77edo
| ← 76edo | 77edo | 78edo → |
77 equal divisions of the octave (abbreviated 77edo or 77ed2), also called 77-tone equal temperament (77tet) or 77 equal temperament (77et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 77 equal parts of about 15.6 ¢ each. Each step represents a frequency ratio of 21/77, or the 77th root of 2.
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 | -4 | +21 | -17 | -38 | +7 | +27 | -9 | -31 | -6 | -47 | |
| 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
| Degree | Cents | Approximate Ratios in the 13-limit |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 15.584 | 81/80, 99/98 |
| 2 | 31.169 | 64/63, 49/48 |
| 3 | 46.753 | 33/32, 36/35 |
| 4 | 62.338 | 28/27, 26/25 |
| 5 | 77.922 | 21/20, 25/24 |
| 6 | 93.506 | 135/128 |
| 7 | 109.091 | 16/15 |
| 8 | 124.675 | 15/14 |
| 9 | 140.260 | 13/12 |
| 10 | 155.844 | 12/11, 11/10 |
| 11 | 171.429 | 72/65 |
| 12 | 187.013 | 10/9 |
| 13 | 202.597 | 9/8 |
| 14 | 218.182 | 256/225 |
| 15 | 233.766 | 8/7 |
| 16 | 249.351 | 15/13 |
| 17 | 264.935 | 7/6 |
| 18 | 280.519 | 33/28 |
| 19 | 296.104 | 32/27, 13/11 |
| 20 | 311.688 | 6/5 |
| 21 | 327.273 | 98/81 |
| 22 | 342.857 | 11/9, 39/32 |
| 23 | 358.442 | 16/13 |
| 24 | 374.026 | 56/45, 26/21 |
| 25 | 389.610 | 5/4 |
| 26 | 405.195 | 33/26, 81/64 |
| 27 | 420.779 | 14/11, 32/25 |
| 28 | 436.364 | 9/7 |
| 29 | 451.948 | 13/10 |
| 30 | 467.532 | 21/16 |
| 31 | 483.117 | 120/91 |
| 32 | 498.701 | 4/3 |
| 33 | 514.286 | 27/20 |
| 34 | 529.870 | 49/36 |
| 35 | 545.455 | 11/8, 15/11 |
| 36 | 561.039 | 18/13 |
| 37 | 576.623 | 7/5 |
| 38 | 592.208 | 45/32 |
| 39 | 607.792 | 64/45 |
| 40 | 623.377 | 10/7 |
| 41 | 638.961 | 13/9 |
| 42 | 654.545 | 16/11, 22/15 |
| 43 | 670.130 | 72/49 |
| 44 | 685.714 | 40/27 |
| 45 | 701.299 | 3/2 |
| 46 | 716.883 | 91/60 |
| 47 | 732.468 | 32/21 |
| 48 | 748.052 | 20/13 |
| 49 | 763.636 | 14/9 |
| 50 | 779.221 | 11/7, 25/16 |
| 51 | 794.805 | 52/33, 128/81 |
| 52 | 810.390 | 8/5 |
| 53 | 825.974 | 45/28, 21/13 |
| 54 | 841.558 | 13/8 |
| 55 | 857.143 | 18/11, 64/39 |
| 56 | 872.727 | 81/49 |
| 57 | 888.312 | 5/3 |
| 58 | 903.896 | 27/16, 22/13 |
| 59 | 919.481 | 56/33 |
| 60 | 935.065 | 12/7 |
| 61 | 950.649 | 26/15 |
| 62 | 966.234 | 7/4 |
| 63 | 981.818 | 225/128 |
| 64 | 997.403 | 16/9 |
| 65 | 1012.987 | 9/5 |
| 66 | 1028.571 | 65/36 |
| 67 | 1044.156 | 11/6, 20/11 |
| 68 | 1059.740 | 24/13 |
| 69 | 1075.325 | 28/15 |
| 70 | 1090.909 | 15/8 |
| 71 | 1106.494 | 256/135 |
| 72 | 1122.078 | 40/21, 48/25 |
| 73 | 1137.662 | 27/14, 25/13 |
| 74 | 1153.247 | 64/33, 35/18 |
| 75 | 1168.831 | 63/32, 96/49 |
| 76 | 1184.416 | 160/81, 196/99 |
| 77 | 1200.000 | 2/1 |
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.