31edo: Difference between revisions

Line 15:

31edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. To get a better approximation to quarter-comma meantone requires going to much higher edo sizes: [[112edo]] (as 112b) to get a major third closer to just in absolute error, or [[174edo]] (as 174b) to get a major third closer to just in relative error. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).

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.

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. For example, it closely approximates the pentad 11:12:15:18:21, where the largest error occurs in the [[6/5]] minor third. This makes 31edo a very efficient 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 [[strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.

Line 22:

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 {{nowrap|[[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 {{nowrap|[[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.

Although 31edo approximates the 11th harmonic very poorly, it closely approximates the otonal pentad 11:12:15:18:21. The chord's largest pairwise tuning error occurs in the [[6/5]] minor, which is 5.96 cents flat. Its utonal counterpart, 1/(21:18:15:12:11), contains the same ascending pairwise interval ratios and therefore has the same tuning errors. In other octave voicings, this interval can appear as its octave complement, the [[5/3]] major sixth, which is correspondingly 5.96 cents sharp. Within an octave, the two pentads together yield 60 distinct triadic voicings, 40 distinct tetradic voicings, and 10 distinct pentadic voicings, counting inversions separately and removing duplicates under transposition.

=== Prime harmonics ===

@@ Line 15: / Line 15: @@
 edo's [[3/2|perfect fifth]] is flat of just by 5.2{{c}}, as befits a tuning of [[meantone]]. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. To get a better approximation to quarter-comma meantone requires going to much higher edo sizes: [[112edo]] (as 112b) to get a major third closer to just in absolute error, or [[174edo]] (as 174b) to get a major third closer to just in relative error. 31edo's approximation of [[7/4]], a cent flat, is also very close to just. Because of the near-just 5/4 and 7/4, 31edo is relatively quite accurate in the [[7-limit]]. Many 7-limit JI scales are well-approximated in 31 (with tempering, of course).
-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.
+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. For example, it closely approximates the pentad 11:12:15:18:21, where the largest error occurs in the [[6/5]] minor third. This makes 31edo a very efficient 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 [[strict zeta edo]], meaning that it is a zeta peak, zeta peak integer, zeta integral, and zeta gap edo all at once.
@@ Line 22: / Line 22: @@
 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 {{nowrap|[[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 {{nowrap|[[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.
-Although 31edo approximates the 11th harmonic very poorly, it closely approximates the otonal pentad 11:12:15:18:21. The chord's largest pairwise tuning error occurs in the [[6/5]] minor, which is 5.96 cents flat. Its utonal counterpart, 1/(21:18:15:12:11), contains the same ascending pairwise interval ratios and therefore has the same tuning errors. In other octave voicings, this interval can appear as its octave complement, the [[5/3]] major sixth, which is correspondingly 5.96 cents sharp. Within an octave, the two pentads together yield 60 distinct triadic voicings, 40 distinct tetradic voicings, and 10 distinct pentadic voicings, counting inversions separately and removing duplicates under transposition.
 === Prime harmonics ===

31edo: Difference between revisions

FloraC (talk | contribs)