4096/4095

4096/4095, the minisma or minic comma, also described as the tridecimal schisma or tridecimal schismina, is an unnoticeable 13-limit superparticular comma of about 0.42 cents. It is the difference between the septimal comma (64/63) and the wilsorma (65/64), and between the septimal quartertone (36/35) and the tridecimal quartertone (1053/1024). It is also a Mersenne comma.

Interval information

Ratio	4096/4095
Factorization	212 × 3-2 × 5-1 × 7-1 × 13-1
Monzo	[12 -2 -1 -1 0 -1⟩
Size in cents	0.4227162¢
Names	minisma, minic comma
Color name	s3urg1, sathurugu 1sn, Sathurugu comma
FJS name	[math]\displaystyle{ \text{P1}_{5,7,13} }[/math]
Special properties	square superparticular, reduced, reduced subharmonic
Tenney norm (log2 nd)	23.9996
Weil norm (log2 max(n, d))	24
Wilson norm (sopfr(nd))	55
Comma size	unnoticeable
S-expression	S64
Open this interval in xen-calc

It is also the smallest superparticular ratio in the 2.3.5.7.13 subgroup.

Temperaments

Tempering out this comma in the full 13-limit defines the minismic temperament, or in the 2.3.5.7.13 subgroup, the minic temperament. In either case, it equates the tridecimal quartertone with the septimal one, and enables the minismic chords, the essentially tempered chords in the 21-odd-limit. You may find a list of good equal temperaments that support these temperaments below.

Minic

Subgroup: 2.3.5.7.13

Mapping:

[⟨	1	0	0	0	0	12	],
⟨	0	1	0	0	0	-2	],
⟨	0	0	1	0	0	-1	],
⟨	0	0	0	1	0	-1	],

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

Optimal tunings:

WE: ~2 = 1199.9720 ¢, ~3/2 = 701.9948 ¢, ~5/4 = 386.3824 ¢, ~7/4 = 968.9004 ¢

CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0114 ¢, ~5/4 = 386.3886 ¢, ~7/4 = 968.9218 ¢

Optimal ET sequence: 31, 41, 46, 53, 77, 84, 87, 99, 118, 130, 140, 171, 224, 270, 441, 935, 1376, 1506, 1947, 2441, 2882, 5323f, 6699df, 8205cdf, 9140cdf, 9581cdff, 12022bcdff, 14904bcddfff, 19957bccdddffff

Badness (Sintel): 0.135

Minismic

Subgroup: 2.3.5.7.11.13

Mapping:

[⟨	1	0	0	0	0	12	],
⟨	0	1	0	0	0	-2	],
⟨	0	0	1	0	0	-1	],
⟨	0	0	0	1	0	-1	],
⟨	0	0	0	0	1	0	]]

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

Optimal tunings:

WE: ~2 = 1199.9720 ¢, ~3/2 = 701.9948 ¢, ~5/4 = 386.3824 ¢, ~7/4 = 968.9004 ¢, ~11/8 = 551.4020 ¢

CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0114 ¢, ~5/4 = 386.3886 ¢, ~7/4 = 968.9218 ¢, ~11/8 = 551.3984 ¢

Optimal ET sequence: 31, 41, 46, 53, 77, 84, 87, 118, 130, 183, 217, 224, 270, 494, 764, 935, 1012, 1236, 1506, 3236, 3323, 3817, 5323f, 9140cdef, 14463bcddeefff, 17786bccddeeefff

Badness (Sintel): 0.479

Etymology

The names minisma and minic comma were given by Flora Canou in 2026 after the general interval size measure knowns as the mina, because 4096/4095 is the default comma represented by a mina (or three tinas) in Sagittal notation. Mina is a clipping of schismina, which describes a range of commas in Sagittal notation, and in turn comes from schisma (the interval region) + -in (diminutive suffix). Eufalesio proposed trinyblisma, for the denominator is the largest integer that can be written in 3 nybbles (1.5 bytes).

See also

• List of superparticular intervals