Misty comma

Interval information

Ratio	67108864/66430125
Factorization	226 × 3-12 × 5-3
Monzo	[26 -12 -3⟩
Size in cents	17.59885¢
Name	misty comma
Color name	ssg33, sasatrigu 3rd, Sasatrigu comma
FJS name	[math]\text{ddd3}_{5,5,5}[/math]
Special properties	reduced, reduced subharmonic
Tenney height (log2 nd)	51.9853
Weil height (log2 max(n, d))	52
Wilson height (sopfr(nd))	103
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~3.40939 bits
Comma size	small
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The misty comma, 67108864/66430125 = [26 -12 -3⟩, is a small comma of 17.599 cents. It is the amount by which a stack of three ptolemaic diminished fourths (512/405) exceed the octave. It can be written as (81/80)/(32805/32768)², (2048/2025)/(32805/32768), (128/125)/(531441/524288). Since these are commas of 12edo, so is the misty comma. It factors into simpler commas as (393216/390625)(1600000/1594323). However, the misty temperament, the 5-limit temperament tempering out the misty comma, is much more accurate than 12 equal can provide, and is also a comma of 87edo and 99edo. This temperament is notably in the schismic–Pythagorean equivalence continuum, with n = 3.

Etymology

The misty comma seems to have been named by Paul Erlich in 2002, referring to the fact that Carl Lumma "missed" this comma in a 5-limit comma search.