639edo - Revision history

FloraC: /* Theory */ expand

2026-05-19T09:19:03Z

Theory: expand

← Older revision		Revision as of 09:19, 19 May 2026
Line 3:		Line 3:

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

	=== Prime harmonics ===		=== Prime harmonics ===
Line 9:		Line 11:

	=== 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 ==

FloraC: Update

2026-05-19T08:49:32Z

Update

← Older revision		Revision as of 08:49, 19 May 2026
Line 17:		Line 17:
	! 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
	\|-		\|-
Line 24:		Line 24:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| 1013 -639 }}		\| {{Monzo\| 1013 -639 }}
	\| {{~~mapping~~\| 639 1013 }}		\| {{Mapping\| 639 1013 }}
	\| −0.1238		\| −0.1238
	\| 0.1238		\| 0.1238
Line 31:		Line 31:
	\|-		\|-
	\| 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
Line 39:		Line 39:
	\| 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
Line 46:		Line 46:
	\| 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
Line 53:		Line 53:
	\| 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
Line 60:		Line 60:
	\| 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
Line 67:		Line 67:
	\| 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
Line 77:		Line 77:
	\|+ 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
	\|-		\|-
Line 96:		Line 96:
	\|-		\|-
	\| 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]] / 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		<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

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:32:48Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:32, 13 March 2026
Line 101:		Line 101:
	\| [[Ennealimmal]] / ennealimmalis		\| [[Ennealimmal]] / ennealimmalis
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<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

ArrowHead294 at 23:09, 20 February 2025

2025-02-20T23:09:05Z

← Older revision		Revision as of 23:09, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|639~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 19:22, 15 January 2025

2025-01-15T19:22:09Z

← Older revision		Revision as of 19:22, 15 January 2025
Line 26:		Line 26:
	\| {{monzo\| 1013 -639 }}		\| {{monzo\| 1013 -639 }}
	\| {{mapping\| 639 1013 }}		\| {{mapping\| 639 1013 }}
	\| ~~−0~~.1238		\| −0.1238
	\| 0.1238		\| 0.1238
	\| 6.59		\| 6.59
Line 33:		Line 33:
	\| {{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
	\| 6.04		\| 6.04
Line 40:		Line 40:
	\| 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
	\| 5.65		\| 5.65
Line 47:		Line 47:
	\| 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
	\| 5.43		\| 5.43
Line 54:		Line 54:
	\| 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
	\| 5.08		\| 5.08
Line 61:		Line 61:
	\| 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
	\| 5.16		\| 5.16
Line 68:		Line 68:
	\| 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
Line 101:		Line 101:
	\| [[Ennealimmal]] / ennealimmalis		\| [[Ennealimmal]] / ennealimmalis
	\|}		\|}
	<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

ArrowHead294: Partial undo

2024-11-16T12:36:30Z

Partial undo

← Older revision		Revision as of 12:36, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~{comma basis begin}}~~		{\| class="wikitable center-4 center-5 center-6"
			\|-
			! rowspan="2" \| [[Subgroup]]
			! rowspan="2" \| [[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
Line 62:		Line 71:
	\| 0.0971		\| 0.0971
	\| 5.17		\| 5.17
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{{rank-2 ~~begin}}~~		{\| class="wikitable center-all left-5"
			\|+ style="font-size: 105%;" \| Table of rank-2 temperaments by generator
			\|-
			! Periods<br />per 8ve
			! Generator*
			! Cents*
			! Associated<br />ratio*
			! Temperaments
	\|-		\|-
	\| 1		\| 1
Line 84:		Line 100:
	\| 6/5<br />(36/35)		\| 6/5<br />(36/35)
	\| [[Ennealimmal]] / ennealimmalis		\| [[Ennealimmal]] / ennealimmalis
	~~{{rank-2 end}~~}		\|}
	~~{{orf}}~~		<nowiki />* [[Normal lists\|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct

ArrowHead294 at 04:20, 16 November 2024

2024-11-16T04:20:07Z

← Older revision		Revision as of 04:20, 16 November 2024
Line 12:		Line 12:

	== 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		\| −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		\| −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		\| −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		\| −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		\| −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		\| −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		\| −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}}

FloraC: Adopt template: Factorization; misc. cleanup

2023-11-02T13:02:50Z

Adopt template: Factorization; misc. cleanup

← Older revision		Revision as of 13:02, 2 November 2023
Line 3:		Line 3:

	== 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 ~~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 ===		=== Prime harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== 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 ==		== Regular temperament properties ==

FloraC: Cleanup; linking; clarify the title row of the rank-2 temp table

2023-10-25T06:35:34Z

Cleanup; linking; clarify the title row of the rank-2 temp table

← Older revision		Revision as of 06:35, 25 October 2023
Line 3:		Line 3:

	== Theory ==		== 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 ===		=== Prime harmonics ===
Line 24:		Line 24:
	\| 2.3		\| 2.3
	\| {{monzo\| 1013 -639 }}		\| {{monzo\| 1013 -639 }}
	\| [{{~~val~~\| 639 1013 }}]		\| {{mapping\| 639 1013 }}
	\| -0.1238		\| -0.1238
	\| 0.1238		\| 0.1238
Line 31:		Line 31:
	\| 2.3.5		\| 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		\| 0.1134
Line 38:		Line 38:
	\| 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 }}
	\| [{{~~val~~\| 639 1013 1484 1794 }}]		\| {{mapping\| 639 1013 1484 1794 }}
	\| -0.1369		\| -0.1369
	\| 0.1062		\| 0.1062
Line 45:		Line 45:
	\| 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
	\| [{{~~val~~\| 639 1013 1484 1794 2211 }}]		\| {{mapping\| 639 1013 1484 1794 2211 }}
	\| -0.1554		\| -0.1554
	\| 0.1020		\| 0.1020
Line 52:		Line 52:
	\| 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
	\| [{{~~val~~\| 639 1013 1484 1794 2211 2365 }}]		\| {{mapping\| 639 1013 1484 1794 2211 2365 }}
	\| -0.1650		\| -0.1650
	\| 0.0955		\| 0.0955
Line 59:		Line 59:
	\| 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
	\| [{{~~val~~\| 639 1013 1484 1794 2211 2365 2612 }}]		\| {{mapping\| 639 1013 1484 1794 2211 2365 2612 }}
	\| -0.1487		\| -0.1487
	\| 0.0970		\| 0.0970
Line 66:		Line 66:
	\| 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
	\| [{{~~val~~\| 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
Line 76:		Line 76:
	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(Reduced)~~		! Generator*
	! Cents~~<br>(Reduced)~~		! Cents*
	! Associated<br>Ratio		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
Line 99:		Line 99:
	\| [[Ennealimmal]] / ennealimmalis		\| [[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

FloraC: 5-limit commas

2023-10-03T13:36:53Z

5-limit commas

← Older revision		Revision as of 13:36, 3 October 2023
Line 3:		Line 3:

	== Theory ==		== 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 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.

	=== Prime harmonics ===		=== Prime harmonics ===