19-comma

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.

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]\displaystyle{ \text{dd}{-2} }[/math]
Special properties	reduced, reduced harmonic
Tenney norm (log2 nd)	60.1143
Weil norm (log2 max(n, d))	60.2286
Wilson norm (sopfr(nd))	117
Open this interval in xen-calc

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