80edo
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[edit]
| 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.