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 the fact that it equates 14/11 with 9/7 and 11/8 with 15/11 could potentially be considered too much tuning damage. 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 the fact that it equates 14/11 with 9/7 and 11/8 with 15/11 could potentially be considered too much tuning damage. 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, 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 | 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]]. 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. | ||
One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (such as [[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. | One step of 31edo, measuring about 38.7¢, is called a [[diesis]] because it stands in for several intervals called "dieses" (such as [[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. | ||
31edo is the 11th [[prime numbers|prime]] | 31edo is the 11th [[prime numbers|prime]] EDO, following [[29edo]] and coming before [[37edo]]. | ||
== Intervals == | == Intervals == | ||
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[[Category:Books]] | [[Category:Books]] | ||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Golden]] | [[Category:Golden meantone]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||
[[Category:Meantone]] | [[Category:Meantone]] | ||