643edo: Difference between revisions

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== 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]]. 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|columns=11}}


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


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| 2.3
| 2.3
| {{monzo| -1019 643 }}
| {{monzo| -1019 643 }}
| [{{val| 643 1019 }}]
| {{mapping| 643 1019 }}
| +0.0771
| +0.0771
| 0.0771
| 0.0771
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| 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
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| 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
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| 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
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| 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
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| 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
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per Octave
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Helmholtz]]
| [[Helmholtz]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


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

Revision as of 06:27, 25 October 2023

← 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 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. 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 it is distinct

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