Posted to the tuning list 4-Jul-2001. Updated 2-Oct-2002 The Miracle temperament gives us a logical way of further extending the Fokker extended-diatonic interval-names http://dkeenan.com/Music/IntervalNaming.htm from 31-EDO to Miracle chains, and hence to 41-EDO, 72-EDO and 11-limit JI. Previously there was no obvious way of deciding which of a pair of nearby intervals (such as the neutral seconds 10:11 and 11:12 or the minor sevenths 5:9 and 9:16) should be called "wide" or "narrow", and which should be unmodified. Now the answer is obvious(?). The unmodified one is the one that is represented within a chain of +-15 Miracle generators. i.e. The intervals available in Miracle-16 should be named the same as in 31-EDO, without using "wide" or "narrow". The table below shows how this scheme names the intervals of 72-EDO and 11-(odd)limit ratios. Legend for interval names: 1 unison 2 second 3 third 4 fourth 5 fifth 6 sixth 7 seventh 8 octave m = minor N = neutral M = major d = diminished P = perfect A = augmented s = sub S = super n = narrow W = wide Legend for 72-ET note names: A,B,C,D,E,F,G,#,b as for 12-tET ^ = quarter-tone up (+50 c) f = sixth-tone up (+33 c) / = twelfth-tone up (+17 c) \ = twelfth-tone down (-17 c) t = sixth-tone down (-33 c) v = quarter-tone down (-50 c) No. Cents Intvl Note Most popular ? gens name frm C 27-limit (19-prime-limit) ratio --------------------------------- 0 0 P1 C 1:1 31 17 W1 C/ -10 33 S1 Cf 21 50 WS1 C^ -20 67 nsm2 C#t 24:25 (27:28 25:26) 11 83 sm2 C#\ 20:21 (21:22) -30 100 nm2 C# 16:17 (17:18) 1 117 m2 Db/ 15:16 (14:15) 32 133 Wm2 Dbf 12:13 (13:14) -9 150 N2 Dv 11:12 22 167 WN2 Dt 10:11 -19 183 nM2 D\ 9:10 12 200 M2 D 8:9 -29 217 nSM2 D/ 15:17 (22:25) 2 233 SM2 Df 7:8 33 250 WSM2 D^ 13:15 -8 267 sm3 D#t 6:7 23 283 Wsm3 D#\ 15:17 (11:13) -18 300 nm3 Eb 27:32 (21:25 16:19) 13 317 m3 Eb/ 5:6 -28 333 nN3 Ebf 14:17 3 350 N3 Ev 9:11 (22:27) 34 367 WN3 Et 13:16 -7 383 M3 E\ 4:5 24 400 WM3 E 19:24 (27:34) -17 417 nSM3 E/ 11:14 14 433 SM3 Ef 7:9 (25:32) -27 450 ns4 Fv 10:13 (17:22) 4 467 s4 Ft 16:21 35 483 Ws4 F\ -6 500 P4 F 3:4 25 517 WP4 F/ 20:27 -16 533 nS4 Ff 11:15 15 550 S4 F^ 8:11 -26 567 nA4 F#t 18:25 (13:18) 5 583 A4 F#\ 5:7 +-36 600 WA4/nd5 F# 12:17 (17:24 19:27) -5 617 d5 Gb/ 7:10 26 633 Wd5 Gbf 9:13 (25:36) -15 650 s5 Gv 11:16 16 667 Ws5 Gt 15:22 -25 683 nP5 G\ 27:40 6 700 P5 G 2:3 -35 717 nS5 G/ -4 733 S5 Gf 21:32 27 750 WS5 G^ 13:20 (11:17) -14 767 sm6 G#t 9:14 (16:25) 17 783 Wsm6 G#\ 7:11 -24 800 nm6 G# 12:19 (17:27) 7 817 m6 Ab/ 5:8 -34 833 nN6 Abf 8:13 -3 850 N6 Av 11:18 (27:44) 28 867 WN6 At 17:28 -13 883 M6 A\ 3:5 18 900 WM6 A 16:27 (25:42 19:16) -23 917 nSM6 A/ 13:22 (17:30) 8 933 SM6 Af 7:12 -33 950 nsm7 A^ 11:19 (15:26) -2 967 sm7 A#t 4:7 29 983 Wsm7 A#\ 17:30 (25:44) -12 1000 m7 Bb 9:16 19 1017 Wm7 Bb/ 5:9 -22 1033 nN7 Bbf 11:20 9 1050 N7 Bv 6:11 -32 1067 nM7 Bt 7:13 (13:24) -1 1083 M7 B\ 8:15 (15:28) 30 1100 WM7 B 8:17 (17:36) -11 1117 SM7 B/ 10:21 (21:44) 20 1133 WSM7 Bf 12:25 (27:56 25:52) -21 1150 ns8 Cv 10 1167 s8 Ct -31 1183 n8 C\ Note that, in the same way as Fokker's system, the same interval may have more than one name consistent with this system. Regards, -- Dave Keenan