77edo: Difference between revisions

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== 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.

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 ===

{{Harmonics in equal|77|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 77edo (continued)}}

{{Harmonics in equal|77|columns=11|start=12|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.

Since 77 factors into primes as {{nowrap| 7 × 11 }}, 77edo contains [[7edo]] and [[11edo]] as subset edos.

== Intervals ==

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== Notation ==

=== Ups and downs notation ===

~~Using [[Helmholtz–Ellis]] accidentals, 77edo can be notated using [[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.

Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:

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== Approximation to JI ==

=== ~~Zeta peak index~~ ===

=== Selected just intervals ===

{~~| 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~~

|}

== Regular temperament properties ==

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|-

| 2.3

| {{~~monzo~~| -122 77 }}

| {{Monzo| -122 77 }}

| {{~~mapping~~| 77 122 }}

| {{Mapping| 77 122 }}

| +0.207

| 0.207

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| 2.3.5

| 32805/32768, 1594323/1562500

| {{~~mapping~~| 77 122 179 }}

| {{Mapping| 77 122 179 }}

| −0.336

| 0.785

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| 2.3.5.7

| 126/125, 1029/1024, 10976/10935

| {{~~mapping~~| 77 122 179 216 }}

| {{Mapping| 77 122 179 216 }}

| −0.021

| 0.872

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| 2.3.5.7.11

| 121/120, 126/125, 176/175, 10976/10935

| {{~~mapping~~| 77 122 179 216 266 }}

| {{Mapping| 77 122 179 216 266 }}

| +0.322

| 1.039

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| 2.3.5.7.11.13

| 121/120, 126/125, 176/175, 196/195, 676/675

| {{~~mapping~~| 77 122 179 216 266 285 }}

| {{Mapping| 77 122 179 216 266 285 }}

| +0.222

| 0.974

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| 498.7<br>(46.8)

| 4/3<br>(36/35)

| [[Hendecatonic]]

| [[Hendecatonic (temperament)|Hendecatonic]]

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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 ==

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/wSZez2KgP2U ''microtonal improvisation in 77edo''] (2025)

; [[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]].

* [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure ''A Seed Planted''], in an [https://web.archive.org/web/20160729174100/http://soonlabel.com/xenharmonic/archives/1391 organ version] of [[Claudi Meneghin]].

; [[Joel Grant Taylor]]

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; [[Chris Vaisvil]]

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

* [https://web.archive.org/web/20201127015309/http://micro.soonlabel.com/star/20120830-77et-star.mp3 ''77et Star'']

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED 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]].
+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]].
 et 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.
+It also does surprisingly well (for its size) in a large range of very high odd-limits (41 to 125 range).
 === Prime harmonics ===
-{{Harmonics in equal|77|columns=9}}
+{{Harmonics in equal|77|columns=11}}
-{{Harmonics in equal|77|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 77edo (continued)}}
+{{Harmonics in equal|77|columns=11|start=12|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.
+Since 77 factors into primes as {{nowrap| 7 × 11 }}, 77edo contains [[7edo]] and [[11edo]] as subset edos.
 == Intervals ==
@@ Line 227: / Line 229: @@
 == Notation ==
 === Ups and downs notation ===
-Using [[Helmholtz–Ellis]] accidentals, 77edo can be notated using [[ups and downs notation]]:
+edo 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}}
@@ Line 257: / Line 262: @@
 == Approximation to JI ==
-=== Zeta peak index ===
+=== Selected just intervals ===
-{| class="wikitable center-all"
+{{Q-odd-limit intervals}}
-|-
-! 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 ==
@@ Line 301: / Line 278: @@
 |-
 | 2.3
-| {{monzo| -122 77 }}
+| {{Monzo| -122 77 }}
-| {{mapping| 77 122 }}
+| {{Mapping| 77 122 }}
 | +0.207
 | 0.207
@@ Line 309: / Line 286: @@
 | 2.3.5
 | 32805/32768, 1594323/1562500
-| {{mapping| 77 122 179 }}
+| {{Mapping| 77 122 179 }}
 | −0.336
 | 0.785
@@ Line 316: / Line 293: @@
 | 2.3.5.7
 | 126/125, 1029/1024, 10976/10935
-| {{mapping| 77 122 179 216 }}
+| {{Mapping| 77 122 179 216 }}
 | −0.021
 | 0.872
@@ Line 323: / Line 300: @@
 | 2.3.5.7.11
 | 121/120, 126/125, 176/175, 10976/10935
-| {{mapping| 77 122 179 216 266 }}
+| {{Mapping| 77 122 179 216 266 }}
 | +0.322
 | 1.039
@@ Line 330: / Line 307: @@
 | 2.3.5.7.11.13
 | 121/120, 126/125, 176/175, 196/195, 676/675
-| {{mapping| 77 122 179 216 266 285 }}
+| {{Mapping| 77 122 179 216 266 285 }}
 | +0.222
 | 0.974
@@ Line 422: / Line 399: @@
 | 498.7<br>(46.8)
 | 4/3<br>(36/35)
-| [[Hendecatonic]]
+| [[Hendecatonic (temperament)|Hendecatonic]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|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
+\77=1/1: string 2 open
+\77=2/1: string 7 fret 11
+\77=3/2: string 2 fret 5
+\77=5/4: string 1 fret 4
+\77=7/4: string 6 fret 2
+\77=11/8: string 4 fret 10
+\77=13/8: string 2 fret 6
+\77=17/16: string 1 fret 2
+\77=19/16: string 5 fret 7
+\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 ==
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/wSZez2KgP2U ''microtonal improvisation in 77edo''] (2025)
 ; [[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]].
+* [https://soundcloud.com/jdfreivald/a-seed-planted-starling-pure ''A Seed Planted''], in an [https://web.archive.org/web/20160729174100/http://soonlabel.com/xenharmonic/archives/1391 organ version] of [[Claudi Meneghin]].
 ; [[Joel Grant Taylor]]
@@ Line 434: / Line 447: @@
 ; [[Chris Vaisvil]]
-* [http://micro.soonlabel.com/star/20120830-77et-star.mp3 ''77et Star'']
+* [https://web.archive.org/web/20201127015309/http://micro.soonlabel.com/star/20120830-77et-star.mp3 ''77et Star'']
 [[Category:Listen]]