31edo: Difference between revisions

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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. This makes 31edo 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]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.

Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[~~the Riemann zeta function and tuning #Zeta EDO lists|~~strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.

Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.

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.

In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds, which is much more than the two varieties in 12edo.

In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds each, which is much more than the two varieties available in 12edo.

=== Prime harmonics ===

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 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. This makes 31edo 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]]. It also maps most [[15-odd-limit]] intervals [[consistent]]ly, the exceptions being [[13/9]], [[13/11]], and their [[octave complement]]s.
-Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[the Riemann zeta function and tuning #Zeta EDO lists|strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.
+Other ways in which 31edo is especially accurate is that it represents a record in [[Pepper ambiguity]] in the [[7-odd-limit|7-]], [[9-odd-limit|9-]], and [[11-odd-limit]], which it is consistent through. It is also a [[strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.
 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.
-In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds, which is much more than the two varieties in 12edo.
+In terms of interval categories, because 31edo is a meantone system, the major and minor seconds, thirds, sixth, and sevenths on the chain of fifths are equated to [[5-limit]] intervals, those being [[16/15]], [[10/9]], [[6/5]], [[5/4]], and their [[octave complement]]s. 31edo maps the chromatic semitone to two steps, meaning there are "[[neutral (interval quality)|neutral]]" intervals between minor and major ones, which are not found in [[12edo]]. They can be represented by [[11-limit]] intervals, with [[11/10]]~[[12/11]] being a neutral second, and [[11/9]]~[[27/22]] a neutral third. One step in the other direction from the classical intervals are the subminor and supermajor intervals, which can be seen as intervals of prime [[7/1|7]]. The subminor second is [[21/20]]~[[28/27]], the supermajor second [[8/7]], the subminor third [[7/6]], and the supermajor third [[9/7]]~[[14/11]]. 31edo thus has five varieties of seconds and thirds each, which is much more than the two varieties available in 12edo.
 === Prime harmonics ===