source file: m1483.txt Date: Fri, 24 Jul 1998 00:21:52 -0700 (PDT) Subject: Re: Nano-temperament From: "M. Schulter" [Please note that this article is a draft which may have some mathematical bugs and glitches, but I hope that the concepts at any rate are interesting -- M.S.] In digital music, as in the digital graphic arts, one must adapt "continuous-tone" concepts to the technical realities of devices limited to a set of discrete colors or tones. Thus a monochrome laser printer can, strictly speaking, produce only black and white dots, not intermediate shades. A synthesizer with 768 or 1024 steps per octave can produce only these tones, not the precise intermediate values required for a given tuning scheme. In the case of the laser printer, we can gain the visual effect of intermediate shades of gray by a process of _halftoning_: grouping black and white dots into repeated patterns, or halftone cells, produces an impression of gray shading. Similarly, with a microtunable synthesizer, it is possible to mix slightly different interval sizes in order to approximate an intermediate value. Here I shall first give a simple example of this technique in approximating Pythagorean tuning with a synthesizer such as the Yahama TX81Z having 768 steps per octave, and then consider the somewhat more intricate example of approximating 17-tone equal temperament (17-tet) on a device with 768 or 1024 steps per octave. A possible term for this technique is _nano-temperament_, the deliberate variation of intervals by very small quantities, typically on the order of one synthesizer tuning unit (e.g. 1.5625 cents on a 768-step device, or 1.171875 cents on a 1024-step device). Since "microtonal" variations are often somewhat larger, for example the 5.38-cent tempering of the fifth in 1/4-comma meantone, the term "nano-temperament" may be descriptive. In fact, nano-temperament may be a useful technique in generating a wide variety of temperaments, for example different shades of historical and contemporary meantone. The following examples, which focus on a Gothic or Xeno-Gothic worldview where fifths and fourths are the primary harmonic intervals, should be taken as a reflection of one side of the author's tastes, not of the scope of the technique itself. ---------------------------------- 1. Nano-temperament: A simple case ---------------------------------- The problem of approximating a classic Pythagorean tuning on a synthesizer provides a ready introduction to nano-temperament. Interestingly, this is a case where in practice nano-temperament seems useful on a 768-step device but superfluous on a 1024-step device. Let us first consider the latter device, where each discrete tuning step is equal to 1200/1024 or 1.171875 cents. As it happens, 599 of these steps produce an interval of 701.953125 cents, or only about 002 cents narrower than a pure 3:2 fifth (about 701.955 cents). While one might suspect that the choice of 1024 steps per octave as a standard has more to do with the binary status of this number as 2^10 (making the octave a convenient digital kilostep) than with a taste for precise Pythagorean tuning, nevertheless this choice generates an extraordinarily close approximation of a just fifth. With the very popular standard of 768 steps per octave, however, the approximation is not quite so close. Taking 449 steps of 1200/768 or 1.5625 cents, we produce a fifth of 701.5625 cents, or about .3925 cents narrow. If we consider only a single fifth, this is a very reasonable approximation, being about five times closer to just than 12-tet, in which fifths are often considered "nearly pure." If we consider other intervals generated from a series of such fifths, however, then this small error becomes cumulative, possibly having tangible effects on the aural quality of such intervals. Let us consider, for example, the Pythagorean major third. In Gothic and related contexts, this is an active interval, ideally having a ratio of 81:64 (about 407.82 cents). Four fifths of 701.5625 cents, however, produce a slightly narrow major third of 406.25 cents, or about 1.57 (.3925 x 4) cents smaller than our ideal size. Is this a distinction with a musical difference? At least one author, Easley Blackwood, suggests a demarcation line at around 406 cents as the point where major thirds become too active to serve pleasingly as points of harmonic repose -- corresponding in regular tunings to a fifth of around 701.5 cents.