89edo
| ← 88edo | 89edo | 90edo → |
89 equal divisions of the octave (abbreviated 89edo or 89ed2), also called 89-tone equal temperament (89tet) or 89 equal temperament (89et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 89 equal parts of about 13.5 ¢ each. Each step represents a frequency ratio of 21/89, or the 89th root of 2.
Theory
89edo has a harmonic 3 less than a cent flat and a harmonic 5 less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out quartisma.
As an equal temperament, it tempers out 32805/32768 (schisma) in the 5-limit; 126/125, 1728/1715, and 2401/2400 in the 7-limit; and 176/175, 243/242, 441/440 and 540/539 in the 11-limit. It is an especially good tuning for the myna temperament, both in the 7-limit, tempering out 126/125 and 1728/1715, and in the 11-limit, where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament thrush, tempering out 126/125 and 176/175.
The 13-limit is a little tricky as 13 is tuned distinctly flat, tempering out 169/168, 364/363, 729/728, 832/825, and 1287/1280. 13/10 and 15/13 are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes 13/8 much sharper, conflating it with 18/11. It tempers out 144/143, 196/195, 351/350, and 352/351 instead, and supports 13-limit myna and thrush. However 58edo is a better tuning for those purposes.
The 17 and 19 are tuned fairly well, making it consistent to the no-13 21-odd-limit. The equal temperament tempers out 256/255 and 561/560 in the 17-limit; and 171/170, 361/360, 513/512, and 1216/1215 in the 19-limit.
89edo is the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic φ (i.e. (φ - 1)×1200 cents) within a fraction of a cent.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.83 | +4.70 | +1.96 | +1.49 | -4.57 | +2.91 | -0.88 | +5.43 | -4.86 | +1.03 |
| Relative (%) | +0.0 | -6.2 | +34.8 | +14.5 | +11.1 | -33.9 | +21.6 | -6.6 | +40.3 | -36.0 | +7.7 | |
| Steps (reduced) |
89 (0) |
141 (52) |
207 (29) |
250 (72) |
308 (41) |
329 (62) |
364 (8) |
378 (22) |
403 (47) |
432 (76) |
441 (85) | |
Subsets and supersets
89edo is the 24th prime edo, following 83edo and before 97edo.
Intervals
| Steps | Cents | Approximate Ratios | Ups and Downs Notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 13.483 | ^D, v6E♭ | |
| 2 | 26.966 | 56/55, 64/63, 65/64, 66/65 | ^^D, v5E♭ |
| 3 | 40.449 | 45/44, 49/48, 77/75 | ^3D, v4E♭ |
| 4 | 53.933 | 33/32, 65/63 | ^4D, v3E♭ |
| 5 | 67.416 | 27/26, 28/27, 80/77 | ^5D, vvE♭ |
| 6 | 80.899 | 21/20, 22/21 | ^6D, vE♭ |
| 7 | 94.382 | ^7D, E♭ | |
| 8 | 107.865 | 16/15 | D♯, v7E |
| 9 | 121.348 | 15/14, 77/72 | ^D♯, v6E |
| 10 | 134.831 | 13/12 | ^^D♯, v5E |
| 11 | 148.315 | 12/11, 49/45 | ^3D♯, v4E |
| 12 | 161.798 | 11/10 | ^4D♯, v3E |
| 13 | 175.281 | 72/65 | ^5D♯, vvE |
| 14 | 188.764 | 49/44 | ^6D♯, vE |
| 15 | 202.247 | 9/8, 55/49 | E |
| 16 | 215.73 | ^E, v6F | |
| 17 | 229.213 | 8/7 | ^^E, v5F |
| 18 | 242.697 | ^3E, v4F | |
| 19 | 256.18 | 65/56 | ^4E, v3F |
| 20 | 269.663 | 7/6 | ^5E, vvF |
| 21 | 283.146 | 33/28 | ^6E, vF |
| 22 | 296.629 | 32/27, 77/65 | F |
| 23 | 310.112 | ^F, v6G♭ | |
| 24 | 323.596 | 65/54, 77/64 | ^^F, v5G♭ |
| 25 | 337.079 | 40/33, 63/52 | ^3F, v4G♭ |
| 26 | 350.562 | 11/9, 27/22, 49/40, 60/49 | ^4F, v3G♭ |
| 27 | 364.045 | 16/13 | ^5F, vvG♭ |
| 28 | 377.528 | 56/45, 81/65 | ^6F, vG♭ |
| 29 | 391.011 | 5/4, 44/35 | ^7F, G♭ |
| 30 | 404.494 | 81/64 | F♯, v7G |
| 31 | 417.978 | 14/11, 33/26, 80/63 | ^F♯, v6G |
| 32 | 431.461 | 9/7, 77/60 | ^^F♯, v5G |
| 33 | 444.944 | ^3F♯, v4G | |
| 34 | 458.427 | 64/49 | ^4F♯, v3G |
| 35 | 471.91 | 21/16, 55/42 | ^5F♯, vvG |
| 36 | 485.393 | 65/49 | ^6F♯, vG |
| 37 | 498.876 | 4/3 | G |
| 38 | 512.36 | 66/49 | ^G, v6A♭ |
| 39 | 525.843 | 65/48 | ^^G, v5A♭ |
| 40 | 539.326 | 15/11 | ^3G, v4A♭ |
| 41 | 552.809 | 11/8 | ^4G, v3A♭ |
| 42 | 566.292 | 18/13 | ^5G, vvA♭ |
| 43 | 579.775 | 7/5 | ^6G, vA♭ |
| 44 | 593.258 | 45/32 | ^7G, A♭ |
| 45 | 606.742 | 64/45 | G♯, v7A |
| 46 | 620.225 | 10/7, 63/44 | ^G♯, v6A |
| 47 | 633.708 | 13/9, 81/56 | ^^G♯, v5A |
| 48 | 647.191 | 16/11 | ^3G♯, v4A |
| 49 | 660.674 | 22/15 | ^4G♯, v3A |
| 50 | 674.157 | 65/44 | ^5G♯, vvA |
| 51 | 687.64 | 49/33 | ^6G♯, vA |
| 52 | 701.124 | 3/2 | A |
| 53 | 714.607 | ^A, v6B♭ | |
| 54 | 728.09 | 32/21 | ^^A, v5B♭ |
| 55 | 741.573 | 49/32, 75/49 | ^3A, v4B♭ |
| 56 | 755.056 | 65/42 | ^4A, v3B♭ |
| 57 | 768.539 | 14/9, 81/52 | ^5A, vvB♭ |
| 58 | 782.022 | 11/7, 52/33, 63/40 | ^6A, vB♭ |
| 59 | 795.506 | ^7A, B♭ | |
| 60 | 808.989 | 8/5, 35/22 | A♯, v7B |
| 61 | 822.472 | 45/28, 77/48 | ^A♯, v6B |
| 62 | 835.955 | 13/8 | ^^A♯, v5B |
| 63 | 849.438 | 18/11, 44/27, 49/30, 80/49 | ^3A♯, v4B |
| 64 | 862.921 | 33/20 | ^4A♯, v3B |
| 65 | 876.404 | ^5A♯, vvB | |
| 66 | 889.888 | ^6A♯, vB | |
| 67 | 903.371 | 27/16 | B |
| 68 | 916.854 | 56/33 | ^B, v6C |
| 69 | 930.337 | 12/7, 77/45 | ^^B, v5C |
| 70 | 943.82 | ^3B, v4C | |
| 71 | 957.303 | ^4B, v3C | |
| 72 | 970.787 | 7/4 | ^5B, vvC |
| 73 | 984.27 | ^6B, vC | |
| 74 | 997.753 | 16/9 | C |
| 75 | 1011.236 | ^C, v6D♭ | |
| 76 | 1024.719 | 65/36 | ^^C, v5D♭ |
| 77 | 1038.202 | 20/11 | ^3C, v4D♭ |
| 78 | 1051.685 | 11/6, 81/44 | ^4C, v3D♭ |
| 79 | 1065.169 | 24/13 | ^5C, vvD♭ |
| 80 | 1078.652 | 28/15 | ^6C, vD♭ |
| 81 | 1092.135 | 15/8 | ^7C, D♭ |
| 82 | 1105.618 | C♯, v7D | |
| 83 | 1119.101 | 21/11, 40/21 | ^C♯, v6D |
| 84 | 1132.584 | 27/14, 52/27, 77/40 | ^^C♯, v5D |
| 85 | 1146.067 | 64/33 | ^3C♯, v4D |
| 86 | 1159.551 | ^4C♯, v3D | |
| 87 | 1173.034 | 55/28, 63/32, 65/33 | ^5C♯, vvD |
| 88 | 1186.517 | ^6C♯, vD | |
| 89 | 1200 | 2/1 | D |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-141 89⟩ | [⟨89 141]] | +0.262 | 0.262 | 1.95 |
| 2.3.5 | 32805/32768, 10077696/9765625 | [⟨89 141 207]] | -0.500 | 1.098 | 8.15 |
| 2.3.5.7 | 126/125, 1728/1715, 32805/32768 | [⟨89 141 207 250]] | -0.550 | 0.955 | 7.08 |
| 2.3.5.7.11 | 126/125, 176/175, 243/242, 16384/16335 | [⟨89 141 207 250 308]] | -0.526 | 0.855 | 6.35 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 13\89 | 175.28 | 72/65 | Sesquiquartififths / sesquart |
| 1 | 21\89 | 283.15 | 13/11 | Neominor |
| 1 | 23\89 | 310.11 | 6/5 | Myna |
| 1 | 29\89 | 391.01 | 5/4 | Amigo |
| 1 | 37\89 | 498.87 | 4/3 | Grackle |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Scales
Music
- Singing Golden Myna (2022) – myna[11] in 89edo