624edo - Revision history

FloraC: /* Rank-2 temperaments */

2026-06-20T15:24:22Z

Rank-2 temperaments

← Older revision		Revision as of 15:24, 20 June 2026
Line 159:		Line 159:
	\| [[Iron]]		\| [[Iron]]
	\|}		\|}
	<nowiki/>* [[Normal forms #Minimal-generator form\|minimal-generator form]] ~~in parentheses if distinct~~		<nowiki/>* In [[Normal forms #Minimal-generator form\|minimal-generator form]]

	== Music ==		== Music ==

FloraC: /* Rank-2 temperaments */ + trivish. Also convert the table to minimal form since we're already reducing it to the first semi-octave

2026-06-20T15:24:07Z

Rank-2 temperaments: + trivish. Also convert the table to minimal form since we're already reducing it to the first semi-octave

← Older revision		Revision as of 15:24, 20 June 2026
Line 118:		Line 118:
	\|-		\|-
	\| 6		\| 6
	\| ~~177\624<br>(~~31\624)		\| 31\624
	\| ~~340.38<br>(~~59.62)		\| 59.62
	\| ~~162/133<br>(~~88/85)		\| 88/85
	\| [[Semiseptichrome]]		\| [[Semiseptichrome]]
			\|-
			\| 6
			\| 37\624
			\| 71.15
			\| 25/24
			\| [[Trivish]]
	\|-		\|-
	\| 12		\| 12
	\| ~~259\624<br>(~~1\624)		\| 1\624
	\| ~~498.08<br>(~~1.92)		\| 1.92
	\| ~~4/3<br>(~~32805/32768)		\| 32805/32768
	\| [[Atomic]]		\| [[Atomic]]
	\|-		\|-
	\| 13		\| 13
	\| ~~259\624<br>(~~19\624)		\| 19\624
	\| ~~498.08<br>(~~36.54)		\| 36.54
	\| ~~4/3<br>(~~?)		\| ?
	\| [[Aluminium]] (5-limit)		\| [[Aluminium]] (5-limit)
	\|-		\|-
	\| 16		\| 16
	\| ~~259\624<br>(~~14\624)		\| 14\624
	\| ~~498~~.08~~<br>(48.077)~~		\| 48.08
	\| ~~4/3<br>(~~?)		\| ?
	\| [[Sulfur]]		\| [[Sulfur]]
	\|-		\|-
	\| 24		\| 24
	\| ~~303\624<br>(~~9\624)		\| 9\624
	\| ~~582.69<br>(~~17.31)		\| 17.31
	\| ~~7/5<br>(~~100/99)		\| 100/99
	\| [[Chromium]]		\| [[Chromium]]
	\|-		\|-
	\| 26		\| 26
	\| ~~259\624<br>(~~19\624)		\| 19\624
	\| ~~498.08<br>(~~36.54)		\| 36.54
	\| ~~4/3<br>(~~?)		\| ?
	\| [[Iron]]		\| [[Iron]]
	\|}		\|}
	<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 #Minimal-generator form\|minimal-generator form]] in parentheses if distinct

	== Music ==		== Music ==

Overthink: Undo revision 232446 by Overthink (talk) nvm it changes the rank

2026-06-19T21:00:20Z

Undo revision 232446 by Overthink (talk) nvm it changes the rank

← Older revision		Revision as of 21:00, 19 June 2026
Line 5:		Line 5:
	624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].		624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].

	It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 no-7 [[~~restriction~~]] thereof.		It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 no-7 [[retraction]] thereof.

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

Overthink: /* Theory */ retractions are for subgroups, not temperaments

2026-06-19T20:59:42Z

Theory: retractions are for subgroups, not temperaments

← Older revision		Revision as of 20:59, 19 June 2026
Line 5:		Line 5:
	624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].		624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].

	It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 no-7 [[~~retraction~~]] thereof.		It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 no-7 [[restriction]] thereof.

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

FloraC: Corrections

2026-06-18T07:09:47Z

Corrections

← Older revision		Revision as of 07:09, 18 June 2026
Line 5:		Line 5:
	624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].		624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].

	It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 ~~2.3.5.11.13.17~~-[[~~subgroup]] [[restriction~~]] thereof.		It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 no-7 [[retraction]] thereof.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 20:		Line 20:
	! 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 27:		Line 27:
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{~~monzo~~\| -989 624 }}		\| {{Monzo\| -989 624 }}
	\| {{~~mapping~~\| 624 989 }}		\| {{Mapping\| 624 989 }}
	\| +0.0101		\| +0.0101
	\| 0.0101		\| 0.0101
Line 34:		Line 34:
	\|-		\|-
	\| 2.3.5		\| 2.3.5
	\| {{~~monzo~~\| 23 6 -14 }}, {{monzo\| -69 45 -1 }}		\| {{Monzo\| 23 6 -14 }}, {{monzo\| -69 45 -1 }}
	\| {{~~mapping~~\| 624 989 1449 }}		\| {{Mapping\| 624 989 1449 }}
	\| −0.0256		\| −0.0256
	\| 0.0510		\| 0.0510
Line 41:		Line 41:
	\|-		\|-
	\| 2.3.5.7		\| 2.3.5.7
	\| 250047/250000, 2460375/2458624, ~~{{monzo\| 27 0 -8 -3 }}~~		\| 250047/250000, 2460375/2458624, 134217728/133984375
	\| {{~~mapping~~\| 624 989 1449 1752 }}		\| {{Mapping\| 624 989 1449 1752 }}
	\| −0.0552		\| −0.0552
	\| 0.0678		\| 0.0678
Line 49:		Line 49:
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 9801/9800, 46656/46585, 131072/130977, 151263/151250		\| 9801/9800, 46656/46585, 131072/130977, 151263/151250
	\| {{~~mapping~~\| 624 989 1449 1752 2159 }}		\| {{Mapping\| 624 989 1449 1752 2159 }}
	\| −0.0792		\| −0.0792
	\| 0.0772		\| 0.0772
Line 56:		Line 56:
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325		\| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
	\| {{~~mapping~~\| 624 989 1449 1752 2159 2309 }}		\| {{Mapping\| 624 989 1449 1752 2159 2309 }}
	\| −0.0595		\| −0.0595
	\| 0.0831		\| 0.0831
Line 63:		Line 63:
	\| 2.3.5.7.11.13.17		\| 2.3.5.7.11.13.17
	\| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011		\| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
	\| {{~~mapping~~\| 624 989 1449 1752 2159 2309 2551 }}		\| {{Mapping\| 624 989 1449 1752 2159 2309 2551 }}
	\| −0.0795		\| −0.0795
	\| 0.0911		\| 0.0911
Line 70:		Line 70:
	\| 2.3.5.7.11.13.17.19		\| 2.3.5.7.11.13.17.19
	\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011		\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011
	\| {{~~mapping~~\| 624 989 1449 1752 2159 2309 2551 2651 }}		\| {{Mapping\| 624 989 1449 1752 2159 2309 2551 2651 }}
	\| −0.0861		\| −0.0861
	\| 0.0870		\| 0.0870
Line 77:		Line 77:
	\| 2.3.5.7.11.13.17.19.23		\| 2.3.5.7.11.13.17.19.23
	\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645		\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645
	\| {{~~mapping~~\| 624 989 1449 1752 2159 2309 2551 2651 2823 }}		\| {{Mapping\| 624 989 1449 1752 2159 2309 2551 2651 2823 }}
	\| −0.0906		\| −0.0906
	\| 0.0830		\| 0.0830
Line 87:		Line 87:
	\|+ 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 118:		Line 118:
	\|-		\|-
	\| 6		\| 6
	\| 177\624<br />(31\624)		\| 177\624<br>(31\624)
	\| 340.38<br />(59.62)		\| 340.38<br>(59.62)
	\| 162/133<br />(88/85)		\| 162/133<br>(88/85)
	\| [[Semiseptichrome]]		\| [[Semiseptichrome]]
	\|-		\|-
	\| 12		\| 12
	\| 259\624<br />(1\624)		\| 259\624<br>(1\624)
	\| 498.08<br />(1.92)		\| 498.08<br>(1.92)
	\| 4/3<br />(32805/32768)		\| 4/3<br>(32805/32768)
	\| [[Atomic]]		\| [[Atomic]]
	\|-		\|-
	\| 13		\| 13
	\| 259\624<br />(19\624)		\| 259\624<br>(19\624)
	\| 498.08<br />(36.54)		\| 498.08<br>(36.54)
	\| 4/3<br />(?)		\| 4/3<br>(?)
	\| [[Aluminium]] (5-limit)		\| [[Aluminium]] (5-limit)
	\|-		\|-
	\| 16		\| 16
	\| 259\624<br />(14\624)		\| 259\624<br>(14\624)
	\| 498.08<br />(48.077)		\| 498.08<br>(48.077)
	\| 4/3<br />(?)		\| 4/3<br>(?)
	\| [[Sulfur]]		\| [[Sulfur]]
	\|-		\|-
	\| 24		\| 24
	\| 303\624<br />(17\624)		\| 303\624<br>(9\624)
	\| 582.~~692~~<br />(32.~~692~~)		\| 582.69<br>(17.31)
	\| 7/5<br />(?)		\| 7/5<br>(100/99)
	\| [[Chromium]]		\| [[Chromium]]
	\|-		\|-
	\| 26		\| 26
	\| 259\624<br />(19\624)		\| 259\624<br>(19\624)
	\| 498.08<br />(36.54)		\| 498.08<br>(36.54)
	\| 4/3<br />(?)		\| 4/3<br>(?)
	\| [[Iron]]		\| [[Iron]]
	\|}		\|}

FloraC: /* Theory */ + second harmonics table

2026-06-05T16:18:13Z

Theory: + second harmonics table

← Older revision		Revision as of 16:18, 5 June 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], 3520/3519, and [[3888/3887]] in the [[23-limit]].		624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and [[151263/151250]] in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], [[3520/3519]], and [[3888/3887]] in the [[23-limit]].

