82944/78125
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 [10 4 -7⟩.
| Interval information |
migmag comma
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 diminished comma and the 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 edos 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 edts 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 minor, zig, major and zag, 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