82944/78125 - Revision history

Overthink: /* Temperaments */ structure

2026-06-08T04:23:26Z

Temperaments: structure

← Older revision		Revision as of 04:23, 8 June 2026
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	== Temperaments ==		== Temperaments ==

	[[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>.		[[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 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.		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.

Contribution at 23:20, 7 June 2026

2026-06-07T23:20:35Z

← Older revision		Revision as of 23:20, 7 June 2026
Line 15:		Line 15:
	[[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>.		[[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>.

	Using octave-equivalent [[patent val]]s in the [[5-limit]], the 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.		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.

	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.		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.

Contribution: /* Temperaments */

2026-06-07T23:19:07Z

Temperaments

← Older revision		Revision as of 23:19, 7 June 2026
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	== Temperaments ==		== 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>.

	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 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 ==		== Etymology ==

Contribution: Created page with "{{Infobox Interval | Ratio = 82944/78125 | Name = migmag, migmag comma | Comma = yes }} '''82944/78125''', proposed as the '''migmag''' or '''migmag comma''', is a large 5-limit comma measuring about 103.624 cents. It is the difference between four classic minor thirds and three classic major thirds: (6/5)⁴/(5/4)³ = 82944/78125. Equivalently, it has monzo {{monzo| 10 4 -7 }}. It may also be gene..."

2026-06-07T21:52:45Z

Created page with "{{Infobox Interval | Ratio = 82944/78125 | Name = migmag, migmag comma | Comma = yes }} '''82944/78125''', proposed as the '''migmag''' or '''migmag comma''', is a large 5-limit comma measuring about 103.624 cents. It is the difference between four classic minor thirds and three classic major thirds: (6/5)4/(5/4)3 = 82944/78125. Equivalently, it has monzo {{monzo| 10 4 -7 }}. It may also be gene..."

New page

{{Infobox Interval
| Ratio = 82944/78125
| Name = migmag, migmag comma
| Comma = yes
}}

'''82944/78125''', proposed as the '''migmag''' or '''migmag comma''', is a [[large comma|large]] [[5-limit]] [[comma]] measuring about 103.624 [[cent]]s. It is the difference between four [[6/5|classic minor thirds]] and three [[5/4|classic major thirds]]: ([[6/5]])4/([[5/4]])3 = 82944/78125. Equivalently, it has monzo {{monzo| 10 4 -7 }}.

It may also be generated by the following alternating chain of thirds: [[6/5]] up, [[5/4]] down, [[6/5]] up, [[5/4]] down, [[6/5]] up, [[5/4]] down, [[6/5]] up. Or in ratio form: ([[6/5]]) · ([[5/4]])−1 · ([[6/5]]) · ([[5/4]])−1 · ([[6/5]]) · ([[5/4]])−1 · ([[6/5]]) = 82944/78125.

Since ([[6/5]])4/([[5/4]])3 = (([[6/5]])4/[[2/1]]) · ([[2/1]]/([[5/4]])3), the migmag comma is also the product of the [[648/625|diminished comma]] and the [[128/125|augmented comma]]: [[648/625]] · [[128/125]] = 82944/78125.

== 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.

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.

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

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

== Etymology ==

The proposed name ''migmag'' is a portmanteau of '''''mi'''nor'', ''z'''ig''''', '''''ma'''jor'' and ''z'''ag''''', referring to the alternating minor-third-up and major-third-down zigzag chain construction of the comma.

== See also ==

* [[6/5]] – classic minor third
* [[5/4]] – classic major third
* [[648/625]] – diminished comma
* [[128/125]] – augmented comma
* [[Large comma]]

[[Category:Commas named for the intervals they stack]]