Mystery comma

Interval information

Ratio	70368744177664/68630377364883
Factorization	246 × 3-29
Monzo	[46 -29⟩
Size in cents	43.30497¢
Names	29-comma, 29-3-comma
Color name	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
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[46 -29⟩, the 29-comma of 43.305 cents, is the difference between 46 octaves and 29 fifths, in other words 2⁴⁶/3²⁹. For EDOs up to 400, the 29-comma is tempered out if and only if 29 divides it, for example 29edo, 58edo or 87edo.

Temperaments

Tempering out the 29-comma splits the octave into 29 equal parts and maps the harmonic 3 to 17\29. It leads to a number of regular temperaments including mystery.

29-commatic (58&87)

Subgroup: 2.3.5

Comma list: [46 -29⟩

Mapping: [⟨29 46 0], ⟨0 0 1]]

Mapping generators: ~531441/524288, ~5

POTE generator: ~5/4 = 387.408

Badness: 1.020556

7-limit 58&87

Subgroup: 2.3.5.7

Comma list: 5120/5103, 50421/50000

Mapping: [⟨29 46 0 14], ⟨0 0 1 1]]

POTE generator: ~5/4 = 388.646

Badness: 0.103734

11-limit 58&87 (Mystery)

Subgroup: 2.3.5.7.11

Comma list: 441/440, 896/891, 3388/3375

Mapping: [⟨29 46 0 14 33], ⟨0 0 1 1 1]]

POTE generator: ~5/4 = 388.460

Vals: Template:Val list

Badness: 0.034291

See also

• Gallery of Just Intervals
• Medium commas