31edo: Difference between revisions

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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).

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.

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]].

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 {{Harmonics in equal|31|prec=2}}
-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).
+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.
 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]].