Phaotic comma

Interval information

Ratio	256000/255879
Factorization	211 × 3-9 × 53 × 13-1
Monzo	[11 -9 3 0 0 -1⟩
Size in cents	0.818472¢
Names	phaotic comma, phaotisma
Color name	s3uy31, Sathutriyo comma
FJS name	[math]\text{A1}^{5,5,5}_{13}[/math]
Special properties	reduced
Tenney height (log2 nd)	35.9309
Weil height (log2 max(n, d))	35.9316
Wilson height (sopfr(nd))	77
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~1.20606 bits
Comma size	unnoticeable
S-expression	S79 * S802
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The phaotic comma, or phaotisma, is an unnoticeable 13-limit (2.3.5.13 subgroup) comma with the ratio 256000/255879, and a size of approximately 0.82 ¢. It separates a stack of two syntonic commas from 40/39, or three 27/20 acute fourths from 16/13 (up an octave). It factors into (4096/4095)(4375/4374) or (6656/6561)(507/500)⁻¹.

Temperaments

Tempering it out splits 40/39 into two equal halves, each representing 81/80. It leads to a form of phaotismic temperament in the full 13-limit and phaotic temperament in the 2.3.5.13 subgroup.

Etymology

This comma was finally named by Aura in 2021 after several failed attempts. Its name comes from the Ancient Greek word pháos ("light")- a name that was chosen in part to reflect an association of certain intervals involved with brightness on the part of Osmium.

See also

• Unnoticeable comma