source file: m1513.txt Date: Sun, 23 Aug 1998 20:26:56 -0700 (PDT) Subject: Re: XH 17: n-tet's and harmony From: "M. Schulter" ---------------------------------- Harmonic styles and n-tet's: A multidimensional approach ---------------------------------- --------------- 1. Introduction --------------- This paper grows out of two germinal articles in _Xenharmonikon_ 17: Paul Erlich's far-ranging presentation on scale theory and 22-tone equal temperament (22-tet) [1], and Brian McLaren's sweeping chronicles of microtonalism in the 20th century [2]. Both articles, in different ways, raise the question of how one might go about evaluating the harmonic possibilities of a given tuning dividing the octave into n number of equal parts -- that is, an n-tone equal temperament (n-tet). While the term "harmonic" can mean many things to many people, I use it here basically in the sense of the "vertical" dimension of music: that is, the aspect of music concerned with sonorities of two or more simultaneous tones, and progressions involving such sonorities. Thus while both 13th-century and 18th-century Western European music are harmonically oriented, they are "harmonic" in very different ways, differing in their historical tunings, stable and unstable interval categories, and directed cadences. Curiously, certain scales such as 53-tet can provide an excellent fit with _either_ harmonic system. In evaluating the harmonic possibilities of n-tet's, we need ways of mapping both the scales themselves and the harmonic systems they might realize in whole or part. This paper represents one approach to this multidimensional perspective. Section 2 considers conceptual maps for n-tet scales themselves, while Section 3 focuses on maps for some harmonic systems, and presents some tentative observations about a few n-tet's including Erlich's intriguing 22-tet. Section 4 proposes directions for further analysis. At the outset, I should offer a vital disclaimer: this paper focuses mainly on Western European tunings and harmonic systems from around 1200 to the present, and thus presents at best a very partial and biased sample of the musical possibilities. Other systems, for example gamelan tunings approximating 5-tet or 7-tet, deserve equal consideration, very possibly calling for a modification of the models which follow. Being aware of the dangers of "universal" models of language or music which prove to be more provincial than their authors suspect, I would strongly caution that this paper is not exempt from such dangers. ------------------------------------------------------- 2. Maps for scales: just intervals and meantone tunings ------------------------------------------------------- In orienting ourselves to a given n-tet, there are at least two familiar conceptual maps available. One approach is to compare the intervals of an n-tet with those of a just intonation system based on integer ratios, such as Pythagorean tuning or Zarlino's tuning for vocal music. Another is to compare an n-tet with meantone systems where all fifths (except possibly one) are tempered by the same amount. Both approaches have a long history. Using the just intonation approach, for example, we can say that 53-tet offers excellent approximations both for the true Pythagorean major third of 81:64 and the Zarlinan major third of 5:4. Using the meantone approach, we can say that 19-tet has a close kinship to 1/3-comma meantone, and 31-tet to 1/4-comma meantone.[3] --------------------------------------------------------- 2.1. Intonational justice: limits, mixes, and curiosities --------------------------------------------------------- In describing just intonation systems, it is common to refer to the most complex prime number used in generating the intervals of a given system as its "n-limit." Thus in a 3-limit or quintal just intonation system -- often known as Pythagorean tuning -- all interval ratios are derived from multiples of 2 and 3, with the pure fifth (3:2) as the generating interval. In a 5-limit system such as Zarlino's, the pure major third (5:4) also becomes a generative ratio. In a 7-limit system, the septimal minor seventh (7:4) also becomes literally and figuratively a "prime factor" in defining the scale, and so on. In doing "intonational justice" to these systems, it is very important to recognize that the prime limit of a given system is not necessarily the "limit" of intervals perceived by users of the tuning to be "concordant" or musically valuable. This point is of crucial importance to the analysis which follows, so illustrations may be helpful. Thus in the complex 3-limit system of Gothic polyphony, the major third (81:64) and minor third (32:27) are ranked as relatively concordant in theory, and in practice play a vital role as mildly unstable intervals in various sonorities. The just 9-based intervals of the major second (9:8), minor seventh (16:9), and major ninth (9:4), although regarded as somewhat more tense, are also credited with some "compatibility" or "concord," adding harmonic color and action to the music. Around 1400, the even more complex schismatic form of major third (8192:6561) plays a pivotal role in keyboard tuning systems and compositions. While the 5-limit just intonation of the Renaissance tends to correlate with less subtly shaded theoretical schemes of concord and discord, nevertheless authors such as Zarlino recognize the vital role of intervals such as the major second (9:8 or 10:9), minor seventh (9:5), and major seventh (15:8) in gracing a music based mainly on 5-limit consonances. Composers of the Manneristic epoch around 1600 such as Monteverdi and Gesualdo exploit such intervals more boldly, as well as the diminished fifth (45:32) already endorsed by Zarlino as pleasing when aptly resolved. Much 20th-century music intended for 12-tet provides an even more dramatic example, where stable sonorities involving mixtures of thirds and minor sevenths may suggest a "7-limit" concept, although the tuning system offers no close approximation of a 7:4 ratio.[4] Further, with just intonation systems as with others, the very _deviations_ of certain intervals from the simplest or "purest" possible ratios can have a musical value all its own. For example, the complexity of a 3-limit major third (81:64) nicely fits the Gothic concept of "imperfect concord," while the analogous tension of a 5-limit minor seventh (9:5) adds an edge to Renaissance suspensions and the bolder formations of a Monteverdi. A just appraisal should strive to recognize such intervals as features, not as the "flaws" they might well be in the context of a different system. Having considered some complications of the "n-limit" concept itself, we face the further complication that a given n-tet may approximate a mixture of intervals from various n-limit systems, sometimes a curiously selective one. For example, 53-tet has excellent 3-limit, 5-limit, and 7-limit approximations, so that it might be used for any style of harmony from Gothic to 7-limit just.. While 12-tet also offers a very good 3-limit approximation, it fits 5-limit intervals rather less accurately, and 7-limit intervals still less so. Other scales, however, may offer a "mixed bag" of approximations not so easily described. Thus 19-tet and 22-tet offer fair approximations of 3-limit intervals (with the fifth in either case about 7 cents from a just 3:2), but much closer approaches to just 5-limit intervals. In 17-tet, we have a good 3-limit approximation of 3:2 and 4:3, and also major and minor thirds not too far from 9:7 and 7:6 respectively -- but no real equivalents for 5-limit intervals. Even more intriguingly, 11-tet has no real equivalents for 3-limit fifths or fourths, but reasonable approximations for the 5-limit minor third (5:4) and major sixth (5:3), as well as the 7-limit or 9-limit major third (9:7) and minor seventh (7:4). Brian McLaren notes that in scales such as 13-tet, 18-tet, and 23-tet, "the thirds are reasonably good but there is nothing like a perfect fifth."