Sensipent family: Difference between revisions

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: mapping generators: ~2, ~125/81

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, 162/125 = 443.058

[[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~162/125 = 443.058

{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}

[[Badness]]: 0.035220

Badness (Dirichlet): 0.826

=== 2.3.5.31 ===

Fascinatingly, essentially the only extension that preserves the occurrence of sensi's 5-limit structure in such large edos as [[539edo|539]] is the one to prime 31 by interpreting the generator accurately as [[40/31]]~[[31/24]] by tempering [[961/960|S31 = 961/960]], so that the large [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].

[[Subgroup]]: 2.3.5.31

[[Comma list]]: 78732/78125, 961/960

=== Sensible ===

An extension of sensipent to prime 11 of dubious canonicity (but significantly higher accuracy than [[sensi]]) interprets the generator as [[165/128]]~[[128/99]] by tempering [[8019/8000|S9/S10]] so that [[11/8]] is reached as ([[10/9]])3. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation: that it is equivalent to tempering the [[semiporwellisma]] which as it is equal to [[961/960|S31]] * [[1024/1023|S32]]2 (thus forming the S-expression-based comma list), which by the equivalence of the aforementioned [[lopsided comma]] also implies that this temperament equates ([[33/32]])2 with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension to prime 17 is reasonably accurate through tempering [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.

[[Subgroup]]: 2.3.5.11

[[Comma list]]: [[8019/8000]], [[16384/16335]]

[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~128/99 = 443.115

Badness (Dirichlet): 0.728

=== 2.3.5.11.17 ===

The aforementioned extension to prime 17 through tempering [[1089/1088|S33]] is equivalent to the one by tempering [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).

[[Subgroup]]: 2.3.5.11.17

[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]

[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~22/17 = 443.188

Badness (Dirichlet): 0.639

=== 2.3.5.11.17.31 ===

Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony without the need for interpreting it as high-damage ~[[9/7]] or ~[[13/10]] intervals. Its S-expression-based comma list is {([[256/255|S16]],) [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495|496/495 = S31 * S32]] and [[528/527|528/527 = S32 * S33]] as well as [[16337/16335|16337/16335 = S31/S33]] = ([[17/15|34/30]])/([[33/31]])2 = ([[17/15]])/([[33/31]])2.

[[Subgroup]]: 2.3.5.11.17.31

[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[]]

[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.185

Badness (Dirichlet): 0.519

== Sensi ==

@@ Line 26: / Line 26: @@
 : mapping generators: ~2, ~125/81
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, 162/125 = 443.058
+[[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~162/125 = 443.058
 {{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
 [[Badness]]: 0.035220
+Badness (Dirichlet): 0.826
+=== 2.3.5.31 ===
+Fascinatingly, essentially the only extension that preserves the occurrence of sensi's 5-limit structure in such large edos as [[539edo|539]] is the one to prime 31 by interpreting the generator accurately as [[40/31]]~[[31/24]] by tempering [[961/960|S31 = 961/960]], so that the large [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
+[[Subgroup]]: 2.3.5.31
+[[Comma list]]: 78732/78125, 961/960
+=== Sensible ===
+An extension of sensipent  to prime 11 of dubious canonicity (but significantly higher accuracy than [[sensi]]) interprets the generator as [[165/128]]~[[128/99]] by tempering [[8019/8000|S9/S10]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation: that it is equivalent to tempering the [[semiporwellisma]] which as it is equal to [[961/960|S31]] * [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list), which by the equivalence of the aforementioned [[lopsided comma]] also implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension to prime 17 is reasonably accurate through tempering [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
+[[Subgroup]]: 2.3.5.11
+[[Comma list]]: [[8019/8000]], [[16384/16335]]
+{{Mapping|legend=1| 1 6 8 -6 | 0 -7 -9 15 }}
+{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
+[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~128/99 = 443.115
+Badness (Dirichlet): 0.728
+=== 2.3.5.11.17 ===
+The aforementioned extension to prime 17 through tempering [[1089/1088|S33]] is equivalent to the one by tempering [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
+[[Subgroup]]: 2.3.5.11.17
+[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]
+{{Mapping|legend=1| 1 6 8 -6 -6 | 0 -7 -9 15 16 }}
+{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
+[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~22/17 = 443.188
+Badness (Dirichlet): 0.639
+=== 2.3.5.11.17.31 ===
+Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony without the need for interpreting it as high-damage ~[[9/7]] or ~[[13/10]] intervals. Its S-expression-based comma list is {([[256/255|S16]],) [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495|496/495 = S31 * S32]] and [[528/527|528/527 = S32 * S33]] as well as [[16337/16335|16337/16335 = S31/S33]] = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>.
+[[Subgroup]]: 2.3.5.11.17.31
+[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[]]
+{{Mapping|legend=1| 1 6 8 -6 -6 | 0 -7 -9 15 16 }}
+{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g, 287cg }}
+[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.185
+Badness (Dirichlet): 0.519
 == Sensi ==