31edo: Difference between revisions

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31edo 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).

31edo'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 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]]. (Another way 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.)

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 {{Odd harmonics in edo|edo=31}}
-edo 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).
+edo'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 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]]. (Another way 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.)