[5] Such scales are in fact an opportunity to realize traditional harmonic systems in new ways, or indeed to construct new systems, an opportunity which, as McLaren urges, should not be missed. -------------------------- 2.2. The meantone spectrum -------------------------- A different paradigm for mapping n-tet's is the meantone spectrum, ranging in historical terms from Pythagorean tuning (0-comma meantone) to around 1/3-comma of tempering. This spectrum may be expanded in both directions to embrace regular tunings with fifths wider than a just 3:2, or narrower than just by more than 1/3 syntonic comma. Certain points on the meantone spectrum coincide with just tunings. Thus Pythagorean tuning is both a just intonation system and a meantone tuning, although not a meantone _temperament_. Curiously, the 5-limit just tuning with a fifth equal to 16384:10935 (precisely a schisma narrower than 3:2) very closely approximates 1/11-comma meantone and 12-tet. Other points offer just ratios for certain intervals: thus 1/4-comma meantone has pure 5-limit major thirds (5:4) and minor sixths (8:5), while 1/3-comma has pure minor thirds (6:5) and major sixths (5:3). At first blush, it might seem both intuitive and easy to place a range of n-tet's on the meantone spectrum based on the size of fifth. Thus 19-tet is much like 1/3-comma temperament with its quite narrow fifths and pure minor thirds, and 31-tet like 1/4-comma with its somewhat more moderate tempering of the fifth and pure major thirds. Less familiarly, for example, 91-tet approximates 1/7-comma meantone, with somewhat restful thirds (vis-a-vis Pythagorean or 12-tet). This kind of orientation based on fifth size can work even for certain tunings beyond the historically familiar range. Thus 17-tet, with a fifth of around 705.88 cents, seems to behave as we might expect, serving as a kind of "ultra-Pythagorean" tuning with "superactive" thirds and sixths urgently inviting resolution to stable 3-limit intervals.[6] Even here, however, the novel element of the neutral third cautions that new systems are more than stimulating variations on old ones. A reading of Erlich's insightful article, even if one has (like myself) only begun to digest the riches there offered, neatly reveals how misleading it can be to judge a tuning in this way on the basis of its fifth size. In 22-tet, the fifth at 709.09 cents is more than 7.13 cents wide, yet the scale offers nice approximations of 5-limit thirds and sixths, as well as the 7-limit and 9-limit intervals present in 11-tet (a subset).[7] In short, the meantone model is a "map" which works fairly well for some n-tet's, but not for others. In this area as in others, 22-tet is a tuning with lots of musical and conceptual surprises to upset such models (at least when applied overbroadly) in a delightful way. -------------------------------------------------------- 3. Harmonic systems: trines, triads, tetrads, and beyond -------------------------------------------------------- During the long interval of Western European composition from around 1200 to 1900, interestingly enough, harmonic systems were based on two structures requiring a mininum of three voices: the medieval 3-limit trine (string ratio 12:8:6, frequency ratio 2:3:4); and the Renaissance to Romantic 5-limit triad (string ratio 15:12:10, frequency ratio 4:5:6). Both threefold sonorities have inspired poetic and even theological allegory.[8] Since the time of Debussy, more complex sonorities have been accepted as stable: for example, tetrads such as the "added sixth" (12:15:18:20 in one possible 5-limit tuning) and "minor seventh" (12:14:18:21 in a 9-limit tuning). One obvious requisite of realizing a given harmonic system with a given n-tet is that "reasonable" approximations of the intervals involved in such stable structures be available. Thus 53-tet seems a superb choice for either trinic or triadic harmony. While 12-tet likewise would be a reasonable choice for either system, its almost pure fifths but rather active thirds might make it more apt for trinic than for triadic styles. In contrast, 17-tet has good fifths and fourths for trinic sonorities, but no intervals close enough to 5:4 or 6:5 to support a conventional triadic system. With 11-tet, we have fair approximations for the triadic concords of the minor third (6:5) and major sixth (5:3) -- but no fifths or fourths of the kind required for either trinic or triadic sonorities of a historical kind.[9] With 22-tet, as Erlich rightly emphasizes, we have a set of intervals which may not fit this or that specific integer ratio as well as some other n-tet's, but which covers all bases, as it were, supporting a very diverse range of trines, triads, and tetrads.