56edo: Difference between revisions

Line 3:

== Theory ==

56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the ~~"shrub region" between those of~~ [[17edo]] and [[~~22edo~~]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.

56edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.

56edo has unambiguous approximations to prime harmonics up to [[19/1|19]]. However, the harmonic [[3/1|3]] is quite sharp, leading harmonic [[9/1|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].

56edo has unambiguous approximations to prime harmonics up to [[19/1|19]], and possibly up to [[29/1|29]]. However, the harmonic [[3/1|3]] is quite sharp, leading harmonic [[9/1|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].

One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.

@@ Line 3: / Line 3: @@
 == Theory ==
-edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th in the "shrub region" between those of [[17edo]] and [[22edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.
+edo shares its near perfect quality of classical major third with [[28edo]], which it doubles, while also adding a superpythagorean 5th that is a convergent towards the [[Metallic harmonic series|bronze metallic mean]], following [[17edo]] and preceding [[185edo]]. Because it contains 28edo's major third and also has a step size very close to the syntonic comma, 56edo contains very accurate approximations of both the classic major third [[5/4]] and the Pythagorean major third [[81/64]]. Unfortunately, this "Pythagorean major third" is not the major third as is stacked by fifths in 56edo. However, this interval represents the pythagorean major third consistently in [[224edo]], which is the quadruple of 56edo.
-edo has unambiguous approximations to prime harmonics up to [[19/1|19]]. However, the harmonic [[3/1|3]] is quite sharp, leading harmonic [[9/1|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].
+edo has unambiguous approximations to prime harmonics up to [[19/1|19]], and possibly up to [[29/1|29]]. However, the harmonic [[3/1|3]] is quite sharp, leading harmonic [[9/1|9]] to be even more so, and causing intervals like [[10/9]], [[9/7]], and [[13/9]] to be inconsistent. Therefore, 56edo is not very popular compared to edos like [[53edo|53]] and [[58edo|58]].
 One step of 56edo is the closest direct approximation to the syntonic comma, [[81/80]], with the number of directly approximated syntonic commas per octave being 55.7976. (However, note that by [[patent val]] mapping, 56edo actually maps the syntonic comma inconsistently, to two steps.) [[Barium]] temperament realizes this proximity through regular temperament theory, and is supported by notable edos like [[224edo]], [[1848edo]], and [[2520edo]], which is a highly composite edo.