31edo: Difference between revisions

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== Theory ==

31edo splits the category of thirds into five different intervals: subminor, minor, neutral, major, supermajor. The minor and major thirds represent the 5-limit intervals [[6/5]] and 5/4, the subminor and supermajor thirds represent the 7-limit intervals [[7/6]] and [[9/7]], and the neutral third represents 11/9 and (slightly less accurately) 16/13. This overall relation is called [[myna]] temperament.

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]]. 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).

~~The neutral third doubles to~~ 31edo's [[3/2|perfect fifth]]~~, which~~ is flat of ~~3/2~~ by 5.2{{c}}~~. The major scale generated by this fifth contains the 5-limit major third~~, ~~and~~ as ~~such 31edo is~~ a tuning of [[meantone]] ~~and therefore supports conventional [[5-limit]] harmony~~. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 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.

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=== Stretched and compressed tunings ===

31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, 13~~, and [[4/3]]~~. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.

31edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.

229ed169 has an octave stretched by 2.23893{{c}}. Since the 13th harmonic is exactly halfway between 114 and 115 steps, this difference is the absolute maximum amount of octave stretch 31edo can tolerate before a discrepancy for the 13th harmonic occurs.

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== Relationship to 12edo ==

31edo’s [[circle of fifths|circle of 31 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This ~~makes sense to do~~ because ~~going up 12 steps~~ on the ~~31edo circle of fifths lands you at~~ the ~~note one diesis below your starting note~~, ~~so moving inward or outward on~~ the ~~spiral can be seen as diesis~~-~~sized adjustments to a 12edo~~ (~~more accurately meantone~~[12]) ~~note~~.

31edo’s [[circle of fifths|circle of 31 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This is possible because 18\31 is on the 7\12 kite in the [[scale tree]]. Stated another way, it is possible because the absolute value of 31edo’s [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) is 1.

This "spiral of fifths" can be a useful construct for introducing 31edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.

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 == Theory ==
-edo splits the category of thirds into five different intervals: subminor, minor, neutral, major, supermajor. The minor and major thirds represent the 5-limit intervals [[6/5]] and 5/4, the subminor and supermajor thirds represent the 7-limit intervals [[7/6]] and [[9/7]], and the neutral third represents 11/9 and (slightly less accurately) 16/13. This overall relation is called [[myna]] temperament.
+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]]. 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).
-The neutral third doubles to 31edo's [[3/2|perfect fifth]], which is flat of 3/2 by 5.2{{c}}. The major scale generated by this fifth contains the 5-limit major third, and as such 31edo is a tuning of [[meantone]] and therefore supports conventional [[5-limit]] harmony. The major third is less than a cent sharp of just [[5/4]], making it slightly sharp of [[quarter-comma meantone]]. 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.
@@ Line 30: / Line 28: @@
 === Stretched and compressed tunings ===
-edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, 13, and [[4/3]]. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.
+edo can benefit from slightly [[stretched and compressed tuning|stretching the octave]], especially when using it as an 11-limit equal temperament. With the right amount of stretch we can find a slightly better 3rd harmonic and significantly better 11th harmonic at the expense of somewhat less accurate approximations of 5, 7, and 13. Tunings such as [[80ed6]] and [[111ed12]] are great demonstrations of this.
 ed169 has an octave stretched by 2.23893{{c}}. Since the 13th harmonic is exactly halfway between 114 and 115 steps, this difference is the absolute maximum amount of octave stretch 31edo can tolerate before a discrepancy for the 13th harmonic occurs.
@@ Line 705: / Line 703: @@
 == Relationship to 12edo ==
-edo’s [[circle of fifths|circle of 31 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This makes sense to do because going up 12 steps on the 31edo circle of fifths lands you at the note one diesis below your starting note, so moving inward or outward on the spiral can be seen as diesis-sized adjustments to a 12edo (more accurately meantone[12]) note.
+edo’s [[circle of fifths|circle of 31 fifths]] can be bent into a [[spiral chart|12-spoked "spiral of fifths"]]. This is possible because 18\31 is on the 7\12 kite in the [[scale tree]]. Stated another way, it is possible because the absolute value of 31edo’s [[sharpness#dodeca-sharpness|dodeca-sharpness]] (edosteps per [[Pythagorean comma]]) is 1.
 This "spiral of fifths" can be a useful construct for introducing 31edo to musicians unfamiliar with microtonal music. It may help composers and musicians to make visual sense of the notation, and to understand what size of a jump is likely to land them where compared to 12edo.