77edo: Difference between revisions

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| 2.3
| 2.3
| {{monzo| -122 77 }}
| {{Monzo| -122 77 }}
| {{mapping| 77 122 }}
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| +0.207
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| {{mapping| 77 122 179 }}
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| 126/125, 1029/1024, 10976/10935
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| {{Mapping| 77 122 179 216 }}
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| 121/120, 126/125, 176/175, 10976/10935
| 121/120, 126/125, 176/175, 10976/10935
| {{mapping| 77 122 179 216 266 }}
| {{Mapping| 77 122 179 216 266 }}
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| 2.3.5.7.11.13
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| 121/120, 126/125, 176/175, 196/195, 676/675
| 121/120, 126/125, 176/175, 196/195, 676/675
| {{mapping| 77 122 179 216 266 285 }}
| {{Mapping| 77 122 179 216 266 285 }}
| +0.222
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| 498.7<br>(46.8)
| 498.7<br>(46.8)
| 4/3<br>(36/35)
| 4/3<br>(36/35)
| [[Hendecatonic]]
| [[Hendecatonic (temperament)|Hendecatonic]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Latest revision as of 12:02, 5 April 2026

← 76edo 77edo 78edo →
Prime factorization 7 × 11
Step size 15.5844 ¢ 
Fifth 45\77 (701.299 ¢)
Semitones (A1:m2) 7:6 (109.1 ¢ : 93.51 ¢)
Consistency limit 9
Distinct consistency limit 9

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

Approximation of prime harmonics in 77edo
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)
Approximation of prime harmonics in 77edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -1.99 +7.30 +2.77 +4.62 -0.78 +0.57 +5.19 -1.38 +7.32 +5.98 -6.10
Relative (%) -12.8 +46.8 +17.8 +29.7 -5.0 +3.6 +33.3 -8.9 +46.9 +38.4 -39.1
Steps
(reduced) 		401
(16) 		413
(28) 		418
(33) 		428
(43) 		441
(56) 		453
(68) 		457
(72) 		467
(5) 		474
(12) 		477
(15) 		485
(23)

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.

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Sharp symbol                
                 
Flat symbol              
                 

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
Sharp symbol  
Flat symbol

Sagittal notation

Evo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Revo flavor

Sagittal notationPeriodic table of EDOs with sagittal notation81/8064/6333/32

Approximation to JI

Selected just intervals

The following tables show how 15-odd-limit intervals are represented in 77edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 77edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.656 4.2
13/8, 16/13 1.031 6.6
9/7, 14/9 1.280 8.2
9/8, 16/9 1.313 8.4
15/13, 26/15 1.610 10.3
13/12, 24/13 1.687 10.8
7/6, 12/7 1.936 12.4
13/10, 20/13 2.266 14.5
13/9, 18/13 2.343 15.0
7/4, 8/7 2.592 16.6
15/8, 16/15 2.640 16.9
11/7, 14/11 3.271 21.0
5/4, 8/5 3.297 21.2
13/7, 14/13 3.623 23.2
5/3, 6/5 3.953 25.4
11/9, 18/11 4.551 29.2
9/5, 10/9 4.609 29.6
11/6, 12/11 5.207 33.4
15/14, 28/15 5.233 33.6
11/8, 16/11 5.863 37.6
7/5, 10/7 5.889 37.8
11/10, 20/11 6.424 41.2
13/11, 22/13 6.894 44.2
15/11, 22/15 7.081 45.4
15-odd-limit intervals in 77edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
3/2, 4/3 0.656 4.2
13/8, 16/13 1.031 6.6
9/7, 14/9 1.280 8.2
9/8, 16/9 1.313 8.4
15/13, 26/15 1.610 10.3
13/12, 24/13 1.687 10.8
7/6, 12/7 1.936 12.4
13/10, 20/13 2.266 14.5
13/9, 18/13 2.343 15.0
7/4, 8/7 2.592 16.6
15/8, 16/15 2.640 16.9
11/7, 14/11 3.271 21.0
5/4, 8/5 3.297 21.2
13/7, 14/13 3.623 23.2
5/3, 6/5 3.953 25.4
11/9, 18/11 4.551 29.2
9/5, 10/9 4.609 29.6
11/6, 12/11 5.207 33.4
15/14, 28/15 5.233 33.6
11/8, 16/11 5.863 37.6
7/5, 10/7 5.889 37.8
13/11, 22/13 6.894 44.2
15/11, 22/15 8.504 54.6
11/10, 20/11 9.160 58.8

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

Table of rank-2 temperaments by generator
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)

Keyboards

Lumatone mappings for 77edo are available.

Music

Bryan Deister
Jake Freivald
Joel Grant Taylor
Chris Vaisvil
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