92edo
| ← 91edo | 92edo | 93edo → |
92 equal divisions of the octave (abbreviated 92edo or 92ed2), also called 92-tone equal temperament (92tet) or 92 equal temperament (92et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 92 equal parts of about 13 ¢ each. Each step represents a frequency ratio of 21/92, or the 92nd root of 2.
Theory
92edo's patent val is contorted through the 17-limit, with the same tuning and commas as 46edo, and hence attracts little interest. That said, the approximation to the 19th harmonic is much improved.
However, the 92def val scores best in the 13-limit and achieves 19-odd-limit monotonicity, which the patent val fails to. Using this val, it tempers out 2430/2401, 3136/3125, and 4000/3969 in the 7-limit; 176/175, 243/242 and 896/891 in the 11-limit; 351/350, 352/351, and 364/363 in the 13-limit; 136/135, 221/220 and 289/288 in the 17-limit; and 190/189 among many others in the 19-limit. It supports and provides an excellent tuning for the unnamed bidia extension 12f & 80.
In either val above, the perfect fifth is approximated by 54\92, about 2.4 ¢ sharp. The alternate fifth 53\92 is a very flat fifth, flatter even than that of 26edo and only 0.102 ¢ sharp of 1/2-comma meantone; the 92bcccd val supports flattone, while the 92bcccdd val supports flattertone.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +2.39 | +4.99 | -3.61 | +4.79 | -3.49 | -5.75 | -5.66 | -0.61 | +2.49 | -1.22 | -2.19 |
| Relative (%) | +18.3 | +38.3 | -27.7 | +36.7 | -26.8 | -44.0 | -43.4 | -4.7 | +19.1 | -9.3 | -16.8 | |
| Steps (reduced) |
146 (54) |
214 (30) |
258 (74) |
292 (16) |
318 (42) |
340 (64) |
359 (83) |
376 (8) |
391 (23) |
404 (36) |
416 (48) | |
Subsets and supersets
Since 92 factors into primes as 22 × 23, 92edo has subset edos 2, 4, 23, and 46.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 13 | ^D, v5E♭ | |
| 2 | 26.1 | ^^D, v4E♭ | |
| 3 | 39.1 | 42/41, 43/42 | ^3D, v3E♭ |
| 4 | 52.2 | 33/32, 34/33, 35/34 | ^4D, vvE♭ |
| 5 | 65.2 | ^5D, vE♭ | |
| 6 | 78.3 | 22/21, 23/22 | v4D♯, E♭ |
| 7 | 91.3 | 19/18, 20/19, 39/37 | v3D♯, ^E♭ |
| 8 | 104.3 | 17/16, 35/33 | vvD♯, ^^E♭ |
| 9 | 117.4 | 31/29 | vD♯, ^3E♭ |
| 10 | 130.4 | 14/13, 41/38 | D♯, ^4E♭ |
| 11 | 143.5 | 38/35 | ^D♯, v5E |
| 12 | 156.5 | 23/21, 35/32 | ^^D♯, v4E |
| 13 | 169.6 | 32/29, 43/39 | ^3D♯, v3E |
| 14 | 182.6 | 10/9 | ^4D♯, vvE |
| 15 | 195.7 | 37/33 | ^5D♯, vE |
| 16 | 208.7 | 35/31 | E |
| 17 | 221.7 | 33/29, 42/37 | ^E, v5F |
| 18 | 234.8 | ^^E, v4F | |
| 19 | 247.8 | ^3E, v3F | |
| 20 | 260.9 | 36/31, 43/37 | ^4E, vvF |
