72edo: Difference between revisions

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'''72~~-tone~~ equal ~~temperament~~''', or '''72-~~edo~~''', divides the octave into 72 steps or ''moria''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in ~~12-tone equal temperament~~. ~~72-tone~~ is also a superset of [[24edo~~|24-tone equal temperament~~]], a common and standard tuning of [[Arabic,~~_Turkish~~,~~_Persian~~|Arabic]] music, and has itself been used to tune Turkish music.

The '''72 equal divisions of the octave''' ('''72edo'''), or '''72-tone equal temperament''' ('''72tet''', '''72et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 72 steps or ''[[moria]]''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian|Arabic]] music, and has itself been used to tune Turkish music.

Composers that used ~~72-tone~~ include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.

Composers that used 72edo include [[Alois Hába]], [[Ivan Wyschnegradsky]], [[Julián Carillo]] (who is better associated with [[96edo]]), [[Iannis Xenakis]], [[Ezra Sims]], [[James Tenney]] and the jazz musician [[Joe Maneri]].

== Theory ==

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~~72-tone equal temperament~~ approximates [[11-limit]] [[just intonation]] exceptionally well, is consistent in the [[17-limit]], and is the ninth [[~~The_Riemann_Zeta_Function_and_Tuning~~ #Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in ~~12-tone~~, 72, 42 and 30 steps respectively, but the major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of ~~12-et~~ while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.

72edo approximates [[11-limit]] [[just intonation]] exceptionally well, is [[consistent]] in the [[17-limit]], and is the ninth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.

72 is an excellent tuning for [[~~Gamelismic_clan #Miracle|~~miracle ~~temperament~~]], especially the 11-limit version, and the related rank ~~three~~ temperament [[~~Marvel_family #Prodigy|~~prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].

72edo is an excellent tuning for the [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].

== Intervals ==

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! Degrees

! Cents

! Approximate Ratios ~~(17-limit)~~

! Approximate Ratios *

! colspan="3" | [[Ups and Downs Notation]]

|-

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| D

|}

<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament

Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors:

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 }}
-'''72-tone equal temperament''', or '''72-edo''', divides the octave into 72 steps or ''moria''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in 12-tone equal temperament. 72-tone is also a superset of [[24edo|24-tone equal temperament]], a common and standard tuning of [[Arabic,_Turkish,_Persian|Arabic]] music, and has itself been used to tune Turkish music.
+The '''72 equal divisions of the octave''' ('''72edo'''), or '''72-tone equal temperament''' ('''72tet''', '''72et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 72 steps or ''[[moria]]''. This produces a twelfth-tone tuning, with the whole tone measuring 200 cents, the same as in [[12edo]]. 72edo is also a superset of [[24edo]], a common and standard tuning of [[Arabic, Turkish, Persian|Arabic]] music, and has itself been used to tune Turkish music.
-Composers that used 72-tone include Alois Hába, Ivan Wyschnegradsky, Julián Carillo (who is better associated with [[96edo]]), Iannis Xenakis, Ezra Sims, James Tenney and the jazz musician Joe Maneri.
+Composers that used 72edo include [[Alois Hába]], [[Ivan Wyschnegradsky]], [[Julián Carillo]] (who is better associated with [[96edo]]), [[Iannis Xenakis]], [[Ezra Sims]], [[James Tenney]] and the jazz musician [[Joe Maneri]].
 == Theory ==
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 {{Primes in edo|72|columns=11}}
--tone equal temperament approximates [[11-limit]] [[just intonation]] exceptionally well, is consistent in the [[17-limit]], and is the ninth [[The_Riemann_Zeta_Function_and_Tuning #Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12-tone, 72, 42 and 30 steps respectively, but the major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12-et while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.
+edo approximates [[11-limit]] [[just intonation]] exceptionally well, is [[consistent]] in the [[17-limit]], and is the ninth [[The Riemann Zeta Function and Tuning #Zeta EDO lists|Zeta integral tuning]]. The octave, fifth and fourth are the same size as they would be in 12edo, 72, 42 and 30 steps respectively, but the classic major third ([[5/4]]) measures 23 steps, not 24, and other [[5-limit]] major intervals are one step flat of 12edo while minor ones are one step sharp. The septimal minor seventh ([[7/4]]) is 58 steps, while the undecimal semiaugmented fourth ([[11/8]]) is 33.
-is an excellent tuning for [[Gamelismic_clan #Miracle|miracle temperament]], especially the 11-limit version, and the related rank three temperament [[Marvel_family #Prodigy|prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
+edo is an excellent tuning for the [[miracle]] temperament, especially the 11-limit version, and the related rank-3 temperament [[prodigy]], and is a good tuning for other temperaments and scales, including [[wizard]], [[harry]], [[catakleismic]], [[compton]], [[unidec]] and [[tritikleismic]].
 == Intervals ==
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 ! Degrees
 ! Cents
-! Approximate Ratios (17-limit)
+! Approximate Ratios *
 ! colspan="3" | [[Ups and Downs Notation]]
 |-
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 | D
 |}
+<nowiki>*</nowiki> based on treating 72edo as a 17-limit temperament
 Combining ups and downs notation with [[Kite's_color_notation|color notation]], qualities can be loosely associated with colors: