77edo: Difference between revisions

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→Intervals: reduce cent values to one decimal place per discussion on Discord
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{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degree
! #
! Cents
! Cents
! Approximate Ratios*
! Approximate ratios*
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| 1/1
|-
|-
| 1
| 1
| 15.584
| 15.6
| 81/80, 91/90, 99/98, 105/104
| 81/80, 91/90, 99/98, 105/104
|-
|-
| 2
| 2
| 31.169
| 31.2
| 49/48, 55/54, 64/63, 65/64, ''100/99''
| 49/48, 55/54, 64/63, 65/64, ''100/99''
|-
|-
| 3
| 3
| 46.753
| 46.8
| 33/32, 36/35, 40/39, ''45/44'', ''50/49''
| 33/32, 36/35, 40/39, ''45/44'', ''50/49''
|-
|-
| 4
| 4
| 62.338
| 62.3
| 26/25, 27/26, 28/27
| 26/25, 27/26, 28/27
|-
|-
| 5
| 5
| 77.922
| 77.9
| 21/20, 22/21, 25/24
| 21/20, 22/21, 25/24
|-
|-
| 6
| 6
| 93.506
| 93.5
| 18/17, 19/18, 20/19
| 18/17, 19/18, 20/19
|-
|-
| 7
| 7
| 109.091
| 109.1
| 16/15, 17/16
| 16/15, 17/16
|-
|-
| 8
| 8
| 124.675
| 124.7
| 14/13, 15/14
| 14/13, 15/14
|-
|-
| 9
| 9
| 140.260
| 140.3
| 13/12
| 13/12
|-
|-
| 10
| 10
| 155.844
| 155.8
| ''11/10'', 12/11
| ''11/10'', 12/11
|-
|-
| 11
| 11
| 171.429
| 171.4
| 21/19
| 21/19
|-
|-
| 12
| 12
| 187.013
| 187.0
| 10/9
| 10/9
|-
|-
| 13
| 13
| 202.597
| 202.6
| 9/8
| 9/8
|-
|-
| 14
| 14
| 218.182
| 218.2
| 17/15
| 17/15
|-
|-
| 15
| 15
| 233.766
| 233.8
| 8/7
| 8/7
|-
|-
| 16
| 16
| 249.351
| 249.4
| 15/13, 22/19
| 15/13, 22/19
|-
|-
| 17
| 17
| 264.935
| 264.9
| 7/6
| 7/6
|-
|-
| 18
| 18
| 280.519
| 280.5
| 20/17
| 20/17
|-
|-
| 19
| 19
| 296.104
| 296.1
| 13/11, 19/16, 32/27
| 13/11, 19/16, 32/27
|-
|-
| 20
| 20
| 311.688
| 311.7
| 6/5
| 6/5
|-
|-
| 21
| 21
| 327.273
| 327.3
| 98/81
| 98/81
|-
|-
| 22
| 22
| 342.857
| 342.9
| 11/9, 17/14
| 11/9, 17/14
|-
|-
| 23
| 23
| 358.442
| 358.4
| 16/13, 21/17
| 16/13, 21/17
|-
|-
| 24
| 24
| 374.026
| 374.0
| 26/21, 56/45
| 26/21, 56/45
|-
|-
| 25
| 25
| 389.610
| 389.6
| 5/4
| 5/4
|-
|-
| 26
| 26
| 405.195
| 405.2
| 19/15, 24/19, 33/26
| 19/15, 24/19, 33/26
|-
|-
| 27
| 27
| 420.779
| 420.8
| 14/11, 32/25
| 14/11, 32/25
|-
|-
| 28
| 28
| 436.364
| 436.4
| 9/7
| 9/7
|-
|-
| 29
| 29
| 451.948
| 451.9
| 13/10
| 13/10
|-
|-
| 30
| 30
| 467.532
| 467.5
| 17/13, 21/16
| 17/13, 21/16
|-
|-
| 31
| 31
| 483.117
| 483.1
| 120/91
| 120/91
|-
|-
| 32
| 32
| 498.701
| 498.7
| 4/3
| 4/3
|-
|-
| 33
| 33
| 514.286
| 514.3
| 27/20
| 27/20
|-
|-
| 34
| 34
| 529.870
| 529.9
| 19/14
| 19/14
|-
|-
| 35
| 35
| 545.455
| 545.5
| 11/8, ''15/11'', 26/19
| 11/8, ''15/11'', 26/19
|-
|-
| 36
| 36
| 561.039
| 561.0
| 18/13
| 18/13
|-
|-
| 37
| 37
| 576.623
| 576.6
| 7/5
| 7/5
|-
|-
| 38
| 38
| 592.208
| 592.2
| 24/17, 38/27, 45/32
| 24/17, 38/27, 45/32
|-
|-
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| …
| …
|}
|}
<nowiki />* As a 19-limit temperament
<nowiki/>* As a 19-limit temperament


== Notation ==
== Notation ==

Revision as of 11:03, 25 January 2025

← 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

Template:EDO intro

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

Approximation of prime harmonics in 77edo
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)
Approximation of prime harmonics in 77edo (continued)
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

# Cents Approximate ratios*
0 0.0 1/1
1 15.6 81/80, 91/90, 99/98, 105/104
2 31.2 49/48, 55/54, 64/63, 65/64, 100/99
3 46.8 33/32, 36/35, 40/39, 45/44, 50/49
4 62.3 26/25, 27/26, 28/27
5 77.9 21/20, 22/21, 25/24
6 93.5 18/17, 19/18, 20/19
7 109.1 16/15, 17/16
8 124.7 14/13, 15/14
9 140.3 13/12
10 155.8 11/10, 12/11
11 171.4 21/19
12 187.0 10/9
13 202.6 9/8
14 218.2 17/15
15 233.8 8/7
16 249.4 15/13, 22/19
17 264.9 7/6
18 280.5 20/17
19 296.1 13/11, 19/16, 32/27
20 311.7 6/5
21 327.3 98/81
22 342.9 11/9, 17/14
23 358.4 16/13, 21/17
24 374.0 26/21, 56/45
25 389.6 5/4
26 405.2 19/15, 24/19, 33/26
27 420.8 14/11, 32/25
28 436.4 9/7
29 451.9 13/10
30 467.5 17/13, 21/16
31 483.1 120/91
32 498.7 4/3
33 514.3 27/20
34 529.9 19/14
35 545.5 11/8, 15/11, 26/19
36 561.0 18/13
37 576.6 7/5
38 592.2 24/17, 38/27, 45/32
… … …

* As a 19-limit temperament

Notation

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

Ups and downs notation

Using Helmholtz–Ellis accidentals, 77edo can be notated using ups and downs notation:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Sharp symbol  
Flat symbol

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
414zpi 76.9918536925042 15.5860645308353 8.194847 1.311364 17.029289 77edo 1200.12696887432 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

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

Music

Jake Freivald
Joel Grant Taylor
Chris Vaisvil
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