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 less flat than that, 77edo represents an excellent tuning choice for both valentine (hence also Carlos Alpha), the 31 & 46 temperament, and starling, the rank-3 temperament tempering out 126/125, giving the optimal patent val for 11-limit valentine and its 13-limit extension valentino, as well as 11-limit starling and oxpecker temperaments. For desirers of purer/more convincing harmonies of 19, it's also a great choice for nestoria (the extension of schismic to prime 19) so that ~16:19:24 can be heard to concord in isolation. 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); it is a very good choice for full-subgroup unicorn. These are 7-limit alicorn and 11- and 13-limit camahueto.
77et tempers out the schisma (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.
It also does surprisingly well (for its size) in a large range of very high odd-limits (41 to 125 range).
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.66 | +3.30 | -2.59 | -5.86 | +1.03 | +4.14 | -1.41 | -4.90 |
| Relative (%) | +0.0 | -4.2 | +21.2 | -16.6 | -37.6 | +6.6 | +26.5 | -9.0 | -31.4 | |
| Steps (reduced) |
77 (0) |
122 (45) |
179 (25) |
216 (62) |
266 (35) |
285 (54) |
315 (7) |
327 (19) |
348 (40) | |
| Harmonic | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.01 | -7.37 | -1.99 | +7.30 | +2.77 | +4.62 | -0.78 | +0.57 | +5.19 | -1.38 |
| Relative (%) | -6.5 | -47.3 | -12.8 | +46.8 | +17.8 | +29.7 | -5.0 | +3.6 | +33.3 | -8.9 | |
| Steps (reduced) |
374 (66) |
381 (73) |
401 (16) |
413 (28) |
418 (33) |
428 (43) |
441 (56) |
453 (68) |
457 (72) |
467 (5) | |
Subsets and supersets
Since 77 factors into primes as 7 × 11, 77edo contains 7edo and 11edo as subset edos.
Intervals
| # | Cents | Approximate ratios* | Ups and downs notation |
|---|---|---|---|
| 0 | 0.0 | 1/1 | D |
| 1 | 15.6 | 81/80, 91/90, 99/98, 105/104 | ^D, ^^E♭♭ |
| 2 | 31.2 | 49/48, 55/54, 64/63, 65/64, 100/99 | ^^D, ^3E♭♭ |
| 3 | 46.8 | 33/32, 36/35, 40/39, 45/44, 50/49 | ^3D, v3E♭ |
| 4 | 62.3 | 26/25, 27/26, 28/27 | v3D♯, vvE♭ |
| 5 | 77.9 | 21/20, 22/21, 25/24 | vvD♯, vE♭ |
| 6 | 93.5 | 18/17, 19/18, 20/19 | vD♯, E♭ |
| 7 | 109.1 | 16/15, 17/16 | D♯, ^E♭ |
| 8 | 124.7 | 14/13, 15/14 | ^D♯, ^^E♭ |
| 9 | 140.3 | 13/12 | ^^D♯, ^3E♭ |
| 10 | 155.8 | 11/10, 12/11 | ^3D♯, v3E |
| 11 | 171.4 | 21/19 | v3D𝄪, vvE |
| 12 | 187.0 | 10/9 | 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, 22/19 | ^3E, v3F |
| 17 | 264.9 | 7/6 | v3E♯, vvF |
| 18 | 280.5 | 20/17 | vvE♯, vF |
| 19 | 296.1 | 13/11, 19/16, 32/27 | F |
| 20 | 311.7 | 6/5 | ^F, ^^G♭♭ |
| 21 | 327.3 | 98/81 | ^^F, ^3G♭♭ |
| 22 | 342.9 | 11/9, 17/14 | ^3F, v3G♭ |
| 23 | 358.4 | 16/13, 21/17 | v3F♯, vvG♭ |
| 24 | 374.0 | 26/21, 56/45 | vvF♯, vG♭ |
| 25 | 389.6 | 5/4 | vF♯, G♭ |
| 26 | 405.2 | 19/15, 24/19, 33/26 | F♯, ^G♭ |
| 27 | 420.8 | 14/11, 32/25 | ^F♯, ^^G♭ |
| 28 | 436.4 | 9/7 | ^^F♯, ^3G♭ |
| 29 | 451.9 | 13/10 | ^3F♯, v3G |
| 30 | 467.5 | 17/13, 21/16 | v3F𝄪, vvG |
| 31 | 483.1 | 120/91 | vvF𝄪, vG |
| 32 | 498.7 | 4/3 | G |
| 33 | 514.3 | 27/20 | ^G, ^^A♭♭ |
| 34 | 529.9 | 19/14 | ^^G, ^3A♭♭ |
| 35 | 545.5 | 11/8, 15/11, 26/19 | ^3G, v3A♭ |
| 36 | 561.0 | 18/13 | v3G♯, vvA♭ |
| 37 | 576.6 | 7/5 | vvG♯, vA♭ |
| 38 | 592.2 | 24/17, 38/27, 45/32 | vG♯, A♭ |
| … | … | … |
* As a 19-limit temperament
Notation
Ups and downs notation
77edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
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Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:
| Step offset | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
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Sagittal notation
Evo flavor
Revo flavor
Approximation to JI
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 414zpi | 76.991854 | 15.586065 | 8.194847 | 8.145298 | 1.311364 | 17.029289 | 1200.126969 | 0.126969 | 10 | 10 |
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 | 62.3 | 28/27 | Unicorn / alicorn (77e) / camahueto (77) / qilin (77) |
| 1 | 5\77 | 77.9 | 21/20 | Valentine |
| 1 | 9\77 | 140.3 | 13/12 | Tsaharuk |
| 1 | 15\77 | 233.8 | 8/7 | Guiron |
| 1 | 16\77 | 249.4 | 15/13 | Hemischis (77e) |
| 1 | 20\77 | 311.7 | 6/5 | Oolong |
| 1 | 23\77 | 358.4 | 16/13 | Restles |
| 1 | 31\77 | 483.1 | 45/34 | Hemiseven |
| 1 | 32\77 | 498.7 | 4/3 | Grackle |
| 1 | 34\77 | 529.9 | 512/375 | Tuskaloosa / muscogee |
| 1 | 36\77 | 561.0 | 18/13 | Demivalentine |
| 7 | 32\77 (1\77) |
498.7 (15.6) |
4/3 (81/80) |
Absurdity |
| 11 | 32\77 (3\77) |
498.7 (46.8) |
4/3 (36/35) |
Hendecatonic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Instruments
Skip fretting
Skip fretting system 77 9 11 is a skip fretting system that tunes strings 11\77 apart, with frets placed at intervals of 9\77, or 8.555...-edo. All examples on this page are for 7-string guitar.
- Intervals
0\77=1/1: string 2 open
77\77=2/1: string 7 fret 11
45\77=3/2: string 2 fret 5
25\77=5/4: string 1 fret 4
62\77=7/4: string 6 fret 2
35\77=11/8: string 4 fret 10
54\77=13/8: string 2 fret 6
7\77=17/16: string 1 fret 2
19\77=19/16: string 5 fret 7
40\77=23/16: string 4 fret 2
- Chords
x00030x: Neutral 9th (saj6, v5)
Music
- [[Bryan Deister]
- A Seed Planted[dead link], in an organ version of Claudi Meneghin.