| 21 | 273.9 | 34/29, 41/35 | ^5E, vF |
| 22 | 287 | 13/11, 33/28 | F |
| 23 | 300 | 19/16 | ^F, v5G♭ |
| 24 | 313 | 6/5 | ^^F, v4G♭ |
| 25 | 326.1 | 29/24, 35/29, 41/34 | ^3F, v3G♭ |
| 26 | 339.1 | 28/23 | ^4F, vvG♭ |
| 27 | 352.2 | 38/31 | ^5F, vG♭ |
| 28 | 365.2 | 21/17 | v4F♯, G♭ |
| 29 | 378.3 | 41/33 | v3F♯, ^G♭ |
| 30 | 391.3 | vvF♯, ^^G♭ | |
| 31 | 404.3 | 24/19, 43/34 | vF♯, ^3G♭ |
| 32 | 417.4 | 14/11 | F♯, ^4G♭ |
| 33 | 430.4 | 41/32 | ^F♯, v5G |
| 34 | 443.5 | 31/24, 40/31 | ^^F♯, v4G |
| 35 | 456.5 | 43/33 | ^3F♯, v3G |
| 36 | 469.6 | 21/16, 38/29 | ^4F♯, vvG |
| 37 | 482.6 | 37/28, 41/31 | ^5F♯, vG |
| 38 | 495.7 | 4/3 | G |
| 39 | 508.7 | ^G, v5A♭ | |
| 40 | 521.7 | 23/17, 27/20 | ^^G, v4A♭ |
| 41 | 534.8 | ^3G, v3A♭ | |
| 42 | 547.8 | ^4G, vvA♭ | |
| 43 | 560.9 | 29/21 | ^5G, vA♭ |
| 44 | 573.9 | 32/23, 39/28 | v4G♯, A♭ |
| 45 | 587 | v3G♯, ^A♭ | |
| 46 | 600 | 41/29 | vvG♯, ^^A♭ |
| 47 | 613 | 37/26 | vG♯, ^3A♭ |
| 48 | 626.1 | 23/16, 33/23 | G♯, ^4A♭ |
| 49 | 639.1 | 42/29 | ^G♯, v5A |
| 50 | 652.2 | 35/24 | ^^G♯, v4A |
| 51 | 665.2 | ^3G♯, v3A | |
| 52 | 678.3 | 34/23, 40/27 | ^4G♯, vvA |
| 53 | 691.3 | ^5G♯, vA | |
| 54 | 704.3 | 3/2 | A |
| 55 | 717.4 | ^A, v5B♭ | |
| 56 | 730.4 | 29/19, 32/21 | ^^A, v4B♭ |
| 57 | 743.5 | 43/28 | ^3A, v3B♭ |
| 58 | 756.5 | 31/20 | ^4A, vvB♭ |
| 59 | 769.6 | ^5A, vB♭ | |
| 60 | 782.6 | 11/7 | v4A♯, B♭ |
| 61 | 795.7 | 19/12 | v3A♯, ^B♭ |
| 62 | 808.7 | vvA♯, ^^B♭ | |
| 63 | 821.7 | 37/23 | vA♯, ^3B♭ |
| 64 | 834.8 | 34/21 | A♯, ^4B♭ |
| 65 | 847.8 | 31/19 | ^A♯, v5B |
| 66 | 860.9 | 23/14 | ^^A♯, v4B |
| 67 | 873.9 | ^3A♯, v3B | |
| 68 | 887 | 5/3 | ^4A♯, vvB |
| 69 | 900 | 32/19, 37/22 | ^5A♯, vB |
| 70 | 913 | 22/13, 39/23 | B |
| 71 | 926.1 | 29/17, 41/24 | ^B, v5C |
| 72 | 939.1 | 31/18 | ^^B, v4C |
| 73 | 952.2 | ^3B, v3C | |
| 74 | 965.2 | ^4B, vvC | |
| 75 | 978.3 | 37/21 | ^5B, vC |
| 76 | 991.3 | 39/22 | C |
| 77 | 1004.3 | ^C, v5D♭ | |
| 78 | 1017.4 | 9/5 | ^^C, v4D♭ |
| 79 | 1030.4 | 29/16 | ^3C, v3D♭ |
| 80 | 1043.5 | 42/23 | ^4C, vvD♭ |
| 81 | 1056.5 | 35/19 | ^5C, vD♭ |
| 82 | 1069.6 | 13/7 | v4C♯, D♭ |
| 83 | 1082.6 | 43/23 | v3C♯, ^D♭ |
| 84 | 1095.7 | 32/17 | vvC♯, ^^D♭ |
| 85 | 1108.7 | 19/10, 36/19 | vC♯, ^3D♭ |
| 86 | 1121.7 | 21/11 | C♯, ^4D♭ |
| 87 | 1134.8 | ^C♯, v5D | |
| 88 | 1147.8 | 33/17 | ^^C♯, v4D |
| 89 | 1160.9 | 41/21, 43/22 | ^3C♯, v3D |
| 90 | 1173.9 | ^4C♯, vvD | |
| 91 | 1187 | ^5C♯, vD | |
| 92 | 1200 | 2/1 | D |
Instruments
A Lumatone mapping for 92edo is available.
Music
- microtonal improvisation in 92edo (2025)
- 92edo waltz (2025)