102edo
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Prime factorization
2 × 3 × 17
Step size
11.7647¢
Fifth
60\102 (705.882¢) (→10\17)
Semitones (A1:m2)
12:6 (141.2¢ : 70.59¢)
Dual sharp fifth
60\102 (705.882¢) (→10\17)
Dual flat fifth
59\102 (694.118¢)
Dual major 2nd
17\102 (200¢) (→1\6)
Consistency limit
5
Distinct consistency limit
5
| ← 101edo | 102edo | 103edo → |
102 equal divisions of the octave (abbreviated 102edo or 102ed2), also called 102-tone equal temperament (102tet) or 102 equal temperament (102et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 102 equal parts of about 11.8 ¢ each. Each step represents a frequency ratio of 21/102, or the 102nd root of 2.
Theory
102edo is enfactored in the 5-limit, where it tempers out the same commas (2048/2025, 15625/15552, 20000/19683) as 34edo. In the 7-limit it tempers out 686/675 and 1029/1024; in the 11-limit 385/384, 441/440 and 4000/3993; in the 13-limit 91/90 and 169/168; in the 17-limit 136/135 and 154/153; and in the 19-limit 133/132 and 190/189. It is the optimal patent val for 13-limit echidnic temperament, and the rank-5 temperament tempering out 91/90.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.93 | +1.92 | -4.12 | -3.91 | +1.62 | -5.23 | +5.85 | +0.93 | -3.40 | -0.19 | -4.74 |
| Relative (%) | +33.4 | +16.3 | -35.0 | -33.2 | +13.8 | -44.5 | +49.7 | +7.9 | -28.9 | -1.6 | -40.3 | |
| Steps (reduced) |
162 (60) |
237 (33) |
286 (82) |
323 (17) |
353 (47) |
377 (71) |
399 (93) |
417 (9) |
433 (25) |
448 (40) |
461 (53) | |
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation (Dual flat fifth 59\102) |
Ups and downs notation (Dual sharp fifth 60\102) |
|---|---|---|---|---|
| 0 | 0 | 1/1 | D | D |
| 1 | 11.8 | ^D, E♭♭♭ | ^D, v5E♭ | |
| 2 | 23.5 | ^^D, ^E♭♭♭ | ^^D, v4E♭ | |
| 3 | 35.3 | vvD♯, ^^E♭♭♭ | ^3D, v3E♭ | |
| 4 | 47.1 | 38/37, 39/38 | vD♯, vvE♭♭ | ^4D, vvE♭ |
| 5 | 58.8 | 30/29 | D♯, vE♭♭ | ^5D, vE♭ |
| 6 | 70.6 | 25/24 | ^D♯, E♭♭ | ^6D, E♭ |
| 7 | 82.4 | 21/20, 22/21, 43/41 | ^^D♯, ^E♭♭ | v5D♯, ^E♭ |
| 8 | 94.1 | 37/35 | vvD𝄪, ^^E♭♭ | v4D♯, ^^E♭ |
| 9 | 105.9 | 17/16 | vD𝄪, vvE♭ | v3D♯, ^3E♭ |
| 10 | 117.6 | D𝄪, vE♭ | vvD♯, ^4E♭ | |
| 11 | 129.4 | 14/13, 41/38 | ^D𝄪, E♭ | vD♯, ^5E♭ |
| 12 | 141.2 | 38/35 | ^^D𝄪, ^E♭ | D♯, v6E |
| 13 | 152.9 | 12/11, 35/32 | vvD♯𝄪, ^^E♭ | ^D♯, v5E |
| 14 | 164.7 | 11/10 | vD♯𝄪, vvE | ^^D♯, v4E |
| 15 | 176.5 | 31/28, 41/37 | D♯𝄪, vE | ^3D♯, v3E |
| 16 | 188.2 | 39/35 | E | ^4D♯, vvE |
| 17 | 200 | ^E, F♭♭ | ^5D♯, vE | |
| 18 | 211.8 | 26/23, 35/31, 43/38 | ^^E, ^F♭♭ | E |
| 19 | 223.5 | 25/22, 33/29 | vvE♯, ^^F♭♭ | ^E, v5F |
| 20 | 235.3 | 39/34 | vE♯, vvF♭ | ^^E, v4F |
| 21 | 247.1 | E♯, vF♭ | ^3E, v3F | |
| 22 | 258.8 | 29/25, 43/37 | ^E♯, F♭ | ^4E, vvF |
| 23 | 270.6 | ^^E♯, ^F♭ | ^5E, vF | |
| 24 | 282.4 | 20/17 | vvE𝄪, ^^F♭ | F |
| 25 | 294.1 | vE𝄪, vvF | ^F, v5G♭ | |
| 26 | 305.9 | 31/26, 37/31 | E𝄪, vF | ^^F, v4G♭ |
| 27 | 317.6 | 6/5 | F | ^3F, v3G♭ |
| 28 | 329.4 | 23/19, 29/24 | ^F, G♭♭♭ | ^4F, vvG♭ |
| 29 | 341.2 | 28/23, 39/32 | ^^F, ^G♭♭♭ | ^5F, vG♭ |
| 30 | 352.9 | 38/31 | vvF♯, ^^G♭♭♭ | ^6F, G♭ |
| 31 | 364.7 | 21/17 | vF♯, vvG♭♭ | v5F♯, ^G♭ |
| 32 | 376.5 | 36/29 | F♯, vG♭♭ | v4F♯, ^^G♭ |
| 33 | 388.2 | 5/4 | ^F♯, G♭♭ | v3F♯, ^3G♭ |
| 34 | 400 | ^^F♯, ^G♭♭ | vvF♯, ^4G♭ | |
| 35 | 411.8 | vvF𝄪, ^^G♭♭ | vF♯, ^5G♭ | |
| 36 | 423.5 | vF𝄪, vvG♭ | F♯, v6G | |
| 37 | 435.3 | F𝄪, vG♭ | ^F♯, v5G | |
| 38 | 447.1 | 22/17 | ^F𝄪, G♭ | ^^F♯, v4G |
| 39 | 458.8 | ^^F𝄪, ^G♭ | ^3F♯, v3G | |
| 40 | 470.6 | 21/16 | vvF♯𝄪, ^^G♭ | ^4F♯, vvG |
| 41 | 482.4 | 33/25, 37/28, 41/31 | vF♯𝄪, vvG | ^5F♯, vG |
| 42 | 494.1 | F♯𝄪, vG | G | |
| 43 | 505.9 | G | ^G, v5A♭ | |
| 44 | 517.6 | 31/23 | ^G, A♭♭♭ | ^^G, v4A♭ |
| 45 | 529.4 | 19/14 | ^^G, ^A♭♭♭ | ^3G, v3A♭ |
| 46 | 541.2 | 26/19 | vvG♯, ^^A♭♭♭ | ^4G, vvA♭ |
| 47 | 552.9 | 11/8 | vG♯, vvA♭♭ | ^5G, vA♭ |
| 48 | 564.7 | 43/31 | G♯, vA♭♭ | ^6G, A♭ |
| 49 | 576.5 | ^G♯, A♭♭ | v5G♯, ^A♭ | |
| 50 | 588.2 | ^^G♯, ^A♭♭ | v4G♯, ^^A♭ | |
| 51 | 600 | vvG𝄪, ^^A♭♭ | v3G♯, ^3A♭ | |
| 52 | 611.8 | 37/26 | vG𝄪, vvA♭ | vvG♯, ^4A♭ |
| 53 | 623.5 | G𝄪, vA♭ | vG♯, ^5A♭ | |
| 54 | 635.3 | ^G𝄪, A♭ | G♯, v6A | |
