81edo
| ← 80edo | 81edo | 82edo → |
81 equal divisions of the octave (abbreviated 81edo or 81ed2), also called 81-tone equal temperament (81tet) or 81 equal temperament (81et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 81 equal parts of about 14.8 ¢ each. Each step represents a frequency ratio of 21/81, or the 81st root of 2.
Theory
81edo is notable as a tuning for meantone and related temperaments and is the optimal patent val for a number of them. In particular it is the optimal patent val for 5-limit meantone, 7-limit meantone, 11-limit meanpop, 13-limit meanpop, and the rank-3 temperament erato. The electronic music pioneer Daphne Oram was interested in 81edo[1]. As a step in the Golden meantone series of edos, 81edo marks the point at which the series ceases to display audible changes to meantone temperament, and is also the edo with the lowest average and most evenly spread Just-error across the scale (though 31edo does have the best harmonic 7th). However, it is no longer consistent in the 9-odd-limit, as the best direct approximations of 9/8 and 10/9 are one step above and below the patent val mapping.
Besides meantone, 81edo is a tuning for the cobalt temperament, since 81 contains 27 as a divisor. It also tunes the unnamed 15 & 51 temperament which divides the octave into 3 equal parts, and is a member of the augmented-cloudy equivalence continuum. The 81bd val is a tuning for the septimal porcupine temperament.
In the higher limits, it is a strong tuning for the 2.5.17.19 subgroup, and also can be used to map 19/17 to the meantone major second resulting from stacking of two patent val fifths (13\81).
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.66 | -1.13 | -5.86 | +3.50 | -3.17 | +3.92 | -6.79 | -1.25 | -1.22 | +3.29 | -6.05 |
| Relative (%) | -38.2 | -7.6 | -39.6 | +23.6 | -21.4 | +26.4 | -45.8 | -8.4 | -8.2 | +22.2 | -40.9 | |
| Steps (reduced) |
128 (47) |
188 (26) |
227 (65) |
257 (14) |
280 (37) |
300 (57) |
316 (73) |
331 (7) |
344 (20) |
356 (32) |
366 (42) | |
Subsets and supersets
Since 81 is equal to 34, a perfect power of 3, 81edo contains subset edos 3, 9, and 27.
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 47\81) |
Ups and downs notation (Dual sharp fifth 48\81) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 14.8 | ^D, vvE♭♭ | ^D, vvE♭ | |
| 2 | 29.6 | ^^D, vE♭♭ | ^^D, vE♭ | |
| 3 | 44.4 | 38/37, 39/38, 40/39 | vvD♯, E♭♭ | ^3D, E♭ |
| 4 | 59.3 | 29/28, 30/29, 31/30 | vD♯, ^E♭♭ | ^4D, ^E♭ |
| 5 | 74.1 | 23/22, 24/23 | D♯, ^^E♭♭ | ^5D, ^^E♭ |
| 6 | 88.9 | 20/19, 39/37 | ^D♯, vvE♭ | ^6D, ^3E♭ |
| 7 | 103.7 | 17/16, 35/33 | ^^D♯, vE♭ | v5D♯, ^4E♭ |
| 8 | 118.5 | 15/14, 31/29 | vvD𝄪, E♭ | v4D♯, ^5E♭ |
| 9 | 133.3 | 40/37 | vD𝄪, ^E♭ | v3D♯, v6E |
| 10 | 148.1 | 12/11, 37/34 | D𝄪, ^^E♭ | vvD♯, v5E |
| 11 | 163 | 11/10, 34/31 | ^D𝄪, vvE | vD♯, v4E |
| 12 | 177.8 | 31/28 | ^^D𝄪, vE | D♯, v3E |
| 13 | 192.6 | 19/17 | E | ^D♯, vvE |
| 14 | 207.4 | 35/31 | ^E, vvF♭ | ^^D♯, vE |
| 15 | 222.2 | 25/22, 33/29 | ^^E, vF♭ | E |
| 16 | 237 | 39/34 | vvE♯, F♭ | ^E, vvF |
| 17 | 251.9 | 22/19, 37/32 | vE♯, ^F♭ | ^^E, vF |
| 18 | 266.7 | 7/6 | E♯, ^^F♭ | F |
| 19 | 281.5 | 20/17, 33/28 | ^E♯, vvF | ^F, vvG♭ |
| 20 | 296.3 | 19/16 | ^^E♯, vF | ^^F, vG♭ |
| 21 | 311.1 | F | ^3F, G♭ | |
| 22 | 325.9 | 29/24, 35/29 | ^F, vvG♭♭ | ^4F, ^G♭ |
| 23 | 340.7 | 28/23, 39/32 | ^^F, vG♭♭ | ^5F, ^^G♭ |
| 24 | 355.6 | 38/31 | vvF♯, G♭♭ | ^6F, ^3G♭ |
| 25 | 370.4 | 31/25 | vF♯, ^G♭♭ | v5F♯, ^4G♭ |
| 26 | 385.2 | 5/4 | F♯, ^^G♭♭ | v4F♯, ^5G♭ |
| 27 | 400 | 29/23, 39/31 | ^F♯, vvG♭ | v3F♯, v6G |
| 28 | 414.8 | 14/11 | ^^F♯, vG♭ | vvF♯, v5G |
| 29 | 429.6 | 32/25 | vvF𝄪, G♭ | vF♯, v4G |
| 30 | 444.4 | 22/17, 31/24, 40/31 | vF𝄪, ^G♭ | F♯, v3G |
| 31 | 459.3 | 30/23 | F𝄪, ^^G♭ | ^F♯, vvG |
| 32 | 474.1 | 25/19 | ^F𝄪, vvG | ^^F♯, vG |
| 33 | 488.9 | ^^F𝄪, vG | G | |
| 34 | 503.7 | G | ^G, vvA♭ | |
| 35 | 518.5 | 31/23 | ^G, vvA♭♭ | ^^G, vA♭ |
| 36 | 533.3 | 34/25 | ^^G, vA♭♭ | ^3G, A♭ |
| 37 | 548.1 | 11/8 | vvG♯, A♭♭ | ^4G, ^A♭ |
| 38 | 563 | vG♯, ^A♭♭ | ^5G, ^^A♭ | |
| 39 | 577.8 | G♯, ^^A♭♭ | ^6G, ^3A♭ | |
| 40 | 592.6 | 31/22 | ^G♯, vvA♭ | v5G♯, ^4A♭ |
| 41 | 607.4 | ^^G♯, vA♭ | v4G♯, ^5A♭ | |
| 42 | 622.2 | 33/23 | vvG𝄪, A♭ | v3G♯, v6A |
| 43 | 637 | vG𝄪, ^A♭ | vvG♯, v5A | |
| 44 | 651.9 | 16/11, 35/24 | G𝄪, ^^A♭ | vG♯, v4A |
| 45 | 666.7 | 25/17 | ^G𝄪, vvA | G♯, v3A |
| 46 | 681.5 | 37/25 | ^^G𝄪, vA | ^G♯, vvA |
| 47 | 696.3 | A | ^^G♯, vA | |
| 48 | 711.1 | ^A, vvB♭♭ | A | |
| 49 | 725.9 | 35/23, 38/25 | ^^A, vB♭♭ | ^A, vvB♭ |
| 50 | 740.7 | 23/15 | vvA♯, B♭♭ | ^^A, vB♭ |
| 51 | 755.6 | 17/11, 31/20 | vA♯, ^B♭♭ | ^3A, B♭ |
| 52 | 770.4 | 25/16, 39/25 | A♯, ^^B♭♭ | ^4A, ^B♭ |
| 53 | 785.2 | 11/7 | ^A♯, vvB♭ | ^5A, ^^B♭ |
| 54 | 800 | ^^A♯, vB♭ | ^6A, ^3B♭ | |
| 55 | 814.8 | 8/5 | vvA𝄪, B♭ | v5A♯, ^4B♭ |
| 56 | 829.6 | vA𝄪, ^B♭ | v4A♯, ^5B♭ | |
| 57 | 844.4 | 31/19 | A𝄪, ^^B♭ | v3A♯, v6B |
| 58 | 859.3 | 23/14 | ^A𝄪, vvB | vvA♯, v5B |
| 59 | 874.1 | ^^A𝄪, vB | vA♯, v4B | |
| 60 | 888.9 | B | A♯, v3B | |
| 61 | 903.7 | 32/19 | ^B, vvC♭ | ^A♯, vvB |
| 62 | 918.5 | 17/10 | ^^B, vC♭ | ^^A♯, vB |
| 63 | 933.3 | 12/7 | vvB♯, C♭ | B |
| 64 | 948.1 | 19/11 | vB♯, ^C♭ | ^B, vvC |
| 65 | 963 | B♯, ^^C♭ | ^^B, vC | |
| 66 | 977.8 | ^B♯, vvC | C | |
| 67 | 992.6 | 39/22 | ^^B♯, vC | ^C, vvD♭ |
| 68 | 1007.4 | 34/19 | C | ^^C, vD♭ |
| 69 | 1022.2 | ^C, vvD♭♭ | ^3C, D♭ | |
| 70 | 1037 | 20/11, 31/17 | ^^C, vD♭♭ | ^4C, ^D♭ |
| 71 | 1051.9 | 11/6 | vvC♯, D♭♭ | ^5C, ^^D♭ |
| 72 | 1066.7 | 37/20 | vC♯, ^D♭♭ | ^6C, ^3D♭ |
| 73 | 1081.5 | 28/15 | C♯, ^^D♭♭ | v5C♯, ^4D♭ |
| 74 | 1096.3 | 32/17 | ^C♯, vvD♭ | v4C♯, ^5D♭ |
| 75 | 1111.1 | 19/10 | ^^C♯, vD♭ | v3C♯, v6D |
| 76 | 1125.9 | 23/12 | vvC𝄪, D♭ | vvC♯, v5D |
| 77 | 1140.7 | 29/15 | vC𝄪, ^D♭ | vC♯, v4D |
| 78 | 1155.6 | 37/19, 39/20 | C𝄪, ^^D♭ | C♯, v3D |
| 79 | 1170.4 | ^C𝄪, vvD | ^C♯, vvD | |
| 80 | 1185.2 | ^^C𝄪, vD | ^^C♯, vD | |
| 81 | 1200 | 2/1 | D | D |
Notation
Tom Winspear's notation
81edo accidentals created and used by Tom Winspear, based on those provided in Scala though with a logic correction. The innermost accidentals represent one edostep, followed by two, then the bracket representing three. Conventional sharp/doublesharp/flat/doubleflat accidentals are reached in steps of five and the pattern repeats itself on them. The chromatic scale can be notated utilizing only six accidentals in either direction—the rest are for enharmonics.
Ups and downs notation
Ups and downs notation uses free-standing arrows, but the arrows can be connected to conventional accidentals using Helmholtz–Ellis accidentals:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
| Flat symbol |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
Here, a sharp raises by five steps, and a flat lowers by five steps, so single and double arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, then in some cases certain notes may be best spelled with triple arrows.
Regular temperament properties
Commas
- 5-limit commas: 81/80, [-48 1 20⟩
- 7-limit commas: 81/80, 126/125, [-24 1 0 8⟩
- 11-limit commas: 81/80, 126/125, 385/384, 12005/11979
- 13-limit commas: 81/80, 105/104, 144/143, 196/195, 6655/6591