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).

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: [⟨2 -4 0], ⟨2 0 1], ⟨3 -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

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: [⟨10 -12], ⟨0 4], ⟨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

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.