Mystery comma

Interval information

Ratio	70368744177664/68630377364883
Factorization	246 × 3-29
Monzo	[46 -29⟩
Size in cents	43.30497¢
Names	mystery comma, 29-comma, Pythagorean quadruple-diminished fourth, Pythagorean tridietic semilimma
Color name	Wa-29, s4w4
FJS name	[math]\displaystyle{ \text{dddd4} }[/math]
Special properties	reduced, reduced subharmonic
Tenney height (log2 nd)	91.9639
Weil height (log2 max(n, d))	92
Wilson height (sopfr(nd))	179
Harmonic entropy (Shannon, [math]\displaystyle{ \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 2⁴⁶/3²⁹. 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 (a stack of two mystery commas differing from the limma by Mercator's comma) 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.

See also

• Medium comma
• 29th-octave temperaments