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).

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]]. 31edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through.

Because of the near-just 5/4 and 7/4 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]]. 31edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through.

31edo's 5\31 neutral third generator generates [[3L 4s|mosh]] and [[7L 3s|dicoid]] MOSes. Its 12\31 generator generates an [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8 and 5/4 better in tune.

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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).
-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]]. 31edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through.
+Because of the near-just 5/4 and 7/4 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]]. 31edo represents a record in [[Pepper ambiguity]] in the 7-, 9- and [[11-odd-limit]], which it is [[consistent]] through.
 edo's 5\31 neutral third generator generates [[3L 4s|mosh]] and [[7L 3s|dicoid]] MOSes. Its 12\31 generator generates an [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8 and 5/4 better in tune.