28812/28561

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 twelfth 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, which makes the tesseract comma a useful extension to the rank-3 octaphore and to rank-2 unicorn temperaments.

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 = 1200.000, ~14/13 = 124.539, ~7/4 = 967.452

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

Badness (Dirichlet): 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. As such, this is also a form of negri.

Subgroup: 2.3.5.7.13

Comma list: 10985/10976, 28812/28561

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

Optimal tuning (CTE): ~2 = 1200.000, ~14/13 = 126.679, ~7/4 = 962.564

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

Badness (Dirichlet): 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: 1029/1024, 28672/28561

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

Optimal tuning (CTE): ~2 = 1200.000, ~13/8 = 841.527

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

Badness (Dirichlet): 1.261

Other temperaments

Temperaments discussed elsewhere that temper out the tesseract comma include:

Tridecimal octaphore → Octaphore

2.3.5.7.13 subgroup unicorn (+126/125 and 351/350) → Unicorn family

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.