77edo
| ← 76edo | 77edo | 78edo → |
Theory
With harmonic 3 less than a cent flat, harmonic 5 a bit over three cents sharp and 7's less flat than that, 77edo represents an excellent tuning choice for both valentine, the 31 & 46 temperament, and starling, the 126/125 planar temperament, giving the optimal patent val for 11-limit valentine and its 13-limit extensions dwynwen and valentino, as well as 11-limit starling and oxpecker temperaments. It also gives the optimal patent val for grackle and various members of the unicorn family, with a generator of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit alicorn and 11- and 13-limit camahueto.
77et tempers out 32805/32768 in the 5-limit, 126/125, 1029/1024 and 6144/6125 in the 7-limit, 121/120, 176/175, 385/384 and 441/440 in the 11-limit, and 196/195, 351/350, 352/351, 676/675 and 729/728 in the 13-limit.
77edo is an excellent edo for Carlos Alpha, since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only -0.042912 cents.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.66 | +3.30 | -2.59 | -5.86 | +1.03 | +4.14 | -1.41 | -4.90 | -1.01 | -7.37 |
| Relative (%) | +0.0 | -4.2 | +21.2 | -16.6 | -37.6 | +6.6 | +26.5 | -9.0 | -31.4 | -6.5 | -47.3 | |
| Steps (reduced) |
77 (0) |
122 (45) |
179 (25) |
216 (62) |
266 (35) |
285 (54) |
315 (7) |
327 (19) |
348 (40) |
374 (66) |
381 (73) | |
Intervals
| Steps | Cents | Approximate ratios | Ups and downs notation |
|---|---|---|---|
| 0 | 0 | 1/1 | D |
| 1 | 15.6 | ^D, ^^E♭♭ | |
| 2 | 31.2 | ^^D, ^3E♭♭ | |
| 3 | 46.8 | 36/35, 37/36, 38/37, 39/38 | ^3D, v3E♭ |
| 4 | 62.3 | 27/26, 28/27, 29/28 | v3D♯, vvE♭ |
| 5 | 77.9 | 22/21, 23/22 | vvD♯, vE♭ |
| 6 | 93.5 | 19/18, 37/35, 39/37 | vD♯, E♭ |
| 7 | 109.1 | 16/15, 33/31 | D♯, ^E♭ |
| 8 | 124.7 | 29/27 | ^D♯, ^^E♭ |
| 9 | 140.3 | 13/12, 38/35 | ^^D♯, ^3E♭ |
| 10 | 155.8 | 23/21, 35/32 | ^3D♯, v3E |
| 11 | 171.4 | 21/19, 32/29 | v3D𝄪, vvE |
| 12 | 187 | 29/26, 39/35 | vvD𝄪, vE |
| 13 | 202.6 | 9/8 | E |
| 14 | 218.2 | 17/15 | ^E, ^^F♭ |
| 15 | 233.8 | 8/7 | ^^E, ^3F♭ |
| 16 | 249.4 | 15/13, 37/32 | ^3E, v3F |
| 17 | 264.9 | 7/6 | v3E♯, vvF |
| 18 | 280.5 | 20/17, 27/23 | vvE♯, vF |
| 19 | 296.1 | 19/16, 32/27 | F |
| 20 | 311.7 | ^F, ^^G♭♭ | |
| 21 | 327.3 | 29/24, 35/29 | ^^F, ^3G♭♭ |
| 22 | 342.9 | 28/23, 39/32 | ^3F, v3G♭ |
| 23 | 358.4 | 16/13 | v3F♯, vvG♭ |
| 24 | 374 | 36/29 | vvF♯, vG♭ |
| 25 | 389.6 | 5/4 | vF♯, G♭ |
| 26 | 405.2 | 24/19 | F♯, ^G♭ |
| 27 | 420.8 | 14/11, 37/29 | ^F♯, ^^G♭ |
| 28 | 436.4 | 9/7 | ^^F♯, ^3G♭ |
| 29 | 451.9 | 13/10, 35/27 | ^3F♯, v3G |
| 30 | 467.5 | 17/13, 21/16, 38/29 | v3F𝄪, vvG |
| 31 | 483.1 | 37/28 | vvF𝄪, vG |
| 32 | 498.7 | 4/3 | G |
| 33 | 514.3 | 31/23, 35/26, 39/29 | ^G, ^^A♭♭ |
| 34 | 529.9 | 19/14, 34/25 | ^^G, ^3A♭♭ |
| 35 | 545.5 | 26/19, 37/27 | ^3G, v3A♭ |
| 36 | 561 | 18/13, 29/21 | v3G♯, vvA♭ |
| 37 | 576.6 | 39/28 | vvG♯, vA♭ |
| 38 | 592.2 | 31/22, 38/27 | vG♯, A♭ |
| 39 | 607.8 | 27/19, 37/26 | G♯, ^A♭ |
| 40 | 623.4 | 33/23 | ^G♯, ^^A♭ |
| 41 | 639 | 13/9 | ^^G♯, ^3A♭ |
| 42 | 654.5 | 19/13, 35/24 | ^3G♯, v3A |
| 43 | 670.1 | 25/17, 28/19 | v3G𝄪, vvA |
| 44 | 685.7 | vvG𝄪, vA | |
| 45 | 701.3 | 3/2 | A |
| 46 | 716.9 | ^A, ^^B♭♭ | |
| 47 | 732.5 | 26/17, 29/19, 32/21 | ^^A, ^3B♭♭ |
| 48 | 748.1 | 20/13, 37/24 | ^3A, v3B♭ |
| 49 | 763.6 | 14/9 | v3A♯, vvB♭ |
| 50 | 779.2 | 11/7 | vvA♯, vB♭ |
| 51 | 794.8 | 19/12 | vA♯, B♭ |
| 52 | 810.4 | 8/5 | A♯, ^B♭ |
| 53 | 826 | 29/18, 37/23 | ^A♯, ^^B♭ |
| 54 | 841.6 | 13/8 | ^^A♯, ^3B♭ |
| 55 | 857.1 | 23/14 | ^3A♯, v3B |
| 56 | 872.7 | v3A𝄪, vvB | |
| 57 | 888.3 | vvA𝄪, vB | |
| 58 | 903.9 | 27/16, 32/19 | B |
| 59 | 919.5 | 17/10 | ^B, ^^C♭ |
| 60 | 935.1 | 12/7 | ^^B, ^3C♭ |
| 61 | 950.6 | 26/15 | ^3B, v3C |
| 62 | 966.2 | 7/4 | v3B♯, vvC |
| 63 | 981.8 | 30/17, 37/21 | vvB♯, vC |
| 64 | 997.4 | 16/9 | C |
| 65 | 1013 | ^C, ^^D♭♭ | |
| 66 | 1028.6 | 29/16, 38/21 | ^^C, ^3D♭♭ |
| 67 | 1044.2 | ^3C, v3D♭ | |
| 68 | 1059.7 | 24/13, 35/19 | v3C♯, vvD♭ |
| 69 | 1075.3 | vvC♯, vD♭ | |
| 70 | 1090.9 | 15/8 | vC♯, D♭ |
| 71 | 1106.5 | 36/19 | C♯, ^D♭ |
| 72 | 1122.1 | 21/11 | ^C♯, ^^D♭ |
| 73 | 1137.7 | 27/14 | ^^C♯, ^3D♭ |
| 74 | 1153.2 | 35/18, 37/19, 39/20 | ^3C♯, v3D |
| 75 | 1168.8 | v3C𝄪, vvD | |
| 76 | 1184.4 | vvC𝄪, vD | |
| 77 | 1200 | 2/1 | D |
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-122 77⟩ | [⟨77 122]] | +0.207 | 0.207 | 1.33 |
| 2.3.5 | 32805/32768, 1594323/1562500 | [⟨77 122 179]] | -0.336 | 0.785 | 5.04 |
| 2.3.5.7 | 126/125, 1029/1024, 10976/10935 | [⟨77 122 179 216]] | -0.021 | 0.872 | 5.59 |
| 2.3.5.7.11 | 121/120, 126/125, 176/175, 10976/10935 | [⟨77 122 179 216 266]] | +0.322 | 1.039 | 6.66 |
| 2.3.5.7.11.13 | 121/120, 126/125, 176/175, 196/195, 676/675 | [⟨77 122 179 216 266 285]] | +0.222 | 0.974 | 6.25 |
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
Associated Ratio (Reduced) |
Temperaments |
|---|---|---|---|---|
| 1 | 4\77 | 62.34 | 28/27 | Unicorn / alicorn / camahueto / qilin |
| 1 | 5\77 | 77.92 | 21/20 | Valentine |
| 1 | 9\77 | 140.26 | 13/12 | Tsaharuk |
| 1 | 15\77 | 233.77 | 8/7 | Guiron |
| 1 | 16\77 | 249.35 | 15/13 | Hemischis (77e) |
| 1 | 20\77 | 311.69 | 6/5 | Oolong |
| 1 | 23\77 | 358.44 | 16/13 | Restles |
| 1 | 31\77 | 483.12 | 45/34 | Hemiseven |
| 1 | 32\77 | 498.70 | 4/3 | Grackle |
| 1 | 34\77 | 529.87 | 512/375 | Tuskaloosa Muscogee |
| 7 | 32\77 (1\77) |
498.70 (15.58) |
4/3 (81/80) |
Absurdity |
| 11 | 32\77 (3\77) |
498.70 (46.75) |
4/3 (36/35) |
Hendecatonic |
Music
- A Seed Planted, in an organ version of Claudi Meneghin.