source file: mills2.txt Date: Fri, 27 Oct 1995 08:21:21 -0700 From: "John H. Chalmers" From: mclaren Subject: A new way of regarding non-octave scales; Jose Wurschmidt's categorization: implications for non-octave composition --- As Erv Wilson has remarked, "The field of microtonal scales is absolutely infinite." The closer one looks at any given category of tunings, the more detail one sees. This is nowhere more apparent than with non-octave scales: specifically, that subset defined by the Nth root of K. As everyone realizes, non-octave scales are so new and so unfamiliar to Western ears that their properties remain largely unexplored. While Enrique Moreno (1992) and myself (1989, 1992, 1993, 1995) have made a start, much theoretical work remains. One of the more striking characteristics of non-octave scales is that they often exhibit a marked disjunction between what Jose Wuerschmidt called the "defining interval" and the "constructing interval." Indeed, this perceptual dissonance can be so great in non-octave scales as to turn the ordinary rules of harmony and melody upside-down and inside- out, and often leads to puzzlement and frustration among those who hope to compose with Nth roots of K. Gary Morrison has alluded to this problem in posts on his non-octave 13.6363/oct scale (what he calls "88 CET," an appelation which while accurate stresses its defining rather its constructing interval; and in the case of 13.6363/oct the constructing intervals are actually the ones with most musical importance). However I propose to discuss the issue on a more general basis, covering all Nth roots of K. First, a word about the terminology. Jose Wuerschmidt was a microtonal theorist who did seminal work during the 1920s: his article "Die Quinten- und Terzengewebe" ("The Web of Fifths and Thirds") had a huge impact on subsequent thinking about scale generation, and to a large degree underlies Rothenberg's, Wilson's, Fokker's and Lucy's approach to xenharmonics. (Although they themselves are often not aware of this; W's ideas diffused far & wide, often at 2nd- 3rd, and 4th-hand.) Wuerschmidt's essential idea was that tunings are characterized by two kinds of intervals: one, which he called "constructing intervals," which generate the tuning via an implied underlying harmonic root progress and which define the tuning's tonality--and a second kind of interval, which Wuerschmidt called "defining intervals." This is the interval which (in what Erv Wilson calls "logarithmic space," as opposed to "ratio space") linearly defines the melodic modes of the scale. In the case of 12-TET, the constructing interval is clearly 2^[7/12]. Underlying this approximation one can reach back to a Pythagorean (one might say archetypal) constructing interval 3/2, which places 12-TET firmly in the camp of the positive scales. (I.e., those whose fifths exceed the 3/2 in size. Technically, Bosanquet's positive/negative classification was originally intended to relate scales to the fifth of 12-TET, as anal-renentive detail-obsessives will doubtless point out, but modern usage relates equal tempered scales to the 3/2. ) Other constructing intervals are equally plausible: Erv Wilson has generated JI tunings based on cycles of an interval given by the harmonic mean twixt 4/3 and 11/8, and he has also generated JI tunings based on cycles of 6/5s and 5/4s. One could equally well imagine JI tunings based on cycles of 7-limit, 11-limit, 13-limit and other constructing intervals; Johnny Reinhard has generated a tuning based on the squares of prime numbers. Other constructing intervals are possible. Returning to equal-tempered scales, clearly harmonic progressions other than the 3/2 have been used in other cultures. The Javanese and Balinese do not appear to use a 3/2 at all, nor do the East Indian srutis. Even in this culture, some instruments favor the 5/4 rather than the 3/2--as for example vibes. In the realm of the Nth roots of K, subdivisions of the 3:1--most notably the Bohlen-Pierce scale, 13th root of 3--tend to preserve the 3:1 as a constructing interval when the division is a small number of scale-steps, but as the number of steps rises, other constructing intervals can appear (depending on the exact Nth root of K). By constrast, the defining interval of 12-TET is the semitone of 100 cents. This interval defines the melodic structures, the leading tones and modes possible in 12-TET. Because of the size of the 12-TET defining intervals, many characteristic melodic structures of antiquity cannot be accurately rendered; in 12-TET there is no distinction between the diatonic and the chromatic semitones, for example--nor can the Hellenic enharmonic genus' characteristic plangent near-quartertone be