the relation of 2 to 1, are strong, clear, powerful, consonant." Harry Partch, "Genesis of a Music," 2nd. Ed., Da Capo Press: New York, 1974, pg. 86.

"It is quite remarkable that musicians seem to prefer too wide or `stretched' intervals... In the case of octaves, the craving for stretching has been noticed for both dyads and melodic intervals. the amount of stretching preferred depends on the mid frequency of the interval, among other things. The average for synthetic, vibrato-free octave tones has been found to be about 15 cents. Thus subjects found a just octave too flat but an octave of 1215 cents [= 2.017403967581/1 = approx. 115/57 or 117/58] just." -- Johan Sundberg, "The Science of Musical Sounds," Academic Press, Inc.: San Diego, 1991, pg. 103.

"The experimental results very convincingly show that, on the average, singers and string players perform the upper notes of the major third and the major sixth with sharp intonation (Ward 1970)...The same experiments revealed that also fifths and fourths and even the almighty octave were played or sung sharp, on the average! (A reciprocal effect exists. Pure octaves are consistently judged by musicians to sound flat!) Rather than revealing a preference for a given scale (the Pythagorean), these experiments point to the existence of a previously unexpected universal tendency to play or sing sharp all musical intervals." - Juan Roederer, "Introduction to The Physics and Psychophysics of Music," 1973, pg. 155.

"These results confirm the view that, for simple tones, tonal consonance is related to interval width. The experiment does not support the hypothesis that the human ear is provided with some sort of frequency-ratio detector." Plomp, R., W. A. Wagenaar and A. M. Mimpen. "Musical Interval Recognition with Simultaneous Tones," Acustica, Vol. 29, 1973, pp. 100-109

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"Consequently, these statements can be conclusively made; the ear consciously or unconsciously classifies intervals according to their comparative consonance or comparative dissonance; this faculty in turn stems directly form the comparative smallness or comparative largeness of the numbers of the vibrational ratio..." Harry Partch, "Genesis of a Music," 2nd. Ed., pg. 87.

"In general, the observation that simple tones can be differentiated according to ratio simplicity is largely an artifact, due to the correlation between ratio simplicity and interval width." Levelt, W. J. M., J. P. van de Geer and R. Plomp, "Triadic Comparisons of Musical Intervals," the British Journal of Mathematical and Statistical Psychology," Vol. 19, part 2, 1966, pp. 163-179

"In fact, the interval that listeners accept as the best octave is not the interval with a precisely two-to-one frequency ratio but one with a slightly larger and thus numerically much more complex ratio (Ward, 1954; also see Burns, 1974; Dowling, 1973b, in press; Elfner, 1964; Sundberg & Lindqvist, 1973; Ward & Martin, 1961 Chapter 10, this volume). And the frequencies of the tones actually produced by musicians or preferred by listeners as representative of other musical intervals, as well, often fail to conform to the predicted most simple ratios (Seashore, 1938; Ward & Martin, 1961; Van Esbroeck & Montfort, cited in Risset, 1978)." [Cazden, Norman, Musical Consonance and Dissonance: A Cultural Criterion," Journal of Aesthetics and Art Criticism, Vol. 4, No. 1, 1945, pg. 9]

"Therefore it must be concluded that even just or pythagorean intonation cannot be considered as ideal. Rather, optimum intonation of a diatonic scale probably depends on the structure of the actual sound in the same manner as has been previously discussed with respect to tempered scales." -- E. Terhard and S. Zick, "Evaluation of the Tempered Tone Scale In Normal, Stretched, and Contracted Intonation," Acustica, Vol. 32, 1975, pg. 273.

"The degree of consonance depends on the quality or spectrum of the component tones, i.e., the relative intensity of dissonant vs. consonant upper harmonics." - Juan Roederer, "Introduction to the Physics and Psychophysics of Music," 1973, pg. 143.

