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77edo: Difference between revisions

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→Theory: correction (Carlos Alpha != 9edf!). -dwynwen (best tuned in sharper tunings)
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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]] 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]].
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 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.  
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.  
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=9}}
{{Harmonics in equal|77|columns=9}}
{{Harmonics in equal|77|columns=9|start=10|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 ===
=== Subsets and supersets ===
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== 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 ====
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default [[File:77-EDO_Revo_Sagittal.svg]]
default [[File:77-EDO_Revo_Sagittal.svg]]
</imagemap>
</imagemap>
=== Ups and downs notation ===
Using [[Helmholtz–Ellis]] accidentals, 77edo can be notated using [[ups and downs notation]]:
{{Sharpness-sharp7}}


== Approximation to JI ==
== Approximation to JI ==
=== Zeta peak index ===
=== Zeta peak index ===
{| class="wikitable center-all"
{{ZPI
|-
| zpi = 414
! colspan="3" | Tuning
| steps = 76.9918536925042
! colspan="3" | Strength
| step size = 15.5860645308353
! colspan="2" | Closest edo
| tempered height = 8.194847
! colspan="2" | Integer limit
| pure height = 8.145298
|-
| integral = 1.311364
! ZPI
| gap = 17.029289
! Steps per octave
| octave = 1200.12696887432
! Step size (cents)
| consistent = 10
! Height
| distinct = 10
! 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 ==
== Regular temperament properties ==
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|}
|}
<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 ===
[[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'']{{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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