77edo: Difference between revisions

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Assessment of a full 19-limit interpretation
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With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|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 [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].
With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|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 [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|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.
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 [[17/1|17]] and [[19/1|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.
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.
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! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<br>in the 13-limit
! Approximate Ratios*
|-
|-
| 0
| 0
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| 6
| 6
| 93.506
| 93.506
| 135/128
| 18/17, 19/18, 20/19
|-
|-
| 7
| 7
| 109.091
| 109.091
| 16/15
| 16/15, 17/16
|-
|-
| 8
| 8
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| 11
| 11
| 171.429
| 171.429
| 72/65
| 21/19
|-
|-
| 12
| 12
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| 14
| 14
| 218.182
| 218.182
| 256/225
| 17/15
|-
|-
| 15
| 15
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| 16
| 16
| 249.351
| 249.351
| 15/13
| 15/13, 22/19
|-
|-
| 17
| 17
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| 18
| 18
| 280.519
| 280.519
| 33/28
| 20/17
|-
|-
| 19
| 19
| 296.104
| 296.104
| 13/11, 32/27
| 13/11, 19/16, 32/27
|-
|-
| 20
| 20
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| 22
| 22
| 342.857
| 342.857
| 11/9, 39/32
| 11/9, 17/14
|-
|-
| 23
| 23
| 358.442
| 358.442
| 16/13, 27/22
| 16/13, 21/17
|-
|-
| 24
| 24
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| 26
| 26
| 405.195
| 405.195
| 33/26, 81/64
| 19/15, 24/19, 33/26
|-
|-
| 27
| 27
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| 30
| 30
| 467.532
| 467.532
| 21/16
| 17/13, 21/16
|-
|-
| 31
| 31
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| 34
| 34
| 529.870
| 529.870
| 49/36
| 19/14
|-
|-
| 35
| 35
| 545.455
| 545.455
| 11/8, ''15/11''
| 11/8, ''15/11'', 26/19
|-
|-
| 36
| 36
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| 38
| 38
| 592.208
| 592.208
| 45/32
| 24/17, 38/27, 45/32
|-
|-
| 39
| 39
| 607.792
| 607.792
| 64/45
| 17/12, 27/19, 64/45
|-
|-
| 40
| 40
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| 42
| 42
| 654.545
| 654.545
| 16/11, ''22/15''
| 16/11, 19/13, ''22/15''
|-
|-
| 43
| 43
| 670.130
| 670.130
| 72/49
| 28/19
|-
|-
| 44
| 44
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| 47
| 47
| 732.468
| 732.468
| 32/21
| 26/17, 32/21
|-
|-
| 48
| 48
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| 51
| 51
| 794.805
| 794.805
| 52/33, 128/81
| 19/12, 30/19, 52/33
|-
|-
| 52
| 52
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| 54
| 54
| 841.558
| 841.558
| 13/8, 44/27
| 13/8, 34/21
|-
|-
| 55
| 55
| 857.143
| 857.143
| 18/11, 64/39
| 18/11, 28/17
|-
|-
| 56
| 56
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| 58
| 58
| 903.896
| 903.896
| 27/16, 22/13
| 22/13, 27/16, 32/19
|-
|-
| 59
| 59
| 919.481
| 919.481
| 56/33
| 17/10
|-
|-
| 60
| 60
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| 61
| 61
| 950.649
| 950.649
| 26/15
| 26/15, 19/11
|-
|-
| 62
| 62
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| 63
| 63
| 981.818
| 981.818
| 225/128
| 30/17
|-
|-
| 64
| 64
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| 66
| 66
| 1028.571
| 1028.571
| 65/36
| 38/21
|-
|-
| 67
| 67
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| 70
| 70
| 1090.909
| 1090.909
| 15/8
| 15/8, 32/17
|-
|-
| 71
| 71
| 1106.494
| 1106.494
| 256/135
| 17/9, 19/10, 36/19
|-
|-
| 72
| 72
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| 2/1
| 2/1
|}
|}
<nowiki>*</nowiki> as a 19-limit temperament


== Regular temperament properties ==
== Regular temperament properties ==

Revision as of 09:00, 27 June 2024

← 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
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)

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
39 607.792 17/12, 27/19, 64/45
40 623.377 10/7
41 638.961 13/9
42 654.545 16/11, 19/13, 22/15
43 670.130 28/19
44 685.714 40/27
45 701.299 3/2
46 716.883 91/60
47 732.468 26/17, 32/21
48 748.052 20/13
49 763.636 14/9
50 779.221 11/7, 25/16
51 794.805 19/12, 30/19, 52/33
52 810.390 8/5
53 825.974 21/13, 45/28
54 841.558 13/8, 34/21
55 857.143 18/11, 28/17
56 872.727 81/49
57 888.312 5/3
58 903.896 22/13, 27/16, 32/19
59 919.481 17/10
60 935.065 12/7
61 950.649 26/15, 19/11
62 966.234 7/4
63 981.818 30/17
64 997.403 16/9
65 1012.987 9/5
66 1028.571 38/21
67 1044.156 11/6, 20/11
68 1059.740 24/13
69 1075.325 28/15
70 1090.909 15/8, 32/17
71 1106.494 17/9, 19/10, 36/19
72 1122.078 21/11, 40/21, 48/25
73 1137.662 25/13, 27/14, 52/27
74 1153.247 35/18, 39/20, 49/25, 64/33, 88/45
75 1168.831 63/32, 96/49, 108/55, 128/65, 99/50
76 1184.416 160/81, 180/91, 196/99, 208/105
77 1200.000 2/1

* 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* 		Temperaments
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

Jake Freivald
Joel Grant Taylor
Chris Vaisvil
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