"It became clear that the fifth (2:3) is not always a consonant interval. A chord of two tones that consists of only odd harmonics, for example, shows much worse consonance at the fifth (2:3) than at the major sixth (3:5) or some other frequency ratios. This was proved true by psychological experiments carried out in another institutes (Sensory Inspection Committee in the Japan Union of Scientists and Engineers) with a different method of scaling. Thus, the fact warns against making a mistake in applying the conventional theory of harmony to synthetic musical tones that can take on a variety of harmonic structures." Akio Kameoka and Mamoru Kuriyagawa, "Consonance Theory Part II: Consonance of Complex Tones and Its Calculation Method," Journal of the Acoustical Society of America, Vol. 45, No. 6, 1969, pg. 1460.

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"Long experience in tuning reeds on the Chromelodeon convinces me that it is preferable to ignore partials as a source of musical materials. The ear is not impressed by partials as such. The faculty--the prime faculty--of the ear is the perception of small-number intervals, 2/1, 3/2, 4/3, etc., etc., and the ear cares not a whit whether these intervals are in or out of the overtone series." Harry Partch, "Genesis of a Music," 2nd. Ed., pg. 87.

"In 1987 IPO issued a wonderful disc by Houtsma, Rossing and Wagenaars...illustrating the effects of a moderate stretching...of scale frequencies and/or partial spacings. Part of a Bach chorale is played with synthesized tones. When neither scale nor partial frequencies are stretched, we hear the intended harmonic effect. When the scale is unstretched but the partial frequencies are stretched, the music sounds awful. Clearly, intervals in the ratio of small whole numbers are in themselves insufficient to give Western harmonic effects." -- John R. Pierce, "The Science of Musical Sound," 2nd Ed., 1992, pp. 91-92.

"Clearly the timbre of an instrument strongly affects what tuning and scale sound best on that instrument." Wendy Carlos, "Tuning: At the Crossroads," Computer Music Journal, 1987.

"For two thousand years music theorists searched for a `natural' explanation of musical pitch systems and syntax. Their point of departure was, as a rule, some sort of acoustical data--the lengths of vibrating strings, the overtones series, or some other natural property of sound. Using such data, an attempt was made to show that this or that system as natural--and hence, by extension, necessary and valid. But the development of new tonal systems in the West, the study of the history of Western music, and research in comparative musicology made it clear that musical styles are not natural forms of communication, but are learned and conventional." -- Leonard B. Meyer, "Music, the Arts and Ideas," 1967, p. 73.

"Most instruments in our music culture produce harmonic spectra, as mentioned. However, in the contemporary computer-aided electroacoustic music studios, is not a necessary constraint any longer. One would then ask if this does not open up quite new possibilities also with respect to harmony. If one decides to use one particular kind of inharmonic spectra for all tones, it should be possible to tailor a new scale and a new harmony to this inharmonicity." - Johan Sundberg, "The Science of Musical Sounds," Academic Press, Inc.: San Diego, 1991, pg. 100.

"By using a digital computer, musical tones with an arbitrary distribution of partials can be generated. Experience shows that, in accord with Plomp's and Levelt's experiments with pairs of sinusoidal tones, when no two successive partials are too close together such tones are consonant rather than dissonant, even though the partials are not harmonics of the fundamental. For such tones, the conditions for consonance of two tones will not in general be the traditional ratios of the frequencies of the fundamentals... [The 8-TET scale] is , of course, only one example. If many possible scales made up of tones whose upper partials are not harmonics of the fundamental and having unconventional intervals, which nonetheless can exhibit consonance and dissonance comparable to that obtained with conventional musical instruments (which have harmonic partials) and the diatonic scale. It appears that, by providing music with tones that have accurately specific but nonharmonic partial structures, the digital computer can release music from the constraint of 12 tones without throwing consonance overboard." John R. Pierce, "Attaining Consonance in Arbitrary Scales," Journal of the Acoustical Society of America, 1966, p. 249.

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"That the inception of this scale was a significant insight is abundantly supported by the fact that the first three of its ratios [2/1, 3/2, 4/3] are the most important scale degrees in nearly every musical system worthy of the name that the world has ever known." Harry Partch, "Genesis of a Music," 2nd. Ed., pg. 87.

