77edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|77}}
{{ED intro}}


== Theory ==
== Theory ==
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]] less flat than that, 77edo represents an excellent tuning choice for both [[valentine]] (hence also [[Carlos Alpha]]), the {{nowrap|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 [[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 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.  


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.
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.
 
It also does surprisingly well (for its size) in a large range of very high odd-limits (41 to 125 range).


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|77}}
{{Harmonics in equal|77|columns=9}}
{{Harmonics in equal|77|columns=10|start=10|collapsed=true|title=Approximation of prime harmonics in 77edo (continued)}}
 
=== Subsets and supersets ===
Since 77 factors into primes as {{nowrap|7 × 11}}, 77edo contains [[7edo]] and [[11edo]] as subset edos.


== Intervals ==
== Intervals ==
{| class="wikitable center-all right-2 left-3"
{| class="wikitable center-all right-2 left-3"
|-
|-
! Degree
! #
! Cents
! Cents
! Approximate Ratios<br>in the 13-limit
! Approximate ratios*
! [[Ups and downs notation]]
|-
|-
| 0
| 0
| 0.000
| 0.0
| 1/1
| 1/1
| {{UDnote|step=0}}
|-
|-
| 1
| 1
| 15.584
| 15.6
| 81/80, 99/98
| 81/80, 91/90, 99/98, 105/104
| {{UDnote|step=1}}
|-
|-
| 2
| 2
| 31.169
| 31.2
| 64/63, 49/48
| 49/48, 55/54, 64/63, 65/64, ''100/99''
| {{UDnote|step=2}}
|-
|-
| 3
| 3
| 46.753
| 46.8
| 33/32, 36/35
| 33/32, 36/35, 40/39, ''45/44'', ''50/49''
| {{UDnote|step=3}}
|-
|-
| 4
| 4
| 62.338
| 62.3
| 28/27, 26/25
| 26/25, 27/26, 28/27
| {{UDnote|step=4}}
|-
|-
| 5
| 5
| 77.922
| 77.9
| 21/20, 25/24
| 21/20, 22/21, 25/24
| {{UDnote|step=5}}
|-
|-
| 6
| 6
| 93.506
| 93.5
| 135/128
| 18/17, 19/18, 20/19
| {{UDnote|step=6}}
|-
|-
| 7
| 7
| 109.091
| 109.1
| 16/15
| 16/15, 17/16
| {{UDnote|step=7}}
|-
|-
| 8
| 8
| 124.675
| 124.7
| 15/14
| 14/13, 15/14
| {{UDnote|step=8}}
|-
|-
| 9
| 9
| 140.260
| 140.3
| 13/12
| 13/12
| {{UDnote|step=9}}
|-
|-
| 10
| 10
| 155.844
| 155.8
| 12/11, 11/10
| ''11/10'', 12/11
| {{UDnote|step=10}}
|-
|-
| 11
| 11
| 171.429
| 171.4
| 72/65
| 21/19
| {{UDnote|step=11}}
|-
|-
| 12
| 12
| 187.013
| 187.0
| 10/9
| 10/9
| {{UDnote|step=12}}
|-
|-
| 13
| 13
| 202.597
| 202.6
| 9/8
| 9/8
| {{UDnote|step=13}}
|-
|-
| 14
| 14
| 218.182
| 218.2
| 256/225
| 17/15
| {{UDnote|step=14}}
|-
|-
| 15
| 15
| 233.766
| 233.8
| 8/7
| 8/7
| {{UDnote|step=15}}
|-
|-
| 16
| 16
| 249.351
| 249.4
| 15/13
| 15/13, 22/19
| {{UDnote|step=16}}
|-
|-
| 17
| 17
| 264.935
| 264.9
| 7/6
| 7/6
| {{UDnote|step=17}}
|-
|-
| 18
| 18
| 280.519
| 280.5
| 33/28
| 20/17
| {{UDnote|step=18}}
|-
|-
| 19
| 19
| 296.104
| 296.1
| 32/27, 13/11
| 13/11, 19/16, 32/27
| {{UDnote|step=19}}
|-
|-
| 20
| 20
| 311.688
| 311.7
| 6/5
| 6/5
| {{UDnote|step=20}}
|-
|-
| 21
| 21
| 327.273
| 327.3
| 98/81
| 98/81
| {{UDnote|step=21}}
|-
|-
| 22
| 22
| 342.857
| 342.9
| 11/9, 39/32
| 11/9, 17/14
| {{UDnote|step=22}}
|-
|-
| 23
| 23
| 358.442
| 358.4
| 16/13
| 16/13, 21/17
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 374.026
| 374.0
| 56/45, 26/21
| 26/21, 56/45
| {{UDnote|step=24}}
|-
|-
| 25
| 25
| 389.610
| 389.6
| 5/4
| 5/4
| {{UDnote|step=25}}
|-
|-
| 26
| 26
| 405.195
| 405.2
| 33/26, 81/64
| 19/15, 24/19, 33/26
| {{UDnote|step=26}}
|-
|-
| 27
| 27
| 420.779
| 420.8
| 14/11, 32/25
| 14/11, 32/25
| {{UDnote|step=27}}
|-
|-
| 28
| 28
| 436.364
| 436.4
| 9/7
| 9/7
| {{UDnote|step=28}}
|-
|-
| 29
| 29
| 451.948
| 451.9
| 13/10
| 13/10
| {{UDnote|step=29}}
|-
|-
| 30
| 30
| 467.532
| 467.5
| 21/16
| 17/13, 21/16
| {{UDnote|step=30}}
|-
|-
| 31
| 31
| 483.117
| 483.1
| 120/91
| 120/91
| {{UDnote|step=31}}
|-
|-
| 32
| 32
| 498.701
| 498.7
| 4/3
| 4/3
| {{UDnote|step=32}}
|-
|-
| 33
| 33
| 514.286
| 514.3
| 27/20
| 27/20
| {{UDnote|step=33}}
|-
|-
| 34
| 34
| 529.870
| 529.9
| 49/36
| 19/14
| {{UDnote|step=34}}
|-
|-
| 35
| 35
| 545.455
| 545.5
| 11/8, 15/11
| 11/8, ''15/11'', 26/19
| {{UDnote|step=35}}
|-
|-
| 36
| 36
| 561.039
| 561.0
| 18/13
| 18/13
| {{UDnote|step=36}}
|-
|-
| 37
| 37
| 576.623
| 576.6
| 7/5
| 7/5
| {{UDnote|step=37}}
|-
|-
| 38
| 38
| 592.208
| 592.2
| 45/32
| 24/17, 38/27, 45/32
|-
| {{UDnote|step=38}}
| 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
|}
|}
<nowiki/>* As a 19-limit temperament
== Notation ==
=== Ups and downs notation ===
77edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}
=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:77-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 591 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:77-EDO_Evo_Sagittal.svg]]
</imagemap>
==== Revo flavor ====
<imagemap>
File:77-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:77-EDO_Revo_Sagittal.svg]]
</imagemap>
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 414
| steps = 76.9918536925042
| step size = 15.5860645308353
| tempered height = 8.194847
| pure height = 8.145298
| integral = 1.311364
| gap = 17.029289
| octave = 1200.12696887432
| consistent = 10
| distinct = 10
}}


== 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|Comma List]]
! rowspan="2" | [[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 346: Line 291:
| 2.3
| 2.3
| {{monzo| -122 77 }}
| {{monzo| -122 77 }}
| [{{val| 77 122 }}]
| {{mapping| 77 122 }}
| +0.207
| +0.207
| 0.207
| 0.207
Line 353: Line 298:
| 2.3.5
| 2.3.5
| 32805/32768, 1594323/1562500
| 32805/32768, 1594323/1562500
| [{{val| 77 122 179 }}]
| {{mapping| 77 122 179 }}
| -0.336
| −0.336
| 0.785
| 0.785
| 5.04
| 5.04
Line 360: Line 305:
| 2.3.5.7
| 2.3.5.7
| 126/125, 1029/1024, 10976/10935
| 126/125, 1029/1024, 10976/10935
| [{{val| 77 122 179 216 }}]
| {{mapping| 77 122 179 216 }}
| -0.021
| −0.021
| 0.872
| 0.872
| 5.59
| 5.59
Line 367: Line 312:
| 2.3.5.7.11
| 2.3.5.7.11
| 121/120, 126/125, 176/175, 10976/10935
| 121/120, 126/125, 176/175, 10976/10935
| [{{val| 77 122 179 216 266 }}]
| {{mapping| 77 122 179 216 266 }}
| +0.322
| +0.322
| 1.039
| 1.039
Line 374: Line 319:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 121/120, 126/125, 176/175, 196/195, 676/675
| 121/120, 126/125, 176/175, 196/195, 676/675
| [{{val| 77 122 179 216 266 285 }}]
| {{mapping| 77 122 179 216 266 285 }}
| +0.222
| +0.222
| 0.974
| 0.974
Line 381: Line 326:


