4320edo: Difference between revisions

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==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 ===

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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.

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 ====

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 ===

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}}.

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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:Atomic]]

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 {{Infobox ET}}
 {{EDO intro|4320}}
 ==Theory==
-edo 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]].
+edo 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 ===
@@ Line 9: / Line 10: @@
 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 ====
-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 ===
-edo 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]].
+edo 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}}.
@@ Line 117: / Line 118: @@
 edo 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:Atomic]]