rendered at all accurately. The sharped leading-tone favored by fretless string players (very well rendered by 17-TET) cannot be faithfully reproduced in 12. Ditto the string player's flatted II in the root of a typical I-IV-V-II-I progression. In fact, string players will tend to make a consistent pitch distinction twixt II and IIb, while recovering pianists (and other musically challenged individuals) will perceive no difference between the two root notes. However, the defining interval of the 13th root of 3 is the single scale-step of 146.304 cents, very close in melodic size (and effect) to a single scale-step of 8-TET. Thus, while 12-TET uses a whole-tone very similiar to the familiar tonal 9/8, 13th root of 3 uses a defining interval not close to anything very tonal. In fact the scale-step of 13th of 3 is a good approximation to 3 scale-steps of 24-TET, which forms an entirely anti-tonal interval, lying as it does halfway between one 24-tone circle of 12 fifths and another; again, 146.304 cents is a reasonable counterfeit of the neutral third formed by the geometric mean between the 6/5 and 5/4, but again this is hardly a tonal interval in the just intonation sense (since 350 cents corresponds to an irrational number 1.224053543). Thus 13th of 3 boasts quite tonal harmonic progressions, especially if one deliberately mis-spells the chords with a 13-scale-steps 3:1 on the outside and a 292.608 2-scale-step third on the inside. But the melodic defining intervals and thus the modes of 13th of 3 are utterly anti-tonal and inharmonic, and produce an interesting clash with the constructing intervals. In the Nth roots of 2, this kind of war twixt defining and constructing intervals is rare. Above 48 tones per octave it does not exist; and below there are only a few examples. 35-TET is one example, 26-TET another. The most notable exemplar is 19-TET, in which the 189.47-cent whole-tone defining interval clashes headlong with the very good 694.7368-cent constructing interval of a fifth, unless purely diatonic progressions are used. (That strategy quickly wears out its welcome unless the listeners harbor a particular love for Christmas carols.) Wendy Carlos' alpha and beta scales are characterized by virtually just constructing intervals almost bang-on the just 3:2, but their defining intervals are nothing like the 200-cent approximation of the 9/8 with which we're familiar. Thus the clash twixt defining and constructing intervals must be considered a particular resource of non-octave Nth root of K scales. Moreover, because many Nth roots of K share identical constructing intervals while boasting entirely different defining intervals, the adroit scale designer can fix the constructing interval and generate new scales by searching by alternate Nth roots of K with a different N but the same K. For example, one might decide one wanted a just 5/4. In that case one would fix the constructing interval (a chain of 5/4s) and use a successive set of Ns to generate alternate non-octave scales and then explore their characteristics. The most obvious example is of course the set of equal divisions of the 5/4 ordinally greater than and ordinally less than the familiar 4-equal-part division of the approximate 5/4 used in 12-TET. Maintaining a just 5/4 and using 5 divisions gives a scale-step of [386.31371/5] cents, which yields a non-octave scale of 1200/77.26274 tones/oct = 15.5314 tones/oct. Using 3 divisions of the 5/4 (one less than the familiar 4, just as we above explored one more than the familiar 4) we obtain a scale-step of [386.31371/3] cents, for a non-octave scale of 1200/128.77123 tones/oct = 9.31885 tones/oct. In both cases the constructing interval will be a 5/4, but the defining intervals are quite different. One result of this procedure is to generate a family of harmonically related non-octave scales among which one can "transfer" (to use Ivor Darreg's, and earlier still, Augusto Novaro's, terminology) in a non-octave analogy to traditional tonal modulation. In the case of standard Western modulation, changing to another key maintains the defining interval while moving by the constructing interval; non-octave "modulation" of the kind described above turns this process on its head by maintaining the constructing interval but often moving by the defining interval. Other obvious elaborations abound, but that must be left for another post. --mclaren Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Sat, 28 Oct 1995 04:08 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id TAA15674; Fri, 27 Oct 1995 19:08:53 -0700 Date: Fri, 27 Oct 1995 19:08:53 -0700 Message-Id: <951027220824_78406650@mail02.mail.aol.com> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu