90edo
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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 | 13.333 | ^D, vE♭♭♭ | ^D, v4E♭ | |
| 2 | 26.667 | 65/64, 66/65, 78/77 | ^^D, E♭♭♭ | ^^D, v3E♭ |
| 3 | 40 | 40/39, 49/48 | ^3D, v3E♭♭ | ^3D, vvE♭ |
| 4 | 53.333 | 33/32, 36/35 | D♯, vvE♭♭ | ^4D, vE♭ |
| 5 | 66.667 | 25/24, 26/25, 80/77 | ^D♯, vE♭♭ | ^5D, E♭ |
| 6 | 80 | ^^D♯, E♭♭ | ^6D, v10E | |
| 7 | 93.333 | 55/52 | ^3D♯, v3E♭ | ^7D, v9E |
| 8 | 106.667 | 16/15, 35/33 | D𝄪, vvE♭ | ^8D, v8E |
| 9 | 120 | 15/14 | ^D𝄪, vE♭ | ^9D, v7E |
| 10 | 133.333 | 13/12, 14/13 | ^^D𝄪, E♭ | ^10D, v6E |
| 11 | 146.667 | 49/45 | ^3D𝄪, v3E | D♯, v5E |
| 12 | 160 | 11/10, 35/32 | D♯𝄪, vvE | ^D♯, v4E |
| 13 | 173.333 | 54/49 | ^D♯𝄪, vE | ^^D♯, v3E |
| 14 | 186.667 | 39/35 | E | ^3D♯, vvE |
| 15 | 200 | 28/25 | ^E, vF♭♭ | ^4D♯, vE |
| 16 | 213.333 | ^^E, F♭♭ | E | |
| 17 | 226.667 | 8/7 | ^3E, v3F♭ | ^E, v4F |
| 18 | 240 | E♯, vvF♭ | ^^E, v3F | |
| 19 | 253.333 | 65/56 | ^E♯, vF♭ | ^3E, vvF |
| 20 | 266.667 | 7/6, 64/55 | ^^E♯, F♭ | ^4E, vF |
| 21 | 280 | 33/28 | ^3E♯, v3F | F |
| 22 | 293.333 | 13/11, 77/65 | E𝄪, vvF | ^F, v4G♭ |
| 23 | 306.667 | ^E𝄪, vF | ^^F, v3G♭ | |
| 24 | 320 | 6/5, 77/64 | F | ^3F, vvG♭ |
| 25 | 333.333 | 40/33 | ^F, vG♭♭♭ | ^4F, vG♭ |
| 26 | 346.667 | 39/32, 60/49 | ^^F, G♭♭♭ | ^5F, G♭ |
| 27 | 360 | 16/13 | ^3F, v3G♭♭ | ^6F, v10G |
| 28 | 373.333 | 56/45 | F♯, vvG♭♭ | ^7F, v9G |
| 29 | 386.667 | 5/4 | ^F♯, vG♭♭ | ^8F, v8G |
| 30 | 400 | 49/39 | ^^F♯, G♭♭ | ^9F, v7G |
| 31 | 413.333 | 33/26 | ^3F♯, v3G♭ | ^10F, v6G |
| 32 | 426.667 | 32/25, 50/39, 77/60 | F𝄪, vvG♭ | F♯, v5G |
| 33 | 440 | 9/7 | ^F𝄪, vG♭ | ^F♯, v4G |
| 34 | 453.333 | 13/10 | ^^F𝄪, G♭ | ^^F♯, v3G |
| 35 | 466.667 | ^3F𝄪, v3G | ^3F♯, vvG | |
| 36 | 480 | 33/25 | F♯𝄪, vvG | ^4F♯, vG |
| 37 | 493.333 | 4/3 | ^F♯𝄪, vG | G |
| 38 | 506.667 | 75/56 | G | ^G, v4A♭ |
| 39 | 520 | 65/48 | ^G, vA♭♭♭ | ^^G, v3A♭ |
| 40 | 533.333 | 49/36 | ^^G, A♭♭♭ | ^3G, vvA♭ |
| 41 | 546.667 | 11/8, 48/35 | ^3G, v3A♭♭ | ^4G, vA♭ |
| 42 | 560 | G♯, vvA♭♭ | ^5G, A♭ | |
| 43 | 573.333 | 39/28 | ^G♯, vA♭♭ | ^6G, v10A |
| 44 | 586.667 | 7/5 | ^^G♯, A♭♭ | ^7G, v9A |
| 45 | 600 | ^3G♯, v3A♭ | ^8G, v8A | |
| 46 | 613.333 | 10/7 | G𝄪, vvA♭ | ^9G, v7A |
| 47 | 626.667 | 56/39 | ^G𝄪, vA♭ | ^10G, v6A |
| 48 | 640 | ^^G𝄪, A♭ | G♯, v5A | |
| 49 | 653.333 | 16/11, 35/24 | ^3G𝄪, v3A | ^G♯, v4A |
| 50 | 666.667 | 72/49 | G♯𝄪, vvA | ^^G♯, v3A |
| 51 | 680 | 65/44, 77/52 | ^G♯𝄪, vA | ^3G♯, vvA |
| 52 | 693.333 | A | ^4G♯, vA | |
| 53 | 706.667 | 3/2 | ^A, vB♭♭♭ | A |
| 54 | 720 | 50/33 | ^^A, B♭♭♭ | ^A, v4B♭ |
| 55 | 733.333 | 75/49 | ^3A, v3B♭♭ | ^^A, v3B♭ |
| 56 | 746.667 | 20/13, 77/50 | A♯, vvB♭♭ | ^3A, vvB♭ |
| 57 | 760 | 14/9 | ^A♯, vB♭♭ | ^4A, vB♭ |
| 58 | 773.333 | 25/16, 39/25 | ^^A♯, B♭♭ | ^5A, B♭ |
| 59 | 786.667 | 52/33 | ^3A♯, v3B♭ | ^6A, v10B |
| 60 | 800 | 78/49 | A𝄪, vvB♭ | ^7A, v9B |
| 61 | 813.333 | 8/5, 77/48 | ^A𝄪, vB♭ | ^8A, v8B |
| 62 | 826.667 | 45/28 | ^^A𝄪, B♭ | ^9A, v7B |
| 63 | 840 | 13/8 | ^3A𝄪, v3B | ^10A, v6B |
| 64 | 853.333 | 49/30, 64/39 | A♯𝄪, vvB | A♯, v5B |
| 65 | 866.667 | 33/20 | ^A♯𝄪, vB | ^A♯, v4B |
| 66 | 880 | 5/3 | B | ^^A♯, v3B |
| 67 | 893.333 | ^B, vC♭♭ | ^3A♯, vvB | |
| 68 | 906.667 | 22/13 | ^^B, C♭♭ | ^4A♯, vB |
| 69 | 920 | 56/33 | ^3B, v3C♭ | B |
| 70 | 933.333 | 12/7, 55/32 | B♯, vvC♭ | ^B, v4C |
| 71 | 946.667 | ^B♯, vC♭ | ^^B, v3C | |
| 72 | 960 | ^^B♯, C♭ | ^3B, vvC | |
| 73 | 973.333 | 7/4 | ^3B♯, v3C | ^4B, vC |
| 74 | 986.667 | B𝄪, vvC | C | |
| 75 | 1000 | 25/14 | ^B𝄪, vC | ^C, v4D♭ |
| 76 | 1013.333 | 70/39 | C | ^^C, v3D♭ |
| 77 | 1026.667 | 49/27 | ^C, vD♭♭♭ | ^3C, vvD♭ |
| 78 | 1040 | 20/11, 64/35 | ^^C, D♭♭♭ | ^4C, vD♭ |
| 79 | 1053.333 | ^3C, v3D♭♭ | ^5C, D♭ | |
| 80 | 1066.667 | 13/7, 24/13 | C♯, vvD♭♭ | ^6C, v10D |
| 81 | 1080 | 28/15 | ^C♯, vD♭♭ | ^7C, v9D |
| 82 | 1093.333 | 15/8, 66/35 | ^^C♯, D♭♭ | ^8C, v8D |
| 83 | 1106.667 | ^3C♯, v3D♭ | ^9C, v7D | |
| 84 | 1120 | C𝄪, vvD♭ | ^10C, v6D | |
| 85 | 1133.333 | 25/13, 48/25, 77/40 | ^C𝄪, vD♭ | C♯, v5D |
| 86 | 1146.667 | 35/18, 64/33 | ^^C𝄪, D♭ | ^C♯, v4D |
| 87 | 1160 | 39/20 | ^3C𝄪, v3D | ^^C♯, v3D |
| 88 | 1173.333 | 65/33, 77/39 | C♯𝄪, vvD | ^3C♯, vvD |
| 89 | 1186.667 | ^C♯𝄪, vD | ^4C♯, vD | |
| 90 | 1200 | 2/1 | D | D |