"As to whether the interval 3:2 is common to all of the world's musical systems, as has occasionally been claimed, Fritz Kuttner asserts that the "fifth" in Chinese music is 20 to 30 cents flat. It is apparently nearly as flat in Siamese music..." M. Joel Mandelbaum, "Multiple Division of the Octave and the Tonal Resources of 19-Tone Temperament," 1960, p. 16.

"Two theoretical systems evolved in China, one derived from the Cyclic Pentatonic and the other from the division of string lengths. They are found combined in the highest form of Ch'in music. (..) Methods of arriving at these fifths included the use of twelve tubes... The fifths produced by these tubes were small compared to Western fifths. Various musicologiests place them between 670 and 680 cents as compared to the Just fifth of 702 cents." [Lentz, Donald A., The Gamelan Music of Java and Bali, 1965, pg. 27]

"There are...a number of musical cultures that apparently employ approximately equally tempered 5- and 7-interval scales (i.e., 240 and 171 cent step-sizes, respectively) in which the fourths and fifths are significantly mistuned form their natural values. Seven-interval scales are usually associated with Southeast Asian cultures (Malm, 1967). For example, Morton (1974) reports measurements (with a Stroboconn) of the tuning of a Thai xylophone that `varied only + or - 5 cents' from an equally tempered 7-interval tuning. (In ethnomusicological studies measurement variability, if reported at all, is generally reported without definition.) Haddon reported (1952) another example of a xylophone tuned in 171-cent steps from the Chopi tribe in Uganda. The 240-cent step-size, 5-interval scales are typically associated with the `gamelan' (tuned gongs and xylophone-type instruments) orchestras of Java and Bali (e., Kunst, 1949). However, measurements of gamelan tuning by Hood (1966) and McPhee (1966) show extremely large variations, so much so that McPhee states: `Deviations in what is considered the same scale are so large that one might with reason state that there are as many scales as there are gamelans.' Another example of a 5-interval, 24--cent step tuning (measured by a stroboconn, 'variations' of 15 cents) was reported by Wachsmann (1950) for a Ugandan harp. Other examples of equally tempered scales are often reported for pre-instrumental cultures... For example, Boiles (1969) reports measurements (with a Stroboconn, `+ or - 5 cents accuracy') of a South American Indian scale with equal intervals of 175 cents, which results in a progressive octave stretch. Ellis (1963), in extensive measurements in Australian aboriginal pre-instrumental cultures, reports pitch distributions that apparently follow arithmetic scales (i.e., equal separation in Hz).

"Thus there seems to be a propensity for scales that do not utilize perfect consonances and that are in many cases highly variable, in cultures that either are pre-instrumental or whose main instruments are of the xylophone type. Instruments of this type produce tones who partials are largely inharmonic (see Rossing, 1976) and whose pitches are often ambiguous (see de Boer, 1976)." [Burns, E. M. and Ward, W. D., "Intervals, Scales and Tuning," in The Psychology of Music, 1982, ed. Diana Deutsch, pg. 258]

"The acoustically 'perfect' consonances are the rule in some musics, but are not inevitable foundations, for nothing close to the ratio 3:2 is found in certain Javanese and Siamese scales. Intervals which bear no resemblance to any in our diatonic system form melodies which to their users seem 'instinctive' and self-evidently natural. Harmony in seconds, which we would consider flagrantly dissonant, seems to be practiced in the South Sea Islands and elsewhere. In the Icelandic 'Tvisoengvar' the third appears to be treated as a dissonance. It could hardly be maintained that the natural laws presumed to determine harmony vary thus geographically in their application." [Cazden, Norman, Musical Consonance and Dissonance: A Cultural Criterion," Journal of Aesthetics and Art Criticism, Vol. 4, No. 1, 1945, pg. 9]

"3. Preference ratings that listeners give for pairs of tones that do stand in the various simple frequency ratios tend to depart systematically from the orderings of those ratios predicted on the basis of what is usually taken to be their numerical simplicities-with, for example, the major third (5:4) often preferred to the numerically simpler perfect fourth (4:3) (e.g., Butler & Daston, 1968; Krumhansl & Shepard, 1979; see also Davies, 1978, p. 158; Fuda, 1975; Van de Geer, Levelt, & Plomp, 1962).

