576edo - Revision history

MisterShafXen: Fixed typo

2026-04-10T15:45:55Z

Fixed typo

← Older revision		Revision as of 15:45, 10 April 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4\|5]] is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the [[septimal ennealimma]], ~~assining~~ [[7/6]] to 2\9, as well as {{monzo\| 99 -66 0 2 }}, {{monzo\| 110 -57 0 -7 }} , and {{monzo\| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val\| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.		576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4\|5]] is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the [[septimal ennealimma]], assigning [[7/6]] to 2\9, as well as {{monzo\| 99 -66 0 2 }}, {{monzo\| 110 -57 0 -7 }} , and {{monzo\| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val\| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.

	In higher limits, the 2.3.7 subgroup can be used with optional additions of [[19/16\|19]] or [[29/16\|29]], or fractional subgroups using [[13/10]].		In higher limits, the 2.3.7 subgroup can be used with optional additions of [[19/16\|19]] or [[29/16\|29]], or fractional subgroups using [[13/10]].

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"

2026-03-13T13:31:58Z

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 13:31, 13 March 2026
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	\| [[Atomic]] (576)		\| [[Atomic]] (576)
	\|}		\|}
	<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 22:54, 20 February 2025

2025-02-20T22:54:04Z

← Older revision		Revision as of 22:54, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|576~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 17:03, 15 January 2025

2025-01-15T17:03:49Z

← Older revision		Revision as of 17:03, 15 January 2025
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 576 factors as {{~~Factorization~~\|576}}, 576edo has subset edos {{EDOs\| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs\|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a [[Highly composite equal division#Highly factorable numbers\|highly factorable]] edo.		Since 576 factors as {{factorization\|576}}, 576edo has subset edos {{EDOs\| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs\|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a [[Highly composite equal division#Highly factorable numbers\|highly factorable]] edo.

	[[1152edo]], which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.		[[1152edo]], which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.
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	\| [[Atomic]] (576)		\| [[Atomic]] (576)
	\|}		\|}
	<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

2024-11-16T13:21:22Z

Partial undo

← Older revision		Revision as of 13:21, 16 November 2024
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	== Regular temperament properties ==		== Regular temperament properties ==
	=== 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
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	\| 4/3<br />(32805/32768)		\| 4/3<br />(32805/32768)
	\| [[Atomic]] (576)		\| [[Atomic]] (576)
	~~{{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 01:30, 16 November 2024

2024-11-16T01:30:57Z

← Older revision		Revision as of 01:30, 16 November 2024
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	== Theory ==		== Theory ==
	576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4\|5]] is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the [[septimal ennealimma]], assining [[7/6]] to 2\9, as well as {{monzo\| 99 -66 0 2 }}, {{monzo\| 110 -57 0 -7 }} , and {{monzo\| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val\| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.		576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4\|5]] is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the [[septimal ennealimma]], assining [[7/6]] to 2\9, as well as {{monzo\| 99 -66 0 2 }}, {{monzo\| 110 -57 0 -7 }} , and {{monzo\| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val\| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.

	In higher limits, the 2.3.7 subgroup can be used with optional additions of [[19/16\|19]] or [[29/16\|29]], or fractional subgroups using [[13/10]].		In higher limits, the 2.3.7 subgroup can be used with optional additions of [[19/16\|19]] or [[29/16\|29]], or fractional subgroups using [[13/10]].
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	== Regular temperament properties ==		== Regular temperament properties ==

	=== 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
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	\|-		\|-
	\| 12		\| 12
	\| 239\576<br>(1\576)		\| 239\576<br />(1\576)
	\| 497.916<br>(2.083)		\| 497.916<br />(2.083)
	\| 4/3<br>(32805/32768)		\| 4/3<br />(32805/32768)
	\| [[Atomic]] (576)		\| [[Atomic]] (576)
	\|}		{{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}}

Eliora: /* Subsets and supersets */

2024-03-11T00:04:45Z

Subsets and supersets

← Older revision		Revision as of 00:04, 11 March 2024
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	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 576 factors as {{Factorization\|576}}, 576edo has subset edos {{EDOs\| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs\|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is ~~also~~ a [[Highly composite equal division#Highly factorable numbers\|highly factorable]] edo.		Since 576 factors as {{Factorization\|576}}, 576edo has subset edos {{EDOs\| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs\|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is a [[Highly composite equal division#Highly factorable numbers\|highly factorable]] edo.

			[[1152edo]], which is also a highly factorable edo, divides the edostep in two and corrects the mapping for 5.

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

Eliora: /* Theory */ cleanup

2024-02-11T12:46:45Z

Theory: cleanup

← Older revision		Revision as of 12:46, 11 February 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
	~~576~~ is ~~equal to 24 squared, which~~ in ~~itself is double~~ the ~~world~~-~~predominant~~ [[~~12edo~~]]. It ~~is known as a~~ [[~~Highly composite equal division #Highly factorable numbers\|highly factorable edo~~]], ~~which enables it~~ to ~~be played through JI~~-~~agnostic approaches that make use~~ of ~~its divisors (see~~ [[~~#Subsets~~ and ~~supersets~~]] ~~section below)~~. ~~This approach may be preferrable since the~~ [[~~patent val~~]] ~~will create sequences that fall aside by 1\576 of each other~~{{~~clarify~~}}, ~~which may not "live up to~~ the ~~spirit" of a composite number like 576~~.		576edo is [[consistent]] in the 7-odd-limit, though the error on harmonic [[5/4\|5]] is quite large. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. It tempers out the [[septimal ennealimma]], assining [[7/6]] to 2\9, as well as {{monzo\| 99 -66 0 2 }}, {{monzo\| 110 -57 0 -7 }} , and {{monzo\| 88 -75 0 11 }}. In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val\| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.

	~~Nonetheless, 576edo does offer simple interpretations. Despite having bad 5/4~~, ~~576edo is [[consistent]] in~~ the ~~7-odd-limit. As a corollary, 576edo is an excellent~~ 2.3.7 subgroup ~~tuning. Using the patent val, it tempers out the~~ [[~~septimal ennealimma]], 40353607~~/~~40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from [[9edo]]. However, other commas being tempered out are far more complex – {{monzo~~\| ~~99 -66 2 }}, {{monzo\| 110 -57 -7 }}, and {{monzo\| 88 -75 11 }}. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.~~		In higher limits, the 2.3.7 subgroup can be used with optional additions of [[19/16\|19]] or [[29/16\|29]], or fractional subgroups using [[13/10]].

	~~In the 5-limit, the patent val of 576edo [[support~~]]~~s the~~ [[~~atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val~~\| ~~576 913 1336 1618 }}, is a tuning for the [[garibaldi~~]] ~~temperament in the 7-limit. In addition~~, ~~in this case 5/4 comes from~~ [[~~72edo]], and 7~~/~~4 comes form 288edo.~~

	~~576edo supports a messed-up variant of the [[rectified hebrew~~]] scale<sup>[which?]</sup>, but with step hardness of 5:3 instead of 3:2, and in which 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.

	=== Prime harmonics ===		=== Prime harmonics ===
Line 15:		Line 11:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	576edo~~'s nontrivial divisors are~~ {{EDOs\| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}~~. Some~~ of ~~these have been put into practical use. 72edo has been used in~~ {{w\|~~Byzantine music\|Byzantine chanting~~}}, ~~has been theoreticized by {{w\|Alois Hába\|Alois Haba}} and [[Ivan Wyschnegradsky]]~~, and ~~has been used by jazz musician~~ [[~~Joe Maneri]]. 96edo has been used by [[Julian Carrillo~~]]~~. Because of the compositeness, it may be preferrable to make references to smaller edos instead of using the best approximation~~.		Since 576 factors as {{Factorization\|576}}, 576edo has subset edos {{EDOs\| 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}, of which {{EDOs\|12, 24, 72, and 96}} are particularly notable. Overall, 576edo contains a number of notable divisions that are multiples of 12, and it is also a [[Highly composite equal division#Highly factorable numbers\|highly factorable]] edo.

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

FloraC: Cleanup; clarify the title row of the rank-2 temp table; -redundant categories; and mark a few more things to clarify

2023-10-29T08:11:19Z

Cleanup; clarify the title row of the rank-2 temp table; -redundant categories; and mark a few more things to clarify

← Older revision		Revision as of 08:11, 29 October 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	576 is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is ~~what's~~ known as a [[~~oeis:A033833~~\|highly factorable edo]] ~~and is best~~ played through JI-agnostic approaches that make use of its divisors (see Subsets and supersets section below). This approach may be preferrable since the patent val will create sequences that fall aside by 1\576 of each other, which may not "live up to the spirit" of a composite number like 576.		576 is equal to 24 squared, which in itself is double the world-predominant [[12edo]]. It is known as a [[Highly composite equal division #Highly factorable numbers\|highly factorable edo]], which enables it to be played through JI-agnostic approaches that make use of its divisors (see [[#Subsets and supersets]] section below). This approach may be preferrable since the [[patent val]] will create sequences that fall aside by 1\576 of each other{{clarify}}, which may not "live up to the spirit" of a composite number like 576.

	Nonetheless, 576edo does offer simple interpretations. Despite having bad 5/4, 576edo is [[consistent]] in the 7-odd-limit. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. Using the patent val, it tempers out the [[septimal ennealimma]], 40353607/40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from [[9edo]]. However, other commas being tempered out are far more complex – {{monzo\| 99 -66 2 }}, {{monzo\| 110 -57 -7 }}, and {{monzo\| 88 -75 11 }}. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.		Nonetheless, 576edo does offer simple interpretations. Despite having bad 5/4, 576edo is [[consistent]] in the 7-odd-limit. As a corollary, 576edo is an excellent 2.3.7 subgroup tuning. Using the patent val, it tempers out the [[septimal ennealimma]], 40353607/40310784, and assigns 7/6 to 2\9 of the octave, property that ultimately derives from [[9edo]]. However, other commas being tempered out are far more complex – {{monzo\| 99 -66 2 }}, {{monzo\| 110 -57 -7 }}, and {{monzo\| 88 -75 11 }}. The associated rank-2 temperaments are 94 & 576, 41 & 535, and 229 & 347.

	In the 5-limit, 576edo ~~supports~~ the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val\| 576 913 1336 1618 }}, is a tuning for the [[~~Garibaldi temperament\|~~garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.		In the 5-limit, the patent val of 576edo [[support]]s the [[atomic]] temperament and the [[amity]] temperament. The 576c val supports [[maquila]]. The 576ccd val, {{val\| 576 913 1336 1618 }}, is a tuning for the [[garibaldi]] temperament in the 7-limit. In addition, in this case 5/4 comes from [[72edo]], and 7/4 comes form 288edo.

			576edo supports a messed-up variant of the [[rectified hebrew]] scale<sup>[which?]</sup>, but with step hardness of 5:3 instead of 3:2, and in which 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.

	576edo supports a messed-up version of the [[Rectified Hebrew]] scale, but with step hardness of 5:3 instead of 3:2, and from regular temperament theory perspective, 5/4 is reached via 359 third-tone generators down instead of 6 generators up. The relationship that 7/4 is 15 generators and 13/8 is 13 steps is still preserved.
	~~=== Subsets and supersets ===~~
	Its xenharmonic divisors (that is, besides 12edo and its subsets) are {{EDOs\| 8, 9, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}. Some of these have been put into practical use. 72edo has been used in [[wikipedia: Byzantine music\|Byzantine chanting]], has been theoreticized by [[wikipedia: Alois Hába\|Alois Haba]] and [[Ivan Wyschnegradsky]], and has been used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]]. Because of composition, it may be preferrable to make references to smaller edos instead of using the best approximation.
	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|576\|columns=11}}		{{Harmonics in equal\|576\|columns=11}}

	[[~~Category:Equal divisions~~ of the ~~octave\|###]]~~		=== Subsets and supersets ===
			576edo's nontrivial divisors are {{EDOs\| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, and 288 }}. Some of these have been put into practical use. 72edo has been used in {{w\|Byzantine music\|Byzantine chanting}}, has been theoreticized by {{w\|Alois Hába\|Alois Haba}} and [[Ivan Wyschnegradsky]], and has been used by jazz musician [[Joe Maneri]]. 96edo has been used by [[Julian Carrillo]]. Because of the compositeness, it may be preferrable to make references to smaller edos instead of using the best approximation.

	== Regular temperament properties ==		== Regular temperament properties ==
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	\|+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
	\|-		\|-
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	\| 339.583		\| 339.583
	\| 243/200		\| 243/200
	\| [[Amity]]		\| [[Amity]] (576)
	\|-		\|-
	\| 12		\| 12
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	\| 497.916<br>(2.083)		\| 497.916<br>(2.083)
	\| 4/3<br>(32805/32768)		\| 4/3<br>(32805/32768)
	\| [[Atomic]]		\| [[Atomic]] (576)
	\|}		\|}
			<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct
	<~~!-- 3-digit number --~~>


	[[~~Category:Amity~~]]
	[[~~Category:Atomic~~]]

BudjarnLambeth at 06:39, 9 July 2023

2023-07-09T06:39:56Z

← Older revision		Revision as of 06:39, 9 July 2023
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	~~{{novelty}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|576}}		{{EDO intro\|576}}