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 ratios*
0 0 1/1
1 15 64/63
2 30 81/80
3 45 34/33, 36/35
4 60 26/25, 28/27, 33/32, 35/34
5 75 22/21, 25/24, 27/26
6 90 19/18, 20/19, 21/20
7 105 16/15, 17/16, 18/17
8 120 14/13, 15/14
9 135 13/12
10 150 12/11
11 165 11/10
12 180 10/9, 21/19
13 195 19/17
14 210 9/8, 17/15
15 225 8/7
16 240
17 255 15/13, 22/19
18 270 7/6
19 285 13/11, 20/17
20 300 19/16, 25/21
21 315 6/5
22 330 17/14
23 345 11/9
24 360 16/13, 21/17
25 375
26 390 5/4
27 405 19/15, 24/19
28 420 14/11
29 435 9/7
30 450 13/10, 22/17
31 465 17/13
32 480 21/16, 25/19
33 495 4/3
34 510
35 525 19/14
36 540 26/19
37 555 11/8
38 570 18/13
39 585 7/5
40 600 17/12, 24/17
  • based on treating 80edo as a 19-limit temperament; other approaches are possible.