631edo: Difference between revisions

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== Theory ==

631edo is [[consistent]] to the [[9-odd-limit]], with all of the odd harmonics having a flat tendency. Using the [[patent val]], ~~the equal temperament~~ [[tempering out|tempers out]] [[~~4375~~/~~4374~~]], [[~~41503~~/~~41472~~]], [[~~32805~~/~~32768~~]] and [[~~12005~~/~~11979~~]] in the [[11-limit]]; [[1575/1573]]~~, 4375/4374~~, [[~~4459~~/~~4455~~]], [[~~4225~~/~~4224~~]] and 83349/83200 in the [[13-limit]].

631edo is [[consistent]] to the [[9-odd-limit]], with all of the odd harmonics having a flat tendency. Using the [[patent val]], it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[4375/4374]] in the 7-limit; [[12005/11979]] and [[41503/41472]] in the [[11-limit]]; [[1575/1573]], [[4225/4224]], [[4459/4455]], and 83349/83200 in the [[13-limit]].

=== Odd harmonics ===

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=== Subsets and supersets ===

631edo is the 115th [[prime ~~EDO~~]].

631edo is the 115th [[prime edo]].

== Intervals ==

== 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|-1000 631}}

| {{mapping|631 1000}}

| 0.0668

| +0.0668

| 0.0668

| 3.51

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| 32805/32768, {{monzo|-50 -71 70}}

| {{mapping|631 1000 1465}}

| 0.0818

| +0.0818

| 0.0585

| 3.08

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| 4375/4374, 32805/32768, 678223072849/675000000000

| {{mapping|631 1000 1465 1771}}

| 0.1361

| +0.1361

| 0.1067

| 5.61

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| 4375/4374, 41503/41472, 32805/32768, 12005/11979

| {{mapping|631 1000 1465 1771 2183}}

| 0.0980

| +0.0980

| 0.1221

| 6.42

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| 1575/1573, 4375/4374, 4459/4455, 4225/4224, 83349/83200

| {{mapping|631 1000 1465 1771 2183 2335}}

| 0.0797

| +0.0797

| 0.1187

| 6.24

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| 1225/1224, 1701/1700, 833/832, 1575/1573, 4459/4455, 4225/4224

| {{mapping|631 1000 1465 1771 2183 2335 2579}}

| 0.0809

| +0.0809

| 0.1099

| 5.78

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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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| 498.257

| 4/3

| [[Helmholtz]] / [[Pontiac]]

| [[Helmholtz (temperament)|Helmholtz]] / [[Pontiac]]

|}

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

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

== Music ==

; [[Francium]]

* "DUMPSTER FIRE" from ''CAPSLOCK'' (2024) – [https://open.spotify.com/track/7noSmEcwOfZpnFsiQ6vYNv Spotify] | [https://francium223.bandcamp.com/track/dumpster-fire Bandcamp] | [https://www.youtube.com/watch?v=PWP2o71Kqes YouTube]

* "Remember To Forget It." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/7h0mRNT7iS8L722566rzlJ Spotify] | [https://francium223.bandcamp.com/track/remember-to-forget-it Bandcamp] | [https://www.youtube.com/watch?v=Jwuqj4BJHwY YouTube]

* "Don't Have a Problem With You" from ''Don't'' (2025) – [https://open.spotify.com/track/2dBYarUaS8dxuHx91zGVEt Spotify] | [https://francium223.bandcamp.com/track/dont-have-a-problem-with-you Bandcamp] | [https://www.youtube.com/watch?v=bxuOZQwOO30 YouTube]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
-edo is [[consistent]] to the [[9-odd-limit]], with all of the odd harmonics having a flat tendency. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[4375/4374]], [[41503/41472]], [[32805/32768]] and [[12005/11979]] in the [[11-limit]]; [[1575/1573]], 4375/4374, [[4459/4455]], [[4225/4224]] and 83349/83200 in the [[13-limit]].
+edo is [[consistent]] to the [[9-odd-limit]], with all of the odd harmonics having a flat tendency. Using the [[patent val]], it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit; [[4375/4374]] in the 7-limit; [[12005/11979]] and [[41503/41472]] in the [[11-limit]]; [[1575/1573]], [[4225/4224]], [[4459/4455]], and 83349/83200 in the [[13-limit]].
 === Odd harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-edo is the 115th [[prime EDO]].
+edo is the 115th [[prime edo]].
+== Intervals ==
+{{Main|Table of 631edo intervals}}
 == 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|-1000 631}}
 | {{mapping|631 1000}}
-| 0.0668
+| +0.0668
 | 0.0668
 | 3.51
@@ Line 32: / Line 36: @@
 | 32805/32768, {{monzo|-50 -71 70}}
 | {{mapping|631 1000 1465}}
-| 0.0818
+| +0.0818
 | 0.0585
 | 3.08
@@ Line 39: / Line 43: @@
 | 4375/4374, 32805/32768, 678223072849/675000000000
 | {{mapping|631 1000 1465 1771}}
-| 0.1361
+| +0.1361
 | 0.1067
 | 5.61
@@ Line 46: / Line 50: @@
 | 4375/4374, 41503/41472, 32805/32768, 12005/11979
 | {{mapping|631 1000 1465 1771 2183}}
-| 0.0980
+| +0.0980
 | 0.1221
 | 6.42
@@ Line 53: / Line 57: @@
 | 1575/1573, 4375/4374, 4459/4455, 4225/4224, 83349/83200
 | {{mapping|631 1000 1465 1771 2183 2335}}
-| 0.0797
+| +0.0797
 | 0.1187
 | 6.24
@@ Line 60: / Line 64: @@
 | 1225/1224, 1701/1700, 833/832, 1575/1573, 4459/4455, 4225/4224
 | {{mapping|631 1000 1465 1771 2183 2335 2579}}
-| 0.0809
+| +0.0809
 | 0.1099
 | 5.78
@@ Line 67: / Line 71: @@
 === 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 78: / Line 83: @@
 | 498.257
 | 4/3
-| [[Helmholtz]] / [[Pontiac]]
+| [[Helmholtz (temperament)|Helmholtz]] / [[Pontiac]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Francium]]
+* "DUMPSTER FIRE" from ''CAPSLOCK'' (2024) – [https://open.spotify.com/track/7noSmEcwOfZpnFsiQ6vYNv Spotify] | [https://francium223.bandcamp.com/track/dumpster-fire Bandcamp] | [https://www.youtube.com/watch?v=PWP2o71Kqes YouTube]
+* "Remember To Forget It." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/7h0mRNT7iS8L722566rzlJ Spotify] | [https://francium223.bandcamp.com/track/remember-to-forget-it Bandcamp] | [https://www.youtube.com/watch?v=Jwuqj4BJHwY YouTube]
+* "Don't Have a Problem With You" from ''Don't'' (2025) – [https://open.spotify.com/track/2dBYarUaS8dxuHx91zGVEt Spotify] | [https://francium223.bandcamp.com/track/dont-have-a-problem-with-you Bandcamp] | [https://www.youtube.com/watch?v=bxuOZQwOO30 YouTube]
+[[Category:Listen]]