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== Theory ==
== Theory ==
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.  
639edo is [[consistency|distinctly consistent]] in the [[17-odd-limit]]. It has a sharp tendency, with [[harmonic]]s 3 to 17 all tuned sharp. The 639h [[val]] gives a reasonable approximation of [[19/1|harmonic 19]], in which the edo is almost consistent up to the [[25-odd-limit]], with the exception of [[19/16]] and [[25/16]] themselves and their [[octave complement]]s.
 
Using this val, the equal temperament [[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]]; and [[1288/1287]], [[1496/1495]] among others in the [[23-limit]]. It [[support]]s [[ennealimmal]] and its 13-limit extension enneabiotic.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|639|columns=11}}
{{Harmonics in equal|639|columns=11}}
{{Harmonics in equal|639|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 639edo (continued)}}


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 639 = {{factorization|639}}, it has subset edos {{EDOs| 3, 9, 71, and 213 }}.
Since 639 factors into primes as {{nowrap| 3<sup>2</sup> × 71 }}, 639edo has subset edos {{EDOs| 3, 9, 71, and 213 }}.


== Regular temperament properties ==
== Regular temperament properties ==
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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
| 5.17
| 5.17
|-
| 2.3.5.7.11.13.17.19.23
| 1156/1155, 1216/1215, 1288/1287, 1445/1444, 1496/1495, 2058/2057, 2080/2079, 2376/2375
| {{Mapping| 639 1013 1484 1794 2211 2365 2612 2715 2891 }} (639h)
| −0.1643
| 0.0919
| 4.89
|}
|}


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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)
| 26\639
| 315.49<br />(48.83)
| 48.83
| 6/5<br />(36/35)
| 36/35
| [[Ennealimmal]] / ennealimmalis
| [[Ennealimmal]] / enneabiotic
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
Retrieved from "https://en.xen.wiki/w/639edo"