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 [[6/1|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 [[28672/28561|Voltage Comma]] and the [[1029/1024|Gamelisma]], which provides the 77 & 87 temperament '''Cubical''' (see below).

'''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 [[6/1|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 [[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 temperament.

== Temperaments ==

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[[Comma list]]: 28812/28561

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

[[Mapping]]: [⟨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

[[Optimization|Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~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]]

[[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 [[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.

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: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[29edo|29]], [[37edo|37b]], [[38edo|38]], [[47edo|47]], [[57edo|57]], [[58edo|58]], [[67edo|67c]], 76, [[87edo|86c]]

Badness: 1.818

=== Cubical ===

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Comma list: 28672/28561, 1029/1024

Mapping: [~~⟨10 -12~~], ⟨0 4], ⟨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 = 1\1, ~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]]

Badness: 1.261

=== 2.3.7.13 Octaphore ===

''See also: [[Octaphore#Tridecimal Octaphore|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: [[19edo|19]], [[58edo|58]], [[77edo|77]], [[135edo|135]], [[96edo|96d]]

== Etymology ==

@@ Line 1: / Line 1: @@
 {{Infobox Interval|Name=Tesseract Comma|Ratio=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 [[6/1|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 [[28672/28561|Voltage Comma]] and the [[1029/1024|Gamelisma]], which provides the 77 & 87 temperament '''Cubical''' (see below).
+'''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 [[6/1|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 [[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 temperament.
 == Temperaments ==
@@ Line 12: / Line 14: @@
 [[Comma list]]: 28812/28561
-[[Mapping]]: [⟨2 -4 0], ⟨2 0 1], ⟨3 -1 1]]
+[[Mapping]]: [⟨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
+[[Optimization|Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~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]]
 [[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 [[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.
+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: [[9edo|9]], [[10edo|10]], [[19edo|19]], [[29edo|29]], [[37edo|37b]], [[38edo|38]], [[47edo|47]], [[57edo|57]], [[58edo|58]], [[67edo|67c]], 76, [[87edo|86c]]
+Badness: 1.818
 === Cubical ===
@@ Line 27: / Line 44: @@
 Comma list: 28672/28561, 1029/1024
-Mapping: [⟨10 -12], ⟨0 4], ⟨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 = 1\1, ~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]]
 Badness: 1.261
+=== 2.3.7.13 Octaphore ===
+''See also: [[Octaphore#Tridecimal Octaphore|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: [[19edo|19]], [[58edo|58]], [[77edo|77]], [[135edo|135]], [[96edo|96d]]
 == Etymology ==