31edo: Difference between revisions

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Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning 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]], although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course).

Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning 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]], although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. (31edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through.) Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course).

Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et 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]].

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== As a regular temperament ==

~~31edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through.~~ In the [[13-limit]] it doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

In the [[13-limit]] 31edo doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.

=== Commas ===

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-Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning 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]], although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course).
+Each step is equivalent to a frequency ratio of the 31st root of 2, or 38.71 [[cents]]. 31's perfect fifth is flat of the just interval 3/2 (over five cents), as befits a tuning 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]], although the fact that it equates 14/11 with 9/7, and 11/8 with 15/11, may be too off for some. (31edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through.) Many [[7-limit]] JI scales are well-approximated in 31 (with tempering, of course).
 Because of these near-just values and because the 11th harmonic is almost twice as flat as the 3rd harmonic, 31-et 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]].
@@ Line 4,201: / Line 4,201: @@
 == As a regular temperament ==
-edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through. In the [[13-limit]] it doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
+In the [[13-limit]] 31edo doesn't do as well, but is the [[optimal patent val]] for the rank five temperament tempering out the 13-limit comma [[66/65]], which equates [[6/5]] and [[13/11]]. It also provides the optimal patent val for mohajira, squares and casablanca in the 11-limit and huygens/meantone, squares, winston, lupercalia and nightengale in the 13-limit.
 === Commas ===