--------------------------------------- A JI Constant Structure in 17 notes Combining Ptolemy's Intense Chromatic And Ibn Sina's "Most Noble Genus" by Margo Schulter --------------------------------------- The tuning I will here describe took shape as a subset of Zeta-24, a spinoff both of Kraig Grady's Centaur and of another Centaur relative, Rod Poole's 17-note guitar tuning. However, this 17-note Constant Structure may be of interest in itself, since it illustrates a technique of superimposing or merging two tetrachords of interest to arrive at a more complex division of the 4/3 fourth. ! zeta24-ptolemy-ibn_sina17.scl ! Constant Structure: Ptolemy's 22:21-12:11-7:6, Ibn Sina's 8:7-13:21-14:13 17 ! 22/21 12/11 8/7 13/11 26/21 9/7 4/3 88/63 16/11 3/2 11/7 104/63 12/7 16/9 13/7 176/91 2/1 ------------------------------------------------------------ 1. From 9 notes to 12: Superimposing two classic tetrachords ------------------------------------------------------------ Here my starting point was Ptolemy's Intense Chromatic, with tetrachords of 22:21-12:11-7:6 or 1/1-22/21-8/7-4/3, which could be expressed in terms of frequency ratios as 21:22:24:28. Taking two disjunct tetrachords with a middle 9:8 tone, we have this 7-note tuning: 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1 0 80.5 231.2 498.0 702.0 782.5 933.1 1200.0 22;21 12:11 7:6 9:8 22:21 12:11 7:6 80.5 150.6 266.9 203.9 80.5 150.6 266.9 The next step is to superimpose Ibn Sina's "most noble genus" which he gives as either 8:7-13:12-14:13 or 8:7-14:13-13:12, that is, either 1/1-8/7-26/21-4/3, or 1/1-8/7-16/13-4/3. Both feature large Zalzalian thirds, with 26/21 (369.7 cents) in the first permutation very bright, and 16/13 (359.5 cents) in the second at what we might describe as the high end of the "central" Zalzalian range (maybe roughly from 39/32 to 16/13, or 342-360 cents). Here we choose the brighter version with a 26/21 third, although the 16/13 version is also delightful, and both well worth exploring (they might correspond roughly to common tunings of Maqam Rast in Aleppo and Damascus). Since Ptolemy's Intense Diatonic has already supplied Ibn Sina's steps 1/1, 8/7, and 4/3, we need only add a 26/21 (or 16/13) step to obtain the desired version of his tetrachord also, with two disjunct tetrachords producing a mode of 1/1 8/7 26/21 4/3 3/2 12/7 13/7 2/1: 1/1 22/21 8/7 26/21 4/3 3/2 11/7 12/7 13/7 2/1 0 80.5 231.2 369.7 498.0 702.0 782.5 933.1 1071.7 1200.0 22;21 12:11 13:12 14:13 9:8 22:21 12:11 13:12 14:13 80.5 150.6 138.6 128.3 203.9 80.5 150.6 138.6 128.3 The result is a nonatonic or 9-note Constant Structure tuning with all-epimoric steps, a union or superset of Ptolemy's and Ibn Sina's tunings (here taken with disjunct tetrachords). To arrive at a decatonic, one approach is to divide the middle 9/8 tone into two intervals: the first a 22:21 step, following Ptolemy's tetrachord; and the second, a remainder equal to the difference between 9:8 and 22:21, or 189:176 (123.4 cents). This step, although not epimoric or superparticular, might be regarded as a variant on 14:13, being narrower by 352:351 (4.9 cents). 1/1 22/21 8/7 26/21 4/3 88/63 3/2 11/7 12/7 13/7 2/1 0 80.5 231.2 369.7 498.0 578.6 702.0 782.5 933.1 1071.7 1200.0 22;21 12:11 13:12 14:13 22:21 189:176 22:21 12:11 13:12 14:13 80.5 150.6 138.6 128.3 80.5 123.4 80.5 150.6 138.6 128.3 This decatonic is again a Constant Structure, and lends itself to expansion into a 12-note tuning, for example, by adding steps at 13/11 and 16/9. At 1/1-4/3, this adds an approximation of the Archytas Diatonic at 1/1-8/7-13/11-4/3 or 8:7-91:88-44:39, while at 3/2-2/1 it yields this exact permutation at 1/1-8/7-32/27-4/3 or 8:7-28:27-9:8 (a string-length or monochord division at 32:28:27:24). This 12-note tuning is also a Constant Structure. 1/1 22/21 8/7 13/11 26/21 4/3 88/63 3/2 11/7 12/7 16/9 13/7 2/1 0 80.5 231.2 289.2 369.7 498.0 578.6 702.0 782.5 933.1 996.1 1071.7 1200.0 22;21 12:11 91:88 22:21 14:13 22:21 189:176 22:21 12:11 28:27 13:12 14:13 80.5 150.6 58.0 80.5 128.3 80.5 123.4 80.5 150.6 63.0 138.6 128.3 --------------------------------------------------------------- 2. Expanding from 12 to 17: Affinities in 4/3 and 9/8 divisions --------------------------------------------------------------- From this point, I added five other steps to arrive at a 17-note tuning: 12/11, 9/7, 16/11, 104/63, and 176/91. These steps were dictated in part by the available choices within Zeta-24 as a superset, e.g. 176/91 as the closest approximation of 27/14, but the general idea is to expand the structure of the smaller sets. A diagram showing the 17-note tuning as two 4/3 divisions plus a middle 9/8 tone may clarify this structure, as well some asymmetries occurring in the Zeta-24 setting: 1/1 22/21 12/11 8/7 13/11 26/21 9/7 4/3 0 80.5 150.6 231.2 289.2 369.7 435.0 498.0 22;21 126:121 22:21 91:88 22:21 27:26 28:27 80.5 70.1 80.5 58.0 80.5 65.3 63.0 4/3 88/63 16/11 3/2 498.0 578.6 648.7 702.0 22:21 126:121 33:32 80.5 70.1 53.3 3/2 11/7 104/63 12/7 16/9 13/7 176/91 2/1 702.0 782.5 867.8 933.1 996.1 1071.7 1142.0 1200.0 22;21 104:99 27:26 28:27 117:112 176:169 91:88 80.5 85.3 65.3 63.0 75.6 70.3 58.0 This diagram focuses on a disjunct tetrachordal structure, in keeping with the design of the smaller sets in combining the genera of Ptolemy and Ibn Sina. However, a diagram showing three tetrachord divisions, at 1/1-4/3, 4/3-16/9, and 3/2-2/1, may more clearly display concordances such as the perfect fourth between the bright Zalzalian third and sixth at 26/21-104/63. 1/1 22/21 12/11 8/7 13/11 26/21 9/7 4/3 0 80.5 150.6 231.2 289.2 369.7 435.0 498.0 22;21 126:121 22:21 91:88 22:21 27:26 28:27 80.5 70.1 80.5 58.0 80.5 65.3 63.0 4/3 88/63 16/11 3/2 11/7 104/63 12/7 16/9 498.0 578.6 648.7 702.0 782.5 867.8 933.1 996.1 22:21 126:121 33:32 22:21 104:99 27:26 28:27 80.5 70.1 53.3 80.5 85.3 65.3 63.0 3/2 11/7 104/63 12/7 16/9 13/7 176/91 2/1 702.0 782.5 867.8 933.1 996.1 1071.7 1142.0 1200.0 22;21 104:99 27:26 28:27 117:112 176:169 91:88 80.5 85.3 65.3 63.0 75.6 70.3 58.0 ------------------------------------ 3. Exploring tetrachord permutations ------------------------------------ This scheme includes various permutations of Ptolemy's and Ibn Sina's tetrachords that set the foundation for these tunings, as well as a variety of Zalzalian modes. One bit of advice from Safi al-Din al-Urmawi in the 13th century is to explore all six permutations or arrangements of a tetrachord -- or, we might say, in a given tuning system, as many permutations as possible. For example, with Ptolemy's Intense Chromatic, we have available all six permutations. In the following diagrams, scale steps like 22/21 show the location of a note in our 17-note tuning, while parenthetical notations like (22:21) show the step within a tetrachord permutation that a note is representing, when the tetrachord begins on a tuning system step other than the 1/1. Ptolemy's original genus 1/1 22/21 8/7 4/3 22:21-12:11-7;6 22:21 12:11 7:6 (21:22:24:28) Variation on Ptolemy upper interval remains 7:6 1/1 12/11 8/7 4/3 12:11-22:21-7:6 12:11 22:21 7:6 (24:22:21:18) Qutb al-Din al-Shirazi's Hijaz (1:1) (12:11) (14:11) (4:3) 12:11-7:6-22:21 22/21 8/7 4/3 88/63 (33:36:42:44) 12:11 7:6 22:21 Qutb al-Din al-Shirazi's Hijaz Inverted variation (1:1) (22:21) (11:9) (4:3) 22:21-7:6-12:11 12/11 8/7 4/3 16/11 (44:42:36:33) 22:21 7:6 12:11 Sazkarlike Variation (1:1) (7:6) (11:9) (4:3) 7:6-22:21-12:11 9/7 3/2 11/7 12/7 (18:21:22:24) 7:6 22:21 12:11 Al-Farabi's Chromatic Inversion of Ptolemy (1:1) (7:6) (14:11) (4:3) 7:6-12:11-22:21) 12/7 1/1 12/11 8/7 (28:24:22:21) 7:6 12:11 22:21 The forms with 7:6 as the upper interval have an ancient Greek flavor, and so are grouped together. Near Eastern music tends to favor permutations with 7:6 as the middle interval, with Qutb al-Din al-Shirazi's Hijaz genus which he describes around 1300 (12:11-7:6-22:21) a shading still popular today in the similar Iranian genus and modal family of Chahargah. Mikhail Mashaqa, in the 19th century, describes a similar tuning of Hijaz in Arab music. Both the inverse form of this Hijaz tetrachord, 22:21-7:6-12:11, and a permutation like Maqam Sazkar at 7:6-22:21-12:11, may be especially effective in settings where an 11/9 or medium-low Zalzalian third is congenial; in Arab music, this is a bit low for a typical Rast, where 27/22 or 16/13 may be typical, and 26/21 favored in regions like northern Syria (e.g. Aleppo). In such an Arab style, other varieties of Hijaz and Sazkar may be more idiomatic.[1] ------------- 4. Conclusion ------------- One advantage of 17 notes as a size for a JI system like this based on primes 2-3-7-11-13 is the convenient structure where each adjacent step is a variety of "thirdtone," with step sizes here ranging from 33:32 at 53.3 cents (actually more of a quartertone) to 104:99 at 85.3 cents (approaching the Pythagorean limma or regular diatonic semitone at 256:243 or 90.2 cents). At the same time, there is a great deal of diversity, drawing on Ibn Sina's "most noble genus" which has enriched Near Eastern music for the last millennium or so (8:7-13:12-14:13), and Ptolemy's Intense Chromatic (22:21-12:11-7:6) from almost another millennium further back. The technique of superimposing tetrachords may be a useful method to generate tunings, combining traditional elements in a new synthesis. ---- Note ---- 1. For example, we might choose the 22/21 step of the 17-note system for a moderate Arab Rast, with conjunct and disjunct forms featuring al-Farabi's middle or Zalzalian third at 27/22, the conjunct form being usual in the 9th-14th centuries, and the disjunct form now being considered the "fundamental scale" of modern Arab theory: Conjunct Rast Rast-4 Rast-4 tone |-------------------------|------------------------|........