source file: mills2.txt Date: Wed, 19 Jun 1996 09:45:41 -0700 Subject: Post from Brian McLaren From: John Chalmers From: mclaren Subject: a possible misunderstanding of the psychoacoustic data --- In Tuning Digest 708, Marion McCoskey stated: "[mclaren's] assertion that people don't prefer small whole number ratios seems to me to rest on a somewhat shaky basis. Upon watching archers shooting a targets, do we assume from the distribution of their arrows that they don't prefer hitting the bull's eye?" I'd like to point out why Marion's analogy is out of kilter. First, Marion's analogy presumes that the psychoacoustic test results for interval preference are distributed like arrows that miss a bull's eye. This is incorrect. Arrows that miss a bull's eye have a symmetric radical distribution centered around the bull's eye. The mean value of the arrows' distribution is thus the center of the bull's eye. But all the psychoacoustic tests done over the last 130 years consistently show that the mean value for melodic interval preference is skewed much higher than the target small integer ratios--even though test subjects consistently *hear the intervals as "pure" and "just."* Vertical interval preference mean values are skewed slightly less high, but still significantly higher than the values predicted by the small integer ratio theory of consonance. So let's change Marion McCoskey's analogy to reflect the actual facts. In that case, all the arrows would always be grouped toward the top half of the bull's eye--but when the archers looked at them, they would see *all* the arrows as being exactly dead center in the bull's eye. The archers could walk up, examine the arrows closely, and they'd still report all the arrows as being dead center of the bull's eye. This correct analogy is bizarre and counterintuitive enough that it captures the striking nature of the psychoacoustic results. But these are not the only problems with the theory of consonance according to small integer ratios. J. Murray Barbour has pointed out other evidence against the small integer theory in his 1938 paper "Just Intonation Confuted," in Music and Letters, Vol. 19, 1938, pp. 48-60, when he adduces evidence that the tonal fusion of which Helmholtz spoke does not occur in the way or for the small integer ratios Helmholtz's theory predicts: "This extraordinary conclusion received complete and necessary confirmation through experiments conducted by Guthrie and Morrill at the University of Washington. They used the impressive number of 753 student observers and tested a far greater number of intervals--forty-four within an interval slightly larger than a fifth. The averages of such an array of judgments compel respect. The best fused of all intervals tested was a major third, approximately the third of equal temperament, 2/3 comma sharper than the just third. The preferred values for fifth and minor third were similarly sharp. This report states emphatically that no preference whatever was shown for just or for equally temered intervals, and that slight resemblance exists between curves of consonance and Helmholtz's theoretical curves. "These experiments prove conclusively that Helmholtz and his followers are wrong, that singers have no predeliction for the so- called natural or just intervals, not even the major third (5/4), the interval which most surely distinguishes just intonation from equal temperament or the Pythagorean tuning." [J. Murray Barbour, "Just Intonation Confuted," Music and Letters, Vol. 19, 1938, pp. 58-63] Further on, Barbour points out: "Singers show no natural preference for these [just] intervals. They are likely to overleap a desired pitch by a comma or more, after having swooped up to it from a whole tone below. They are very fond of portamento. "Athough the great lack of accuracy revealed in artistic singing is mentioned here to destroy the myth that signers do use or can use just intonation, it by no means follows that the popular antithesis 'singers and musicians' is a truism. It is pointed out in [Carl Stumpf's] Iowa study that although 'singers in the physical sense are never on pitch, in another sense, the perceptual, they are heard as on pitch.'" [J. Murray Barbour, "Just Intonation Confuted," Music and Letters, Vol. 19, 1938, pp. 58-63] Other experiments from the 1990s show that "waves with the same period can lead to average octave judgments that differ consistently by more than half an octave, and that a substantial component of many timbre differences is actually a tone-height difference." [Patterson, R. D., Robert Milroy and Michael Allerhand, "What is the octave of a harmonically rich note?" Contemporary Music Review, 1993, Vol. 9, parts I & 2, pp. 69-81] This flagrantly violates