31edo: Difference between revisions

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31edo's perfect fifth is flat of the just interval [[3/2]] by 5.2{{c}}, 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 most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.

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. 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 [[odd prime sum limit|11-odd-prime-sum-limit]]. It is also a prominent [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]].

Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents, however intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. 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 [[odd prime sum limit|11-odd-prime-sum-limit]]. It is also a prominent [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]].

One step of 31edo, measuring about 38.7{{c}}, 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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 edo's perfect fifth is flat of the just interval [[3/2]] by 5.2{{c}}, 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 most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.
-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. 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 [[odd prime sum limit|11-odd-prime-sum-limit]]. It is also a prominent [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]].
+Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Prime 11 is somewhat less accurate, making intervals like [[11/8]] off by about 9 cents, however intervals like [[11/9]] and [[11/6]] are approximated quite well because the errors cancel out. 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 [[odd prime sum limit|11-odd-prime-sum-limit]]. It is also a prominent [[the Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]].
 One step of 31edo, measuring about 38.7{{c}}, 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.