[1] Thus if we want the extra bit of "edge" that would make our major third unequivocally Pythagorean and active, some adjustment might not be out of place. Fortunately, it is possible to make this adjustment simply by making one fifth out of four an extra tuning step large: 450 steps rather than the usual 499. This yields a fifth of 703.125 cents, or about 1.17 cents wide -- almost exactly correcting the error of the other three fifths, each .3925 cents narrow. Nano-tempering one fifth out of each four in our tuning chain thus produces a major third equal to three 449-step fifths plus one 450-step fifth. From this sum of 1797 steps, we factor out two octaves (1536 steps), arriving at a size of 261 steps, or 407.8125 cents, only about 0.0075 cents from an ideal 81:64. Curiously, this is the same result we get with four equal fifths of 599 steps on a 1024-step device, which yields a major third with a size of (2396 - 2048) or 348 steps, also 407.8125 cents. Such an outcome is not so surprising, since three steps on a 768-step device are equal to four steps on a 1024-step device. Both results, in fact, are equivalent to 87 steps in 256-tet. --------------------------------------------- 1.1. Pythagorean nano-temperament in practice --------------------------------------------- Applying these techniques to an actual chain of fifths, we might choose the following scheme for 768-step devices, with "s" and "l" meaning small and large fifths respectively: l s s s l s s s l s s 768 Eb Bb F C G D A E B F# C# G# s = 449#768 = 701.56 cents l = 450#768 = 703.125 cents A nano-temperament of this kind involves a slight compromise in the quality of the fifths intentionally tuned a bit further from just in order to compensate for cumulative errors in the other direction. Such compromises, fortunately, are typically much milder than those encountered in traditional kinds of temperament. Here each wide fifth of 703.125 cents, although three times as far from pure as the other very slightly narrow fifths, is still only about 1.17 cents or about 1/20 Pythagorean comma from just. At c'-g' (the fifth on middle C), this tempering would produce one beat each 1.88 seconds or so. It should also be noticed that while all major thirds will have as generating intervals three small fifths and one large fifth, thus all sharing the same nearly-ideal Pythagorean size, other intervals will vary somewhat. For example, major sixths or minor thirds may result either from three small fifths, or from two small fifths plus one large one. The former arrangement results in a slightly "subdued" interval about 1.17 cents narrower than 27:16 in the case of M6, or wider in the case of m3. The latter arrangement produces an even more slightly "super-vibrant" interval just .3925 cents narrower than a precise Pythagorean tuning in the case of M6, and the same amount narrower for m3. ------------------------------ Topic No. 3 2. Nano-temperament for 17-tet ------------------------------ Topic No. 4 One way of viewing 17-tet is as a kind of exaggerated Pythagorean tuning. Fifths are equal to ten scale steps of about 70.588 cents, or 705.88 cents. Major thirds are equal to six steps, or 423.53 cents, roughly midway between the Pythagorean 81:64 (407.82 cents) and the 9:7 form of some extended just intonation systems (435.08 cents). While the question of what to do with the latter interval might have many answers, an obvious one from a medievalist viewpoint is "Let it expand to a fifth by conjunct contrary motion." This scale also provides a minor seventh (14 steps) of 988.24 cents, about a third of the way from the Pythagorean 16:9 (996.09 cents) to a 7:4 (968.83 cents). The major second (3 steps) of 211.76 cents is likewise about a third of the way from a Pythagorean 9:8 (203.91 cents) to 8:7 (231.17 cents). Translating a scale step in 17-tet into sythesizer tuning units, we find that one scale step equals about 45.176 units on a 768-step device, or 60.235 units on a 1024-step device. Thus the closest approximation to a scale step is 45 tuning units on a 768-step device or 60 units on a 1024-step device, in either case 70.3125 cents (about 276 cents narrow).[2] To correct for this error, we might add an extra tuning step for every four scale steps on a 1024-step device, or every six steps on a 768-step device: ---------------------------------------------------------------------- 17-tet 768-tet 1024-tet interval steps cents steps cents +/- steps cents +/- ---------------------------------------------------------------------- Unison 0 0.000 0 0.000 0.000 0 0.00 0.000 m2 1 70.588 46* 71.875 +1.287 61* 71.484 +0.896 A1 2 141.176 91 142.188 +1.012 121 141.797 +0.621 M2 3 211.765 136 212.500 +0.735 181 212.109 +0.346 m3 4 282.353 181 282.813 +0.460 241 282.422 +0.069 A2 5 352.941 226 353.125 +0.184 302* 353.906 +0.965 M3 6 423.529 271 423.437 -0.092 362 424.186 +0.657 4 7 494.118 317* 495.313 +1.195 422 494.531 +0.413 d5 8 564.706 362 565.625 +0.919 482 564.844 +0.138 A4 9 635.294 407 635.938 +0.644 543* 636.328 +1.034 5 10 705.882 452 706.250 +0.368 603 706.641 +0.759 m6 11 776.471 497 776.563 +0.092 663 776.953 +0.482 d7 12 847.059 542 846.875 -0.184 723 847.266 +0.207 M6 13 917.647 588* 918.750 +1.103 784* 918.750 +1.103 m7 14 988.235 633 989.063 +0.827 844 989.063 +0.827 A6 15 1058.824 678 1059.375 +0.551 904 1059.375 +0.551 M7 16 1129.412 723 1129.688 +0.276 964 1129.688 +0.276 8 17 1200.000 768 1200.000 0.000 1024 1200.000 0.000 ----------------------------------------------------------------------- For both 768-tet and 1024-tet devices, an asterisk (*) indicates that the last scale step has an extra tuning unit: 46 units (71.875 cents) in 768-tet, and 61 units (71.484 cents) in 1024-tet. Adding what might be called an "intercalary" tuning unit every six steps in 768-tet or four steps in 1024-tet (rather like a leap year) gives a close approximation of 17-tet.[3] However, this simple pattern does cause some intervals to vary from their ideal size by more than half of a tuning unit. Making a correction only when an error of at least half a tuning unit would otherwise occur results in a scheme like this: ---------------------------------------------------------------------- 17-tet 768-tet 1024-tet interval steps cents steps cents +/- steps cents +/- ---------------------------------------------------------------------- Unison 0 0.000 0 0.000 0.000 0 0.00 0.000 m2 1 70.588 45 70.313 -0.275 60 70.313 -0.275 A1 2 141.176 90 140.625 -0.551 120 140.625 -0.551 M2 3 211.765 136* 212.500 +0.735 181* 212.109 +0.346 m3 4 282.353 181 282.813 +0.460 241 282.422 +0.069 A2 5 352.941 226 353.125 +0.184 301 352.734 -0.207 M3 6 423.529 271 423.437 -0.092 361 423.047 -0.482 4 7 494.118 316 493.750 -0.368 422* 494.531 +0.413 d5 8 564.706 361 564.063 -0.643 482 564.844 +0.138 A4 9 635.294 407* 635.938 +0.644 542 635.156 -0.138 5 10 705.882 452 706.250 +0.368 602 705.469 -0.413 m6 11 776.471 497 776.563 +0.092 663* 776.953 +0.482 d7 12 847.059 542 846.875 -0.184 723 847.266 +0.207 M6 13 917.647 587 917.188 -0.459 783 917.578 -0.069 m7 14 988.235 632 987.500 -0.735 843 987.891 -0.344 A6 15 1058.824 678* 1059.375 +0.551 904* 1059.375 +0.551 M7 16 1129.412 723 1129.688 +0.276 964 1129.688 +0.276 8 17 1200.000 768 1200.000 0.000 1024 1200.000 0.000 ----------------------------------------------------------------------- ------------- 3. Conclusion ------------- The method of nano-temperament seems useful in a range of musical contexts, for example in approximating various shades of meantone. Since a meantone major third is derived from four fifths, it is possible by varying the size of _some_ of these fifths by one tuning unit to achieve shades of _average_ temperament only 1/4 tuning unit apart. The major thirds will vary by an amount four times this size: a single tuning unit. In many ways, nano-temperament presents a kinder and gentler version of usual tuning dilemmas: some intervals are might slightly less pure in order to avoid larger discrepancies or inconsistencies. As the use of synthesizers for historical and new scales and temperaments becomes more common, this technique invites further exploration. --------------------- Notes --------------------- 1. Easley Blackwood, _The Structure of Recognizable Diatonic Tunings_ (Princeton: Princeton University Press, 1985), pp. 202-203. Blackwood sets "the largest permissible fifth" for music where thirds are stable intervals at about 701.5 cents. 2. Pythagorean tuning enthusiasts may take note that a 17-tet step of 70.588 cents is not far from three Pythagorean commas of around 23.46 cents, or 70.38 cents. As it happens, both 768-tet and 1024-tet give an excellent approximation of this comma with 15 and 20 tuning units respectively, either yielding 23.4375 cents. 3. This "intercalary" metaphor is much indebted to Guido d'Arezzo, who compares an octave to a week: "Just as when seven days have elapsed we repeat the same ones, so that we always name the first and eighth days the same; so we always represent and name the first and eighth notes the same way..." See Guido's _Micrologus_ (c. 1030?), translated in Warren Badd, tr., Claude V. Palisca, ed., _Hucbald, Guido, and John on Music: Three Medieval Treatises_ (Yale University Press: New Haven, 1978), at p. 61. Guido's metaphor has a special appeal because the term "octave" is indeed used in the liturgical calendar to signify the interval of a week. Given the complexities of calendar reform, possibly comparing an interval tuned slighter wider (or narrower) than the closest approximation on a digital device to a "leap year" is not so inapposite. Margo Schulter mschulter@value.net 22 July 1998