643edo: Difference between revisions

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The '''643 equal division of the octave''' (643edo) divides the octave into 643 equal parts of 1.86625 cents each. It is uniquely [[consistent]] to the 21-limit, with a generally flat tendency, but the 5th harmonic is
{{Infobox ET}}
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 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 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 three 13-limit temperament [[Breed_family#Vili|vili temperament]].
{{ED intro}}


== 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 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 ===
643edo is the 117th [[prime edo]].
643edo is the 117th [[prime edo]].


[[Category:sesquiquartififths]]
== Regular temperament properties ==
[[Category:vili]]
{| class="wikitable center-4 center-5 center-6"
[[Category:Prime EDO]]
|-
! 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]]

Latest revision as of 02:30, 17 April 2025

← 642edo 643edo 644edo →
Prime factorization 643 (prime)
Step size 1.86625 ¢ 
Fifth 376\643 (701.711 ¢)
Semitones (A1:m2) 60:49 (112 ¢ : 91.45 ¢)
Consistency limit 21
Distinct consistency limit 21

643 equal divisions of the octave (abbreviated 643edo or 643ed2), also called 643-tone equal temperament (643tet) or 643 equal temperament (643et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 643 equal parts of about 1.87 ¢ each. Each step represents a frequency ratio of 21/643, or the 643rd root of 2.

Theory

643edo is distinctly consistent to the 21-odd-limit, with a generally flat tendency, but the 5th harmonic is only 0.000439 cents sharp as the denominator of a convergent to log25, after 146 and before 4004. As an equal temperament, it tempers out 32805/32768 in the 5-limit and 2401/2400 in the 7-limit, so that it supports 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

Approximation of prime harmonics in 643edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.244 +0.000 -0.241 -0.774 -0.714 -0.445 -0.779 +0.653 +0.594 +0.843
Relative (%) +0.0 -13.1 +0.0 -12.9 -41.5 -38.3 -23.9 -41.7 +35.0 +31.8 +45.2
Steps
(reduced) 		643
(0) 		1019
(376) 		1493
(207) 		1805
(519) 		2224
(295) 		2379
(450) 		2628
(56) 		2731
(159) 		2909
(337) 		3124
(552) 		3186
(614)

Subsets and supersets

643edo is the 117th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1019 643⟩ [⟨643 1019]] +0.0771 0.0771 4.13
2.3.5 32805/32768, [1 99 -68⟩ [⟨643 1019 1493]] +0.0513 0.7270 3.90
2.3.5.7 2401/2400, 32805/32768, [9 21 -17 -1⟩ [⟨643 1019 1493 1805]] +0.0600 0.0647 3.47
2.3.5.7.11 2401/2400, 3025/3024, 32805/32768, 391314/390625 [⟨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 [⟨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 [⟨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 [⟨643 1019 1493 1805 2224 2379 2628 2731]] +0.1186 0.0801 4.29

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 94\643 175.43 448/405 Sesquiquartififths
1 267\643 498.29 4/3 Helmholtz

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
  • "Bobson Dugnutt" from Don't Give Your Kids These Names! (2025) − Spotify | Bandcamp | YouTube − in Botolphic, 643edo tuning
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