[10] ------------------------------------------------------- 3.2. Broader views: quintal/quartal, tertian, and other ------------------------------------------------------- To make our map of harmonic systems more inclusive, and thus more accommodating to the special potentials of a variety of n-tet's, we must note that the trinic-triadic-tetradic paradigm of composed Western European and related music is only one possible approach. More generally, we might view Gothic trinicism as one subset of _quintal/quartal_ harmony where fifths and fourths serve as the most euphonious simple intervals, and Renaissance-Romantic triadicism likewise as a subset of _tertian_ harmony based on thirds and sixths as the most favored intervals. A given n-tet might support some of these intervals but not others, as in the case of 11-tet. We should not ignore the tertian possibilities of this scale because it happens not to have anything like a triadic fifth. -------------------------------------------- 3.3. Instability, system, and expressiveness -------------------------------------------- Also, even in focusing on the historical trinic and triadic systems, we should note the vital role of _unstable_ sonorities in defining the qualities of a system. For example, in Gothic style, complex 3-limit sonorities such as 4:6:9 or 9:12:16 may have a rather concordant quality; in traditional 5-limit harmony, as Erlich notes [11], these sonorites serve as "suspended" dissonances; and in 20th-century quartal/quintal harmony they may form stable chords of the fifth and fourth. Similarly, in Gothic style, a combination of three superimposed thirds (e.g. e-g-b-d') may resolve to a stable 3-limit sonority (e.g. f-c'); in triadic harmony, it may resolve to a 5-limit sonority; in 7-limit harmony, it may serve as a stable tetrad. Looking _only_ at the stable intervals of a given harmonic system, we might overlook some crucial ways in which a "kindred" n-tet can alter and distort the subtle "balance of power" between the intervals, possibly in creative ways. For example, while I have referred to 17-tet as a "neo-Gothic" tuning, its distortions of the Gothic harmonic spectrum might well warrant, at the least, an accent upon the _neo-_ portion of this description. In theory and practice, Gothic music is premised on a tuning system where major and minor thirds (81:64, 32:27) are somewhat more concordant than bare major seconds and minor sevenths (9:8, 16:9) -- although these intervals alike are often regarded as neither stable nor sharply discordant. In 17-tet, with its major thirds of around 423.53 cents, this order is arguably reversed, altering the harmonic color of the system radically although many of the standard trinic resolutions (e.g. those involving M3-5 and M6-8 by stepwise contrary motion) remain quite practical. In fact, these progressions may take on an extra quantum of force from the increased level of harmonic tension and the "supernarrow" 70.59-cent leading tone; but we have moved into the realm of Xeno-Gothic as opposed to Gothic style.[12] If our purpose is a more or less faithful approximation of the actual Gothic interval system, then Pythagorean just intonation would be ideal, and 53-tet -- or, yes, 12-tet -- a wiser choice than 17-tet. If our purpose is instead a _really_ novel distortion of the modified Pythagorean system in vogue around 1400, with its prominent mixture of regular and schismatic thirds and sixths, then Erlich's 22-tet has some unique possibilities. While the fifth -- or "perfect seventh" in Erlich's decatonic system -- is about 7.14 cents wide of 3:2, it is closer to just than in 1/3-comma meantone or 19-tet, and might lend an interesting flavor to stable trines. The striking "neo-late-Gothic" feature of 22-tet is its drastically contrasting versions of intervals I will here call in traditional fashion thirds and sixths, although Erlich's decatonic nomenclature is well worth studying and may give a better insight into the scale's structure[13]: --------------------------------------------------------------------- Late Gothic keyboard tuning 22-tet --------------------------------------------------------------------- interval ratio cents steps cents approx m3 (regular) 32:27 294.13 5 272.73 ~7:6 m3 (schisma) 19683:16384 317.60 6 327.27 ~6:5 M3 (schisma) 8192:6561 384.36 7 381.82 ~5:4 M3 (regular) 81:64 407.82 8 436.36 ~9:7 m6 (regular) 128:81 792.18 14 763.64 ~14:9 m6 (schisma) 6561:4096 815.64 15 818.18 ~8:5 M6 (schisma) 32768:19683 882.40 