	It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.		It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|624}}		{{Harmonics in equal\|624\|columns=11}}
			{{Harmonics in equal\|624\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 624edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 624 factors into 2<sup>4</sup> × 3 × 13, 624edo has subset edos {{EDOs\| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.		Since 624 factors into primes as {{nowrap\| 2<sup>4</sup> × 3 × 13 }}, 624edo has subset edos {{EDOs\| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.

	== Regular temperament properties ==		== Regular temperament properties ==

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"

2026-03-13T13:32:44Z

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 152:		Line 152:
	\| [[Iron]]		\| [[Iron]]
	\|}		\|}
	<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

	== Music ==		== Music ==

ArrowHead294 at 22:49, 20 February 2025

2025-02-20T22:49:38Z

← Older revision		Revision as of 22:49, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|624~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2025-01-17T16:29:36Z

Theory: cleanup

← Older revision		Revision as of 16:29, 17 January 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	624edo is [[consistent]] to the [[27-odd-limit]]. It [[tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], 3520/3519, and [[3888/3887]] in the [[23-limit]].		624edo is [[consistent]] to the [[27-odd-limit]]. As an equal temperament, it [[tempering out\|tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], 3520/3519, and [[3888/3887]] in the [[23-limit]].

	It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.		It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|624~~\|columns=11~~}}		{{Harmonics in equal\|624}}

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 624 factors into ~~{{factorization\|624}}~~, 624edo has subset edos {{EDOs\| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.		Since 624 factors into 2<sup>4</sup> × 3 × 13, 624edo has subset edos {{EDOs\| 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 156, and 312 }}.

	== Regular temperament properties ==		== Regular temperament properties ==

ArrowHead294 at 15:01, 16 January 2025

2025-01-16T15:01:58Z

← Older revision		Revision as of 15:01, 16 January 2025
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	== Theory ==		== Theory ==
	624edo is [[consistent]] to the [[27-odd-limit]]. ~~As an equal temperament, it~~ [[~~tempering out\|~~tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], 3520/3519, and [[3888/3887]] in the [[23-limit]].		624edo is [[consistent]] to the [[27-odd-limit]]. It [[tempers out]] {{monzo\| 23 6 -14 }} ([[vishnuzma]]) and {{monzo\| -69 45 -1 }} ([[counterschisma]]) in the [[5-limit]]; [[250047/250000]], [[2460375/2458624]], and 134217728/133984375 in the [[7-limit]]; [[9801/9800]], 46656/46585, [[131072/130977]], and 151263/151250 in the [[11-limit]]; [[1716/1715]], [[2080/2079]], [[4096/4095]], 34398/34375, and 39366/39325 in the [[13-limit]]; [[936/935]], [[1701/1700]], [[2025/2023]], and [[2058/2057]] in the [[17-limit]]; [[1521/1520]], [[2376/2375]], [[2432/2431]], and 3328/3325 in the [[19-limit]]; [[2024/2023]], [[2025/2024]], [[2646/2645]], 3520/3519, and [[3888/3887]] in the [[23-limit]].

