31edo: Difference between revisions

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== ~~Basic theory~~ ==

== Theory ==

~~{{Harmonics in equal|31|prec=2|columns=18}}~~

31edo's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning [[support~~|supporting~~]] [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates 9/7 with 14/11 and 11/8 with 15/11. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 13/9, 13/11, 18/13, and 22/13.

31edo's perfect fifth is flat of the just interval [[3/2]] (over five cents), as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].

Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap ~~EDO~~, a zeta peak ~~EDO~~, and a zeta peak integer ~~EDO~~]], meaning it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|strict zeta ~~EDO~~]]. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[Trivial temperament|non-trivial]] ~~EDO~~ to be consistent in the 11-[[~~Odd~~ prime sum limit|odd-prime-sum-limit]].

Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo, a zeta peak edo, and a zeta peak integer edo]], meaning it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|strict zeta edo]]. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 11-[[odd prime sum limit|odd-prime-sum-limit]].

One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.

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31edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.

31edo is the 11th [[prime ~~numbers|prime~~]] ~~EDO~~, following [[29edo]] and coming before [[37edo]].

=== Prime harmonics ===

{{Harmonics in equal|31|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}

=== Subsets and supersets ===

31edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]].

== Intervals ==

@@ Line 11: / Line 11: @@
 {{Wikipedia| 31 equal temperament }}
-== Basic theory ==
+== Theory ==
-{{Harmonics in equal|31|prec=2|columns=18}}
-edo's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning [[support|supporting]] [[meantone]], but the major third is less than a cent sharp (of just 5/4), making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of 7/4, a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates 9/7 with 14/11 and 11/8 with 15/11. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of 13/9, 13/11, 18/13, and 22/13.
+edo's perfect fifth is flat of the just interval [[3/2]] (over five cents), as befits a tuning [[support]]ing [[meantone]], but the major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 31's approximation of [[7/4]], a cent flat, is also very close to just. It is a very tone-efficient melodic approximation of the [[11-limit]] (and specifically the [[11-odd-limit]]), although it conflates [[9/7]] with [[14/11]] and [[11/8]] with [[15/11]]. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course). It also maps all [[15-odd-limit]] intervals consistently, with the sole exceptions of [[13/9]], [[13/11]], [[18/13]], and [[22/13]].
-Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap EDO, a zeta peak EDO, and a zeta peak integer EDO]], meaning it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|strict zeta EDO]]. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[Trivial temperament|non-trivial]] EDO to be consistent in the 11-[[Odd prime sum limit|odd-prime-sum-limit]].
+Because of the near-just 5/4 and 7/4 and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31edo is relatively quite accurate and is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 6th zeta integral edo, the 7th zeta gap edo, a zeta peak edo, and a zeta peak integer edo]], meaning it is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|strict zeta edo]]. Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the 7-, 9-, and [[11-odd-limit]], which it is [[consistent]] through, and that it is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 11-[[odd prime sum limit|odd-prime-sum-limit]].
 One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (most notably, [[128/125]] and [[648/625]]) which are tempered out in [[12edo]]. The diesis is a defining sound of 31edo; when it does not appear directly in a scale, it often shows up as the difference between two or more intervals of a similar size. The diesis is demonstrated in [[SpiralProgressions]]. [[Zhea Erose]]'s 31edo music uses the interval frequently.
@@ Line 22: / Line 21: @@
 edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.
-edo is the 11th [[prime numbers|prime]] EDO, following [[29edo]] and coming before [[37edo]].
+=== Prime harmonics ===
+{{Harmonics in equal|31|columns=12}}
+{{Harmonics in equal|31|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 31edo (continued)}}
+=== Subsets and supersets ===
+edo is the 11th [[prime edo]], following [[29edo]] and coming before [[37edo]].
 == Intervals ==