19-comma

Interval information

Factorization	2-30 × 319
Monzo	[-30 19⟩
Size in cents	137.145¢
Names	19-comma, Pythagorean kleisma, Pythagorean inverse double-diminished second
Color name	L3w-2, trilawa negative 2nd
FJS name	[math]\text{dd}{-2}[/math]
Special properties	reduced, reduced harmonic
Tenney height (log2 nd)	60.1143
Weil height (log2 max(n, d))	60.2286
Wilson height (sopfr(nd))	117
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math])	~4.25307 bits
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The 19-comma, otherwise known as the Pythagorean kleisma (monzo: [-30 19⟩, ratio: 1162261467/1073741824), is an interval of about 137.1 ¢. It is the amount by which nineteen perfect fifths exceed eleven octaves, or (3/2)¹⁹/2¹¹. If used as an interval in its own right, it is the Pythagorean inverse double-diminished second. Treating it as a comma, tempering out this comma gives rise to graywood, which is supported by edos 19, 38, 57, and 76 in their patent vals.

Terminology

The term Pythagorean kleisma seems to be first used by Flora Canou in 2024, for this is the moskleisma of the Pythagorean diatonic scale, where kleisma (adjective: kleismic) refers to the inverse double-diminished 1-step i.e. |2L - 3s|. It is equated with the 5-limit kleisma of 15625/15552 along with many other intervals in meantone. It can also be reasoned as a fitting name as by tempering out this comma alongside the meantone comma (81/80), we get 19edo, which supports kleismic.

See also

• Large comma