4320edo - Revision history

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"

2026-03-13T13:32:42Z

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
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	\| [[Mercury]]		\| [[Mercury]]
	\|}		\|}
	<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

	== Miscellany ==		== Miscellany ==

FloraC: Update linking

2025-02-26T16:53:36Z

Update linking

← Older revision		Revision as of 16:53, 26 February 2025
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	[[Eliora]] proposes that 1 step of 4320edo be called a '''click''' as an [[interval size measure]]. This is because 4320 kilometers per hour equals 1200 meters per second, and "clicks" or "clicks" is a slang name for kilometers per hour. A [[cent]] is equal to 3.6 steps of 4320edo, just as 1 m/s = 3.6 km/h. For example, a perfect fifth is 701.955 cents. Since 701.955 m/s = 2527.038 km/h, this means that perfect fifth in 4320edo is 2527 steps. And checking the harmonics table, it does match the actual value.		[[Eliora]] proposes that 1 step of 4320edo be called a '''click''' as an [[interval size measure]]. This is because 4320 kilometers per hour equals 1200 meters per second, and "clicks" or "clicks" is a slang name for kilometers per hour. A [[cent]] is equal to 3.6 steps of 4320edo, just as 1 m/s = 3.6 km/h. For example, a perfect fifth is 701.955 cents. Since 701.955 m/s = 2527.038 km/h, this means that perfect fifth in 4320edo is 2527 steps. And checking the harmonics table, it does match the actual value.

	A [[semitone]] therefore is 360 clicks, a [[quartertone]] is 180 clicks, minutes period is 72 clicks, a [[morion]] is 60 clicks, mercury period is 54 clicks, the Dröbisch angle is 12 clicks.		A [[semitone (interval size measure)\|semitone]] therefore is 360 clicks, a [[quartertone]] is 180 clicks, minutes period is 72 clicks, a [[morion]] is 60 clicks, mercury period is 54 clicks, the Dröbisch angle is 12 clicks.

	Since 4320edo is consistent in the 23-odd-limit, this means that the values of the 23-odd-limit intervals in clicks can be found by simply applying the [[patent val]].		Since 4320edo is consistent in the 23-odd-limit, this means that the values of the 23-odd-limit intervals in clicks can be found by simply applying the [[patent val]].

ArrowHead294 at 23:12, 20 February 2025

2025-02-20T23:12:19Z

Show changes

BudjarnLambeth: Add links

2024-12-24T07:17:01Z

Add links

← Older revision		Revision as of 07:17, 24 December 2024
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	{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|4320}}		{{EDO intro\|4320}}

	==Theory==		==Theory==
	4320edo is distinctly consistent in the [[23-odd-limit]] and it is an excellent no-29s 37-limit tuning. While the consistency fact is not remarkable in its own right ([[282edo]] is the first such EDO), what's remarkable is the relationship that 4320edo offers to fractions of the octave, given that it is also a [[Highly composite equal division#Largely composite numbers\|largely composite EDO]]. It is the first largely composite EDO with a greater consistency limit since [[72edo]].		4320edo is [[distinctly consistent]] in the [[23-odd-limit]] and it is an excellent no-29s [[37-limit]] tuning. While the consistency fact is not remarkable in its own right ([[282edo]] is the first such EDO), what's remarkable is the relationship that 4320edo offers to fractions of the octave, given that it is also a [[Highly composite equal division#Largely composite numbers\|largely composite EDO]]. It is the first largely composite EDO with a greater consistency limit since [[72edo]].

	=== Subsets ===		=== Subsets ===
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	Out of the harmonics in the 23-limit approximated by 4320edo, only 3 and 5 have step sizes coprime with the number 4320. The 7th harmonic comes from [[135edo]], 11th harmonic comes from [[864edo]], 13th harmonic derives from [[2160edo]], 17th harmonic derives from [[80edo]], 19th harmonic derives from [[480edo]], and the 23rd harmonic comes from [[720edo]]. Beyond that, 31st harmonic comes from [[240edo]], and the 37th comes from 864edo.		Out of the harmonics in the 23-limit approximated by 4320edo, only 3 and 5 have step sizes coprime with the number 4320. The 7th harmonic comes from [[135edo]], 11th harmonic comes from [[864edo]], 13th harmonic derives from [[2160edo]], 17th harmonic derives from [[80edo]], 19th harmonic derives from [[480edo]], and the 23rd harmonic comes from [[720edo]]. Beyond that, 31st harmonic comes from [[240edo]], and the 37th comes from 864edo.

	Other notable divisors 4320edo has are [[12edo]], the dominant tuning system in the world today, [[15edo]], known for use by [[Easley Blackwood Jr]]., [[72edo]], which has found usage in Byzantine chanting and various other applications, [[96edo]] notable for its use by [[Julian Carrillo]], [[270edo]], notable for its excellent closed representation of the 13-limit relative to its size, [[360edo]], notable for being a number of degrees in a circle and carrying the interval size measure Dröbisch angle.		Other notable divisors 4320edo has are [[12edo]], the dominant tuning system in the world today, [[15edo]], known for use by [[Easley Blackwood Jr]]., [[72edo]], which has found usage in Byzantine chanting and various other applications, [[96edo]] notable for its use by [[Julian Carrillo]], [[270edo]], notable for its excellent closed representation of the [[13-limit]] relative to its size, [[360edo]], notable for being a number of degrees in a circle and carrying the interval size measure ''Dröbisch angle''.

