639edo: Difference between revisions

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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
m Update
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! rowspan="2" | [[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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 1013 -639 }}
| {{Monzo| 1013 -639 }}
| {{mapping| 639 1013 }}
| {{Mapping| 639 1013 }}
| −0.1238
| −0.1238
| 0.1238
| 0.1238
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }}
| {{Monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }}
| {{mapping| 639 1013 1484 }}
| {{Mapping| 639 1013 1484 }}
| −0.1601
| −0.1601
| 0.1134
| 0.1134
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 4375/4374, {{monzo| 58 -14 -13 -2 }}
| 2401/2400, 4375/4374, {{monzo| 58 -14 -13 -2 }}
| {{mapping| 639 1013 1484 1794 }}
| {{Mapping| 639 1013 1484 1794 }}
| −0.1369
| −0.1369
| 0.1062
| 0.1062
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 4375/4374, 5632/5625, 161280/161051
| 2401/2400, 4375/4374, 5632/5625, 161280/161051
| {{mapping| 639 1013 1484 1794 2211 }}
| {{Mapping| 639 1013 1484 1794 2211 }}
| −0.1554
| −0.1554
| 0.1020
| 0.1020
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449
| 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449
| {{mapping| 639 1013 1484 1794 2211 2365 }}
| {{Mapping| 639 1013 1484 1794 2211 2365 }}
| −0.1650
| −0.1650
| 0.0955
| 0.0955
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625
| 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625
| {{mapping| 639 1013 1484 1794 2211 2365 2612 }}
| {{Mapping| 639 1013 1484 1794 2211 2365 2612 }}
| −0.1487
| −0.1487
| 0.0970
| 0.0970
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400
| 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400
| {{mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h)
| {{Mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h)
| −0.1618
| −0.1618
| 0.0971
| 0.0971
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 9
| 9
| 168\639<br />(26\639)
| 168\639<br>(26\639)
| 315.49<br />(48.83)
| 315.49<br>(48.83)
| 6/5<br />(36/35)
| 6/5<br>(36/35)
| [[Ennealimmal]] / ennealimmalis
| [[Ennealimmal]] / enneabiotic
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Revision as of 08:49, 19 May 2026

← 638edo 639edo 640edo →
Prime factorization 32 × 71
Step size 1.87793 ¢ 
Fifth 374\639 (702.347 ¢)
Semitones (A1:m2) 62:47 (116.4 ¢ : 88.26 ¢)
Consistency limit 17
Distinct consistency limit 17

639 equal divisions of the octave (abbreviated 639edo or 639ed2), also called 639-tone equal temperament (639tet) or 639 equal temperament (639et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 639 equal parts of about 1.88 ¢ each. Each step represents a frequency ratio of 21/639, or the 639th root of 2.

Theory

639edo is distinctly consistent in the 17-odd-limit. It has a sharp tendency, with harmonics of 3 to 17 all tuned sharp. The 639h val gives a reasonable approximation of harmonic 19, where it tempers out [1 27 -18⟩ (ennealimma) and [55 -1 -23⟩ (counterwürschmidt comma) in the 5-limit; 2401/2400 and 4375/4374 in the 7-limit; 5632/5625 and 19712/19683 in the 11-limit; 2080/2079 and 4459/4455 in the 13-limit; 1156/1155, 2058/2057, and 2601/2600 in the 17-limit; 1216/1215, 1445/1444, 1540/1539, 2376/2375, and 2926/2925 in the 19-limit. It supports 11-limit ennealimmal and its 13-limit extension ennealimmalis.

Prime harmonics

Approximation of prime harmonics in 639edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.392 +0.541 +0.188 +0.795 +0.787 +0.209 -0.799 +0.834 -0.469 +0.504
Relative (%) +0.0 +20.9 +28.8 +10.0 +42.3 +41.9 +11.1 -42.6 +44.4 -25.0 +26.9
Steps
(reduced) 		639
(0) 		1013
(374) 		1484
(206) 		1794
(516) 		2211
(294) 		2365
(448) 		2612
(56) 		2714
(158) 		2891
(335) 		3104
(548) 		3166
(610)

Subsets and supersets

Since 639 = 32 × 71, it has subset edos 3, 9, 71, and 213.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [1013 -639⟩ [⟨639 1013]] −0.1238 0.1238 6.59
2.3.5 [1 -27 18⟩, [55 -1 -23⟩ [⟨639 1013 1484]] −0.1601 0.1134 6.04
2.3.5.7 2401/2400, 4375/4374, [58 -14 -13 -2⟩ [⟨639 1013 1484 1794]] −0.1369 0.1062 5.65
2.3.5.7.11 2401/2400, 4375/4374, 5632/5625, 161280/161051 [⟨639 1013 1484 1794 2211]] −0.1554 0.1020 5.43
2.3.5.7.11.13 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449 [⟨639 1013 1484 1794 2211 2365]] −0.1650 0.0955 5.08
2.3.5.7.11.13.17 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625 [⟨639 1013 1484 1794 2211 2365 2612]] −0.1487 0.0970 5.16
2.3.5.7.11.13.17.19 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400 [⟨639 1013 1484 1794 2211 2365 2612, 2715]] (639h) −0.1618 0.0971 5.17

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 53\639 99.53 18/17 Quindro
1 206\639 386.85 5/4 Counterwürschmidt
9 168\639
(26\639) 		315.49
(48.83) 		6/5
(36/35) 		Ennealimmal / enneabiotic

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

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