41-comma
Ratio | 36 893 488 147 419 103 232 / 36 472 996 377 170 786 403 |
Factorization | 2^{65} × 3^{-41} |
Monzo | [65 -41⟩ |
Size in cents | 19.844965¢ |
Names | 41-comma, Pythagorean countercomma, countercomp comma |
Color name | Tribisawa 5th, Wa-41 |
FJS name | [math]\text{6d5}[/math] |
Special properties | reduced, reduced subharmonic |
Tenney height (log_{2} nd) | 129.983 |
Weil height (log_{2} max(n, d)) | 130 |
Wilson height (sopfr (nd)) | 253 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~3.73929 bits |
Comma size | small |
open this interval in xen-calc |
The 41-comma, Pythagorean countercomma, or countercomp comma (monzo: [65 -41⟩), is a 3-limit interval of 19.845 cents. It is the amount by which a stack of 41 perfect fifths (3/2) falls short of 24 octaves, in other words 2^{24}/(3/2)^{41}. It can also be described as the difference between a perfect fifth and a stack of six apotomes, or equivalently as the small gap between the Pythagorean triple-augmented unison and the triple-diminished fifth.
Temperaments
Tempering out this comma leads to the countercomp temperament, which splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. For equal divisions N up to 1230, the comma is tempered out if and only if 41 divides N. Examples are 41edo, 164edo, 205edo, 246edo, 328edo and 369edo. See countercomp family for a number of rank-2 temperaments where it is tempered out.
Terminology
The names were given by Flora Canou in 2022. Pythagorean countercomma was derived by analogy to Pythagorean comma as part of a series of Pythagorean interval names. Countercomp comma was derived from the temperament name, countercomp, which was changed from counterpyth as in earlier materials, where this comma was also called counterpyth comma. It was renamed after the convention was established that the temperament of the Pythagorean comma should be compton and never Pythagorean, for fear of confusion with Pythagorean tuning.