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 {{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 supports 11-limit [[ennealimmal]] and its 13-limit extension ennealimmalis.

=== Prime harmonics ===

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| 2.3

| {{monzo| 1013 -639 }}

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

| {{mapping| 639 1013 }}

| -0.1238

| 0.1238

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| 2.3.5

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

| [{{~~val~~| 639 1013 1484 }}]

| {{mapping| 639 1013 1484 }}

| -0.1601

| 0.1134

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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

| 0.1062

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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

| 0.1020

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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

| 0.0955

Line 59:

| 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

| 0.0970

Line 66:

| 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 }}] (639h)

| {{mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h)

| -0.1618

| 0.0971

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|+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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| [[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

@@ Line 3: / Line 3: @@
 == 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 {{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 supports 11-limit [[ennealimmal]] and its 13-limit extension ennealimmalis.
 === Prime harmonics ===
@@ Line 24: / Line 24: @@
 | 2.3
 | {{monzo| 1013 -639 }}
-| [{{val| 639 1013 }}]
+| {{mapping| 639 1013 }}
 | -0.1238
 | 0.1238
@@ Line 31: / Line 31: @@
 | 2.3.5
 | {{monzo| 1 -27 18 }}, {{monzo| 55 -1 -23 }}
-| [{{val| 639 1013 1484 }}]
+| {{mapping| 639 1013 1484 }}
 | -0.1601
 | 0.1134
@@ Line 38: / Line 38: @@
 | 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
 | 0.1062
@@ Line 45: / Line 45: @@
 | 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
 | 0.1020
@@ Line 52: / Line 52: @@
 | 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
 | 0.0955
@@ Line 59: / Line 59: @@
 | 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
 | 0.0970
@@ Line 66: / Line 66: @@
 | 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 }}] (639h)
+| {{mapping| 639 1013 1484 1794 2211 2365 2612, 2715 }} (639h)
 | -0.1618
 | 0.0971
@@ Line 76: / Line 76: @@
 |+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 99: / Line 99: @@
 | [[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