761edo: Difference between revisions

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== Theory ==

761edo is [[consistent]] to the [[9-odd-limit]]. It [[tempers out]] [[32805/32768]], [[420175/419904]] and {{monzo|3 13 -15 4}} in the [[7-limit]]. The equal temperament is strong in the 2.3.5.13.29.31 [[subgroup]], tempering out 32805/32768, 21141/21125, 3627/3625, [[140625/140608]] and 45349632/45287125.

761edo is [[consistent]] to the [[9-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[420175/419904]] and {{monzo| 3 13 -15 4 }} in the [[7-limit]]. The equal temperament is strong in the 2.3.5.13.29.31 [[subgroup]], tempering out 32805/32768, 21141/21125, 3627/3625, [[140625/140608]] and 45349632/45287125.

=== Odd harmonics ===

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== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" |[[Subgroup]]

|-

! rowspan="2" |[[Comma list~~|Comma List~~]]

! rowspan="2" | [[Subgroup]]

! rowspan="2" |[[Mapping]]

! rowspan="2" | [[Comma list]]

! rowspan="2" |Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | [[Mapping]]

! colspan="2" |Tuning ~~Error~~

! rowspan="2" | Optimal 8ve stretch (¢)

|-

! colspan="2" | Tuning error

![[TE error|Absolute]] (¢)

|-

![[TE simple badness|Relative]] (%)

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{monzo|-1206 761}}

| {{mapping|761 1206}}

| 0.0778

| +0.0778

| 0.0778

| 4.93

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| 32805/32768, {{monzo|69 81 -85}}

| {{mapping|761 1206 1767}}

| 0.0490

| +0.0490

| 0.0755

| 4.79

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| 32805/32768, 420175/419904, {{monzo|3 13 -15 4}}

| {{mapping|761 1206 1767 2136}}

| 0.0925

| +0.0925

| 0.0998

| 6.33

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| 498.292

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|}

<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

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

== Music ==

; [[Francium]]

* "shortcrust" from ''wiloliquy'' (2025) – [https://open.spotify.com/track/7eX41UXoJhmlzFVe3hWP7d Spotify] | [https://francium223.bandcamp.com/track/shortcrust Bandcamp] | [https://www.youtube.com/watch?v=0vPnUo6X9IY YouTube]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|761}}
+{{ED intro}}
 == Theory ==
-edo is [[consistent]] to the [[9-odd-limit]]. It [[tempers out]] [[32805/32768]], [[420175/419904]] and {{monzo|3 13 -15 4}} in the [[7-limit]]. The equal temperament is strong in the 2.3.5.13.29.31 [[subgroup]], tempering out 32805/32768, 21141/21125, 3627/3625, [[140625/140608]] and 45349632/45287125.
+edo is [[consistent]] to the [[9-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[420175/419904]] and {{monzo| 3 13 -15 4 }} in the [[7-limit]]. The equal temperament is strong in the 2.3.5.13.29.31 [[subgroup]], tempering out 32805/32768, 21141/21125, 3627/3625, [[140625/140608]] and 45349632/45287125.
 === Odd harmonics ===
@@ Line 13: / Line 13: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+|-
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Comma list]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | [[Mapping]]
-! colspan="2" |Tuning Error
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-|-
+! colspan="2" | Tuning error
-![[TE error|Absolute]] (¢)
+|-
-![[TE simple badness|Relative]] (%)
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
 | {{monzo|-1206 761}}
 | {{mapping|761 1206}}
-| 0.0778
+| +0.0778
 | 0.0778
 | 4.93
@@ Line 32: / Line 33: @@
 | 32805/32768, {{monzo|69 81 -85}}
 | {{mapping|761 1206 1767}}
-| 0.0490
+| +0.0490
 | 0.0755
 | 4.79
@@ Line 39: / Line 40: @@
 | 32805/32768, 420175/419904, {{monzo|3 13 -15 4}}
 | {{mapping|761 1206 1767 2136}}
-| 0.0925
+| +0.0925
 | 0.0998
 | 6.33
@@ Line 46: / Line 47: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+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*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 57: / Line 59: @@
 | 498.292
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |}
+<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
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+== Music ==
+; [[Francium]]
+* "shortcrust" from ''wiloliquy'' (2025) – [https://open.spotify.com/track/7eX41UXoJhmlzFVe3hWP7d Spotify] | [https://francium223.bandcamp.com/track/shortcrust Bandcamp] | [https://www.youtube.com/watch?v=0vPnUo6X9IY YouTube]