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{{Infobox ET}}
{{Infobox ET}}
The '''77 equal divisions of the octave''' ('''77edo'''), or the '''77(-tone) equal temperament''' ('''77et''', '''77tet''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 77 steps of size about 15.6 [[cent]]s each.
{{EDO intro|77}}


== Theory ==
== Theory ==
With fifths less than a cent flat, major thirds a bit over three cents sharp and [[7/4]]'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.


77edo is an excellent EDO for [[Carlos Alpha]] scale, 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.


=== Prime harmonics ===
=== Prime harmonics ===
Line 335: Line 335:
== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 541: Line 541:


== Music ==
== Music ==
'''[[Jake Freivald]]'''
; [[Jake Freivald]]
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Freivald-J.-A-Seed-Planted-2nd-Version-77edo.mp3 A Seed Planted], in an [http://soonlabel.com/xenharmonic/archives/1391 organ version] of Claudi Meneghin.
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Freivald-J.-A-Seed-Planted-2nd-Version-77edo.mp3 ''A Seed Planted''], in an [http://soonlabel.com/xenharmonic/archives/1391 organ version] of [[Claudi Meneghin]].
'''[[Joel Grant Taylor]]'''
 
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/star_1-GrimaldiA+Bmod.mp3 Star 1-GrimaldiA+Bmod]
; [[Joel Grant Taylor]]
'''[[Chris Vaisvil]]'''
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/star_1-GrimaldiA+Bmod.mp3 ''Star 1-GrimaldiA+Bmod'']
* [http://micro.soonlabel.com/star/20120830-77et-star.mp3 77et Star]
 
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/star/20120830-77et-star.mp3 ''77et Star'']


[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Listen]]
[[Category:Listen]]
[[Category:Star]]
[[Category:Star]]
[[Category:Starling]]
[[Category:Starling]]
[[Category:Valentine]]
[[Category:Valentine]]

Revision as of 12:02, 1 May 2023

← 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.

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
in the 13-limit
0 0.000 1/1
1 15.584 81/80, 99/98
2 31.169 64/63, 49/48
3 46.753 33/32, 36/35
4 62.338 28/27, 26/25
5 77.922 21/20, 25/24
6 93.506 135/128
7 109.091 16/15
8 124.675 15/14
9 140.260 13/12
10 155.844 12/11, 11/10
11 171.429 72/65
12 187.013 10/9
13 202.597 9/8
14 218.182 256/225
15 233.766 8/7
16 249.351 15/13
17 264.935 7/6
18 280.519 33/28
19 296.104 32/27, 13/11
20 311.688 6/5
21 327.273 98/81
22 342.857 11/9, 39/32
23 358.442 16/13
24 374.026 56/45, 26/21
25 389.610 5/4
26 405.195 33/26, 81/64
27 420.779 14/11, 32/25
28 436.364 9/7
29 451.948 13/10
30 467.532 21/16
31 483.117 120/91
32 498.701 4/3
33 514.286 27/20
34 529.870 49/36
35 545.455 11/8, 15/11
36 561.039 18/13
37 576.623 7/5
38 592.208 45/32
39 607.792 64/45
40 623.377 10/7
41 638.961 13/9
42 654.545 16/11, 22/15
43 670.130 72/49
44 685.714 40/27
45 701.299 3/2
46 716.883 91/60
47 732.468 32/21
48 748.052 20/13
49 763.636 14/9
50 779.221 11/7, 25/16
51 794.805 52/33, 128/81
52 810.390 8/5
53 825.974 45/28, 21/13
54 841.558 13/8
55 857.143 18/11, 64/39
56 872.727 81/49
57 888.312 5/3
58 903.896 27/16, 22/13
59 919.481 56/33
60 935.065 12/7
61 950.649 26/15
62 966.234 7/4
63 981.818 225/128
64 997.403 16/9
65 1012.987 9/5
66 1028.571 65/36
67 1044.156 11/6, 20/11
68 1059.740 24/13
69 1075.325 28/15
70 1090.909 15/8
71 1106.494 256/135
72 1122.078 40/21, 48/25
73 1137.662 27/14, 25/13
74 1153.247 64/33, 35/18
75 1168.831 63/32, 96/49
76 1184.416 160/81, 196/99
77 1200.000 2/1

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 Octave 		Generator Temperaments
1 1\77
1 2\77
1 3\77
1 4\77 Unicorn / alicorn / camahueto / qilin
1 5\77 Valentine
1 6\77
1 8\77
1 9\77 Tsaharuk
1 10\77
1 12\77
1 13\77
1 15\77 Guiron
1 16\77 Hemischis
1 17\77
1 18\77
1 19\77
1 20\77 Oolong
1 23\77 Restles
1 24\77
1 25\77
1 26\77
1 27\77
1 29\77
1 30\77
1 31\77 Hemiseven
1 32\77 Helmholtz / grackle
1 34\77 Mabila
1 36\77
1 37\77
1 38\77
7 1\77 Absurdity
7 2\77
7 3\77
7 4\77
7 5\77
11 1\77 Hendecatonic
11 2\77
11 3\77

Music

Jake Freivald
Joel Grant Taylor
Chris Vaisvil
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