31edo: Difference between revisions
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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.) | 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.) | ||
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. | ||
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== MOS scales == | == MOS scales == | ||
The fact that 31edo has meantone scales is well-known, but some other [[MOS]]es and MOS chains are also useful: | |||
* 31edo's 9\31 neutral third generator generates [[Step ratio|ultrasoft]] [[3L 4s|mosh]] and [[Step ratio|superhard]] [[7L 3s|dicoid]] MOSes. | |||
* Its 12\31 generator generates a [[Step ratio|semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth closer to just. | |||
* A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo. | |||
* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[Step ratio|ultrasoft]] 9-tone orwelloid ([[4L 5s]]) MOS could be treated as a 9-tone well temperament. | |||
* It supports [[6edf]] (miracle) and [[9edf]] (Carlos alpha), fifth-equivalent equal temperaments that hit many good JI approximations. | |||
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