639edo: Difference between revisions

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== 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 ~~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 ===

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=== Subsets and supersets ===

Since 639 = ~~32 × 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 ==

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

| {{mapping| 639 1013 }}

| -0.1238

| −0.1238

| 0.1238

| 6.59

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| {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }}

| {{mapping| 639 1013 1484 }}

| -0.1601

| −0.1601

| 0.1134

| 6.04

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| 2401/2400, 4375/4374, {{monzo| 58 -14 -13 -2 }}

| {{mapping| 639 1013 1484 1794 }}

| -0.1369

| −0.1369

| 0.1062

| 5.65

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| 2401/2400, 4375/4374, 5632/5625, 161280/161051

| {{mapping| 639 1013 1484 1794 2211 }}

| -0.1554

| −0.1554

| 0.1020

| 5.43

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| 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449

| {{mapping| 639 1013 1484 1794 2211 2365 }}

| -0.1650

| −0.1650

| 0.0955

| 5.08

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| 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625

| {{mapping| 639 1013 1484 1794 2211 2365 2612 }}

| -0.1487

| −0.1487

| 0.0970

| 5.16

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| 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400

| {{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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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|-

| 9

| 168\639 (26\639)

| 168\639 (26\639)

| 315.49 (48.83)

| 315.49 (48.83)

| 6/5 (36/35)

| 6/5 (36/35)

| [[Ennealimmal]] / ennealimmalis

|}

<nowiki>*~~</nowiki>~~ [[Normal lists|~~octave~~-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|639}}
+{{ED intro}}
 == Theory ==
-edo 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 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 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 ===
@@ Line 9: / Line 9: @@
 === 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 ==
 {| 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 25: / Line 26: @@
 | {{monzo| 1013 -639 }}
 | {{mapping| 639 1013 }}
-| -0.1238
+| −0.1238
 | 0.1238
 | 6.59
@@ Line 32: / Line 33: @@
 | {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }}
 | {{mapping| 639 1013 1484 }}
-| -0.1601
+| −0.1601
 | 0.1134
 | 6.04
@@ Line 39: / Line 40: @@
 | 2401/2400, 4375/4374, {{monzo| 58 -14 -13 -2 }}
 | {{mapping| 639 1013 1484 1794 }}
-| -0.1369
+| −0.1369
 | 0.1062
 | 5.65
@@ Line 46: / Line 47: @@
 | 2401/2400, 4375/4374, 5632/5625, 161280/161051
 | {{mapping| 639 1013 1484 1794 2211 }}
-| -0.1554
+| −0.1554
 | 0.1020
 | 5.43
@@ Line 53: / Line 54: @@
 | 2080/2079, 2401/2400, 4375/4374, 5632/5625, 20480/20449
 | {{mapping| 639 1013 1484 1794 2211 2365 }}
-| -0.1650
+| −0.1650
 | 0.0955
 | 5.08
@@ Line 60: / Line 61: @@
 | 1156/1155, 2058/2057, 2080/2079, 2401/2400, 4375/4374, 5632/5625
 | {{mapping| 639 1013 1484 1794 2211 2365 2612 }}
-| -0.1487
+| −0.1487
 | 0.0970
 | 5.16
@@ Line 67: / Line 68: @@
 | 1156/1155, 1216/1215, 1445/1444, 2058/2057, 2080/2079, 2376/2375, 2401/2400
 | {{mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h)
-| -0.1618
+| −0.1618
 | 0.0971
 | 5.17
@@ Line 74: / Line 75: @@
 === 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
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 94: / Line 96: @@
 |-
 | 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
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct