643edo: Difference between revisions

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-The ''643 equal division'' divides the octave into 643 equal parts of 1.866 cents each. It is uniquely [[consistent|consistent]] to the 21-limit, with a generally flat tendency. It tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports [[Schismatic_family#Sesquiquartififths|sesquiquartififths temperament]]. In the 11-limit it tempers out 3025/3024 and | -1 3 -4 5 -2 &gt;; 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|optimal patent val]] for the rank three 13-limit temperament [[Breed_family#Vili|vili temperament]].
+{{Infobox ET}}
-[[Category:sesquiquartififths]]
+{{ED intro}}
-[[Category:vili]]
+== 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]]. 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}}
+=== Subsets and supersets ===
+edo is the 117th [[prime edo]].
+== Regular temperament properties ==
+{| 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
+| {{monzo| -1019 643 }}
+| {{mapping| 643 1019 }}
+| +0.0771
+| 0.0771
+| 4.13
+|-
+| 2.3.5
+| 32805/32768, {{monzo| 1 99 -68 }}
+| {{mapping| 643 1019 1493 }}
+| +0.0513
+| 0.7270
+| 3.90
+|-
+| 2.3.5.7
+| 2401/2400, 32805/32768, {{monzo| 9 21 -17 -1 }}
+| {{mapping| 643 1019 1493 1805 }}
+| +0.0600
+| 0.0647
+| 3.47
+|-
+| 2.3.5.7.11
+| 2401/2400, 3025/3024, 32805/32768, 391314/390625
+| {{mapping| 643 1019 1493 1805 2224 }}
+| +0.0927
+| 0.0874
+| 4.68
+|-
+| 2.3.5.7.11.13
+| 1001/1000, 1716/1715, 3025/3024, 4225/4224, 32805/32768
+| {{mapping| 643 1019 1493 1805 2224 2379 }}
+| +0.1094
+| 0.0881
+| 4.72
+|-
+| 2.3.5.7.11.13.17
+| 1001/1000, 1089/1088, 1701/1700, 1716/1715, 2601/2600, 4225/4224
+|{{mapping| 643 1019 1493 1805 2224 2379 2628 }}
+| +0.1094
+| 0.0816
+| 4.37
+|-
+| 2.3.5.7.11.13.17.19
+| 1001/1000, 1089/1088, 1521/1520, 1701/1700, 1716/1715, 1729/1728, 2601/2600
+| {{mapping| 643 1019 1493 1805 2224 2379 2628 2731 }}
+| +0.1186
+| 0.0801
+| 4.29
+|}
+=== Rank-2 temperaments ===
+{| 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
+| 94\643
+| 175.43
+| 448/405
+| [[Sesquiquartififths]]
+|-
+| 1
+| 267\643
+| 498.29
+| 4/3
+| [[Helmholtz (temperament)|Helmholtz]]
+|}
+<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]]