16 872.73 ~5:3 M6 (regular) 27:16 905.87 17 936.95 ~12:7 _____________________________________________________________________ --------------------------------------------------------------------- In either system, we have contrasting pairs of more active and more blending thirds and sixths -- but with the "active" forms much more dramatically so in 22-tet, to say the least. One attraction of n-tet's is their ability to transform the familiar into the new, and Erlich's tuning does so in ways a musician might especially relish. ----------------------------- 4. Conclusions and directions ----------------------------- The above analysis should not invalidate any of the historically realized harmonic possibilities, nor the xenharmonic possibilities already being realized in practice and theory. For example, McLaren's engaging chronicles sum up one outlook on n-tet's: "Scales without fifths -- viz., 9, 11, or 13 tones per octave -- favor a fast-paced contrapuntal style with brisk percussive timbres, while scales with excellent fifths, like 31, 41, 53, or 118 tones per octave, favor a triadic homophonic style of composition."[14] The concepts I have sketched might lead us to take this conventional wisdom -- if one can use this term in such an experimental context -- and season it with some further questions. For example, does "triadic" here mean specifically "5-limit" or "tertian," or might it have Partch's wider sense of any three-tone combination (e.g. 1:3:9)? A tuning such as 53-tet can nicely support trinic, triadic (in the tertian sense), or tetradic harmony, and all these potentials merit exploration. Also, where would a tuning such as 17-tet fall in such a dichotomy? It has an "excellent" fifth, closer to a just 3:2 than 31-tet, for example -- but is hardly ideal for "triadic" harmony, at least in a tertian sense. However, it might nicely support a "homophonic style of composition" based on Xeno-Gothic progressions from unstable 7-limit or 9-limit sonorities for three or more voices to stable 3-limit sonorities -- and possibly "reverse suspensions" where a third resolves to a fourth or a major second.[15] Finally, although this may be a curious comment for one of Pythagorean inclinations, I would like to join McLaren in questioning whether the fifth need be treated as the only generative interval for scales or harmonic systems.[16] If a tuning such as 11-tet or 13-tet happens not to have any equivalent of 3:2, why not focus on the various 5-limit, 7-limit, 9-limit, and other intervals it _does_ approximate, and develop a concept of harmonic style -- possibly in some cases "homophonic" -- from there? As McLaren shows in this and other portions of his history, timbre is indeed a crucial factor. Personally, I have found that intervals near 9:7, for example, can take on a pleasant effect if given a certain kind of "choirlike" voicing available on some synthesizers, lending themselves to slower textures with resolutions to fifths in a Gothic-like fashion. Of course, one might add that "a fast-paced contrapuntal style" can suit a wide variety of n-tet's, as it has often suited composers of the Gothic, Renaissance, and Baroque traditions based more or less on the ideal of some kind of just intonation. Both McLaren and Erlich have helped show the way to new realms of theory and practice, while very fruitfully inviting more open questions. ----- Notes ----- 1. Paul Erlich, "Tuning, Tonality and Twenty-Two Tone Temperament," _Xenharmonikon_ 17 (Spring 1998), 12-40. 2. Brian McLaren, "A Brief History of Microtonality in the Twentieth Century," _Xenharmonikon_ 17 (Spring 1998), 57-110. 3. Here it may be wise to explain that meantone tunings are commonly described by specifying the fraction of a syntonic comma (81:80, about 21.51 cents) by which 11 of the 12 fifths of a familiar 12-tone chromatic scale are tempered. Conventionally, it is assumed that this tempering is in the _narrow_ direction. Thus in 1/4-comma meantone, each fifth is narrowed by 1/4 syntonic comma or about 5.38 cents, being reduced from a just 3:2 (about 701.955 cents) to around 697.58 cents. In the special case of 12-tet, the 12th fifth (traditionally often g#-eb, for example) is tempered (in theory) by precisely the same amount as the other 11, all being equal to 700 cents. 