28812/28561: Difference between revisions

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| Comma = yes

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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 ~~fall short of~~ 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 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.

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

[[Badness]] (~~Dirichlet~~): 2.528

[[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 [[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

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

Badness (~~Dirichlet~~): 1.818

Badness (Sintel): 1.818

=== Cubical ===

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

Badness (~~Dirichlet~~): 1.261

Badness (Sintel): 1.261

=== Other temperaments ===

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 | 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 fall short of 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 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.
@@ Line 23: / Line 23: @@
 {{Optimal ET sequence|legend=1| 9, 10, 19, 29, 37b, 48, 49f, 58, 67, 68, 77, 87 }}
-[[Badness]] (Dirichlet): 2.528
+[[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 [[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
@@ Line 38: / Line 38: @@
 {{Optimal ET sequence|legend=0| 9, 10, 19, 29, 37b, 38, 47, 57, 58, 67c, 76, 86c }}
-Badness (Dirichlet): 1.818
+Badness (Sintel): 1.818
 === Cubical ===
@@ Line 53: / Line 53: @@
 {{Optimal ET sequence|legend=0| 10, 37b, 47, 57, 67, 77, 87, 97, 107, 124b, 144 }}
-Badness (Dirichlet): 1.261
+Badness (Sintel): 1.261
 === Other temperaments ===