639edo: Difference between revisions

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== 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 [[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 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. It [[support]]s [[ennealimmal]] and its 13-limit extension enneabiotic.

=== Prime harmonics ===

=== ~~Miscellaneous properties~~ ===

=== Subsets and supersets ===

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

Since 639 factors into primes as {{nowrap| 32 × 71 }}, 639edo has subset edos {{EDOs| 3, 9, 71, and 213 }}.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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|-

| 2.3

| {{~~monzo~~| 1013 -639 }}

| {{Monzo| 1013 -639 }}

| [{{~~val~~| 639 1013 }}]

| {{Mapping| 639 1013 }}

| -0.1238

| −0.1238

| 0.1238

| 6.59

|-

| 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

| 6.04

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| 2.3.5.7

| 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

| 5.65

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| 2.3.5.7.11

| 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

| 5.43

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| 2.3.5.7.11.13

| 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

| 5.08

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| 2.3.5.7.11.13.17

| 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

| 5.16

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| 2.3.5.7.11.13.17.19

| 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

| 5.17

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=== Rank-2 temperaments ===

{| 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 per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

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| 18/17

| [[Quindro]]

|-

| 1

| 206\639

| 386.85

| 5/4

| [[Counterwürschmidt]]

|-

| 9

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| 315.49 (48.83)

| 6/5 (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

[[~~Category:Equal divisions of the octave|~~#~~##]] <!~~-- 3-~~digit number~~ -->

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|639}}
+{{ED intro}}
 == Theory ==
-edo 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 [[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.
+edo 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. It [[support]]s [[ennealimmal]] and its 13-limit extension enneabiotic.
 === Prime harmonics ===
 {{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 factors into primes as {{nowrap| 3<sup>2</sup> × 71 }}, 639edo has subset edos {{EDOs| 3, 9, 71, and 213 }}.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 23: / Line 26: @@
 |-
 | 2.3
-| {{monzo| 1013 -639 }}
+| {{Monzo| 1013 -639 }}
-| [{{val| 639 1013 }}]
+| {{Mapping| 639 1013 }}
-| -0.1238
+| −0.1238
 | 0.1238
 | 6.59
 |-
 | 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
 | 6.04
@@ Line 38: / Line 41: @@
 | 2.3.5.7
 | 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
 | 5.65
@@ Line 45: / Line 48: @@
 | 2.3.5.7.11
 | 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
 | 5.43
@@ Line 52: / Line 55: @@
 | 2.3.5.7.11.13
 | 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
 | 5.08
@@ Line 59: / Line 62: @@
 | 2.3.5.7.11.13.17
 | 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
 | 5.16
@@ Line 66: / Line 69: @@
 | 2.3.5.7.11.13.17.19
 | 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
 | 5.17
@@ Line 74: / Line 77: @@
 === Rank-2 temperaments ===
 {| 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)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 86: / Line 90: @@
 | 18/17
 | [[Quindro]]
+|-
+| 1
+| 206\639
+| 386.85
+| 5/4
+| [[Counterwürschmidt]]
 |-
 | 9
@@ Line 91: / Line 101: @@
 | 315.49<br>(48.83)
 | 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
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->