639edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.3
| {{monzo| 1013 -639 }}
| {{monzo| 1013 -639 }}
| {{mapping| 639 1013 }}
| {{mapping| 639 1013 }}
| -0.1238
| &minus;0.1238
| 0.1238
| 0.1238
| 6.59
| 6.59
Line 32: Line 24:
| {{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
| &minus;0.1601
| 0.1134
| 0.1134
| 6.04
| 6.04
Line 39: Line 31:
| 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
| &minus;0.1369
| 0.1062
| 0.1062
| 5.65
| 5.65
Line 46: Line 38:
| 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
| &minus;0.1554
| 0.1020
| 0.1020
| 5.43
| 5.43
Line 53: Line 45:
| 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
| &minus;0.1650
| 0.0955
| 0.0955
| 5.08
| 5.08
Line 60: Line 52:
| 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
| &minus;0.1487
| 0.0970
| 0.0970
| 5.16
| 5.16
Line 67: Line 59:
| 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
| &minus;0.1618
| 0.0971
| 0.0971
| 5.17
| 5.17
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 94: Line 80:
|-
|-
| 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
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 04:20, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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