| 55 | 647.1 | 16/11 | ^^G𝄪, ^A♭ | ^G♯, v5A |
| 56 | 658.8 | 19/13, 41/28 | vvG♯𝄪, ^^A♭ | ^^G♯, v4A |
| 57 | 670.6 | 28/19 | vG♯𝄪, vvA | ^3G♯, v3A |
| 58 | 682.4 | G♯𝄪, vA | ^4G♯, vvA | |
| 59 | 694.1 | A | ^5G♯, vA | |
| 60 | 705.9 | ^A, B♭♭♭ | A | |
| 61 | 717.6 | ^^A, ^B♭♭♭ | ^A, v5B♭ | |
| 62 | 729.4 | 32/21 | vvA♯, ^^B♭♭♭ | ^^A, v4B♭ |
| 63 | 741.2 | 43/28 | vA♯, vvB♭♭ | ^3A, v3B♭ |
| 64 | 752.9 | 17/11 | A♯, vB♭♭ | ^4A, vvB♭ |
| 65 | 764.7 | ^A♯, B♭♭ | ^5A, vB♭ | |
| 66 | 776.5 | ^^A♯, ^B♭♭ | ^6A, B♭ | |
| 67 | 788.2 | 41/26 | vvA𝄪, ^^B♭♭ | v5A♯, ^B♭ |
| 68 | 800 | vA𝄪, vvB♭ | v4A♯, ^^B♭ | |
| 69 | 811.8 | 8/5 | A𝄪, vB♭ | v3A♯, ^3B♭ |
| 70 | 823.5 | 29/18, 37/23 | ^A𝄪, B♭ | vvA♯, ^4B♭ |
| 71 | 835.3 | 34/21 | ^^A𝄪, ^B♭ | vA♯, ^5B♭ |
| 72 | 847.1 | 31/19 | vvA♯𝄪, ^^B♭ | A♯, v6B |
| 73 | 858.8 | 23/14 | vA♯𝄪, vvB | ^A♯, v5B |
| 74 | 870.6 | 38/23, 43/26 | A♯𝄪, vB | ^^A♯, v4B |
| 75 | 882.4 | 5/3 | B | ^3A♯, v3B |
| 76 | 894.1 | ^B, C♭♭ | ^4A♯, vvB | |
| 77 | 905.9 | ^^B, ^C♭♭ | ^5A♯, vB | |
| 78 | 917.6 | 17/10 | vvB♯, ^^C♭♭ | B |
| 79 | 929.4 | vB♯, vvC♭ | ^B, v5C | |
| 80 | 941.2 | B♯, vC♭ | ^^B, v4C | |
| 81 | 952.9 | ^B♯, C♭ | ^3B, v3C | |
| 82 | 964.7 | ^^B♯, ^C♭ | ^4B, vvC | |
| 83 | 976.5 | 44/25 | vvB𝄪, ^^C♭ | ^5B, vC |
| 84 | 988.2 | 23/13 | vB𝄪, vvC | C |
| 85 | 1000 | 41/23 | B𝄪, vC | ^C, v5D♭ |
| 86 | 1011.8 | C | ^^C, v4D♭ | |
| 87 | 1023.5 | ^C, D♭♭♭ | ^3C, v3D♭ | |
| 88 | 1035.3 | 20/11 | ^^C, ^D♭♭♭ | ^4C, vvD♭ |
| 89 | 1047.1 | 11/6 | vvC♯, ^^D♭♭♭ | ^5C, vD♭ |
| 90 | 1058.8 | 35/19 | vC♯, vvD♭♭ | ^6C, D♭ |
| 91 | 1070.6 | 13/7 | C♯, vD♭♭ | v5C♯, ^D♭ |
| 92 | 1082.4 | 43/23 | ^C♯, D♭♭ | v4C♯, ^^D♭ |
| 93 | 1094.1 | 32/17 | ^^C♯, ^D♭♭ | v3C♯, ^3D♭ |
| 94 | 1105.9 | vvC𝄪, ^^D♭♭ | vvC♯, ^4D♭ | |
| 95 | 1117.6 | 21/11, 40/21 | vC𝄪, vvD♭ | vC♯, ^5D♭ |
| 96 | 1129.4 | C𝄪, vD♭ | C♯, v6D | |
| 97 | 1141.2 | 29/15 | ^C𝄪, D♭ | ^C♯, v5D |
| 98 | 1152.9 | 37/19 | ^^C𝄪, ^D♭ | ^^C♯, v4D |
| 99 | 1164.7 | vvC♯𝄪, ^^D♭ | ^3C♯, v3D | |
| 100 | 1176.5 | vC♯𝄪, vvD | ^4C♯, vvD | |
| 101 | 1188.2 | C♯𝄪, vD | ^5C♯, vD | |
| 102 | 1200 | 2/1 | D | D |