28812/28561: Difference between revisions
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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 [[12/1|twelfth harmonic]], and the amount by which four [[14/13]] semitones | '''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 [[12/1|twelfth harmonic]], and the amount by which four [[14/13]] semitones exceed the [[4/3]] perfect fourth. | ||
It can be factored into the [[28672/28561|voltage comma]] and the [[1029/1024|gamelisma]], which provides the 77 & 87 temperament '''cubical''' (see below); it can also be factored into the [[octaphore]] plus four [[729/728|squbemas]], which makes the tesseract comma a useful extension to the rank-3 octaphore and to rank-2 unicorn temperaments. | It can be factored into the [[28672/28561|voltage comma]] and the [[1029/1024|gamelisma]], which provides the 77 & 87 temperament '''cubical''' (see below); it can also be factored into the [[octaphore]] plus four [[729/728|squbemas]], which makes the tesseract comma a useful extension to the rank-3 octaphore and to rank-2 unicorn temperaments. | ||
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{{Optimal ET sequence|legend=1| 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87 }} | {{Optimal ET sequence|legend=1| 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87 }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 2.528 | ||
==== 2.3.5.7.13 subgroup ==== | ==== 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 [[105/104|animist comma]] by noticing that the difference between 4/3 and 5/4 (that is, 16/15) is equivalent in mapping to 14/13. | 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 [[105/104|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 | Subgroup: 2.3.5.7.13 | ||
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{{Optimal ET sequence|legend=0| 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c }} | {{Optimal ET sequence|legend=0| 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c }} | ||
Badness ( | Badness (Sintel): 1.818 | ||
=== Cubical === | === Cubical === | ||
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{{Optimal ET sequence|legend=0| 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144 }} | {{Optimal ET sequence|legend=0| 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144 }} | ||
Badness ( | Badness (Sintel): 1.261 | ||
=== Other temperaments === | === Other temperaments === | ||
Latest revision as of 08:07, 19 July 2025
| Interval information |
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 exceed 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 (Sintel): 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 (Sintel): 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 (Sintel): 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.