72edo: Difference between revisions

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== Theory ==

72edo approximates [[11-limit]] [[just intonation]] exceptionally well, is [[consistent]] in the [[17-odd-limit]], is the first [[trivial temperament|non-trivial]] [[edo]] to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]], and is the ~~ninth~~ [[~~zeta integral edo~~]]. 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.

72edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is [[consistent]] in the [[17-odd-limit]] and is the ninth [[zeta integral edo. It is the first [[trivial temperament|non-trivial]] [[edo]] to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]]. It is the first edo to be [[Minimal consistent EDOs|accurately consistent]]{{idiosyncratic}} and [[Minimal consistent EDOs|distinctly accurate]]{{idiosyncratic}} in the [[11-odd-limit]].

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.

72edo is the only regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s, ''and'' tunes the octave just. It is also an excellent tuning for [[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]].

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 == Theory ==
-edo approximates [[11-limit]] [[just intonation]] exceptionally well, is [[consistent]] in the [[17-odd-limit]], is the first [[trivial temperament|non-trivial]] [[edo]] to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]], and is the ninth [[zeta integral edo]]. 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.
+edo approximates [[11-limit]] [[just intonation]] exceptionally well. It is [[consistent]] in the [[17-odd-limit]] and is the ninth [[zeta integral edo. It is the first [[trivial temperament|non-trivial]] [[edo]] to be consistent in the 12- and 13-[[odd prime sum limit|odd-prime-sum-limit]]. It is the first edo to be [[Minimal consistent EDOs|accurately consistent]]{{idiosyncratic}} and [[Minimal consistent EDOs|distinctly accurate]]{{idiosyncratic}} in the [[11-odd-limit]].
+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.
 edo is the only regular temperament which treats harmonics 24 to 28 as being equidistant in pitch, splits [[25/24]] into two equal [[49/48]][[~]][[50/49]]'s, splits [[28/27]] into two equal [[55/54]]~[[56/55]]'s, ''and'' tunes the octave just. It is also an excellent tuning for [[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]].