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, 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]].

One step of 31edo, measuring about 38.7¢, 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.

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.

31edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L 3s) [[mos]] accurate 13:17:19 chords.

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* 12\31 generator generates a [[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 close to [[19/13]].

* 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 [[ultrasoft]] 9-tone orwelloid [[4L 5s]] MOS could be treated as a 9-tone well temperament.

* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0{{c}}) subminor third generator. The [[ultrasoft]] 9-tone orwelloid [[4L 5s]] MOS could be treated as a 9-tone well temperament.

* It has close approximations to [[6edf]] (→ [[miracle]]) and [[9edf]] (→ [[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.

@@ Line 16: / Line 16: @@
 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]].
-One step of 31edo, measuring about 38.7¢, 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.
+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.
 edo is close to a circle made by stacking 31 pure [[17/13]] subfourths. A circle of 31 pure 17/13's closes with an error of only 2.74 cents ([[relative error]] 7.1%). Remarkably, 31edo tempers out [[83521/83486]], the 0.7-cent difference between a stack of four 17/13's and a stack of one 19/13 and one 2/1, giving 31edo's [[oneirotonic]] (5L&nbsp;3s) [[mos]] accurate 13:17:19 chords.
@@ Line 713: / Line 713: @@
 * 12\31 generator generates a [[semihard]] [[oneirotonic]] scale, similar to the 5L&nbsp;3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth close to [[19/13]].
 * 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 [[ultrasoft]] 9-tone orwelloid [[4L&nbsp;5s]] MOS could be treated as a 9-tone well temperament.
+* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0{{c}}) subminor third generator. The [[ultrasoft]] 9-tone orwelloid [[4L&nbsp;5s]] MOS could be treated as a 9-tone well temperament.
 * It has close approximations to [[6edf]] (&rarr; [[miracle]]) and [[9edf]] (&rarr; [[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.