41-comma

Interval information

Ratio	36 893 488 147 419 103 232 / 36 472 996 377 170 786 403
Factorization	265 × 3-41
Monzo	[65 -41⟩
Size in cents	19.84496¢
Names	41-comma, counterpyth comma
FJS name	[math]\displaystyle{ \text{6d5} }[/math]
Special properties	reduced, reduced subharmonic
Tenney norm (log2 nd)	129.983
Weil norm (log2 max(n, d))	130
Wilson norm (sopfr(nd))	253
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[65 -41⟩, the 41-comma or counterpyth comma, is an interval of 19.845 cents. It is the amount by which 41 fifths (3/2) fall short of 24 octaves, in other words 2²⁴/(3/2)⁴¹.

Temperaments

Tempering out this comma splits the octave into 41 equal parts and maps the harmonic 3 to 24\41. It leads to a number of regular temperaments in the counterpyth family. 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.

Counterpyth (41&205)

Subgroup: 2.3.5

Comma list: [65 -41⟩

Mapping: [⟨41 65 0], ⟨0 0 1]]

Mapping generators: ~531441/524288, ~5

POTE generator: ~5/4 = 386.668

Badness: 0.934310

7-limit 41&123

Subgroup: 2.3.5.7

Comma list: 1029/1024, 537824/531441

Mapping: [⟨41 65 0 115], ⟨0 0 1 0]]

POTE generator: ~5/4 = 385.731

Badness: 0.161056

7-limit 41&205

Subgroup: 2.3.5.7

Comma list: 5120/5103, 2500000/2470629

Mapping: [⟨41 65 0 20], ⟨0 0 1 1]]

POTE generator: ~5/4 = 385.667

Badness: 0.142344

7-limit 41&369

Subgroup: 2.3.5.7

Comma list: 2401/2400, 52613349376/52301766015

Mapping: [⟨41 65 95 115], ⟨0 0 2 1]]

POTE generator: ~8/7 = 231.045

Badness: 0.134559

See also

• Small comma
• Schismic-counterpyth equivalence continuum