77edo: Difference between revisions

(15 intermediate revisions by 7 users not shown)

Line 1:

== 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 &~~amp;~~ 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.

~~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 ===

{{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 ==

Line 225:

Line 228:

== 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.

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

=== Sagittal notation ===

==== Evo flavor ====

Line 249:

Line 261:

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"~~

{{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

|}

== Regular temperament properties ==

Line 424:

Line 414:

|}

<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 ==

; [[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]].

@@ 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]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the {{nowrap|31 &amp; 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]].
-et 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.
+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.
-edo 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 ===
-{{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 ==
@@ Line 225: / Line 228: @@
 == Notation ==
+=== 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}}
 === Sagittal notation ===
 ==== Evo flavor ====
@@ Line 249: / Line 261: @@
 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"
+{{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 ==
@@ Line 424: / Line 414: @@
 |}
 <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
+\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]].