"The mean ratings of consonance indicate that there are large differences between consonance as defined by earlier judgments of musicians steeped in beat lore and by laymen in the present experiment. The octave, that most consonant of intervals to the music theoretician, got only an average consonance rating of 4.6, being exceeded by 9 others. The intervals judged most consonant, at 5.7, were the ratios 5:7 and 5:8 -- a 583-cent tritone and an 814-cent minor sixth, respectively. The cellar (2.0) was occupied by the four smallest intervals (8:9, 9:10, 10:11, and 11:12 -- 204, 182, 165, and 150 cents, respectively). ...It seems clear that these results do not agree with the rules of consonance laid down by the small-integer proponents. It is therefore unnecessary, in explaining consonance, to appeal to neuromythological structures such as a series of brain circuits tuned to each frequency, so that 'in the case of the fifth, 3:2, every third wave of the higher note and every second wave of the lower could use the same circuit, and so on' (Boomsliter and Creel, 1961, pg. 20)."[Ward, W.D., "Musical Perception," in Foundations of Modern Auditory Theory, ed. Jerry V. Tobias, Vol. I. Academic Press: New York, 1970, pg. 435]

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"In any given range of pitch the comparative consonance of an interval is determined by the relative frequency of the wave period in the sounding of the interval." Harry Partch, "Genesis of Music," 2nd. Ed., Da Capo Press: New York, 1974, pg. 151.

"Systematic measurements show that people tend to find that an interval traditionally classified as `consonant' sounds progressively dissonant the farther down in the bass it is played...In the very low bass, even the octave sounds dissonant!" Johan Sundberg, "The Science of Musical Sounds," Academic Press, Inc.: San Diego, 1991, pg. 73.

"Even the order in which two instruments define a musical interval is relevant. For instance, if a clarinet and a violin sound a major third, with the clarinet playing the lower note, the first dissonant pair of harmonics will be the 7th harmonic of the clarinet with the 6th harmonic of the violin (because only odd harmonics of the clarinet are present). This interval sounds smooth. If, on the other hand, the clarinet is playing the upper tone, the 3rd harmonic of the latter will collide with the 4th harmonic of the violin tone, and the interval will sound harsh.'" - Juan Roederer, "Introduction to The Physics and Psychophysics of Music," 1973, pg. 143.

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"[The ear] can determine almost immediately, exactly or approximately, the relationship in vibrations per second, or cycles, of two tones sounded simultaneously; it can say instantly whether the two tones are in the correct ratio (in tune) or not in the correct ratio (out of tune)." Harry Partch, "Genesis of a Music," 2nd. Ed., pg. 86.

"Time is a fundamental limitation on the ability to perceive pitch. When a tone sounds, a certain time must pass before the listener develops a sensation of pitch. The length of this time depends upon the frequency of the tone. In order to establish a pitch, a listener must receive a number of cycles of a tone. Thus it takes a longer time to perceive the pitch of a tone at a lower frequency because that tone has a longer period. For example, a tone must last at least 40 msec at 100 Hz whereas a tone at 1000 Hz must last only 13 msec." - Charles Dodge & Thomas A. Jerse, "Computer Music: Synthesis, Performance and Composition," 1985, pg. 46.

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"The handling and consideration of tones is, by virtue of their vibrations, an exact mathematical process. If a tone makes 200 cycles, the 2/1 ("octave") above it makes 400 cycles, a doubling of 200, or 200 more cycles." [Genesis of a Music, 2nd. Ed., 1974, pg. 77]

"In fact, the interval that listeners accept as the best octave is not the interval with a precisely two-to-one frequency ratio but one with a slightly larger and thus numerically much more complex ratio (Ward, 1954; also see Burns, 1974; Dowling, 1973b, in press; Elfner, 1964; Sundberg and Lindqvist, 1973; Ward & Martin, 1961, Chapter 10, this volume). And the frequencies of the tones actually produced by musicians or preferred by listeners as representative of other musical intervals, as well, often fail to conform to the predicted most simple ratios (Seashore, 1938; Ward & Martin, 1961; Van Esbroeck & Montfort--cited in Risset, 1978). [R. N. Shepard, "Structural Representations of Musical Pitch," pg. 347, in "The Psychology of Music," Ed. Diana Deutsch, 1982]

