Sensipent family: Difference between revisions

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

[[Badness]]: 0.035220

[[Badness]]:

* Smith: 0.035220

* Dirichlet: 0.826

~~Badness (Dirichlet): 0.826~~

=== 2.3.5.31 subgroup ===

Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one with prime 31 by interpreting the generator accurately as [[31/24]]~[[40/31]], tempering out [[961/960|S31 = 961/960]], so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. 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]].

=== 2.3.5.31 ===

Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[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, as sensipent splits [[16/15]] into two equal parts. 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~~

[[Comma list]]: 961/960, 2511/2500

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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050

Badness (Dirichlet): 0.243

[[Badness]] (Dirichlet): 0.243

=== Sendai ===

Sendai is an accurate extension of (2.3.5.31) [[#Sensipent|sensipent]] to primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).

Sendai is an accurate extension of (2.3.5.31) [[#Sensipent|sensipent]] with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).

[[Subgroup]]: 2.3.5.23.29.31

Line 65:

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989

Badness (Dirichlet): 0.283

[[Badness]] (Dirichlet): 0.283

=== 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 is equal to [[961/960|S31]] * [[1024/1023|S32]]2 (thus forming the S-expression-based comma list). 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.

Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] 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 also tempers out the [[semiporwellisma]], which is equal to [[961/960|S31]] × [[1024/1023|S32]]2 (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], 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 with prime 17 is reasonably accurate through tempering out [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.

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

Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its S-expression-based comma list is {([[256/255|S16]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering out [[496/495]] = S31 × S32 and [[528/527]] = S32 × S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])2 = ([[17/15]])/([[33/31]])2. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].

[[Subgroup]]: 2.3.5.11

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

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

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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115

Badness (Dirichlet): 0.728

[[Badness]] (Dirichlet): 0.728

==== 2.3.5.11.17 ====

==== 2.3.5.11.17 subgroup ====

~~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]]~~

[[Comma list]]: 256/255, 1089/1088, 1377/1375

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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.188

Badness (Dirichlet): 0.639

[[Badness]] (Dirichlet): 0.639

==== 2.3.5.11.17.23 ====

==== 2.3.5.11.17.23 subgroup ====

[[Subgroup]]: 2.3.5.11.17.23

[[Comma list]]: ~~[[8019~~/~~8000]]~~, ~~[[16384~~/~~16335]]~~, ~~[[256~~/~~255]]~~, ~~[[576~~/~~575]]~~

[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375

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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185

Badness (Dirichlet): 0.555

[[Badness]] (Dirichlet): 0.555

~~==== 2.3.5.11.17.23.31 ====~~

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]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495]] = S31 * S32 and [[528/527]] = S32 * S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])2 = ([[17/15]])/([[33/31]])2. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].

==== 2.3.5.11.17.23.31 subgroup ====

[[Subgroup]]: 2.3.5.11.17.23.31

[[Comma list]]: ~~[[8019/8000]], [[16384/16335]], [[~~256/255]], [[576/575]], [[961/960]]

[[Comma list]]: 256/255, 576/575, 961/960, 1089/1088, 1377/1375

Line 127:

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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.183

Badness (Dirichlet): 0.490

[[Badness]] (Dirichlet): 0.490

== Sensi ==

@@ Line 30: / Line 30: @@
 {{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
-[[Badness]]: 0.035220
+[[Badness]]:
+* Smith: 0.035220
+* Dirichlet: 0.826
-Badness (Dirichlet): 0.826
+=== 2.3.5.31 subgroup ===
+Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one with prime 31 by interpreting the generator accurately as [[31/24]]~[[40/31]], tempering out [[961/960|S31 = 961/960]], so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. 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]].
-=== 2.3.5.31 ===
-Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[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, as sensipent splits [[16/15]] into two equal parts. 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
+[[Comma list]]: 961/960, 2511/2500
 {{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
@@ Line 49: / Line 49: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
-Badness (Dirichlet): 0.243
+[[Badness]] (Dirichlet): 0.243
 === Sendai ===
-{{ See also | User:VIxen/Table_of_sensipent_intervals }}
+{{ See also | User:VIxen/Table of sensipent intervals }}
-Sendai is an accurate extension of (2.3.5.31) [[#Sensipent|sensipent]] to primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).
+Sendai is an accurate extension of (2.3.5.31) [[#Sensipent|sensipent]] with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).
 [[Subgroup]]: 2.3.5.23.29.31
@@ Line 65: / Line 65: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989
-Badness (Dirichlet): 0.283
+[[Badness]] (Dirichlet): 0.283
 === 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 is equal to [[961/960|S31]] * [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). 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.
+Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] 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 also tempers out the [[semiporwellisma]], which is equal to [[961/960|S31]] × [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], 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 with prime 17 is reasonably accurate through tempering out [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
+The aforementioned extension with prime 17 through tempering out [[1089/1088|S33]] is equivalent to the one by tempering out [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
+Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its S-expression-based comma list is {([[256/255|S16]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering out [[496/495]] = S31 × S32 and [[528/527]] = S32 × S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
 [[Subgroup]]: 2.3.5.11
-[[Comma list]]: [[8019/8000]], [[16384/16335]]
+[[Comma list]]: 8019/8000, 16384/16335
 {{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
@@ Line 82: / Line 86: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115
-Badness (Dirichlet): 0.728
+[[Badness]] (Dirichlet): 0.728
-==== 2.3.5.11.17 ====
+==== 2.3.5.11.17 subgroup ====
-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]]
+[[Comma list]]: 256/255, 1089/1088, 1377/1375
 {{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
@@ Line 99: / Line 102: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.188
-Badness (Dirichlet): 0.639
+[[Badness]] (Dirichlet): 0.639
-==== 2.3.5.11.17.23 ====
+==== 2.3.5.11.17.23 subgroup ====
 [[Subgroup]]: 2.3.5.11.17.23
-[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]]
+[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375
 {{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
@@ Line 112: / Line 115: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185
-Badness (Dirichlet): 0.555
+[[Badness]] (Dirichlet): 0.555
-==== 2.3.5.11.17.23.31 ====
-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]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495]] = S31 * S32 and [[528/527]] = S32 * S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
+==== 2.3.5.11.17.23.31 subgroup ====
 [[Subgroup]]: 2.3.5.11.17.23.31
-[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]], [[961/960]]
+[[Comma list]]: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
 {{Mapping|legend=1| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
@@ Line 127: / Line 128: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.183
-Badness (Dirichlet): 0.490
+[[Badness]] (Dirichlet): 0.490
 == Sensi ==