View source for 77edo

{{Infobox ET}}
{{EDO intro}}

== 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]] 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. 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 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. 

=== Prime harmonics ===
{{Harmonics in equal|77|columns=9}}
{{Harmonics in equal|77|columns=9|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 ==
{| class="wikitable center-all right-2 left-3"
|-
! #
! Cents
! Approximate ratios*
! [[Ups and downs notation]]
|-
| 0
| 0.0
| 1/1
| {{UDnote|step=0}}
|-
| 1
| 15.6
| 81/80, 91/90, 99/98, 105/104
| {{UDnote|step=1}}
|-
| 2
| 31.2
| 49/48, 55/54, 64/63, 65/64, ''100/99''
| {{UDnote|step=2}}
|-
| 3
| 46.8
| 33/32, 36/35, 40/39, ''45/44'', ''50/49''
| {{UDnote|step=3}}
|-
| 4
| 62.3
| 26/25, 27/26, 28/27
| {{UDnote|step=4}}
|-
| 5
| 77.9
| 21/20, 22/21, 25/24
| {{UDnote|step=5}}
|-
| 6
| 93.5
| 18/17, 19/18, 20/19
| {{UDnote|step=6}}
|-
| 7
| 109.1
| 16/15, 17/16
| {{UDnote|step=7}}
|-
| 8
| 124.7
| 14/13, 15/14
| {{UDnote|step=8}}
|-
| 9
| 140.3
| 13/12
| {{UDnote|step=9}}
|-
| 10
| 155.8
| ''11/10'', 12/11
| {{UDnote|step=10}}
|-
| 11
| 171.4
| 21/19
| {{UDnote|step=11}}
|-
| 12
| 187.0
| 10/9
| {{UDnote|step=12}}
|-
| 13
| 202.6
| 9/8
| {{UDnote|step=13}}
|-
| 14
| 218.2
| 17/15
| {{UDnote|step=14}}
|-
| 15
| 233.8
| 8/7
| {{UDnote|step=15}}
|-
| 16
| 249.4
| 15/13, 22/19
| {{UDnote|step=16}}
|-
| 17
| 264.9
| 7/6
| {{UDnote|step=17}}
|-
| 18
| 280.5
| 20/17
| {{UDnote|step=18}}
|-
| 19
| 296.1
| 13/11, 19/16, 32/27
| {{UDnote|step=19}}
|-
| 20
| 311.7
| 6/5
| {{UDnote|step=20}}
|-
| 21
| 327.3
| 98/81
| {{UDnote|step=21}}
|-
| 22
| 342.9
| 11/9, 17/14
| {{UDnote|step=22}}
|-
| 23
| 358.4
| 16/13, 21/17
| {{UDnote|step=23}}
|-
| 24
| 374.0
| 26/21, 56/45
| {{UDnote|step=24}}
|-
| 25
| 389.6
| 5/4
| {{UDnote|step=25}}
|-
| 26
| 405.2
| 19/15, 24/19, 33/26
| {{UDnote|step=26}}
|-
| 27
| 420.8
| 14/11, 32/25
| {{UDnote|step=27}}
|-
| 28
| 436.4
| 9/7
| {{UDnote|step=28}}
|-
| 29
| 451.9
| 13/10
| {{UDnote|step=29}}
|-
| 30
| 467.5
| 17/13, 21/16
| {{UDnote|step=30}}
|-
| 31
| 483.1
| 120/91
| {{UDnote|step=31}}
|-
| 32
| 498.7
| 4/3
| {{UDnote|step=32}}
|-
| 33
| 514.3
| 27/20
| {{UDnote|step=33}}
|-
| 34
| 529.9
| 19/14
| {{UDnote|step=34}}
|-
| 35
| 545.5
| 11/8, ''15/11'', 26/19
| {{UDnote|step=35}}
|-
| 36
| 561.0
| 18/13
| {{UDnote|step=36}}
|-
| 37
| 576.6
| 7/5
| {{UDnote|step=37}}
|-
| 38
| 592.2
| 24/17, 38/27, 45/32
| {{UDnote|step=38}}
|-
| …
| …
| …
|}
<nowiki/>* As a 19-limit temperament

== Notation ==
=== 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>

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

{{Sharpness-sharp7}}

== Approximation to JI ==
=== Zeta peak index ===
{| class="wikitable center-all"
|-
! 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
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -122 77 }}
| {{mapping| 77 122 }}
| +0.207
| 0.207
| 1.33
|-
| 2.3.5
| 32805/32768, 1594323/1562500
| {{mapping| 77 122 179 }}
| −0.336
| 0.785
| 5.04
|-
| 2.3.5.7
| 126/125, 1029/1024, 10976/10935
| {{mapping| 77 122 179 216 }}
| −0.021
| 0.872
| 5.59
|-
| 2.3.5.7.11
| 121/120, 126/125, 176/175, 10976/10935
| {{mapping| 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
| {{mapping| 77 122 179 216 266 285 }}
| +0.222
| 0.974
| 6.25
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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<br>(1\77)
| 498.7<br>(15.6)
| 4/3<br>(81/80)
| [[Absurdity]]
|-
| 11
| 32\77<br>(3\77)
| 498.7<br>(46.8)
| 4/3<br>(36/35)
| [[Hendecatonic]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Music ==
; [[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]].

; [[Joel Grant Taylor]]
* [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]]
* [http://micro.soonlabel.com/star/20120830-77et-star.mp3 ''77et Star'']

[[Category:Listen]]
[[Category:Star]]
[[Category:Starling]]
[[Category:Valentine]]