80edo
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
Script error: No such module "primes_in_edo".
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.
| Periods per Octave |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio (Reduced) |
Temperament |
|---|---|---|---|---|
| 1 | 1\80 | 15 | 121/120 | Yarman |
| 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 |