Punctisma

The punctisma, with a ratio of 43904/43875, is an unnoticeable 13-limit (specifically 2.3.5.7.13-subgroup) comma measuring about 1.14 cents. It is the difference between three 15/14 semitones and a 16/13 neutral third.

Interval information

Ratio	43904/43875
Factorization	27 × 3-3 × 5-3 × 73 × 13-1
Monzo	[7 -3 -3 3 0 -1⟩
Size in cents	1.143913¢
Name	punctisma
Color name	s3uz3g33, sathutrizogu 3rd
FJS name	[math]\displaystyle{ \text{dd3}^{7,7,7}_{5,5,5,13} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	30.8432
Weil norm (log2 max(n, d))	30.8441
Wilson norm (sopfr(nd))	72
Comma size	unnoticeable
S-expression	S14/S15
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In terms of other commas, it is the difference between 196/195 and 225/224. It factors into (2401/2400)⋅(4096/4095). In the 19-limit it further factors into (2926/2925)⋅(3136/3135).

Temperaments

Tempering out this comma in the full 13-limit leads to the rank-5 punctismic temperament, or in the 2.3.5.17.13 subgroup, the rank-4 punctic temperament. It is characteristic of the 10n family of edos, including decoid tunings 130edo, 140edo, and 270edo, which approximate both 15/14 and 16/13 to within a fraction of a cent. It is also tempered out by any temperament which tempers out both 2401/2400 and 4096/4095, such as 31edo, 41edo, 99edo, 171edo, 311edo, and 441edo. Perhaps surprisingly, 111edo and 159edo temper it out as well.

Punctic

Subgroup: 2.3.5.7.13

Subgroup-val mapping: [⟨1 0 0 0 7], ⟨0 1 0 0 -3], ⟨0 0 1 0 -3], ⟨0 0 0 1 1]]

mapping generators: ~2, ~3, ~5, ~7

Optimal tunings:

WE: ~2 = 1199.9609 ¢, ~3/2 = 702.0362 ¢, ~5/4 = 386.4823 ¢, ~7/4 = 968.7720 ¢

CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0380 ¢, ~5/4 = 386.4679 ¢, ~7/4 = 968.7382 ¢

Optimal ET sequence: 12f, 19, 31, 41, 50, 60, 80, 99, 111, 121, 130, 140, 171, 251, 270, 441, 1152, 1453, 1593, 2034, 2164, 2605, 3046, 4179cf, 4620cf, 5191cff

Badness (Sintel): 0.467

Punctismic

Subgroup: 2.3.5.7.11.13

Mapping:

[⟨	1	0	0	0	0	7	],
⟨	0	1	0	0	0	-3	],
⟨	0	0	1	0	0	-3	],
⟨	0	0	0	1	0	3	],
⟨	0	0	0	0	1	1	]]

mapping generators: ~2, ~3, ~5, ~7, ~11

Optimal tunings:

WE: ~2 = 1199.9609 ¢, ~3/2 = 702.0362 ¢, ~5/4 = 386.4823 ¢, ~7/4 = 968.7720 ¢, ~11/8 = 551.4353 ¢

CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0380 ¢, ~5/4 = 386.4679 ¢, ~7/4 = 968.7382 ¢, ~11/8 = 551.3948 ¢

Optimal ET sequence: 12f, …, 19, 22f, 29, 31, 41, 60e, 72f, 80, 99e, 109, 111, 121, 130, 190, 270, 552, 581, 711, 742, 882, 1012, 1323, 1593, 2605, 4198cf, 4750cef

Badness (Sintel): 1.77

Etymology

This comma was named by Jerdle and Aura in 2023, from Latin pūnctum ("point") based on both its relation to the 10n family of edos as well as its small size.