80edo

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80edo
Prime factorization 24 × 5
Step size 15.0000¢
Fifth 47\80 (705.0¢)
Major 2nd 14\80 (210.0¢)
Semitones (A1:m2) 9:5 (135.0¢ : 75.0¢)
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 harmonics in 80edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43
Error absolute (¢) +0.00 +3.04 +3.69 +6.17 +3.68 -0.53 +0.04 +2.49 +1.73 +5.42 -5.04 +3.66 +5.94 -1.52
relative (%) +0 +20 +25 +41 +25 -4 +0 +17 +12 +36 -34 +24 +40 -10
Steps
(reduced)
80
(0)
127
(47)
186
(26)
225
(65)
277
(37)
296
(56)
327
(7)
340
(20)
362
(42)
389
(69)
396
(76)
417
(17)
429
(29)
434
(34)

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, 29/28, 30/29
5 75 25/24, 22/21, 27/26, 24/23, 23/22
6 90 21/20, 19/18, 20/19
7 105 16/15, 17/16, 18/17
8 120 15/14
9 135 27/25, 13/12, 14/13
10 150 12/11
11 165 11/10
12 180 10/9, 21/19
13 195 28/25, 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, 29/25
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 32/25, 14/11, 23/18
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