4. Not so surprisingly, given its affinities to Pythagorean tuning, 12-tet offers a better approximation of just intonation for sonorities built in fourths or fifths (e.g. d-g-c' or d-a-e', calling ideally for 3:2, 4:3, 16:9, 9:4, etc.) than for sonorities built with 5-limit or higher thirds; 20th-century harmony draws on both possibilities. 5. McLaren, see n. 2 above, at p. 87. 6. Of course, such expectations assume a familiarity with the Gothic harmonic styles of the 13th and 14th centuries. 7. Erlich, see n. 1 above, explores the potentials of 22-tet vis-a-vis other n-tet's at some length; see especially pp. 13-22. 8. Thus the trine (outer octave, lower fifth, upper fourth) for Johannes de Grocheio (c. 1300), and likewise the triad for later theorists such as Johannes Lippius, Johannes Kepler, and Andreas Werckmeister, was a "perfect harmony" representing in music the harmony of the Trinity. In traditional theories based on string ratios, these two sonorities represent the harmonic divisions of the octave (12:8:6) and fifth (15:12:10) respectively. From the perspective of frequency ratios, each presents three adjacent tones in the harmonic series (2:3:4, 4:5:6) beginning with a tone equal to 2^n of the fundamental. 9. For McLaren's comments on other n-tet's presenting a similar situation of "no fifths but good thirds," see n. 7 above. Systems with fifths varying from a just 3:2 by as much as around 7 cents on the narrow side (1/3-comma meantone, 19-tet) or the wide side (22-tet) can clearly support a "triadic" style of harmony; some systems of unequal well-temperament narrow certain fifths in a triadic context by as much as 1/3 Pythagorean comma, or about 7.82 cents. 10. Erlich, see n. 1 above, at p. 25, offers a very handy table showing the intervals of 22-tet and the many integer ratios nicely approximated by the tuning. 11. Ibid., at pp. 12-13 and 26. 12. "Xeno-Gothic" can refer to a specific just tuning based on a chain of 23 pure fifths generating a 24-note octave, or to a set of Gothic and "Gothic-like" progressions supported by this scale, or to a general attitude. A Xeno-Gothic tuning includes the medieval 12-note chromatic Pythagorean scale as a subset, but additionally introduces new forms of unstable intervals a Pythagorean comma wider or narrower than usual. The "Xeno-" aspect may relate especially to such intervals, which for example can form unstable combinations resembling 7-limit or 9-limit tetrads (e.g. ~12:14:18:21) resolving to stable 3-limit trinic harmonies. A scale such as 17-tet or 22-tet, which includes both reasonable approximations of 3:2 and 4:3 and intervals resembling 9:7, 12:7, or 7:4, for example, can therefore elicit a "Xeno-Gothic" response: "Let's try having that near-9:7 expand to a fifth, and that near-12:7 to an octave -- or having that near-7:4 contract to a fifth, etc." 13. Erlich, see n. 1 above, e.g. at p. 25, Table 1. This scheme is indeed a radically different "map" for the intervals: thus 17 steps is a "minor 9th" (about 927.3 cents), while 18 steps is a "major ninth" (about 981.8 cents). Much of the surrounding analysis is breathtaking, the musical equivalent of walking around in some higher-level geometry. Interestingly, as discussed in recent Tuning List posts, 22-tet with its decatonic scale may invite some Jewish allegories, adding another view to the metaphors discussed in n. 8 above. 14. McLaren, see n. 2 above, at p. 81. Here the author's emphasis is on the realization that "_all_ equal temperaments are equally useful for composing beautiful music" (emphasis in original). 15. Here I am warmly indebted to John Chalmers, Jr. and Gary Morrison for many discussions about the possibilities of systems such as 17-tet, but of course any dubious conclusions expressed in this paper are solely my responsibility. 16. McLaren, see n. 2 above, at pp. 86-87, specifically raises questions about models of "diatonicism" presented in classic works of Joseph Yasser and Easley Blackwood as the best basis for approaching tunings such as 13-tet, 18-tet, and 23-tet -- and also many of the 5*n-tet's (e.g. 10-tet, 15-tet, etc.) which violate Blackwood's requirements for a "recognizable diatonic" tuning. It seems to me that such questioning is in the best tradition of these two theorists. Most respectfully, Margo Schulter mschulter@value.net