| 1/1 273/242 27/22 4/3 3/2 18/11 39/22 2/1 0 208.7 354.5 498.0 702.0 852.6 991.2 1200.0 273:242 99:91 88:81 9:8 12:11 13:12 44:39 208.7 145.9 143.5 203.9 150.6 138.6 208.8 Disjunct Rast Rast-4 tone Rast-4 |-------------------------|.........|-----------------------| 1/1 273/242 27/22 4/3 3/2 56/33 24/13 2/1 0 208.7 354.5 498.0 702.0 915.6 1061.4 1200.0 273:242 99:91 88:81 9:8 112:99 99:91 13:12 208.7 145.9 143.5 203.9 213.6 145.9 138.6 In a maqam related to Rast and known as Hijazkar, the lower Rast tetrachord has only its second step altered from a tone somewhere around 9/8 to a semitone, forming a genus which the Lebanese composer and theorist Amine Beyhom calls Zirkula, the name in Arabic for the characteristic step of thus tetrachord at a semitone above the 1/1 of Rast. Hijazkar also has a disjunct Hijaz tetrachord on the 3/2 where Beyhom favors a higher third step, here realized in Qutb al-Din's tuning of 12:11-7:6-22:21, where this step is at 14/11: Hijazkar Hijaz-4 tone Hijaz-4 |-------------------------|.........|-----------------------| 1/1 126/121 27/22 4/3 3/2 18/11 21/11 2/1 0 70.1 354.5 498.0 702.0 852.6 1119.5 1200.0 126:121 33:28 88:81 9:8 12:11 7:6 22:21 70.1 284.4 143.5 203.9 150.6 266.9 80.5 Maqam Sazkar, or at least the "textbook" form of the principal notes, can also be derived from Rast, and here yet more simply, by replacing the step of a tone at around 9/8 in the lower tetrachord with a smallish minor third -- or, from another point of view, a "plus-tone" preferably not too far from 7/6 or so. This genus is characteristic of ascending Sazkar, with the regular Rast tetrachord often used also, for example in descending: Sazkar (ascending form) Sazkar-4 tone Rast-4 |-------------------------|.........|-----------------------| 1/1 13/11 27/22 4/3 3/2 56/33 24/13 2/1 0 208.7 354.5 498.0 702.0 915.6 1061.4 1200.0 13:11 27:26 88:81 9:8 112:99 99:91 13:12 289.2 65.3 143.5 203.9 213.6 145.9 138.6 These forms of Hijazkar and Sazkar are not necessarily ideal. In the first Hijaz tetrachord of Hijazkar, we might prefer a slightly wider semitone than 126:121 (70.1 cents) and a somewhat narrower middle step than 33:28 (284.4) cents. In Sazkar, likewise, an ideal form of the lower tetrachord might have a narrower first step than 13:11, and closer to 7:6 (which would allow for a slightly larger middle semitone step than 27:26). Thus, in this 17-note tuning, a Hijazkar tetrachord like that found on the 11/7 step, 1/1-104/99-16/13-4/3 (0-85.3-359.5-498.0 cents) or 104:99-198:169-13:12 (85.3-274.2-138.6 cents) might more nearly approach the ideal, with the middle step quite close to 7:6. Likewise, for Sazkar, a tuning of the lower tetrachord like that found on the 13/7 step of our 17-note system, at 1/1-168/143-16/13-4/3 (0-278.9-359.5-498.0 cents) or 168:143-22:21-13:12 (278.9-80.5-138.6 cents) might more closely approach the ideal conveyed by Beyhom. Part of the adventure with a tuning system is exploring the different locations and rotations, each of which may have its own flavor. The structure of JI based on primes 2-3-7-11-13 may be especially intriguing because of small commas such as 352:351 and 364:363 which promote this diversity and variation. Margo Schulter 20 January 2014