643edo: Difference between revisions

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== Theory ==

643edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with a generally flat tendency, but the [[5/1|5th harmonic]] is only 0.000439 cents sharp as the denominator of a convergent to log25, after [[146edo|146]] and before [[4004edo|4004]]. It [[tempering out|tempers out]] [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], [[2431/2430]] and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.

643edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with a generally flat tendency, but the [[5/1|5th harmonic]] is only 0.000439 cents sharp as the denominator of a convergent to log25, after [[146edo|146]] and before [[4004edo|4004]]. As an equal temperament, it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], [[2431/2430]] and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.

=== Prime harmonics ===

=== Subsets and supersets ===

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== Regular temperament properties ==

{~~{comma basis begin}}~~

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

|-

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3

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| 0.0801

| 4.29

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

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

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

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| 498.29

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

~~{{rank~~-~~2 end}}~~

|}

~~{{orf}}~~

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

== Music ==

; [[Francium]]

* "Bobson Dugnutt" from ''Don't Give Your Kids These Names!'' (2025) − [https://open.spotify.com/track/1ROUQlzxJR7pDpM8GLujol Spotify] | [https://francium223.bandcamp.com/track/bobson-dugnutt Bandcamp] | [https://www.youtube.com/watch?v=Bg2w1__AW4k YouTube] − in Botolphic, 643edo tuning

[[Category:Sesquiquartififths]]

[[Category:Vili]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|643}}
+{{ED intro}}
 == Theory ==
-edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with a generally flat tendency, but the [[5/1|5th harmonic]] is only 0.000439 cents sharp as the denominator of a convergent to log<sub>2</sub>5, after [[146edo|146]] and before [[4004edo|4004]]. It [[tempering out|tempers out]] [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], [[2431/2430]] and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.
+edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with a generally flat tendency, but the [[5/1|5th harmonic]] is only 0.000439 cents sharp as the denominator of a convergent to log<sub>2</sub>5, after [[146edo|146]] and before [[4004edo|4004]]. As an equal temperament, it [[tempering out|tempers out]] [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], [[2431/2430]] and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.
 === Prime harmonics ===
-{{Harmonics in equal|643|columns=11}}
+{{Harmonics in equal|643}}
 === Subsets and supersets ===
@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
@@ Line 62: / Line 71: @@
 | 0.0801
 | 4.29
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 77: / Line 93: @@
 | 498.29
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Francium]]
+* "Bobson Dugnutt" from ''Don't Give Your Kids These Names!'' (2025) − [https://open.spotify.com/track/1ROUQlzxJR7pDpM8GLujol Spotify] | [https://francium223.bandcamp.com/track/bobson-dugnutt Bandcamp] | [https://www.youtube.com/watch?v=Bg2w1__AW4k YouTube] − in Botolphic, 643edo tuning
 [[Category:Sesquiquartififths]]
 [[Category:Vili]]