90edo
Jump to navigation
Jump to search
Prime factorization
2 × 32 × 5
Step size
13.3333¢
Fifth
53\90 (706.667¢)
Semitones (A1:m2)
11:5 (146.7¢ : 66.67¢)
Dual sharp fifth
53\90 (706.667¢)
Dual flat fifth
52\90 (693.333¢) (→26\45)
Dual major 2nd
15\90 (200¢) (→1\6)
Consistency limit
7
Distinct consistency limit
7
| ← 89edo | 90edo | 91edo → |
90 equal divisions of the octave (abbreviated 90edo or 90ed2), also called 90-tone equal temperament (90tet) or 90 equal temperament (90et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 90 equal parts of about 13.3 ¢ each. Each step represents a frequency ratio of 21/90, or the 90th root of 2.
Theory
As an equal temperament, it tempers out 2048/2025 (diaschisma) in the 5-limit, 245/243 and 3125/3087 in the 7-limit, 121/120 and 176/175 in the 11-limit, and 169/168 and 275/273 in the 13-limit. It provides the optimal patent val for the 31 & 90 temperament in the 7-, 11- and 13-limit. Notably, it is the second lowest in a series of four consecutive edos to temper out the quartisma.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +4.71 | +0.35 | +4.51 | -3.91 | -4.65 | -0.53 | +5.06 | +1.71 | -4.18 | -4.11 | -1.61 |
| Relative (%) | +35.3 | +2.6 | +33.8 | -29.3 | -34.9 | -4.0 | +38.0 | +12.8 | -31.3 | -30.9 | -12.1 | |
| Steps (reduced) |
143 (53) |
209 (29) |
253 (73) |
285 (15) |
311 (41) |
333 (63) |
352 (82) |
368 (8) |
382 (22) |
395 (35) |
407 (47) | |
Interval table
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 52\90) |
Ups and downs notation (Dual sharp fifth 53\90) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 13.3 | ^D, vE♭♭♭ | ^D, v4E♭ | |
| 2 | 26.7 | ^^D, E♭♭♭ | ^^D, v3E♭ | |
| 3 | 40 | 41/40 | vD♯, ^E♭♭♭ | ^3D, vvE♭ |
| 4 | 53.3 | 32/31, 33/32, 34/33 | D♯, vvE♭♭ | ^4D, vE♭ |
| 5 | 66.7 | 26/25 | ^D♯, vE♭♭ | ^5D, E♭ |
| 6 | 80 | ^^D♯, E♭♭ | v5D♯, ^E♭ | |
| 7 | 93.3 | 39/37 | vD𝄪, ^E♭♭ | v4D♯, ^^E♭ |
| 8 | 106.7 | 17/16, 33/31 | D𝄪, vvE♭ | v3D♯, ^3E♭ |
| 9 | 120 | 15/14 | ^D𝄪, vE♭ | vvD♯, ^4E♭ |
| 10 | 133.3 | 40/37, 41/38 | ^^D𝄪, E♭ | vD♯, ^5E♭ |
| 11 | 146.7 | 25/23, 37/34 | vD♯𝄪, ^E♭ | D♯, v5E |
| 12 | 160 | 34/31 | D♯𝄪, vvE | ^D♯, v4E |
| 13 | 173.3 | ^D♯𝄪, vE | ^^D♯, v3E | |
| 14 | 186.7 | 29/26, 39/35 | E | ^3D♯, vvE |
| 15 | 200 | 37/33 | ^E, vF♭♭ | ^4D♯, vE |
| 16 | 213.3 | 26/23 | ^^E, F♭♭ | E |
| 17 | 226.7 | vE♯, ^F♭♭ | ^E, v4F | |
| 18 | 240 | 23/20, 39/34 | E♯, vvF♭ | ^^E, v3F |
| 19 | 253.3 | 22/19, 37/32 | ^E♯, vF♭ | ^3E, vvF |
| 20 | 266.7 | 7/6 | ^^E♯, F♭ | ^4E, vF |
| 21 | 280 | 20/17 | vE𝄪, ^F♭ | F |
| 22 | 293.3 | E𝄪, vvF | ^F, v4G♭ | |
| 23 | 306.7 | 31/26, 37/31 | ^E𝄪, vF | ^^F, v3G♭ |
| 24 | 320 | F | ^3F, vvG♭ | |
| 25 | 333.3 | 17/14, 23/19, 40/33 | ^F, vG♭♭♭ | ^4F, vG♭ |
| 26 | 346.7 | ^^F, G♭♭♭ | ^5F, G♭ | |
| 27 | 360 | 16/13 | vF♯, ^G♭♭♭ | v5F♯, ^G♭ |
| 28 | 373.3 | 31/25, 41/33 | F♯, vvG♭♭ | v4F♯, ^^G♭ |
| 29 | 386.7 | 5/4 | ^F♯, vG♭♭ | v3F♯, ^3G♭ |
| 30 | 400 | 29/23, 39/31 | ^^F♯, G♭♭ | vvF♯, ^4G♭ |
| 31 | 413.3 | 33/26 | vF𝄪, ^G♭♭ | vF♯, ^5G♭ |
| 32 | 426.7 | 32/25, 41/32 | F𝄪, vvG♭ | F♯, v5G |
| 33 | 440 | 40/31 | ^F𝄪, vG♭ | ^F♯, v4G |
| 34 | 453.3 | 13/10 | ^^F𝄪, G♭ | ^^F♯, v3G |
| 35 | 466.7 | 17/13, 38/29 | vF♯𝄪, ^G♭ | ^3F♯, vvG |
| 36 | 480 | 29/22, 33/25, 37/28 | F♯𝄪, vvG | ^4F♯, vG |
| 37 | 493.3 | ^F♯𝄪, vG | G | |
| 38 | 506.7 | G | ^G, v4A♭ | |
| 39 | 520 | ^G, vA♭♭♭ | ^^G, v3A♭ | |
| 40 | 533.3 | 34/25 | ^^G, A♭♭♭ | ^3G, vvA♭ |
| 41 | 546.7 | vG♯, ^A♭♭♭ | ^4G, vA♭ | |
| 42 | 560 | G♯, vvA♭♭ | ^5G, A♭ | |
| 43 | 573.3 | 32/23, 39/28 | ^G♯, vA♭♭ | v5G♯, ^A♭ |
| 44 | 586.7 | ^^G♯, A♭♭ | v4G♯, ^^A♭ | |
| 45 | 600 | 41/29 | vG𝄪, ^A♭♭ | v3G♯, ^3A♭ |
| 46 | 613.3 | 37/26 | G𝄪, vvA♭ | vvG♯, ^4A♭ |
| 47 | 626.7 | 23/16, 33/23 | ^G𝄪, vA♭ | vG♯, ^5A♭ |
| 48 | 640 | ^^G𝄪, A♭ | G♯, v5A | |
| 49 | 653.3 | 35/24 | vG♯𝄪, ^A♭ | ^G♯, v4A |
| 50 | 666.7 | 25/17 | G♯𝄪, vvA | ^^G♯, v3A |
| 51 | 680 | 37/25 | ^G♯𝄪, vA | ^3G♯, vvA |
| 52 | 693.3 | A | ^4G♯, vA | |
| 53 | 706.7 | ^A, vB♭♭♭ | A | |
| 54 | 720 | ^^A, B♭♭♭ | ^A, v4B♭ | |
| 55 | 733.3 | 26/17, 29/19 | vA♯, ^B♭♭♭ | ^^A, v3B♭ |
| 56 | 746.7 | 20/13, 37/24 | A♯, vvB♭♭ | ^3A, vvB♭ |
| 57 | 760 | 31/20 | ^A♯, vB♭♭ | ^4A, vB♭ |
| 58 | 773.3 | 25/16 | ^^A♯, B♭♭ | ^5A, B♭ |
| 59 | 786.7 | 41/26 | vA𝄪, ^B♭♭ | v5A♯, ^B♭ |
| 60 | 800 | A𝄪, vvB♭ | v4A♯, ^^B♭ | |
| 61 | 813.3 | 8/5 | ^A𝄪, vB♭ | v3A♯, ^3B♭ |
| 62 | 826.7 | ^^A𝄪, B♭ | vvA♯, ^4B♭ | |
| 63 | 840 | 13/8 | vA♯𝄪, ^B♭ | vA♯, ^5B♭ |
| 64 | 853.3 | A♯𝄪, vvB | A♯, v5B | |
| 65 | 866.7 | 28/17, 33/20, 38/23 | ^A♯𝄪, vB | ^A♯, v4B |
| 66 | 880 | B | ^^A♯, v3B | |
| 67 | 893.3 | ^B, vC♭♭ | ^3A♯, vvB | |
| 68 | 906.7 | ^^B, C♭♭ | ^4A♯, vB | |
| 69 | 920 | 17/10 | vB♯, ^C♭♭ | B |
| 70 | 933.3 | 12/7 | B♯, vvC♭ | ^B, v4C |
| 71 | 946.7 | 19/11 | ^B♯, vC♭ | ^^B, v3C |
| 72 | 960 | 40/23 | ^^B♯, C♭ | ^3B, vvC |
| 73 | 973.3 | vB𝄪, ^C♭ | ^4B, vC | |
| 74 | 986.7 | 23/13 | B𝄪, vvC | C |
| 75 | 1000 | 41/23 | ^B𝄪, vC | ^C, v4D♭ |
| 76 | 1013.3 | C | ^^C, v3D♭ | |
| 77 | 1026.7 | ^C, vD♭♭♭ | ^3C, vvD♭ | |
| 78 | 1040 | 31/17 | ^^C, D♭♭♭ | ^4C, vD♭ |
| 79 | 1053.3 | vC♯, ^D♭♭♭ | ^5C, D♭ | |
| 80 | 1066.7 | 37/20 | C♯, vvD♭♭ | v5C♯, ^D♭ |
| 81 | 1080 | 28/15, 41/22 | ^C♯, vD♭♭ | v4C♯, ^^D♭ |
| 82 | 1093.3 | 32/17 | ^^C♯, D♭♭ | v3C♯, ^3D♭ |
| 83 | 1106.7 | vC𝄪, ^D♭♭ | vvC♯, ^4D♭ | |
| 84 | 1120 | C𝄪, vvD♭ | vC♯, ^5D♭ | |
| 85 | 1133.3 | 25/13 | ^C𝄪, vD♭ | C♯, v5D |
| 86 | 1146.7 | 31/16, 33/17 | ^^C𝄪, D♭ | ^C♯, v4D |
| 87 | 1160 | vC♯𝄪, ^D♭ | ^^C♯, v3D | |
| 88 | 1173.3 | C♯𝄪, vvD | ^3C♯, vvD | |
| 89 | 1186.7 | ^C♯𝄪, vD | ^4C♯, vD | |
| 90 | 1200 | 2/1 | D | D |