80edo

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The 80 equal temperament, often abbreviated 80-tET, 80-EDO, or 80-ET, is the scale derived by dividing the octave into 80 equally-sized steps. Each step represents a frequency ratio of exactly 15 cents. 80et is the first equal temperament that represents the 19-limit tonality diamond consistently (it barely manages to do so).

80 et tempers out 136/135, 169/168, 176/175, 190/189, 221/220, 256/255, 286/285, 289/288, 325/324, 351/350, 352/351, 361/360, 364/363, 400/399, 456/455, 476/475, 540/539, 561/560, 595/594, 715/714, 936/935, 969/968, 1001/1000, 1275/1274, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.

80 supports a profusion of 19-limit (and lower) rank two temperaments which have mostly not been explored. We might mention:

31&80 <<7 6 15 27 -24 -23 -20 ... ||

72&80 <<24 30 40 24 32 24 0 ... ||

34&80 <<2 -4 -50 22 16 2 -40 ... ||

46&80 <<2 -4 30 22 16 2 40 ... ||

29&80 <<3 34 45 33 24 -37 20 ... ||

12&80 <<4 -8 -20 -36 32 4 0 ... ||

22&80 <<6 -10 12 -14 -32 6 -40 ... ||

58&80 <<6 -10 12 -14 -32 6 40 ... ||

41&80 <<7 26 25 -3 -24 -33 20 ... ||

In each case, the numbers joined by an ampersand represent 19-limit patent vals (meaning obtained by rounding to the nearest integer) and the first and most important part of the wedgie is given.

Intervals of 80edo

degrees cents pions 7mus ratios*
0 1/1
1 15 15.9 19.2 (13.316) 64/63
2 30 31.8 38.4 (26.616) 81/80
3 45 47.7 57.6 (39.A16) 34/33, 36/35
4 60 63.6 76.8 (4C.D16) 26/25, 28/27, 33/32, 35/34
5 75 79.5 96 (6016) 22/21, 25/24, 27/26
6 90 95.4 115.2 (73.316) 19/18, 20/19, 21/20
7 105 111.3 134.4 (86.616) 16/15, 17/16, 18/17
8 120 127.2 153.6 (99.A16) 14/13, 15/14
9 135 143.1 172.8 (AC.D16) 13/12
10 150 159 192 (C016) 12/11
11 165 174.9 211.2 (D3.316) 11/10
12 180 190.8 230.4 (E6.616) 10/9, 21/19
13 195 206.7 249.6 (F9.A16) 19/17
14 210 222.6 268.8 (10C.D16) 9/8, 17/15
15 225 238.5 288 (12016) 8/7
16 240 254.4 307.2 (133.316)
17 255 270.3 326.4 (146.616) 15/13, 22/19
18 270 286.2 345.6 (159.A16) 7/6
19 285 301.1 364.8 (16C.D16) 13/11, 20/17
20 300 318 384 (18016) 19/16, 25/21
21 315 333.9 403.2 (193.316) 6/5
22 330 349.8 422.4 (1A6.616) 17/14
23 345 365.7 441.6 (1B9.A16) 11/9
24 360 381.6 460.8 (1CC.D16) 16/13, 21/17
25 375 397.5 480 (1E016)
26 390 413.4 499.2 (1F3.316) 5/4
27 405 429.3 518.4 (206.616) 19/15, 24/19
28 420 445 537.6 (219.A16) 14/11
29 435 461.1 556.8 (22C.D16) 9/7
30 450 477 576 (24016) 13/10, 22/17
31 465 492.9 595.2 (253.316) 17/13
32 480 508.8 614.4 (266.616) 21/16, 25/19
33 495 524.7 633.6 (279.A16) 4/3
34 510 540.6 652.8 (28C.D16)
35 525 556.5 672 (2A016) 19/14
36 540 572.4 691.2 (2B3.316) 26/19
37 555 588.3 710.4 (2C6.616) 11/8
38 570 604.2 729.6 (2D9.A16) 18/13
39 585 620.1 748.8 (2EC.D16) 7/5
40 600 636 768 (30016) 17/12, 24/17
  • based on treating 80edo as a 19-limit temperament; other approaches are possible.