"It is instructive to examine a few intervals between pure tones that occur in different octaves. If C4 (262 Hz) and G4 (392 Hz) are sounded together, the difference frequency is 130 Hz, which is 40 percent greater than the critical bandwidth (approximately 90 Hz in this octave). Thus they sound consonant. However, an octave lower on the scale, a perfect fifth is not quite so consonant, since the frequency difference between C3 (131 Hz) and G3 (196 Hz) is 65 Hz, which is less than the critical bandwidth. Another octave or so lower, the frequency difference approaches 1/4 of the critical bandwidth, the criterion for maximum roughness or dissonance. So the degree of dissonance of the interval between two pure tones is strongly dependent on their location on the musical scale." [Rossing, "The Science of Musical Sound," 2nd. Ed., 1990, pg. 157]

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"There is just one element in all the study of music which does not change: the ratio. And the reason the ratio does not change is simply and wholly because physiologically the ear does not change except over a period of thousands or millions of years." [Partch, "Genesis of a Music," 2nd. Ed., 1974, pg. 97]

"Experiments with inharmonic partials (Slaymaker, 1970; Pierce, 1966) have shown that consonance is indeed dependent on the coincidence of partials and not necessarily on the simple frequency ratio between the fundamental frequencies (which is usually the cause of the coincidence)." [R.A. Rasch & R. Plomp, "The Perception of Musical Tones," pg. 21, in "the Psychology of Music," Ed. Diana Deutsch, 1982]

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"The scale of musical intervals begin with absolute consonance (1 to 1) and gradually progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd number of their ratios increase." [Partch, "Genesis of a Music," 2nd. Ed., 1974, pg. 87]

"As Risset (1978) has remarked, ratios that are very close to simple ratios (e.g., 29,998:20,000) are highly complex and yet indiscriminable from the corresponding simple ratios (in this case 3:2)... Preference ratings that listeners give for pairs of tones that do stand in the various simple frequency ratios tend to depart systematically from the orderings of those ratios predicted on the basis of what is usually taken to be their numerical simplicities--with, for example, the major third (5:4) often preferred to the numerically simpler perfect fourth (4:34) (e.g., Butler & Daston, 1968; Krumhansl & Shepard, 1979; see also Davies, 1978, p. 158; Fyda, 1975; Van de Geer, Levelt & Plomp, 1962). [R. N. Shepard, "Structural Representations of Musical Pitch," pg. 347, in "The Psychology 9f Music," Ed. Diana Deutsch, 1982]

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Again let me point out that Partch was a genius. This does not change the fact that almost everything he wrote about psychoacoustics was dead wrong.

Still--even though Partch's arguments in favor of just intonation are based on a set of numerological canards and psychoacoustic falsehoods--this does NOT make Partch's tuning system useless or ugly-sounding or pointless. Just intonation remains a beautiful and musically interesting "angle of hearing." Much uniquely interesting music can be composed using just intonation; this alone is reason to use JI in composition, and excellent reason to listen to music composed with small integer ratios.

Just intonation offers many advantages which other tuning systems do not: for example, each transposition of a JI melody to another pitch produces a new and unique distribution of intervals, and this makes iterated sequences of melodic notes, which would sound repetitious in an equal temperament, extremely interesting in just intonation.

Using small integer ratios to generate a scale also allows the composer to produce startling and dramatic effects when modulating to another key.

Integer ratios also allow a composer to climb the harmonic series directly, exploring radically new and non-western musical intervals: the 11/8, the 13/8, the 17/11, the 19/16, the 31/16, and so on ad infinitum.

Most of all, just intonation allows the composer to use exotic and hitherto-undreamt-of mathematical techniques to generate radically new tunings of breathtaking beauty. For example, Erv Wilson's combination-product-set scale generation techniques, John Chalmers' tritriadic methods, juggler numbers, Farey series, Pascal's Triangle, hailstone numbers, and many other numerical series and methods.

