-------------------------------------- Zeta-20 JI tuning An offshoot of Kraig Grady's Centaur and Rod Poole's 17-note tuning Primes 2-3-7-11-13 -------------------------------------- Keyboard mapping with BRILLO (Basically Regularized Interval Locations Logically Organized -- more or less) With inspiration from Fr. Scipione Stella and Fabio Colonna Note reduplication of 1/1, 9/8, 4/3, and 3/2 on both manuals C* C#* D* Eb* E* F* F#* G* G#* A* Bb* B* C* 91/88 9/8 7/6 11/9 4/3 11/8 3/2 14/9 27/16 7/4 11/6 2/1 91/44 58 204 267 347 498 551 702 765 906 969 1049 1200 1258 99:91 28:27 22:21 12:11 33:32 12:11 28:27 99:91 91:88 22:21 12:11 91:88 146 63 81 151 53 151 63 146 58 81 151 58 C C# D Eb E F F# G G# A Bb B C 1/1 13/12 9/8 13/11 14/11 4/3 13/9 3/2 13/8 22/13 16/9 21/11 2/1 0 139 204 289 418 498 637 702 841 911 996 1119 1200 13:12 27:26 104:99 12:11 22:21 13:12 27:26 13:12 176:169 91:88 12:11 22:21 139 65 85 151 81 139 65 139 70 58 151 81 -------------- Chains of fifths and Zalzalian or middle thirds Dashed lines --- show pure fifths (3/2) Dotted lines ... show "virtually tempered" fifths (176/117 or 182/121) Note option of either 22/13 (A) or 27/16 (G#*) 702.0 702.0 706.9 702.0 702.0 706.7 706.9 702.0 702.0 3:2 3:2 176:117 3:2 3:2 182:121 176:117 3:2 3:2 13/9 --- 13/12 --- 13/8 ... 11/9 --- 11/6 --- 11/8 ... 91/88 ... 14/9 ----- 7/6 --- 7/4 636.6 138.6 840.5 347.4 1049.4 551.3 58.0 764.9 266.9 968.8 | | | | | | | | | | | | | | | 11:9 | | | | | | | | 347.4 | | 11:9 39:32 39:32 11:9 11:9 11:9 364:297 11:9 11:9 347.4 342.5 342.5 347.4 347.4 347.4 352.2 347.4 347.4 | | | | | | | | | | | | | | | 176:117 | 182:121 | 3:2 | | | | | | | 706.9 | 706.7 | 702.0 | | | | | | | .... 22/13 ... 14/11 --- 21/11 | | | | | | 910.8 417.5 1119.5 13/11 ... 16/9 ---- 4/3 ---- 1/1 ---- 3/2 --- 9/8 ---- 27/16 289.2 996.1 498.0 0.0 702.0 203.9 905.9 176:117 3:2 3:2 3:2 3:2 3:2 706.9 702.0 702.0 702.0 702.0 702.0 There are, in this view, two chains of fifths: At 13/11-21/11, 8 fifths all within 5 cents of just (using 9/8-22/13-14/11) At 13/9-7/4, 9 fifths all within 5 cents of just in theory -- or 5.9 cents in 1024-ed2 We can also look at the system as similar to George Secor's tuning (in a tempered form) which he recalls using within his 17-note well-temperament (17-WT) in 1978, see his "The 17-note Puzzle -- And the Neo-medieval Key That Unlocks It," _Xenharmonikon_ 18 (Spring, 2006), pp. 55-80 at 71, available at . In a JI form, Secor's tuning is as follows: 13/12 4/3 13/8 1/1 11/9 3/2 11/6 138.6 498.0 840.5 0.0 347.4 702.0 1049.4 16:13 39:32 16:13 11:9 27:22 11:9 359.5 342.5 359.5 347.4 354.5 347.4 Erv Wilson's Rast/Bayyati Matrix based on al-Farabi's Zalzalian thirds of 27/22 and 11/9 , and Jacques Dudon's Mohajira tunings based on various JI or tempered ratios (e.g. 1/1-13/12-59/48-4/3-3/2, or 48:52:59:64:72, see Scala archive, dudon_mohajira_r.scl), are two other examples of this kind of technique with chains of Zalzalian thirds, a form of Dudon's entrelacs or an "interlacing" of two chains of fifths. Margo Schulter 12 December 2013