Mystery comma
Ratio | 70368744177664/68630377364883 |
Factorization | 2^{46} × 3^{-29} |
Monzo | [46 -29⟩ |
Size in cents | 43.304975¢ |
Names | mystery comma, 29-comma, Pythagorean quadruple-diminished fourth, Pythagorean tridietic semilimma |
Color name | Wa-29, s^{4}w4 |
FJS name | [math]\text{dddd4}[/math] |
Special properties | reduced, reduced subharmonic |
Tenney height (log_{2} nd) | 91.9639 |
Weil height (log_{2} max(n, d)) | 92 |
Wilson height (sopfr (nd)) | 179 |
Harmonic entropy (Shannon, [math]\sqrt{nd}[/math]) |
~4.51083 bits |
Comma size | medium |
open this interval in xen-calc |
[46 -29⟩, the mystery comma or 29-comma of 43.305 cents, is the difference between 46 octaves and 29 fifths, in other words 2^{46}/3^{29}. When treated as an interval in its own right, there are actually two very good descriptive names for this interval. Firstly, and most obviously, it can be called the Pythagorean quadruple-diminished fourth, however, because it is also almost exactly half of the traditional Pythagorean limma in addition to being displaced from the unison's scale degree by three, one can also call this interval the Pythagorean tridietic semilimma.
Temperaments
Tempering out this comma splits the octave into 29 equal parts and maps the harmonic 3 to 17\29, leading to the 5-limit version of mystery temperament. For EDOs up to 400, the 29-comma is tempered out if and only if 29 divides it, for example 29edo, 58edo or 87edo.