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.
The 17 and 19 are tuned fairly well, making it consistent to the no-11 21-odd-limit. The equal temperament tempers out 256/255 in the 17-limit; and 171/170, 361/360, 513/512, and 1216/1215 in the 19-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 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.66 | +3.30 | -2.59 | -5.86 | +1.03 | +4.14 | -1.41 | -4.90 | -1.01 | -7.37 | -1.99 |
| Relative (%) | +0.0 | -4.2 | +21.2 | -16.6 | -37.6 | +6.6 | +26.5 | -9.0 | -31.4 | -6.5 | -47.3 | -12.8 | |
| Steps (reduced) |
77 (0) |
122 (45) |
179 (25) |
216 (62) |
266 (35) |
285 (54) |
315 (7) |
327 (19) |
348 (40) |
374 (66) |
381 (73) |
401 (16) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.30 | +2.77 | +4.62 | -0.78 | +0.57 | +5.19 | -1.38 | +7.32 | +5.98 | -6.10 | +1.90 | +5.74 |
| Relative (%) | +46.8 | +17.8 | +29.7 | -5.0 | +3.6 | +33.3 | -8.9 | +46.9 | +38.4 | -39.1 | +12.2 | +36.9 | |
| Steps (reduced) |
413 (28) |
418 (33) |
428 (43) |
441 (56) |
453 (68) |
457 (72) |
467 (5) |
474 (12) |
477 (15) |
485 (23) |
491 (29) |
499 (37) | |
Intervals
| Degree | Cents | Approximate Ratios* |
|---|---|---|
| 0 | 0.000 | 1/1 |
| 1 | 15.584 | 81/80, 91/90, 99/98, 105/104 |
| 2 | 31.169 | 49/48, 55/54, 64/63, 65/64, 100/99 |
| 3 | 46.753 | 33/32, 36/35, 40/39, 45/44, 50/49 |
| 4 | 62.338 | 26/25, 27/26, 28/27 |
| 5 | 77.922 | 21/20, 22/21, 25/24 |
| 6 | 93.506 | 18/17, 19/18, 20/19 |
| 7 | 109.091 | 16/15, 17/16 |
| 8 | 124.675 | 14/13, 15/14 |
| 9 | 140.260 | 13/12 |
| 10 | 155.844 | 11/10, 12/11 |
| 11 | 171.429 | 21/19 |
| 12 | 187.013 | 10/9 |
| 13 | 202.597 | 9/8 |
| 14 | 218.182 | 17/15 |
| 15 | 233.766 | 8/7 |
| 16 | 249.351 | 15/13, 22/19 |
| 17 | 264.935 | 7/6 |
| 18 | 280.519 | 20/17 |
| 19 | 296.104 | 13/11, 19/16, 32/27 |
| 20 | 311.688 | 6/5 |
| 21 | 327.273 | 98/81 |
| 22 | 342.857 | 11/9, 17/14 |
| 23 | 358.442 | 16/13, 21/17 |
| 24 | 374.026 | 26/21, 56/45 |
| 25 | 389.610 | 5/4 |
| 26 | 405.195 | 19/15, 24/19, 33/26 |
| 27 | 420.779 | 14/11, 32/25 |
| 28 | 436.364 | 9/7 |
| 29 | 451.948 | 13/10 |
| 30 | 467.532 | 17/13, 21/16 |
| 31 | 483.117 | 120/91 |
| 32 | 498.701 | 4/3 |
| 33 | 514.286 | 27/20 |
| 34 | 529.870 | 19/14 |
| 35 | 545.455 | 11/8, 15/11, 26/19 |
| 36 | 561.039 | 18/13 |
| 37 | 576.623 | 7/5 |
| 38 | 592.208 | 24/17, 38/27, 45/32 |
| … | … | … |
* As a 19-limit temperament
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* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 4\77 | 62.34 | 28/27 | Unicorn / alicorn (77e) / camahueto (77) / qilin (77) |
| 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- A Seed Planted [dead link], in an organ version of Claudi Meneghin.