Sensipent family: Difference between revisions

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Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.

~~=== 2.3.5.31 subgroup ===~~

The generator can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.

~~For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].~~

~~Subgroup: 2.3.5.31~~

~~Comma list: 961/960, 2511/2500~~

~~Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}

~~{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}~~

~~Badness (Sintel): 0.243~~

~~=== Sendai ===~~

~~{{See also| Sensipent #Sendai interval table }}~~

Sendai is an accurate extension of 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~~

~~Comma list: 465/464, 576/575, 621/620, 900/899~~

~~Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}~~

~~Optimal tunings:~~

* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}

~~{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}~~

~~Badness (Sintel): 0.283~~

=== Sensible ===

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.

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 [[S-expression|S31⋅S322]] (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 {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, 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]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.

The aforementioned extension with prime 17 through tempering out [[1089/1088~~|S33]]~~ implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.

The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 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]].

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 {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[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]])/([[33/31]])2. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].

Subgroup: 2.3.5.11

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Comma list: 55/54, 66/65, 77/75, 143/140

Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}

Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}

Optimal tunings:

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Badness (Sintel): 1.12

== Other subgroup extensions ==

=== Sensipent (2.3.5.31 subgroup) ===

The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.

For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].

Subgroup: 2.3.5.31

Comma list: 961/960, 2511/2500

Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}

Optimal tunings:

* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}

{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}

Badness (Sintel): 0.243

=== Sendai ===

Sendai is an accurate extension of 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

Comma list: 465/464, 576/575, 621/620, 900/899

Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}

Optimal tunings:

* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}

Badness (Sintel): 0.283

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Sensipent family| ]]

[[Category:Rank 2]]

@@ Line 35: / Line 35: @@
 Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
-=== 2.3.5.31 subgroup ===
-The generator can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.
-For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
-Subgroup: 2.3.5.31
-Comma list: 961/960, 2511/2500
-Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
-Optimal tunings:
-* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
-{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
-Badness (Sintel): 0.243
-=== Sendai ===
-{{See also| Sensipent #Sendai interval table }}
-Sendai is an accurate extension of 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
-Comma list: 465/464, 576/575, 621/620, 900/899
-Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
-Optimal tunings:
-* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
-{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
-Badness (Sintel): 0.283
 === Sensible ===
 {{See also| Sensipent #Sensible interval table }}
-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.
+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 [[S-expression|S31⋅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 {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, 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]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.
-The aforementioned extension with prime 17 through tempering out [[1089/1088|S33]] implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.
+The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 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]].
+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 {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[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]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
 Subgroup: 2.3.5.11
@@ Line 337: / Line 299: @@
 Comma list: 55/54, 66/65, 77/75, 143/140
-Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}
+Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}
 Optimal tunings:
@@ Line 772: / Line 734: @@
 Badness (Sintel): 1.12
+== Other subgroup extensions ==
+=== Sensipent (2.3.5.31 subgroup) ===
+The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.
+For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
+Subgroup: 2.3.5.31
+Comma list: 961/960, 2511/2500
+Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
+Optimal tunings:
+* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
+{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
+Badness (Sintel): 0.243
+=== Sendai ===
+{{See also| Sensipent #Sendai interval table }}
+Sendai is an accurate extension of 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
+Comma list: 465/464, 576/575, 621/620, 900/899
+Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
+Optimal tunings:
+* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
+{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
+Badness (Sintel): 0.283
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Sensipent family| ]] <!-- main article -->
 [[Category:Rank 2]]