643edo: Difference between revisions

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{{Infobox ET
| Prime factorization = 643 (prime)
| Step size = 1.86625¢
| Fifth = 376\643 (701.71¢)
| Semitones = 60:49 (111.98¢ : 91.45¢)
| Consistency = 21
}}
{{EDO intro|643}}
{{EDO intro|643}}


== 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 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.


=== Prime harmonics ===
{{Harmonics in equal|643|columns=11}}
=== Miscellaneous properties ===
643edo is the 117th [[prime edo]].
643edo is the 117th [[prime edo]].


{{Harmonics in equal|643|columns=11}}
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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 }}
| [{{val| 643 1019 }}]
| +0.0771
| 0.0771
| 4.13
|-
| 2.3.5
| 32805/32768, {{monzo| 1 99 -68 }}
| [{{val| 643 1019 1493 }}]
| +0.0513
| 0.7270
| 3.90
|-
| 2.3.5.7
| 2401/2400, 32805/32768, {{monzo| 9 21 -17 -1 }}
| [{{val| 643 1019 1493 1805 }}]
| +0.0600
| 0.0647
| 3.47
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 32805/32768, 391314/390625
| [{{val| 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
| [{{val| 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
| [{{val| 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
| [{{val| 643 1019 1493 1805 2224 2379 2628 2731 }}]
| +0.1186
| 0.0801
| 4.29
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per Octave
! Generator<br>(Reduced)
! Cents<br>(Reduced)
! Associated<br>Ratio
! Temperaments
|-
| 1
| 94\643
| 175.43
| 448/405
| [[Sesquiquartififths]]
|-
| 1
| 267\643
| 498.29
| 4/3
| [[Helmholtz]]
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 16:17, 29 September 2022

← 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

Template:EDO intro

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 log25, after 146 and before 4004. 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)

Miscellaneous properties

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 Octave 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 94\643 175.43 448/405 Sesquiquartififths
1 267\643 498.29 4/3 Helmholtz
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