643edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|643}}
{{ED intro}}


== Theory ==
== Theory ==
643edo is uniquely [[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 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 [[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 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 ===
=== Prime harmonics ===
{{Harmonics in equal|643|columns=11}}
{{Harmonics in equal|643}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
643edo is the 117th [[prime edo]].
643edo is the 117th [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 24: Line 25:
| 2.3
| 2.3
| {{monzo| -1019 643 }}
| {{monzo| -1019 643 }}
| [{{val| 643 1019 }}]
| {{mapping| 643 1019 }}
| +0.0771
| +0.0771
| 0.0771
| 0.0771
Line 31: Line 32:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 1 99 -68 }}
| 32805/32768, {{monzo| 1 99 -68 }}
| [{{val| 643 1019 1493 }}]
| {{mapping| 643 1019 1493 }}
| +0.0513
| +0.0513
| 0.7270
| 0.7270
Line 38: Line 39:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 32805/32768, {{monzo| 9 21 -17 -1 }}
| 2401/2400, 32805/32768, {{monzo| 9 21 -17 -1 }}
| [{{val| 643 1019 1493 1805 }}]
| {{mapping| 643 1019 1493 1805 }}
| +0.0600
| +0.0600
| 0.0647
| 0.0647
Line 45: Line 46:
| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 32805/32768, 391314/390625
| 2401/2400, 3025/3024, 32805/32768, 391314/390625
| [{{val| 643 1019 1493 1805 2224 }}]
| {{mapping| 643 1019 1493 1805 2224 }}
| +0.0927
| +0.0927
| 0.0874
| 0.0874
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1001/1000, 1716/1715, 3025/3024, 4225/4224, 32805/32768
| 1001/1000, 1716/1715, 3025/3024, 4225/4224, 32805/32768
| [{{val| 643 1019 1493 1805 2224 2379 }}]
| {{mapping| 643 1019 1493 1805 2224 2379 }}
| +0.1094
| +0.1094
| 0.0881
| 0.0881
Line 59: Line 60:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1001/1000, 1089/1088, 1701/1700, 1716/1715, 2601/2600, 4225/4224
| 1001/1000, 1089/1088, 1701/1700, 1716/1715, 2601/2600, 4225/4224
| [{{val| 643 1019 1493 1805 2224 2379 2628 }}]
|{{mapping| 643 1019 1493 1805 2224 2379 2628 }}
| +0.1094
| +0.1094
| 0.0816
| 0.0816
Line 66: Line 67:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 1001/1000, 1089/1088, 1521/1520, 1701/1700, 1716/1715, 1729/1728, 2601/2600
| 1001/1000, 1089/1088, 1521/1520, 1701/1700, 1716/1715, 1729/1728, 2601/2600
| [{{val| 643 1019 1493 1805 2224 2379 2628 2731 }}]
| {{mapping| 643 1019 1493 1805 2224 2379 2628 2731 }}
| +0.1186
| +0.1186
| 0.0801
| 0.0801
Line 74: Line 75:
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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 Octave
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 91: Line 93:
| 498.29
| 498.29
| 4/3
| 4/3
| [[Helmholtz]]
| [[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:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Sesquiquartififths]]
[[Category:Sesquiquartififths]]
[[Category:Vili]]
[[Category:Vili]]
[[Category:Prime EDO]]

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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