	==== Proposal for an interval size measure ====		==== Proposal for an interval size measure ====
	Eliora proposes that 1 step of 4320edo be called a '''click'''. This is because 4320 kilometers per hour equals 1200 meters per second, and "clicks" or "clicks" is a slang name for kilometers per hour. A [[cent]] is equal to 3.6 steps of 4320edo, just as 1 m/s = 3.6 km/h. For example, a perfect fifth is 701.955 cents. Since 701.955 m/s = 2527.038 km/h, this means that perfect fifth in 4320edo is 2527 steps. And checking the harmonics table, it does match the actual value.		[[Eliora]] proposes that 1 step of 4320edo be called a '''click''' as an [[interval size measure]]. This is because 4320 kilometers per hour equals 1200 meters per second, and "clicks" or "clicks" is a slang name for kilometers per hour. A [[cent]] is equal to 3.6 steps of 4320edo, just as 1 m/s = 3.6 km/h. For example, a perfect fifth is 701.955 cents. Since 701.955 m/s = 2527.038 km/h, this means that perfect fifth in 4320edo is 2527 steps. And checking the harmonics table, it does match the actual value.

	A semitone therefore is 360 clicks, a quartertone is 180 clicks, minutes period is 72 clicks, a [[morion]] is 60 clicks, mercury period is 54 clicks, the Dröbisch angle is 12 clicks.		A [[semitone]] therefore is 360 clicks, a [[quartertone]] is 180 clicks, minutes period is 72 clicks, a [[morion]] is 60 clicks, mercury period is 54 clicks, the Dröbisch angle is 12 clicks.

	Since 4320edo is consistent in the 23-odd-limit, this means that the values of the 23-odd-limit intervals in clicks can be found by simply applying the patent val.		Since 4320edo is consistent in the 23-odd-limit, this means that the values of the 23-odd-limit intervals in clicks can be found by simply applying the [[patent val]].

	=== Regular temperament theory ===		=== Regular temperament theory ===
	4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the atomic temperament, it supports period-60 temperament [[minutes]]. It also provides the optimal patent val for the period-80 temperament [[mercury]].		4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the [[atomic]] temperament, it supports period-60 temperament [[minutes]]. It also provides the optimal patent val for the period-80 temperament [[mercury]].

	In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them. In the 13-limit, it tempers out [[6656/6655]], 67392/67375, 151263/151250. In the 17-limit, it tempers out [[12376/12375]], 14400/14399, 28561/28560, and also commas associated with 80edo, such as [[80-17-comma]] and 80-11/10-comma, that is {{monzo\|-91 0 -80 0 80}}.		In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them. In the 13-limit, it tempers out [[6656/6655]], 67392/67375, 151263/151250. In the 17-limit, it tempers out [[12376/12375]], 14400/14399, 28561/28560, and also commas associated with 80edo, such as [[80-17-comma]] and 80-11/10-comma, that is {{monzo\|-91 0 -80 0 80}}.
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	4320edo is the 69th highly abundant EDO. Nice.		4320edo is the 69th highly abundant EDO. Nice.

	When it comes to interval size measures, a curious observation is also that 4320 km/h is close enough to whole integer to equal to 2684 mph, and [[2684edo]] is a zeta peak EDO.		When it comes to interval size measures, a curious observation is also that 4320 km/h is close enough to whole integer to equal to 2684 mph, and [[2684edo]] is a [[zeta]] peak EDO.
	[[Category:Equal divisions of the octave\|####]]		[[Category:Equal divisions of the octave\|####]]
	[[Category:Atomic]]		[[Category:Atomic]]

BudjarnLambeth at 07:07, 9 July 2023

2023-07-09T07:07:22Z

← Older revision		Revision as of 07:07, 9 July 2023
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	~~{{novelty}}~~{{Infobox ET}}		{{Infobox ET}}
	{{EDO intro\|4320}}		{{EDO intro\|4320}}
	==Theory==		==Theory==

BudjarnLambeth at 04:25, 9 July 2023

2023-07-09T04:25:22Z

← Older revision		Revision as of 04:25, 9 July 2023
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	{{Infobox ET}}		{{novelty}}{{Infobox ET}}
	{{EDO intro\|4320}}		{{EDO intro\|4320}}
	==Theory==		==Theory==

Eliora: /* Regular temperament theory */

2023-05-27T00:20:43Z

Regular temperament theory

← Older revision		Revision as of 00:20, 27 May 2023
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	4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the atomic temperament, it supports period-60 temperament [[minutes]]. It also provides the optimal patent val for the period-80 temperament [[mercury]].		4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the atomic temperament, it supports period-60 temperament [[minutes]]. It also provides the optimal patent val for the period-80 temperament [[mercury]].