the small-integer doctrine. You can try some simple experiments that destroy the credibility of the theory of consonance according to small integer ratios on your own if you have a precise tunable synthesizer or Csound: [1] Tune up the dyad 1: (sqrt(3/2) i.e., the ratio 1: 1.2247449 Starting at 320 Hz this would give a dyad with note frequencies of 320 Hz, 391.91836 Hz. Now tune up the dyad 45/32 and compare the two. (Starting at 450 Hz this would give a dyad with note frequencies of 320 Hz, 450 Hz.) Because the ratio 45/32 is a much smaller ratio than the irrational number (anathema to the mystic numerology of small whole numbers) square root of 3/2, the small-number consonance myth predicts that the just intonation 45/32 tritone will sound more consonant than the geometric-mean neutral third. However, the irrational-ratio neutral third clearly sounds much more consonant. [2] Tune up 11/9 and compare it to 9/8. Which sounds more consonant? The 9/8 uses smaller numbers and should please the ear more, according to the smaller integer theory of consonance. But the 11/9 is clearly a more consonant interval. (Various forum subscribers will predictably claim that the ear hears the 11/9 as a 5/4 or a 6/5 in this context. That conflicts drastically with Harry Partch's experience and prescriptions--namely, that 11-limit odentities and udentities are clearly and unmistakably heard as unique intervals with distinctive sonic characteristics entirely apart from the 5-limit odentities and udentities. Anyone who has listened to Partch's music knows this to be true. If you want to argue with Partch, you've got a real problem, because you're now arguing against the acoustic reality of limits higher than the 5-limit.) [3] Tune up a second inversion 4:5:6 chord--that is, a just major chord with a 4/3 in the bass, i.e., a 9:12:15 triad. Now compare it with a 10:12:15 minor chord. The 9:12:15 chord clearly wants to resolve to *something* because the 4/3 in the bass sounds discordant, whereas the 10:12:15 triad sits happily where it is, not requiring any resolution. Harmonic theory for more than four hundred years has recognized that a second inversion chord demands resolution--and yet the progression of a 10:12:15 triad to a 9:12:15 triad clearly sounds like a progression from consonance to dissonance, while the progression of a 9:12:15 to a 10:12:15 triad clearly sounds like a progression from dissonance to consonance. Yet this violates the myth of consonance by small integer ratios. (If you want to *really* blow away the small whole number theory, resolve a 9:12:15 second inversion major triad to an 18:22:27 root-position neutral triad. Again, a dissonance audibly goes to a consonance--yet the numbers go from small to pretty damn large.) [4] Tune up the ratio 3:5:7 and then tune up the ratio 18:22:27. Which one sounds more consonant? Clearly the neutral triad 18:22:27 sounds quite consonant, while the 3:5:7 just triad sounds quite dissonant. Again, however, this violates the theory of consonance by small integer ratios. [5] Tune up the 7:5 and then tune up the ratio 11:9. Which dyad sounds more consonant? Clearly the 7:5 tritone sounds much less consonant than the 11:9 neutral third, yet this too violates the theory of consonance by small integers. Many forum subscribers will claim that the ear approximates large ratios by smaller ones automatically--this does NOT solve the problem, for it simply puts us on the slippery slope. What's a "large" ratio? And why doesn't the ear reduce ad infinitum? Where does the process of reductive approximation stop, and why? This also sidesteps the dilemma of the just perfect fourth. In fact, the ratio 4/3 has bedeviled music theorists for thousands of years. Lippius and Zarlino classed the just perfect fourth as a vertical consonance, while Marchettus of Padua classed it as a vertical dissonance. Franco of Cologne, along with Boethius, classed the diatessaron (perfect fourth) as a consonance--yet Rameau and Mersenne classed it as a dissonance. Even today the disagreement continues, with the perfect fourth sometimes classed as a consonance, sometimes as a dissonance. My own music teacher told us "it depends on the circumstances, and it's very complicated." Basically he threw his hands up in the air. Yet according to the consonance theory of small whole numbers, the perfect just fourth should always be the third most consonant interval (2:1 first, then 3/2, then 4/3). Worse yet, Boomsliter and Creel reported in 1970 that "Incidentally, all of them [i.e., the test subjects] find the misnamed true [i.e., just intonation] scale to be dull and lifeless for all melodies. They find it inferior to the tempered scale, while the tuning chosen for each melody sounded better than the tempered scale." [Boomsliter, Paul C. and Warren Creel, "Hearing With Ears Instead of Instruments," Journal of the Audio Engineering Society, August 9170, 18(4), pg. 407] Yet Boomsliter and Creel's referential search organ produced "preferred" pitches with far more complex ratios than the simple small whole numbers of the natural (that is, JI) scale. Again, this violates (for melodies, rather than for vertical harmonies) the theory of consonance according to small integer ratios. So what have we got? Over and over again, small integer ratios are heard as dissonant both melodically and harmonically when the theory of small whole numbers predicts that they should be heard as consonant. Too, as Norman Cazden points out: "Traditional rules of harmony and counterpoint explicitly proscribe parallel successions of perfect intervals, though with the unaccountable excpetion of that same troublesome fourth when it is not in the lowest placement. That this notorious ruling relates to some more abstract quality than is presented concretely in sonorous events is plain from its inapplicability to unison or octave parallels used `merely' for 'doublings.' It is difficult to reconcile the standing proscription with the purportedly superior 'consonant' status of favored combinations. "The traditional rules for harmony and counterpoint alike regularly proscribe that dissonances be resolved, and further that they be resolved wherever possible to imperfect consonances [i.e., triads], rather than to the supposedly 'better' perfect combinations. Now the Law of Nature seems quite unable to account either for the permissibility of dissonance in music, or, if it be permitted, the need for such dissonances to be resolved. That such resolution is then best satisfied by second-rank or imperfect consonances therefore compounds the difficulties extremely. "Indeed, the recognized need for resolution appears thus to arise independetly of that Natural Law. At the least, it indicates that that Law cannot stand as the sole determinant of auditory judgment. The resolution relation also flatly contradicts the Law's evaluation of the relative status of desirable intervals. Thus at least some of the fundamental instructions for musical procedures, all of them demonstrably derived from the practices of leading composers over many centuries, contravene the premise of an inherently superior status for those special relationships expressible in the simplest number ratios. " [Cazden, Norman, "The Definition of Consonance and Dissonance," International Review of the Aesthetics and Sociology of Music, 1980, Vol. 2, pp. 123-168] Worse yet, the small whole number theory of consonance has an shady and very unsavory past. "In its most general axiomatic form, the Natural Law theory states that consonance results from the ratios of small whole numbers. "Crystallization of that axiom was ascribed by his disciples to Pythagoras, though undoubtedly its basis has been widely known to acient civilizations. However, inseparably from the strictly musical relevance of the principles, the magic of 'numbers come to life as music' has long been linked more to the marvels of mystic cosmologies and numerological speculations than to any indications bearing on the practice of music as an art. It is accordingly noteworthy that for the purposes of speculative ventures devoted to wonderment over the 'harmony of the spheres,' the musical relevance tends ever to remain remote and elementary. Typically their relevations have applied more to the raw ingredients that are later transformed into music, to tuning formulas or to the derivation and naming of scale degrees, than to musical relationships proper. So tenuous and curiously pre-musical a handling is retained down to the present day in the fanciful use of musical metaphor that inspires the still extensive literature of number mysticism, such as occasionally also entices students with musical training and experience." [Cazden, Norman, "The Definition of Consonance and Dissonance," International Review of the Aesthetics and Sociology of Music, 1980, Vol. 2, pp. 123-168] As J. Murray Barbour put it, the theory of small whole number ratios "has always been a beautiful theory. Its devotees have been drawn chiefly from the ranks of mystics and philosophers-- mathematicians who knew no music and musicians who knew no mathematics." [Barbour, J. M., "Just Intonation Confuted," Music and Letters, Vol. 19, 1938, pg. 60] --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Wed, 19 Jun 1996 21:33 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id MAA03772; Wed, 19 Jun 1996 12:33:45 -0700 Date: Wed, 19 Jun 1996 12:33:45 -0700 Message-Id: Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu