-------------------------------------------------- A Quasi-Wilsonian "Septendecene" in Zest-24: A septimal genus 17 in a tempered setting -------------------------------------------------- One of the treasures of Erv Wilson's writings generously made available on the Anaphoria Embassy website is a paper on _Some Basic Patterns Underlying Genus 12 & 17_ (1980, revised 1981, 1983). While this paper is rich with many ideas, one which captured my imagination was generating a 17-note tuning or "genus 17" by taking a 7-note diatonic mode in a just intonation (JI) system such as Ptolemy's Intensive Diatonic and then transposing it to each of the six steps of a Pythagorean hexachord (1/1 9/8 81/64 4/3 3/2 27/16). This concept appears at the very opening of the presentation (p. 1), with a focus on Pythagorean (2-3 prime) and 5-limit (2-3-5 prime) tunings. What would happen if I applied the same basic idea to a septimal system based on ratios of primes 2-3-7 -- or, indeed, a temperament approximating these ratios like Zest-24? That was an exciting question, and I suspected that the answers might be not only interesting but musically inspiring. Quickly I should note that using the very attractive term "septendecene" for such a 17-note grouping generated by hexachord transposition may be something of a poetic license, since Wilson himself uses it on some unnumbered pages dated and copyrighted 1977 to describe a different method of generating a 17-note tuning by tetrachord rotation. However, just as the term "duodene" for a 12-note set can have various uses (often referring specifically to a style of JI tuning, but sometimes applied more generally, as by Bosanquet), so "septendecene" might freely apply to various types of 17-note sets, including those generated by the hexachord transposition method. And now to my adventure. -------------------------------------- 1. Hexachord transposition: a JI model -------------------------------------- To start with, I focused on this JI scale as a starting point, one of my favorites in either pure or tempered realizations. We could call it a septimal neomedieval Dorian mode since it follows the basic step pattern of a medieval European Dorian mode, here colored by the delightful 7:6 minor third and 7:4 minor seventh: |--------------------| |-------------------------| 1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1 0 204 267 498 702 906 968 1200 9:8 28:27 8:7 9:8 28:27 8:7 204 63 231 204 63 231 As this diagram shows, there are two symmetrical tetrachords of 1:1-9:8:7:6-4:3 or 24:27:28:32. The step sizes of 9:8, 8:7, and 28:27 are identical to those of the diatonic of Archytas, called by Ptolemy the Tonic Diatonic, albeit here with a different order or permutation of steps (here we have 9:8-28:27-8:7, and there 8:7-28:27-9:8). Now for a step with great medieval resonance for me: we simply transpose this 7-note septimal Dorian mode to each of the six steps of a Pythagorean hexachord: 1/1-9/8-81/64-4/3-3/2-27/16. People interested in the hexachord side of things might visit: For our present purposes, I might just remark that the hexachord has six notes and thus five adjacent melodic steps or intervals. Four of these are 9:8 tones, while the centrally located semitonne has a size of 256:243 (about 90 cents). Transposing our septimal mode to each step gives this arrangement: 1/1: 1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1 9/8: 9/8 81/64 21/16 3/2 27/16 243/128 63/32 9/8 81/64 81/64 729/512 189/128 27/16 243/128 2187/2048 567/512 32/27 4/3: 4/3 3/2 14/9 16/9 2/1 9/8 7/6 4/3 3/2: 3/2 27/16 7/4 2/1 9/8 81/64 21/16 3/2 27/16: 27/16 243/128 63/32 9/8 81/64 729/512 189/128 27/16 Ordering this resulting set of 17 notes to an octave, we might arrange these notes on a lattice with 3:2 for the horizontal dimension and the septimal minor third at 7:6 for the vertical dimension. In this type of lattice, each square or quadrangle (e.g. 1/1-7/6-3/2-7/4) shows a complete 12:14:18:21 tetrad (a rounded 0-267-702-969 cents), with the lowest note of the tetrad at the lower left corner. Here are there are six such sonorities: 765 267 969 471 1173 675 177 14/9 -- 7/6 -- 7/4 - 21/16--63/32-189/128-567/512 | | | | | | | 16/9 - 4/3 -- 1/1 -- 3/2 -- 9/8-- 27/16- 81/64--729/512 - 2187/2048 996 498 0 702 204 906 408 612 114 --------------------------------- 2. In a tempered context: Zest-24 --------------------------------- My idea was now to apply this kind of scale generation to the crafting of a 17-note subset or "genus 17" in Zest-24. Briefly I should explain that Zest-24, the Zarlino Extraordinaire Spectrum Tuning, consists of two irregularly tempered 12-note circles each based on Gioseffo Zarlino's renowned 2/7-comma meantone temperament of 1558. Each circle has eight fifths (F-C#) in this regular temperament, with the other four fifths tuned equally wide to close the circle. The two circles are placed at a distance of about 50.28 cents, the enharmonic diesis of Zarlino's regular tuning. While we may find within each circle some intervals in the Pythagorean-septimal portion of the spectrum, the interweaving of the circles provides septimal approximations in many more locations. At the same time, the element of irregular temperament causes intervals and sonorities to shift more or less subtly in color as we move around the system: sonorities often mix distinctively "septimal" intervals with others shading into neighboring regions of the spectrum. My "genus 17" project may illustrate these points as well as some other quirks of the temperament. As a starting point, I sought out a reasonably close approximation of the neomedieval septimal Dorian mode discussed above. Here an asterisk (*) shows a note in the upper 12-note circle of fifths raised by the 50.28-cent diesis: Db Eb Eb* Gb Ab Bb Bb* Db 0 217 267 504 708 925 975 1200 217 50 237 204 217 50 225 Here, as in the 2-3-7 JI version, the septimal minor third and seventh come from a different chain of fifths than the other notes of the mode -- respectively the upper and lower 12-note circles. The near-7:6 (or ~7:6 for short) step at Eb* is very slightly wide of pure, and the ~7:4 step at Bb* is wide by this tiny amount plus the temperament of the large fifth Eb*-Bb*, about 6.42 cents, or a bit less than seven cents in all. While the melodic step Gb-Ab at a rounnded 204 cents is virtually identical to the 9:8 step of the JI version, we find two steps of 217 cents: Db-Eb and Ab-Bb. These tempered steps are almost equal to the mean or average of 9:8 (204 cents) and 8:7 (231 cents); we might call them "septimal eventones." As we'll see, two of these eventones yield a near-just 9:7 major third, a feature useful in our 17-note system. The major sixth Db-Bb, from a chain of three large fifths, is around 925 cents -- and yields a fair approximation of 12:7 (933 cents). The scale locations in our 2-3-7 JI version of the mode where 8:7 steps or large tones appear also have ratios fairly close to 8:7 in this tempered version: Eb*-Gb at 237 cents and Bb*-D at 225 cents. These intervals, as in the just version, involve two notes from different chains of fifths. These sizes of 225 and 237 cents, or more precisely about 224.59 and 237.16 cents, illustrate a characteristic feature of this irregular temperament: interval sizes often graduated in steps of 12.57 cents, or a quarter of the 50.28-cent diesis. Melodically notable are the two 50-cent semitones Eb-Eb* and Bb-Bb*, actually diesis steps considerably smaller than the JI step of 28/27, the semitone or thirdtone of Archytas at 63 cents. These 50-cent semitones are a routine feature of Zest-24 in septimal contexts, for example, and rather narrower even than the diatonic semitone of 22-EDO at 54.54 cents or 1/22 octave. ----------------------------- 3. Generating a 17-tone genus ----------------------------- To generate a 17-tone set, I somewhat modified Wilson's method of transposing a mode to the steps of a Pythagorean hexachord. Rather I sought a Zest-24 subset that would provide approximate transpositions to each of the first six steps of the tempered version: that is, to the steps of 0-217-267-504-708-925 cents. I knew that this was altering the parameters for Wilson's method, since these steps are at once tempered and taken from two chains of fifths. However, with assistance from Manuel Op de Coul's free software Scala available on the Internet , I arrived at this solution, shown in rounded cents: 0: 0 217 267 504 708 925 975 1200 217: 217 434 484 708 925 1129 1179 217 267: 267 484 554 759 975 1180 50 267 504: 504 708 759 1008 1200 217 267 504 708: 708 925 975 1200 217 434 484 708 925: 925 1129 1200 217 434 625 708 925 As I found, this 17-note set provided septimally flavored transpositions not only to these six steps of the original mode, but also to the seventh step at 975 cents, thus covering all of its degrees -- and also to the step of the 17-note system at 484 cents, a narrow fourth which could be considerd a tempered equivalent of 21:16 (about 471 cents). Here are mappings and interval sizes for these eight transpositions: Db Eb Eb* Gb Ab Bb Bb* Db 0 217 267 504 708 925 975 1200 217 50 237 204 217 50 225 Eb F F* Ab Bb C C* Eb 0 217 267 492 708 913 963 1200 217 50 225 217 204 71 237 Eb* F* Gb* Ab* Bb* C* Db* Eb* 0 217 287 492 708 913 983 1200 217 71 204 217 204 71 217 F* G* Ab* Bb* C* Eb Eb* Eb* 0 192 274 492 696 933 983 1200 217 83 217 217 204 50 217 Gb Ab Ab* Cb Db Eb Eb* Gb 0 204 254 504 708 925 975 1200 217 50 250 204 217 50 225 Ab Bb Bb* Db Eb F F* Ab 0 217 267 492 708 925 975 1200 217 50 225 217 217 50 225 Bb C Db Eb F G Db Eb 0 204 275 492 708 900 983 1200 204 71 217 217 192 83 217 Bb* C* Db* Eb* F* G* Ab* Bb* 0 204 275 492 708 900 983 1200 204 71 217 217 192 83 217 Quickly surveying these interval sizes may suggest why due emphasis should be given to the concept of _approximate_ transpositions, with a dramatic degree of what be called, depending on one's editorial viewpoint, either variegation or inaccuracy! Here a "septimal" flavor of our Dorian is considered to be one with a minor seventh at a rounded size of 963, 975, or 983 cents. While minor thirds representing 7:6 are often virtually just (267 cents) or reasonably close (275 cents), we also find sizes of 287 cents (over 20 cents wide, and much closer to 13:11 at about 289 cents), and also 254 cents (over 12 cents narrow, and very close to 22:19 at a rounded 254 cents). ---------------------------- 4. Shadings and alternatives ---------------------------- I might add that while septimal flavors of the mode occur at eight locations of the 17-note genus, there are actually two versions available at Eb (225 cents above our "1/1" at Db), the first already listed above, and the second an alternative form similar to that found at Eb* (267 cents above Db). Eb F F* Ab Bb C C* Eb 0 217 267 492 708 913 963 1200 217 50 225 217 204 50 237 Eb F Gb Ab Bb C Db Eb 0 217 287 492 708 913 983 1200 217 71 204 217 217 71 217 These two forms differ in the tuning of the third and seventh degrees. The first approach, closer to JI, derives the near-pure minor third from the 217-cent "eventone" Eb-F plus a 50-cent diesis or small semitone, thus Eb-F*. Likewise the minor seventh Eb-C* at 963 cents, narrow of 7:4 by a bit less than six cents, is equal to the major sixth Eb-C at 913 cents plus a 50-cent step, Eb-C*. The second approach generates all modal steps from a single chain of fifths, with two large fifths (around 708.38 cents) or small fourths (around 491.62 cents) producing a 983-cent minor seventh (Eb-Ab-Db). A chain of three of these large fifths down or small fourths up would produce a minor third such as Bb-Db or Bb*-Db* at 275 cents, or about eight cents wide of 7:6, as can be seen in the modes on Bb and Bb*. Here, however, the chain Eb-Ab-Db-Gb has a large fourth at Db-Gb (the usual meantone size of 504.19 cents), so that the third is 287 cents rather than 275 cents. This need not be seen as a flaw, but does result in a different cast or shading than the near-just version of the first approach. The first version also has the JI-like feature of the comma: a distinction between a usual fourth (here 492 or 504 cents, and in JI the just 4:3 at ~498 cents) and the special narrow fourth at Ab-C*, almost identical to the just 21:16 (~471 cents) which would occur in a pure tuning between the corresponding steps 4/3 and 7/4. In contrast, the second version has the usual fourth Ab-Db (492 cents) at this same location. Thus the near-JI version can be chosen if one seeks the most accurate septimal representations, or wants to use the beautiful 21:16 (for example in a 16:21:24:28 sonority, Ab-C*-Eb-F*, which could resolve to Bb-F, all within the notes of the mode). The alternative version offers a less "eventful" realization which tempers out the comma, rather as in George Secor's 17-note well-temperament. While the first version more closely models JI, in fact both versions show the "variegation" we have emphasized in representing an interval such as the 7:6 minor third. This becomes clear when we compare the sizes of each of the four minor thirds in a JI version and in these tempered approaches: JI: 1/1 - 7/6 9/8 - 4/3 3/2-7/4 27/16-2/1 7:6 32:27 7:6 32:27 267 294 267 294 Eb - F* F - Ab Bb - C* C - Eb Tempered 1: 0 - 267 217 - 492 708 - 963 913 - 1200 267 275 254 287 Eb - Gb F - Ab Bb - Db C - Eb Tempered 2: 0 - 287 217 - 492 708 - 983 913 - 1200 287 275 275 287 The JI version has two 7:6 minor thirds and also two Pythagorean minor thirds at 32:27 or ~294 cents -- with our 287-cent or near-13:11 thirds not too far from this ratio. The second tempered version with two minor thirds at 275 cents and the other two at 287 cents thus has a pattern somewhat comparable to that of the JI version, but with these large and small thirds in different positions. In JI, we find small 7:6 thirds at 1/1-7/6 and 3/2-7/4, steps 1-3 and 5-7 of the scale; and larger 32:27 thirds at 9/8-4/3 and 27/16-2/1, steps 2-4 and 6-8. Here, however, we have small 275-cent thirds at 2-4 and 5-7, and larger 287-cent thirds at 1-3 and 6-8. The first tempered version has no fewer than four minor third sizes, one for each of these intervals: a near-pure 267 cents, the 275-cent and 287-cent sizes we have noted in the other tempered version, and also a narrow 254 cents at Bb-C*. While temperament is often presented as a strategy for "regularizing" or "evening out" some of the variations in step and interval sizes occur in JI based on multiple primes (2-3-5, 2-3-7, etc.), here our irregular temperament increases the range of interval diversity -- or unpredictability, again depending on one's viewpoint. Melodically, the second tempered version offers closer approximations of the 28:27 thirdtone of Archytas, using 71-cent semitones (actually a just 25:24, ~70.67 cents) to represent this 63-cent JI step. The first version, in contrast, offers striking 50-cent semitones. --------------------- 5. A lattice overview --------------------- The choice of two tempered versions of our Dorian mode at Eb might serve as an introduction to a general overview of the genus-17 system. We can map it as lattice with the horizontal dimension showing regular fifths at 492 or 504 cents, and vertical lines (here slanted from left to right) showing minor thirds. When these slanted vertical lines are dashed, they show thirds belonging to what are regarded as "septimal flavor" tetrads approximating 12:14:18:21, with outer minor sevenths at 963, 975, or 983 cents. Dotted lines show minor thirds belonging to other types of tetrads moving into neighboring regions with 996-cent minor sevenths almost identical to a just Pythagorean 16:9; or 950-cent minor sevenths in the fascinating "interseptimal" region between the 7-based ratios of 12:7 (933 cents) and 7:4 (969 cents). The numbers within each square or quadrangle show the sizes of a tetrad's minor seventh, two minor thirds, and major third in cents: Gb* Db* Ab* Eb* Bb* 554 ------ 50 ------ 759 ----- 267 ----- 975 / 983 / 983 / 983 / 996 . / 287 275 / 275 275 / 275 287 / 287 300 . / 421 / 434 / 421 / 408 . Db* ----- Ab* ----- Eb* ---- Bb* ----- F* ------ C* ------ G* 50 ------ 759 ----- 267 ---- 975 ----- 483 ---- 1179 ----- 675 . 950 / 963 / 975 / 975 / 963 / 950 . . 242 250 / 254 267 / 267 267 / 267 267 / 267 254 / 250 242 . . 454 / 441 / 441 / 441 / 441 / 454 . Cb ----- Gb ------ Db ------ Ab ------ Eb ------ Bb ------- F 1008 ----- 504 ----- 0 ------ 708 ----- 217 ------ 925 ----- 434 . 996 / 983 / 983 / 983 / 996 . . 300 287 / 287 275 / 275 275 / 275 287 / 287 300 . . 408 / 421 / 434 / 421 / 408 . Ab ------ Eb ------ Bb ------ F ------- C ------- G 708 ----- 217 ----- 925 ----- 434 ---- 1129 ----- 625 Within the set of "septimal" tetrads as here defined, we find minor thirds ranging from 254-287 cents, and major thirds at 421-441 cents; in a just 12:14:18:21, these intervals would have pure sizes of 7:6 and 9:7 (267 and 435 cents). The 254-cent minor third moves well into the interseptimal region intervening between the neighborhoods of 8:7 (231 cents) and 7:6, while the 287-cent third, as we have seen, is very close to 13:11 and approaches the Pythagorean region of 32:27. As our diagram shows, septimal tetrads featuring these small or large sizes also have minor thirds at the middle sizes of 267 or 275 cents closest to 7:6. Among the major thirds of these tetrads, 421 cents is rather larger than 14:11 (~418 cents) and approaches 17-EDO (423.53 cents), while 441 cents can be regarded as a temperament of 9:7 (about 6 cents larger than pure), but verges on the interseptimal region between this just major third and the narrow fourth at 21:16 (471 cents). The middle size of 434 cents is very close to a just 9:7. Interestingly, this best approximation of 9:7 occurs within a single fifth circle, where it results from a chain of all four large fifths at 708.38 cents (Db-Ab-Eb-Bb-F or Db*-Ab*-Eb*-Bb*-F*), and is formed from two equal "eventones" of 217 cents each (Db-Eb-F or Db*-Eb*-F*). Septimal and other tetrads on the lattice are generated by two basic methods: using fifths within a single circle, or mixing notes from both circles. The diagram thus has three rows of tetrads. In the lowest row, tetrads are generated within the lower circle of fifths; in the middle row, by combining notes from both circles; and in the upper row, within the upper circle. In the lower row, tetrads shade symmetrically from the closest approximation of 12:14:18:21 in the center position at Bb-Db-F-Ab (0-275-708-983 cents) to a Pythagorean flavor nicely represented by Ab-Cb-Eb-Gb and C-Eb-G-Bb (0-300-708-996 or 0-287-696-996 cents). The intermediate Eb-Gb-Bb-Db and F-Ab-C-Eb (0-287-708-983 cents or 0-275-696-983 cents) are considered within a "septimal" flavor, but have some 287-cent and 421-cent thirds shading somewhat in the direction of the Pythagorean region and typical of a 17-note circulating temperament, for example. In the middle row, the center two tetrads have 975-cent minor sevenths and pairs of near-just 267-cent minor thirds (0-267-708-975 cents); this symmetrically shades towards the fine interseptimal tetrads at Cb-Db*-Gb-Ab* and Bb-C*-F-G* (0-242-696-950 or 0-242-708-950 cents). The intermediate Gb-Ab*-Db-Eb* and Eb-F*-Bb-C* (0-254-696-963 or 0-267-708-963 cents) are regarded as "septimal" and have the closest approximations of 7:4 available in Zest-24 (trivially closer on the narrow side than a 975-cent seventh almost equally impure on the wide side), but also have a 254-cent minor third shading them toward the interseptimal region. The highest row is much like the lowest, except that there is an asymmetry because our 17-note genus includes a chain of nine fifths from the lower circle (Cb-G, with Cb at 1008 cents serving as an equivalent to 16/9 to Db in the quartal sonority Db-Gb-Cb, for example), but only eight from the upper circle (Gb-G). Thus there are four tetrads on this row as compared to five on the lowest row, with Bb*-Db*-F*-Ab* in the central septimal region. Moving to the right, we have the same transition as on the lower row to the intermediate F*-A*-C*-Eb* and the Pythagorean-like C-Eb-G-Bb. To the left, however, we have only the intermediate Eb*-Gb*-Bb*-Db*; adding to the system the note Cb* would produce a Pythagorean-like Ab*-Cb*-Eb*-Gb* and make this row identical to the lower one. The