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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro}}
{{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 {{nowrap|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.  


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.  
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.
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|columns=12}}
{{Harmonics in equal|77|columns=9}}
{{Harmonics in equal|77|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 77edo (continued)}}
{{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*
! 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, 91/90, 99/98, 105/104
| 81/80, 91/90, 99/98, 105/104
| {{UDnote|step=1}}
|-
|-
| 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''
| {{UDnote|step=2}}
|-
|-
| 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''
| {{UDnote|step=3}}
|-
|-
| 4
| 4
| 62.338
| 62.3
| 26/25, 27/26, 28/27
| 26/25, 27/26, 28/27
| {{UDnote|step=4}}
|-
|-
| 5
| 5
| 77.922
| 77.9
| 21/20, 22/21, 25/24
| 21/20, 22/21, 25/24
| {{UDnote|step=5}}
|-
|-
| 6
| 6
| 93.506
| 93.5
| 18/17, 19/18, 20/19
| 18/17, 19/18, 20/19
| {{UDnote|step=6}}
|-
|-
| 7
| 7
| 109.091
| 109.1
| 16/15, 17/16
| 16/15, 17/16
| {{UDnote|step=7}}
|-
|-
| 8
| 8
| 124.675
| 124.7
| 14/13, 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
| ''11/10'', 12/11
| ''11/10'', 12/11
| {{UDnote|step=10}}
|-
|-
| 11
| 11
| 171.429
| 171.4
| 21/19
| 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
| 17/15
| 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, 22/19
| 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
| 20/17
| 20/17
| {{UDnote|step=18}}
|-
|-
| 19
| 19
| 296.104
| 296.1
| 13/11, 19/16, 32/27
| 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, 17/14
| 11/9, 17/14
| {{UDnote|step=22}}
|-
|-
| 23
| 23
| 358.442
| 358.4
| 16/13, 21/17
| 16/13, 21/17
| {{UDnote|step=23}}
|-
|-
| 24
| 24
| 374.026
| 374.0
| 26/21, 56/45
| 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
| 19/15, 24/19, 33/26
| 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
| 17/13, 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
| 19/14
| 19/14
| {{UDnote|step=34}}
|-
|-
| 35
| 35
| 545.455
| 545.5
| 11/8, ''15/11'', 26/19
| 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
| 24/17, 38/27, 45/32
| 24/17, 38/27, 45/32
| {{UDnote|step=38}}
|-
|-
| …
| …
Line 182: Line 225:
| …
| …
|}
|}
<nowiki />* As a 19-limit temperament
<nowiki/>* As a 19-limit temperament


== Notation ==
== 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 ===
=== Sagittal notation ===
==== Evo flavor ====
==== Evo flavor ====
Line 210: Line 262:
</imagemap>
</imagemap>


=== Ups and downs notation ===
== Approximation to JI ==
Using [[Helmholtz–Ellis]] accidentals, 77edo can be notated using [[ups and downs notation]]:
=== Zeta peak index ===
 
{{ZPI
{{Sharpness-sharp7}}
| 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 ==
Line 221: Line 283:
! rowspan="2" | [[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
|-
|-
Line 267: Line 329:
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperament
! Temperament
|-
|-
| 1
| 1
| 4\77
| 4\77
| 62.34
| 62.3
| 28/27
| 28/27
| [[Unicorn]] / alicorn (77e) / camahueto (77) / qilin (77)
| [[Unicorn]] / alicorn (77e) / camahueto (77) / qilin (77)
Line 281: Line 343:
| 1
| 1
| 5\77
| 5\77
| 77.92
| 77.9
| 21/20
| 21/20
| [[Valentine]]
| [[Valentine]]
Line 287: Line 349:
| 1
| 1
| 9\77
| 9\77
| 140.26
| 140.3
| 13/12
| 13/12
| [[Tsaharuk]]
| [[Tsaharuk]]
Line 293: Line 355:
| 1
| 1
| 15\77
| 15\77
| 233.77
| 233.8
| 8/7
| 8/7
| [[Guiron]]
| [[Guiron]]
Line 299: Line 361:
| 1
| 1
| 16\77
| 16\77
| 249.35
| 249.4
| 15/13
| 15/13
| [[Hemischis]] (77e)
| [[Hemischis]] (77e)
Line 305: Line 367:
| 1
| 1
| 20\77
| 20\77
| 311.69
| 311.7
| 6/5
| 6/5
| [[Oolong]]
| [[Oolong]]
Line 311: Line 373:
| 1
| 1
| 23\77
| 23\77
| 358.44
| 358.4
| 16/13
| 16/13
| [[Restles]]
| [[Restles]]
Line 317: Line 379:
| 1
| 1
| 31\77
| 31\77
| 483.12
| 483.1
| 45/34
| 45/34
| [[Hemiseven]]
| [[Hemiseven]]
Line 323: Line 385:
| 1
| 1
| 32\77
| 32\77
| 498.70
| 498.7
| 4/3
| 4/3
| [[Grackle]]
| [[Grackle]]
Line 329: 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]]
|-
|-
| 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
<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 ===


== Zeta properties ==
[[Lumatone mapping for 77edo|Lumatone mappings for 77edo]] are available.
=== Zeta peak index ===
{| class="wikitable"
|-
! colspan="3" | Tuning
! colspan="3" | Strength
! colspan="2" | Closest EDO
! colspan="2" | 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
|}


== 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'']{{dead link}}, in an [https://web.archive.org/web/20190412162407/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]].
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