	It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.		It provides an excellent [[optimal patent val]] for the rank-6 temperament tempering out 936/935, as well as the rank-5 2.3.5.11.13.17-[[subgroup]] [[restriction]] thereof.
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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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	\| {{monzo\| 23 6 -14 }}, {{monzo\| -69 45 -1 }}		\| {{monzo\| 23 6 -14 }}, {{monzo\| -69 45 -1 }}
	\| {{mapping\| 624 989 1449 }}		\| {{mapping\| 624 989 1449 }}
	\| ~~−0~~.0256		\| −0.0256
	\| 0.0510		\| 0.0510
	\| 2.65		\| 2.65
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	\| 250047/250000, 2460375/2458624, {{monzo\| 27 0 -8 -3 }}		\| 250047/250000, 2460375/2458624, {{monzo\| 27 0 -8 -3 }}
	\| {{mapping\| 624 989 1449 1752 }}		\| {{mapping\| 624 989 1449 1752 }}
	\| ~~−0~~.0552		\| −0.0552
	\| 0.0678		\| 0.0678
	\| 3.52		\| 3.52
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	\| 9801/9800, 46656/46585, 131072/130977, 151263/151250		\| 9801/9800, 46656/46585, 131072/130977, 151263/151250
	\| {{mapping\| 624 989 1449 1752 2159 }}		\| {{mapping\| 624 989 1449 1752 2159 }}
	\| ~~−0~~.0792		\| −0.0792
	\| 0.0772		\| 0.0772
	\| 4.02		\| 4.02
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	\| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325		\| 1716/1715, 2080/2079, 4096/4095, 34398/34375, 39366/39325
	\| {{mapping\| 624 989 1449 1752 2159 2309 }}		\| {{mapping\| 624 989 1449 1752 2159 2309 }}
	\| ~~−0~~.0595		\| −0.0595
	\| 0.0831		\| 0.0831
	\| 4.32		\| 4.32
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	\| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011		\| 936/935, 1701/1700, 1716/1715, 2025/2023, 4096/4095, 11016/11011
	\| {{mapping\| 624 989 1449 1752 2159 2309 2551 }}		\| {{mapping\| 624 989 1449 1752 2159 2309 2551 }}
	\| ~~−0~~.0795		\| −0.0795
	\| 0.0911		\| 0.0911
	\| 4.74		\| 4.74
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	\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011		\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2025/2023, 2376/2375, 11016/11011
	\| {{mapping\| 624 989 1449 1752 2159 2309 2551 2651 }}		\| {{mapping\| 624 989 1449 1752 2159 2309 2551 2651 }}
	\| ~~−0~~.0861		\| −0.0861
	\| 0.0870		\| 0.0870
	\| 4.53		\| 4.53
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	\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645		\| 936/935, 1521/1520, 1701/1700, 1716/1715, 2024/2023, 2025/2023, 2376/2375, 2646/2645
	\| {{mapping\| 624 989 1449 1752 2159 2309 2551 2651 2823 }}		\| {{mapping\| 624 989 1449 1752 2159 2309 2551 2651 2823 }}
	\| ~~−0~~.0906		\| −0.0906
	\| 0.0830		\| 0.0830
	\| 4.32		\| 4.32
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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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	\|-		\|-
	\| 6		\| 6
	\| 177\624<br>(31\624)		\| 177\624<br />(31\624)
	\| 340.38<br>(59.62)		\| 340.38<br />(59.62)
	\| 162/133<br>(88/85)		\| 162/133<br />(88/85)
	\| [[Semiseptichrome]]		\| [[Semiseptichrome]]
	\|-		\|-
	\| 12		\| 12
	\| 259\624<br>(1\624)		\| 259\624<br />(1\624)
	\| 498.08<br>(1.92)		\| 498.08<br />(1.92)
	\| 4/3<br>(32805/32768)		\| 4/3<br />(32805/32768)
	\| [[Atomic]]		\| [[Atomic]]
	\|-		\|-
	\| 13		\| 13
	\| 259\624<br>(19\624)		\| 259\624<br />(19\624)
	\| 498.08<br>(36.54)		\| 498.08<br />(36.54)
	\| 4/3<br>(?)		\| 4/3<br />(?)
	\| [[Aluminium]] (5-limit)		\| [[Aluminium]] (5-limit)
	\|-		\|-
	\| 16		\| 16
	\| 259\624<br>(14\624)		\| 259\624<br />(14\624)
	\| 498.08<br>(48.077)		\| 498.08<br />(48.077)
	\| 4/3<br>(?)		\| 4/3<br />(?)
	\| [[Sulfur]]		\| [[Sulfur]]
	\|-		\|-
	\| 24		\| 24
	\| 303\624<br>(17\624)		\| 303\624<br />(17\624)
	\| 582.692<br>(32.692)		\| 582.692<br />(32.692)
	\| 7/5<br>(?)		\| 7/5<br />(?)
	\| [[Chromium]]		\| [[Chromium]]
	\|-		\|-
	\| 26		\| 26
	\| 259\624<br>(19\624)		\| 259\624<br />(19\624)
	\| 498.08<br>(36.54)		\| 498.08<br />(36.54)
	\| 4/3<br>(?)		\| 4/3<br />(?)
	\| [[Iron]]		\| [[Iron]]
	\|}		\|}
	<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

	== Music ==		== Music ==
	; [[Eliora]]		; [[Eliora]]
	* [https://www.youtube.com/watch?v=vEDajIHqRUw~~&t=3s~~&pp=ygUGNjI0ZWRv ''Etude in Iron''] (2024)		* [https://www.youtube.com/watch?v=vEDajIHqRUw&pp=ygUGNjI0ZWRv ''Etude in Iron''] (2024)

	[[Category:Ainismic]]		[[Category:Ainismic]]
	[[Category:Listen]]		[[Category:Listen]]