80edo

From Xenharmonic Wiki
(Redirected from 80et)
Jump to: navigation, search

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 is exactly 15 cents.

Theory

80et is the first equal temperament that represents the 19-limit tonality diamond consistently, though it barely manages to do so.

80et tempers out 176/175 and 540/539 in the 11-limit, 169/168, 325/324, 351/350, 352/351, 364/363 and 1001/1000 in the 13-limit, 136/135, 221/220, 256/255, 289/288, 561/560, 595/594, 715/714, 936/935, 1275/1274 in the 17-limit, 190/189, 286/285, 361/360, 400/399, 456/455, 476/475, 969/968, 1331/1330, 1445/1444, 1521/1520, 1540/1539 and 1729/1728 in the 19-limit, not to mention such important non-superparticular commas as 2048/2025, 4000/3969, 1728/1715 and 3136/3125.

80et provides the optimal patent val for 5-limit diaschismic, for 13-limit srutal, and for 7-, 11- and 13-limit bidia. It is a good tuning for various temperaments in canou family, especially in higher limits.

80et 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

Degree Cents Approximate Ratios*
0 0 1/1
1 15 64/63
2 30 81/80, 50/49
3 45 36/35, 49/48, 34/33
4 60 28/27, 33/32, 26/25, 35/34
5 75 25/24, 22/21, 27/26
6 90 21/20, 19/18, 20/19
7 105 16/15, 17/16, 18/17
8 120 15/14, 14/13
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 81/70, 15/13, 22/19
18 270 7/6
19 285 13/11, 20/17
20 300 25/21, 19/16
21 315 6/5
22 330 17/14
23 345 11/9
24 360 16/13
25 375 21/17
26 390 5/4
27 405 24/19, 19/15
28 420 14/11
29 435 9/7
30 450 35/27, 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.

Just approximation

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23 prime 29 prime 31
Error absolute (¢) 0.00 +3.04 +3.69 +6.17 +3.68 -0.53 +0.04 +2.49 +1.73 +5.42 -5.04
relative (%) 0.0 +20.3 +24.6 +41.1 +24.5 -3.5 +0.3 +16.6 +11.5 +36.2 -33.6