	In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them.		In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them. In the 13-limit, it tempers out [[6656/6655]], 67392/67375, 151263/151250. In the 17-limit, it tempers out [[12376/12375]], 14400/14399, 28561/28560, and also commas associated with 80edo, such as [[80-17-comma]] and 80-11/10-comma, that is {{monzo\|-91 0 -80 0 80}}.

	Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.		Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.

Eliora: /* Regular temperament theory */ this is the opv for 1\80 Hg

2023-05-04T19:57:38Z

Regular temperament theory: this is the opv for 1\80 Hg

← Older revision		Revision as of 19:57, 4 May 2023
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	=== Regular temperament theory ===		=== Regular temperament theory ===
	4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the atomic temperament, it supports period-60 temperament [[minutes]]. It also ~~supports~~ the period-80 temperament [[mercury]].		4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the atomic temperament, it supports period-60 temperament [[minutes]]. It also provides the optimal patent val for the period-80 temperament [[mercury]].

	In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them.		In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them.

	Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.		Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.

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

Eliora: /* Theory */ no mention of supersets so just subsets

2023-03-18T22:58:56Z

Theory: no mention of supersets so just subsets

← Older revision		Revision as of 22:58, 18 March 2023
Line 4:		Line 4:
	4320edo is distinctly consistent in the [[23-odd-limit]] and it is an excellent no-29s 37-limit tuning. While the consistency fact is not remarkable in its own right ([[282edo]] is the first such EDO), what's remarkable is the relationship that 4320edo offers to fractions of the octave, given that it is also a [[Highly composite equal division#Largely composite numbers\|largely composite EDO]]. It is the first largely composite EDO with a greater consistency limit since [[72edo]].		4320edo is distinctly consistent in the [[23-odd-limit]] and it is an excellent no-29s 37-limit tuning. While the consistency fact is not remarkable in its own right ([[282edo]] is the first such EDO), what's remarkable is the relationship that 4320edo offers to fractions of the octave, given that it is also a [[Highly composite equal division#Largely composite numbers\|largely composite EDO]]. It is the first largely composite EDO with a greater consistency limit since [[72edo]].

	=== ~~Divisors~~ ===		=== Subsets ===
	4320's divisors are {{EDOs\|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 108, 120, 135, 144, 160, 180, 216, 240, 270, 288, 360, 432, 480, 540, 720, 864, 1080, 1440, 2160}}. In addition to being largely composite, it is [[oeis:A002093\|highly abundant]] (although not superabundant)~~. It's~~ abundancy index is 2.5 = exactly 5/2.		4320's divisors are {{EDOs\|1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 32, 36, 40, 45, 48, 54, 60, 72, 80, 90, 96, 108, 120, 135, 144, 160, 180, 216, 240, 270, 288, 360, 432, 480, 540, 720, 864, 1080, 1440, 2160}}. In addition to being largely composite, it is [[oeis:A002093\|highly abundant]] (although not superabundant), with an abundancy index of 2.5 = exactly 5/2, as well as [[highly composite equal division#highly factorable numbers\|highly factorable EDO]], with a total of 382 ways of being split into subset EDOs.

	Out of the harmonics in the 23-limit approximated by 4320edo, only 3 and 5 have step sizes coprime with the number 4320. The 7th harmonic comes from [[135edo]], 11th harmonic comes from [[864edo]], 13th harmonic derives from [[2160edo]], 17th harmonic derives from [[80edo]], 19th harmonic derives from [[480edo]], and the 23rd harmonic comes from [[720edo]]. Beyond that, 31st harmonic comes from [[240edo]], and the 37th comes from 864edo.		Out of the harmonics in the 23-limit approximated by 4320edo, only 3 and 5 have step sizes coprime with the number 4320. The 7th harmonic comes from [[135edo]], 11th harmonic comes from [[864edo]], 13th harmonic derives from [[2160edo]], 17th harmonic derives from [[80edo]], 19th harmonic derives from [[480edo]], and the 23rd harmonic comes from [[720edo]]. Beyond that, 31st harmonic comes from [[240edo]], and the 37th comes from 864edo.

Eliora: /* Regular temperament theory */ odin is landscape + kalisma together

2023-03-09T21:29:59Z

Regular temperament theory: odin is landscape + kalisma together

← Older revision		Revision as of 21:29, 9 March 2023
Line 21:		Line 21:
	4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the atomic temperament, it supports period-60 temperament [[minutes]]. It also supports the period-80 temperament [[mercury]].		4320edo tempers out the [[Kirnberger's atom]], and aside from tuning the atomic temperament, it supports period-60 temperament [[minutes]]. It also supports the period-80 temperament [[mercury]].

	In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]]~~. It~~ is a tuning for the rank-3 temperament [[odin]].		In the 7-limit, 4320edo tempers out the [[landscape comma]], and in the 11-limit, the [[kalisma]], and as such it is a tuning for the rank-3 temperament [[odin]] tempering out both of them.

	Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.		Higher harmonics it represents well past the 23-limit are 31, 37, 47, 59, 61, 71.