Just intonation also suffers many unique disadvantages. Modulation to other keys demands a scale with many more notes than the usual 12 of western music. There are also no rule-books or instructional texts for dealing with ratios made up of higher members of the harmonic series: is the 19/16, for example, a consonance or a dissonance? How ought the 7:9:11 chord to be used in a musical context? And so on.

While it remains apodictically true that non-equal non-just tunings better fit the human tendency to hear stretched intervals as "pure" than other types of tunings, this does not render other types of tunings musically invalid. As John R. Pierce, Wendy Carlos, myself, Bill Sethares, Johan Sundberg and many others have pointed, consonance and dissonance are determined by musical context, the tuning used, and the precise spectral and phase characteristics of the timbres in question.

Witch-hunts for "good" and "bad" tunings are futile. All three general classes of tunings [just, equal tempered, non-just non-equal-tempered] sound dramatically different, and provide equally limitless musical resources for the xenharmonic composer.

While Harry Partch's statements about psychoacoustics uniformly conflict with the facts, this does not mean that his music is worthless. Forget Partch's book...celebrate his music. The music is what matters.

APPENDIX

Predictably, this article will spark the usual firestorm of protests from just intonation enthusiasts unwilling to accept the proven facts of the human ear/brain system. Such protests are symptoms of the appalling ignorance of today's purportedly "educated" musicians. Contemporary musicians are not to blame: their ignorance results from the disgracefully inadequate state of musical "education" throughout the western world. In so-called institutions of "higher learning," music is still taught as an intellectually toxic witch's brew of numerology, superstition and acoustic fairytales. The startling and fascinating results of the psychoacoustic research carried out over the last 40 years are uniformly ignored by textbooks on so-called "contemporary" music, with the inevitable consequence that graduates from the world's most presitigious musical institutions remain shockingly ignorant of how their own ears work

Those who doubt these statements are advised to study the following extremely abbreviated list of references:

Allen, D. "Octave discriminability of musical and non-musical subjects," Psychonomic Science, Vol. 7, 1967, pp. 421-422

Appleton, J., "Machine Songs III: Music in the Service of Science--Science in the Service of Music," Computer Music Journal, 13(3), Fall 1992, pp. 17-21

Attneave, F. and Olson, R. "Pitch as a medium: A new approach to psychophysical scalaing," Am. J. of Psychology, Vol. 84, 1971, pp. 146-166

Balzano, G. "On the bases of similarity of musical intervals: A chronometric analysis," J. Acoust. Soc. Am., Vol. 61, 1977, S51(A)

Balzano, G. J. "The pitch set as a level of description for studying musical pitch perception," in M. CLynes (Ed.,), Music, mind, and brain: the neurophysiology of music. New York: Plenum, 1982, pp. 321-351

Bastiaans, M. J., "Gabor's expansion of a signal into Gaussian elementary signals," Proceedings of the IEEE, Vol. 68, 1980, pp. 538-539

Becker, Judith. "Is western art music superior?" Musical Quarterly, Vol. 72, No. 2, 1986, pp. 341-359

Bekesy, G. von, "Three Experiments Concerned with Pitch Perception," J. Acoust. Soc. Am., 35, 1963, pp. 1722-1724.

Bekesy, G. von, "Hearing Theories and Complex Sounds," J. Acoust. Soc. Am., Vol. 35, No. 4, pg. 589, April 1963

Bekesy, G. von, "Ueber die nichtlinearen Verzerungen des Ohres," Ann. Phys., Vol. 20, pp. 809-827, 1934

Bekesy, G. von, "Sensations on the Skin Similar to Directional Hearing, Beats, and Harmonics of the Ear," J. Acoust. Soc. Am., Vol. 29, pp. 489-501, 1957

Bekesy, G. von, "Ueber akustische Rauhigkeit, " Z. techn. Phys., Vol. 16, pp. 276-282, 1935

Bekesy, G. von, "Concerning the Fundamental Component of Periodic Pulse Patterns and MOudlated Vibrations Observed on the Cohlear Model with Nerve Supply," J. Acooust. Soc. Am., Vol. 33, 1961, pp. 888-896

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