80edo

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80edo
Prime factorization 24 × 5
Step size 15.00000¢
Fifth 47\80 (705.00¢)
Major 2nd 14\80 (210¢)
Semitones (A1:m2) 9:5 (135¢ : 75¢)
Consistency limit 19

The 80 equal divisions of the octave (80edo), or the 80(-tone) equal temperament (80tet, 80et) when viewed from a regular temperament perspective, is the tuning system 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-odd-limit tonality diamond consistently, though it barely manages to do so. Despite this, a large number of intervals in higher odd limits in the 29-prime-limit are consistent, and its patent val generally does well at approximating (29-prime-limited) harmonic series segments, such as modes 16 through 30 but especially modes 8 through 15, with inconsistencies usually caused by not cancelling the over-sharpness of compound harmonics 21, 27, 35, 45 and 49 (and their octave-equivalents), which may be seen as an interesting limitation. It achieves this much consistency because all primes in the 29-limit except 13 are sharp of just. This means it can be used as a general-purpose approximate 29-prime-limit system with a relatively manageable number of tones, with some care taken around inconsistency.

80et tempers out 2048/2025, 3136/3125, 1728/1715, 4375/4374 and 4000/3969 in the 7-limit, 176/175, 540/539 and 4000/3993 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 and 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, 208/207, 253/252, 323/322 and 460/459 in the 23-limit and 320/319 in the 29-limit, equating a sharp 29/16 with a near-perfect 20/11, although this equivalence begins to make more sense when you consider the error cancellations with other sharp harmonics and as a way to give more reasonable interpretations to otherwise questionably mapped intervals.

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.

Prime harmonics

Approximation of prime intervals in 80 EDO
Prime number 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0.00 +3.04 +3.69 +6.17 +3.68 -0.53 +0.04 +2.49 +1.73 +5.42
relative (%) +0 +20 +25 +41 +25 -4 +0 +17 +12 +36
Steps (reduced) 80 (0) 127 (47) 186 (26) 225 (65) 277 (37) 296 (56) 327 (7) 340 (20) 362 (42) 389 (69)

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
9 135 13/12, 14/13
10 150 12/11
11 165 11/10
12 180 10/9, 21/19
13 195 19/17
14 210 9/8, 17/15, 26/23
15 225 8/7
16 240 23/20
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, 23/19, 29/24
23 345 11/9
24 360 16/13
25 375 21/17, 36/29
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, 29/22
33 495 4/3
34 510 51/38, 75/56
35 525 19/14, 23/17
36 540 15/11, 26/19
37 555 11/8
38 570 18/13, 32/23
39 585 7/5
40 600 17/12, 24/17

* based on treating 80edo as a 29-limit temperament; other approaches are possible. Inconsistent interpretations in italic.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢)
Tuning error
Absolute (¢) Relative (%)
2.3 [127 -80 [80 127]] -0.961 0.960 6.40
2.3.5 2048/2025, 390625000/387420489 [80 127 186]] -1.169 0.837 5.59
2.3.5.7 1728/1715, 2048/2025, 3136/3125 [80 127 186 225]] -1.426 0.851 5.68
2.3.5.7.11 176/175, 540/539, 896/891, 1331/1323 [80 127 186 225 277]] -1.353 0.775 5.17
2.3.5.7.11.13 169/168, 176/175, 325/324, 364/363, 540/539 [80 127 186 225 277 296]] -1.105 0.901 6.01
2.3.5.7.11.13.17 136/135, 169/168, 176/175, 221/220, 364/363, 540/539 [80 127 186 225 277 296 327]] -0.949 0.917 6.12
2.3.5.7.11.13.17.19 136/135, 169/168, 176/175, 190/189, 221/220, 364/363, 400/399 [80 127 186 225 277 296 327 340]] -0.903 0.867 5.78

Rank-2 temperaments

80et supports a profusion of 19-limit (and lower) rank-2 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.

Rank-2 temperaments by generator
Periods
per Octave
Generator
(Reduced)
Cents
(Reduced)
Associated Ratio
(Reduced)
Temperament
1 1\80 15 121/120 Yarman I
1 3\80 45 36/35~40/39 Quartonic
1 9\80 135 27/25 Superlimmal
1 21\80 315 6/5 Parakleismic / parkleismic / paradigmic
1 29\80 435 9/7 Supermajor
1 31\80 465 17/13 Semisept
1 39\80 585 7/5 Pluto
2 21\80
(19\80)
315
(285)
6/5
(33/28)
Semiparakleismic
2 29\80
(11\80)
435
(165)
9/7
(11/10)
Echidna
Semisupermajor
2 33\80
(7\80)
495
(105)
4/3
(17/16)
Srutal
4 33\80
(7\80)
495
(105)
4/3
(17/16)
Bidia
5 15\80
(1\80)
225
(15)
8/7
(64/63)
Pentorwell
5 37\80
(5\80)
555
(75)
11/8
(25/24)
Trisedodge / countdown
8 39\80
(1\80)
585
(15)
7/5
(99/98~100/99)
Octoid / octopus
10 21\80
(3\80)
315
(45)
6/5
(40/39)
Deca
20 33\80
(1\80)
495
(15)
4/3
(99/98~100/99)
Degrees