639edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|639}}
{{ED intro}}


== Theory ==
== Theory ==
639edo is [[consistent]] to the [[17-odd-limit]], but the 639h val gives a reasonable approximation of harmonic 19, where it tempers out [[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.  
639edo is [[consistency|distinctly consistent]] in the [[17-odd-limit]]. It has a sharp tendency, with [[harmonic]]s of 3 to 17 all tuned sharp. The 639h [[val]] gives a reasonable approximation of [[19/1|harmonic 19]], where it [[tempering out|tempers out]] {{monzo| 1 27 -18 }} ([[ennealimma]]) and {{monzo| 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 [[support]]s 11-limit [[ennealimmal]] and its 13-limit extension ennealimmalis.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|639|columns=11}}
{{Harmonics in equal|639|columns=11}}


=== Miscellaneous properties ===
=== Subsets and supersets ===
Since 639 = 3<sup>2</sup> × 71, it has subset edos {{EDOs| 3, 9, 71, and 213 }}.  
Since 639 = {{factorization|639}}, it has subset edos {{EDOs| 3, 9, 71, and 213 }}.


== 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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 24: Line 25:
| 2.3
| 2.3
| {{monzo| 1013 -639 }}
| {{monzo| 1013 -639 }}
| [{{val| 639 1013 }}]
| {{mapping| 639 1013 }}
| -0.1238
| −0.1238
| 0.1238
| 0.1238
| 6.59
| 6.59
Line 31: Line 32:
| 2.3.5
| 2.3.5
| {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }}
| {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }}
| [{{val| 639 1013 1484 }}]
| {{mapping| 639 1013 1484 }}
| -0.1601
| −0.1601
| 0.1134
| 0.1134
| 6.04
| 6.04
Line 38: Line 39:
| 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 }}
| [{{val| 639 1013 1484 1794 }}]
| {{mapping| 639 1013 1484 1794 }}
| -0.1369
| −0.1369
| 0.1062
| 0.1062
| 5.65
| 5.65
Line 45: Line 46:
| 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
| [{{val| 639 1013 1484 1794 2211 }}]
| {{mapping| 639 1013 1484 1794 2211 }}
| -0.1554
| −0.1554
| 0.1020
| 0.1020
| 5.43
| 5.43
Line 52: Line 53:
| 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
| [{{val| 639 1013 1484 1794 2211 2365 }}]
| {{mapping| 639 1013 1484 1794 2211 2365 }}
| -0.1650
| −0.1650
| 0.0955
| 0.0955
| 5.08
| 5.08
Line 59: Line 60:
| 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
| [{{val| 639 1013 1484 1794 2211 2365 2612 }}]
| {{mapping| 639 1013 1484 1794 2211 2365 2612 }}
| -0.1487
| −0.1487
| 0.0970
| 0.0970
| 5.16
| 5.16
Line 66: Line 67:
| 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
| [{{val| 639 1013 1484 1794 2211 2365 2612, 2715 }}]
| {{mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h)
| -0.1618
| −0.1618
| 0.0971
| 0.0971
| 5.17
| 5.17
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 8ve
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 86: Line 88:
| 18/17
| 18/17
| [[Quindro]]
| [[Quindro]]
|-
| 1
| 206\639
| 386.85
| 5/4
| [[Counterwürschmidt]]
|-
|-
| 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]] / ennealimmalis
|}
|}
 
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Latest revision as of 23:09, 20 February 2025

← 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 / ennealimmalis

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

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