reader may have noticed that there are actually two tetrads shown with the lowest note of Eb: Eb-G-Bb-Db in the lowest row from the single lower circle of fifths, or Eb-F*-Bb-C* in the middle row mixing notes from the two circles. These alternative forms may be found in the two septimal Dorian modal transpositions to Eb which we discussed above. There are thus 10 "septimal" tetrads in our 17-note set available at nine distinct locations (two at the location of Eb), shown on the lattice with dashed lines on all sides. Additionally we see two fine Pythagorean region tetrads on either end of the lower row; two interseptimal tetrads at either end of the middle row; and another Pythagorean region tetrad at the right end of the upper row. These five tetrads have a slanting dotted line on one side. We might ask how an equivalent JI lattice might look for a tuning based on pure ratios of 2-3-7 supplying tetrads corresponding to our tempered septimal ones. Here is one such possible system, with dotted lines indicating tetrads that in our tempered version move clearly into the Pythagorean or interseptimal regions, but in this just tuning would, of course, also be pure 12:14:18:21 sonorities: 534 36 738 240 942 49/36 - 49/48 - 49/32- 147/128--441/256 | | | | . | | | | . 63 765 267 969 471 1173 675 28/27 - 14/9 --- 7/6 --- 7/4 -- 21/16 - 63/32 - 189/128 . | | | | | . . | | | | | . 16/9 --- 4/3 --- 1/1 --- 3/2 --- 9/8 -- 27/16 -- 81/64 996 498 0 702 204 906 408 . | | | | . . | | | | . 32/21 -- 8/7 --- 12/7 -- 9/7 -- 27/14 - 81/56 729 231 933 435 1137 639 This 25-note system, in addition to the indicated 15 pure 12:14:18:21 tetrads, would also offer a wealth of Pythagorean sonorities, for example, as a more intricate diagram of tetrads would show. ----------------------------------------- 6. Keyboard layout: a regularized mapping ----------------------------------------- The ideal synthesizer implementation of our tempered 17-tone genus might be on a generalized keyboard of the kind for which Erv Wilson has offered many mapping of just or tempered tunings. More modestly, however, my realization uses a 24-note "regularized keyboard" with two 12-note manuals, one for each circle of fifths. The term "regularized" means that each 12-note manual has the same sequence of steps and intervals. Here we might say that the two manuals share the same variations and irregularities of the 12-note circle. The mapping for our 17-genus is often rather like that of a 2-3-7 JI lattice, but with some important distinctions also: 50 267 483 554 675 759 975 1180 1250 Db* Eb* F* Gb* G* Ab* Bb* C* Db* 0 217 434 504 625 708 925 1008 1129 1200 Db Eb F Gb G Ab Bb Cb C Db Unlike a _generalized_ keyboard, of course, there is the usual Halberstadt distinction between tones and semitones on either manual. There is further the element of irregular temperament: thus Db-F or Db*-F* forms a near-just 9:7, while Eb-G or Eb*-G* forms a near-just 81:64 at some 25 cents smaller. Certain subsets do have a certain lattice-like structure, for example a tempered version this fine tuning by Carter Scholz identified in the Scala scale archive (scholz.scl) as from "Simple Tune #1": 28/27 14/9 7/6 7/4 63 - 765 - 267 - 969 | | | 4/3 - 1/1 - 3/2 498 0 702 | 8/7 231 and realized as follows: 50 267 759 975 Db* Eb* Ab* Bb* Db Eb Gb Ab Db 0 217 504 708 1200 Here the steps at 50, 267, 759, and 975 cents on the upper keyboard all represent septimal ratios (28/27, 7/6, 14/9, 7/4), and 504, 708, and 1200 cents on the lower keyboard the Pythagorean 4/3, 3/2 and 2/1. The complicating factor is 217 cents on the lower keyboard, which might stand for either 9/8 or, as here, 8/7. Mathematically it is very slightly closer to 9/8, and acts musically rather like 9/8 (or more precisely 44/39 at around 209 cents) in forming a minor third with Gb not too far from the Pythagorean 32:27 (and very close to 13:11). However, it acts rather like 8/7 in forming a "septimal flavor" minor seventh with Db at 983.24 cents, numerically very slightly closer to 16:9 but leaning in color toward 7:4, and approaching the 981.82 cents of 22-EDO. The net result might be kind of moderate variation on Carter Scholz's JI tuning -- with the usual inaccuracies and compromises of temperament, of course. I might close by briefly mentioning another kind of just tuning which this 17-genus can emulate: the 1-3-7-9 hexany, with six notes forming two pure 12:14:18:21 tetrads: 267 969 471 7/6 7/4 21/16 1.7 - 3.7 - 7.9 | | | 1.3 - 1.9 - 3.9 1/1 3/2 9/8 0 702 204 In our 17-genus, one tempered version is: Ab Bb Bb* C* Eb F* Ab 0 217 267 471 708 975 1200 Following the above lattice arrangement for the JI version, this could be written: 267 975 471 Bb* F* C* 1.7 - 3.7 - 7.9 | | | 1.3 - 1.9 - 3.9 Ab Eb Bb 0 708 217 While the tempered version often corresponds closely with the just tuning, there are notable variations: Bb-C* is an interseptimal interval of 254 cents representing a 7:6 third; and Bb-Ab, representing 16:9, is a 983-cent minor seventh with a 7:4-like flavor. Both of these touches somewhat alter the balance and color of the hexany, hopefully as a stimulating ramification of the original JI version with its own unique appeal. Another variation on the hexany involves similar variations or compromises: Gb Ab Ab* Bb* Db Eb* Gb 0 204 254 471 696 963 1200 254 963 471 Ab* Eb* Bb* 1.7 - 3.7 - 7.9 | | | 1.3 - 1.9 - 3.9 Gb Db Ab 0 696 204 This time we have a small 254-cent interseptimal third representing 7:6 above the lowest step of the hexany, and a 983-cent minor seventh at Bb*-Ab* representing the 16:9 interval of 21/16-7/3 in the just hexany. A subtle distinction of color is that here the 7/4 step is represented as a narrow 963 cents (Gb-Eb*) rather than a wide 975 cents (Ab-F* in the last example) -- these being the closest approximations in Zest-24, about equally accurate. These two tempered versions both feature a near-just 21:16. Another version involves more of a compromise involving this interval: Db Eb Eb* F* Ab Bb* Db 0 217 267 484 708 975 1200 267 975 484 Eb* Bb* F* 1.7 - 3.7 - 7.9 | | | 1.3 - 1.9 - 3.9 Db Ab Eb 0 708 217 If seeking a hexany with the most accurate representations of 7:6, we could hardly do much better in a tempered system: all three locations for this interval on the lattice are at a virtually just 267.034 cents. Further, if one wants to maximize the number of minor sevenths close to 7:4, we have not only the two 975-cent realizations where this ratio is expected on the lattice, but also two 983-cent intervals where the JI version has 16:9. Of course, if one relishes a just 16:9 color -- or a contrast between 7:4 and 16:9 sevenths like that of the JI version -- then the other transpositions can oblige. However, this plethora of 7:6 and 7:4 approximations does have a consequence for the 21:16 approximation. Combining a near-pure 267-cent minor third (Db-Eb*) with a large 217-cent eventone (the octave complement or inversion of a 983-cent minor seventh) results in an interval of about 484 cents, or 13 cents wide of a just 21:16. This variation could be regarded as another facet of unequal temperament, nicely illustrating the 12.57-cent distinction in interval sizes that often occurs (here from 471.22 cents in the previous examples to 483.79 cents, as compared with 21/16 at 470.71 cents). The tuning of the near-21:16 might be of special interest when using the tempered equivalent of the hexany's 16:21:24:28 sonority found at 1/1-21/16-3/2-7/4. Here we compare the just version and our three tempered emulations: JI: 1/1 21/16 3/2 7/4 0 471 702 969 21:16 8:7 7:6 471 231 267 Ab: Ab C* Eb F* 0 471 708 975 471 237 267 Gb: Gb Bb* Db Eb* 0 471 696 963 471 225 267 Db: Db F* Ab Bb* 0 484 708 975 484 225 267 The transpositions on Ab and Gb give a close rendition of the just 21:16, while the Db transposition offers a different shading at 484 cents, interestingly close to the complex JI ratio of 189:143 that occurs in a 24-note superset of the 1-3-7-9-11-13 eikosany that I use (482.85 cents). Each subtle gradation has its own appeal, not to mention the JI original. Most appreciatively, Margo Schulter mschulter@calweb.com