82944/78125

Interval information

Ratio	82944/78125
Factorization	210 × 34 × 5-7
Monzo	[10 4 -7⟩
Size in cents	103.624¢
Names	migmag, migmag comma
FJS name	[math]\displaystyle{ \text{ddd3}_{5,5,5,5,5,5,5} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	32.5933
Weil norm (log2 max(n, d))	32.6797
Wilson norm (sopfr(nd))	67
Comma size	large
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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⟩.

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 terms of its monzo [10 4 -7⟩, this means imposing the relation: 2¹⁰ · 3⁴ = 5⁷. It splits the perfect fifth into 7 semitones, three of which reach 6/5, and four semitones equals 5/4.

Using octave-equivalent patent vals 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.

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

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