28812/28561

Interval information

Ratio	28812/28561
Factorization	22 × 3 × 74 × 13-4
Monzo	[2 1 0 4 0 -4⟩
Size in cents	15.14798¢
Name	Tesseract Comma
FJS name	[math]\displaystyle{ \text{A2}^{7,7,7,7}_{13,13,13,13} }[/math]
Special properties	reduced
Tenney height (log2 nd)	29.6161
Weil height (log2 max(n, d))	29.6288
Wilson height (sopfr(nd))	87
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28812/28561 (the Tesseract Comma) is a small comma in the 2.3.7.13 subgroup. It is the amount by which four 13/7 sevenths fall short of the sixth harmonic, and the amount by which four 14/13 semitones fall short of the 4/3 perfect fourth.

It can be factored into the Voltage Comma and the Gamelisma, which provides the 77 & 87 temperament Cubical (see below); it can also be factored into the Octaphore plus four Squbemas (squbemae?), which makes the Tesseract Comma a useful extension to the rank-3 Octaphore temperament.

Temperaments

Tesseract

Tempering out the Tesseract Comma in its minimal subgroup, 2.3.7.13, yields the rank-3 Tesseract temperament.

Subgroup: 2.3.7.13

Comma list: 28812/28561

Mapping: [⟨1 2 2 3], ⟨0 -4 0 -1], ⟨0 0 1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~14/13 = 124.539, ~7/4 = 967.452

Optimal ET sequence: 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87

Badness: 2.528

2.3.5.7.13 subgroup

By noticing that three generators is almost exactly 5/4, we can add prime 5 to the subgroup by tempering out the Cantonisma. We can equivalently temper out the Animist comma by noticing that the difference between 4/3 and 5/4 (that is, 16/15) is equivalent in mapping to 14/13.

Subgroup: 2.3.5.7.13

Comma list: 28812/28561, 10985/10976

Mapping: [⟨1 2 2 2 3], ⟨0 -4 3 0 -1], ⟨0 0 0 1 1]]

Optimal tuning (CTE): ~2 = 1\1, ~14/13 = 126.679, ~7/4 = 962.564

Optimal ET sequence: 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c

Badness: 1.818

Cubical

By factoring the Tesseract comma into the Voltage Comma and Gamelisma, we get the rank-2 temperament Cubical. This temperament is so named because its lattice is the same as Tesseract, but with one dimension collapsed; similarly, a cube can be thought of as a Tesseract with one of its dimensions collapsed.

Subgroup: 2.3.7.13

Comma list: 28672/28561, 1029/1024

Mapping: [⟨1 10 0 3], ⟨0 0 4], ⟨0 3 1]]

Optimal tuning (CTE): ~2 = 1\1, ~13/8 = 841.527

Optimal ET sequence: 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144

Badness: 1.261

2.3.7.13 Octaphore

See also: Tridecimal Octaphore

By factoring the Tesseract Comma into the Octaphore and the Squbesma, we get a 13-limit extension of the rank-2 Octaphore temperament.

Subgroup: 2.5.7.13

Comma list:

Mapping: [⟨1 2 2 4 4 5], ⟨0 -8 0 -23 2 -25], ⟨0 0 1 0 -2 0]]

Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 62.281, ~5/4 = 386.512

Optimal ET sequence: 19, 58, 77, 135, 96d

Etymology

The name Tesseract Comma was chosen by Unque in 2025. This name was chosen because tempering the comma cleaves the Perfect Fourth into four parts, and a tesseract is the 4D regular polytope made from four-sided regular polygons.