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 factors into (6656/6655)⋅(10648/10647). 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 rank-5 minismic temperament, or in the 2.3.5.7.13 subgroup, the rank-4 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 known 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
• Tridecimal schisma (disambiguation page)