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 [[~~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.

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 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]].

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]].

The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.

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=== Prime harmonics ===

{{Harmonics in equal|72|start=13|~~columns~~=12|title=Approximation of prime harmonics in 72edo (continued)}}

{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}

=== Subsets and supersets ===

@@ Line 14: / Line 14: @@
 == 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 [[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.
+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 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]].
+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]].
 The 13th harmonic (octave reduced) is so closely mapped on [[acoustic phi]] that 72edo could be treated as a 2.3.5.7.11.ϕ.17 temperament.
@@ Line 23: / Line 23: @@
 === Prime harmonics ===
-{{Harmonics in equal|72|columns=12}}
+{{Harmonics in equal|72|columns=9}}
-{{Harmonics in equal|72|start=13|columns=12|title=Approximation of prime harmonics in 72edo (continued)}}
+{{Harmonics in equal|72|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 72edo (continued)}}
 === Subsets and supersets ===