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all right-3 left-5"
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated Ratio<br>(Reduced)
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| 1
| 1
| 4\77
| 4\77
| 62.34
| 62.3
| 28/27
| 28/27
| [[Unicorn]] / alicorn / camahueto / qilin
| [[Unicorn]] / alicorn (77e) / camahueto (77) / qilin (77)
|-
|-
| 1
| 1
| 5\77
| 5\77
| 77.92
| 77.9
| 21/20
| 21/20
| [[Valentine]]
| [[Valentine]]
Line 403: Line 349:
| 1
| 1
| 9\77
| 9\77
| 140.26
| 140.3
| 13/12
| 13/12
| [[Tsaharuk]]
| [[Tsaharuk]]
Line 409: Line 355:
| 1
| 1
| 15\77
| 15\77
| 233.77
| 233.8
| 8/7
| 8/7
| [[Guiron]]
| [[Guiron]]
Line 415: Line 361:
| 1
| 1
| 16\77
| 16\77
| 249.35
| 249.4
| 15/13
| 15/13
| [[Hemischis]] (77e)
| [[Hemischis]] (77e)
Line 421: Line 367:
| 1
| 1
| 20\77
| 20\77
| 311.69
| 311.7
| 6/5
| 6/5
| [[Oolong]]
| [[Oolong]]
Line 427: Line 373:
| 1
| 1
| 23\77
| 23\77
| 358.44
| 358.4
| 16/13
| 16/13
| [[Restles]]
| [[Restles]]
Line 433: Line 379:
| 1
| 1
| 31\77
| 31\77
| 483.12
| 483.1
| 45/34
| 45/34
| [[Hemiseven]]
| [[Hemiseven]]
Line 439: Line 385:
| 1
| 1
| 32\77
| 32\77
| 498.70
| 498.7
| 4/3
| 4/3
| [[Grackle]]
| [[Grackle]]
Line 445: Line 391:
| 1
| 1
| 34\77
| 34\77
| 529.87
| 529.9
| 512/375
| 512/375
| [[Tuskaloosa]]<br>[[Muscogee]]
| [[Tuskaloosa]] / [[muscogee]]
|-
| 1
| 36\77
| 561.0
| 18/13
| [[Demivalentine]]
|-
|-
| 7
| 7
| 32\77<br>(1\77)
| 32\77<br>(1\77)
| 498.70<br>(15.58)
| 498.7<br>(15.6)
| 4/3<br>(81/80)
| 4/3<br>(81/80)
| [[Absurdity]]
| [[Absurdity]]
Line 457: Line 409:
| 11
| 11
| 32\77<br>(3\77)
| 32\77<br>(3\77)
| 498.70<br>(46.75)
| 498.7<br>(46.8)
| 4/3<br>(36/35)
| 4/3<br>(36/35)
| [[Hendecatonic]]
| [[Hendecatonic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|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 mapping for 77edo|Lumatone mappings for 77edo]] are available.


== Music ==
== Music ==
; [[Bryan Deister]
* [https://www.youtube.com/shorts/wSZez2KgP2U ''microtonal improvisation in 77edo''] (2025)
; [[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'']{{dead link}}, in an [https://web.archive.org/web/20190412162407/http://soonlabel.com/xenharmonic/archives/1391 organ version] of [[Claudi Meneghin]].


; [[Joel Grant Taylor]]
; [[Joel Grant Taylor]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/star_1-GrimaldiA+Bmod.mp3 ''Star 1-GrimaldiA+Bmod'']
* [https://web.archive.org/web/20201127015546/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/star_1-GrimaldiA+Bmod.mp3 ''Star 1-GrimaldiA+Bmod'']


; [[Chris Vaisvil]]
; [[Chris Vaisvil]]

Latest revision as of 09:38, 2 July 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

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

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

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

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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