82944/78125: Difference between revisions

(2 intermediate revisions by one other user not shown)

Line 13:

== Temperaments ==

[[Tempering out]] the migmag comma equates a stack of four classic minor thirds with a stack of three classic major thirds. In ~~other words~~, the ~~amount by which four~~ [[6/5]] ~~minor thirds exceed an octave is made to cancel the amount by~~ which ~~three [[~~5/4~~]] major thirds fall short of an octave~~.

[[Tempering out]] the migmag comma equates a stack of four classic minor thirds with a stack of three classic major thirds. In terms of its [[monzo]] {{monzo| 10 4 -7 }}, this means imposing the relation: 210 · 34 = 57. It splits the [[3/2|perfect fifth]] into 7 semitones, three of which reach 6/5, and four semitones equals 5/4.

Using ~~patent~~-~~val approximations to~~ [[~~6/5~~]] ~~and~~ [[5/4]], the ~~nontrivial~~ [[edo]]s which ~~temper~~ out the migmag comma ~~are [[11edo]],~~ [[12edo]]~~, [[13edo]],~~ [[24edo]] and [[36edo]]~~. In [[11edo]], [[12edo]] and [[13edo]]~~, for ~~example, [[6/5]] maps to 3 steps and [[5/4]] maps to 4 steps, so four minor thirds and three major thirds both map to 12 steps~~.

Using octave-equivalent [[patent val]]s in the [[5-limit]], the only primitive [[edo]] which tempers out the migmag comma is [[12edo]]. Its multiples [[24edo]] and [[36edo]] also temper it out, but do not give new primitive edo temperaments for this comma.

~~For equal divisions of the~~ [[~~tritave~~]], ~~the corresponding~~ [[~~edt~~]]~~s are [[18edt]], [[19edt]], [[20edt]], [~~[~~21edt]],~~ [~~[37edt~~]], ~~[[38edt]], [[39edt]]~~ and ~~[[57edt]]~~.

In [[12edo]], [[6/5]] maps to 3 steps and [[5/4]] maps to 4 steps, so four minor thirds and three major thirds both map to 12 steps: 4 · 3 = 3 · 4.

~~Among odd equal divisions~~ of ~~[[4/1]], excluding~~ the ~~even cases equivalent to octave edos, the corresponding systems include~~ [[~~23ed4~~]], [[~~25ed4~~]], [[~~47ed4~~]] and [[~~49ed4~~]].

Thus the migmag comma is tempered out in 12edo by equating the chain of four minor thirds with the chain of three major thirds.

It should be noted that [[11edo]] and [[13edo]] may appear to close the same relation if one rounds the sizes of [[6/5]] and [[5/4]] directly as intervals, but they do not temper out 82944/78125 under the 5-limit patent-val mapping.

== Etymology ==

@@ Line 13: / Line 13: @@
 == Temperaments ==
-[[Tempering out]] the migmag comma equates a stack of four classic minor thirds with a stack of three classic major thirds. In other words, the amount by which four [[6/5]] minor thirds exceed an octave is made to cancel the amount by which three [[5/4]] major thirds fall short of an octave.
+[[Tempering out]] the migmag comma equates a stack of four classic minor thirds with a stack of three classic major thirds. In terms of its [[monzo]] {{monzo| 10 4 -7 }}, this means imposing the relation: 2<sup>10</sup> · 3<sup>4</sup> = 5<sup>7</sup>. It splits the [[3/2|perfect fifth]] into 7 semitones, three of which reach 6/5, and four semitones equals 5/4.
-Using patent-val approximations to [[6/5]] and [[5/4]], the nontrivial [[edo]]s which temper out the migmag comma are [[11edo]], [[12edo]], [[13edo]], [[24edo]] and [[36edo]]. In [[11edo]], [[12edo]] and [[13edo]], for example, [[6/5]] maps to 3 steps and [[5/4]] maps to 4 steps, so four minor thirds and three major thirds both map to 12 steps.
+Using octave-equivalent [[patent val]]s in the [[5-limit]], the only primitive [[edo]] which tempers out the migmag comma is [[12edo]]. Its multiples [[24edo]] and [[36edo]] also temper it out, but do not give new primitive edo temperaments for this comma.
-For equal divisions of the [[tritave]], the corresponding [[edt]]s are [[18edt]], [[19edt]], [[20edt]], [[21edt]], [[37edt]], [[38edt]], [[39edt]] and [[57edt]].
+In [[12edo]], [[6/5]] maps to 3 steps and [[5/4]] maps to 4 steps, so four minor thirds and three major thirds both map to 12 steps: 4 · 3 = 3 · 4.
-Among odd equal divisions of [[4/1]], excluding the even cases equivalent to octave edos, the corresponding systems include [[23ed4]], [[25ed4]], [[47ed4]] and [[49ed4]].
+Thus the migmag comma is tempered out in 12edo by equating the chain of four minor thirds with the chain of three major thirds.
+It should be noted that [[11edo]] and [[13edo]] may appear to close the same relation if one rounds the sizes of [[6/5]] and [[5/4]] directly as intervals, but they do not temper out 82944/78125 under the 5-limit patent-val mapping.
 == Etymology ==