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four 14/13 semitones exceed the 4/3 perfect fourth, not fall short of it
 
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{{Infobox Interval|Name=Tesseract Comma|Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma|Ratio=28812/28561|comma=yes}}
{{Infobox Interval
| Name= Tesseract comma
| Color name = 3u<sup>4</sup>z<sup>4</sup>2, Quadthuzo comma
| Ratio = 28812/28561
| Comma = yes
}}
'''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.


'''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 fall short of 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]] (squbemae?), which makes the Tesseract Comma a useful extension to the rank-3 Octaphore and to rank-2 Unicorn temperaments.


== Temperaments ==
== Temperaments ==
=== Tesseract ===
=== Tesseract ===
Tempering out the Tesseract Comma in its minimal subgroup, 2.3.7.13, yields the rank-3 '''Tesseract''' temperament.
Tempering out the tesseract comma in its minimal subgroup, 2.3.7.13, yields the rank-3 '''tesseract''' temperament.


[[Just intonation subgroup|Subgroup]]: 2.3.7.13
[[Subgroup]]: 2.3.7.13


[[Comma list]]: 28812/28561
[[Comma list]]: 28812/28561


[[Mapping]]: [⟨1 2 2 3], ⟨0 -4 0 -1], ⟨0 0 1 1]]
{{Mapping|legend=1| 1 2 2 3 | 0 -4 0 -1 | 0 0 1 1 }}


[[Optimization|Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~14/13 = 124.539, ~7/4 = 967.452
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~14/13 = 124.539, ~7/4 = 967.452


[[Optimal ET sequence]]: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[29edo|29]], [[37edo|37b]], [[48edo|48]], [[49edo|49f]], [[58edo|58]], [[67edo|67]], [[68edo|68]], [[77edo|77]], [[87edo|87]]
{{Optimal ET sequence|legend=1| 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87 }}


[[Badness]]: 2.528
[[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


Comma list: 28812/28561, 10985/10976
Comma list: 10985/10976, 28812/28561


Mapping: [⟨1 2 2 2 3], ⟨0 -4 3 0 -1], ⟨0 0 0 1 1]]
Mapping: {{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 tuning (CTE): ~2 = 1200.000, ~14/13 = 126.679, ~7/4 = 962.564


Optimal ET sequence: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[29edo|29]], [[37edo|37b]], [[38edo|38]], [[47edo|47]], [[57edo|57]], [[58edo|58]], [[67edo|67c]], 76, [[87edo|86c]]
{{Optimal ET sequence|legend=0| 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c }}


Badness: 1.818
Badness (Sintel): 1.818


=== Cubical ===
=== 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.
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
Subgroup: 2.3.7.13


Comma list: 28672/28561, 1029/1024
Comma list: 1029/1024, 28672/28561


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


Optimal tuning (CTE): ~2 = 1\1, ~13/8 = 841.527
Optimal tuning (CTE): ~2 = 1200.000, ~13/8 = 841.527


Optimal ET sequence: [[10edo|10]], [[37edo|37b]], [[47edo|47]], [[57edo|57]], [[67edo|67]], [[77edo|77]], [[87edo|87]], [[97edo|97]], [[107edo|107]], [[124edo|124b]], [[144edo|144]]
{{Optimal ET sequence|legend=0| 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144 }}


Badness: 1.261
Badness (Sintel): 1.261


=== Other temperaments ===
=== Other temperaments ===
Temperaments discussed elsewhere that temper out the Tesseract comma include:
Temperaments discussed elsewhere that temper out the tesseract comma include:


Tridecimal Octaphore → [[Octaphore#Tridecimal%20Octaphore|Octaphore]]
Tridecimal octaphore → [[Octaphore #Tridecimal%20Octaphore|Octaphore]]


2.3.5.7.13 subgroup Unicorn (+351/350 +126/125) → [[Unicorn family#2.3.5.7.13%20subgroup|Unicorn Family]]
2.3.5.7.13 subgroup unicorn (+126/125 and 351/350) → [[Unicorn family #2.3.5.7.13%20subgroup|Unicorn family]]


== Etymology ==
== Etymology ==
The name Tesseract Comma was chosen by [[User:Unque|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.
The name tesseract comma was chosen by [[User:Unque|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.
 
[[Category:Commas named for how they divide the fourth]]
[[Category:Commas named for how they divide the fourth]]
[[Category:Commas named for the intervals they stack]]
[[Category:Commas named for the intervals they stack]]
[[Category:Rational intervals]]
[[Category:13-limit intervals]]
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