Sensipent family: Difference between revisions

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[[Badness]] (Sintel): 0.826

~~=== 2.3.5.31 ===~~

The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts, as sqrt(5/3) is the midpoint of 5/4 and 4/3, which it is the difference of. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Other subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.

This extension can be applied to many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 31/24~40/31 such that 24:31:40 is a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.

~~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~~

==== Subgroup extensions ====

[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai|Sendai]] is perhaps more accurate but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one ~~might~~ consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and [[111edo]] are good general tunings, ~~and as is their sum,~~ [[~~176edo~~]]. All ~~of these have in common that they~~ support sensible by patent val.

[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai|Sendai]] is perhaps more accurate but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and [[111edo]] are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap| 65 + 111 = 176g }} is almost patent val.

=== Full 7-limit extensions ===

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

==== Subgroup extensions ====

The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.

This extension can be applied to many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 31/24~40/31 such that 24:31:40 is a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.

[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy.

== Sensi ==

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Badness (Sintel): 1.12

== ~~Other subgroup~~ extensions ==

== Subgroup extensions ==

=== ~~Sendai~~ ===

=== Sensipent (2.3.5.31) ===

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

Subgroup: 2.3.5.31

~~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). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence.

Comma list: 961/960, 2511/2500

~~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}}

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

* CWE: ~2 = 1200.0000{{c}}, ~31/24 = ~~442~~.~~9762~~{{c}}

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

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

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

Badness (Sintel): 0.~~283~~

Badness (Sintel): 0.243

=== Sensible ===

=== Sensible (2.3.5.11) ===

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.

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 for 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 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 {{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]]. A notable non-patent val is [[84edo|84g]].

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Subgroup: 2.3.5.11

Comma list: 8019/8000~~, 16384/16335~~

Comma list: 6912/6875, 8019/8000

~~Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}~~

~~: mapping generators: ~~~2~~, ~128/99~~

Optimal tunings:

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

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}

~~: mapping generators: ~2, ~22/17~~

Optimal tunings:

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

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}

Optimal tunings:

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

~~Subgroup-val mapping:~~ {{~~mapping~~| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}

Optimal tunings:

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Badness (Sintel): 0.490

=== Sendai (2.3.5.23) ===

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). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence. It is ideal in tuning for ambiguating the generator's 2.3.5.31 interpretation as 40/31~31/24, being between these (and hence also a good tuning of 2.3.5.31 sensipent at the cost of some 5-limit accuracy).

{{Todo|inline=1|complete section|comment=Add the data for the intermediate subgroups. }}

==== 2.3.5.23.29.31 subgroup ====

Subgroup: 2.3.5.23.29.31

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

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:Sensipent family| ]]

[[Category:Temperament families]]

[[Category:~~Sensipent family| ]] <!~~-~~- main article -->~~

[[Category:Catalogs of rank-2 temperaments]]

~~[[Category:Rank~~ 2]]

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 [[Badness]] (Sintel): 0.826
-=== 2.3.5.31 ===
-The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts, as sqrt(5/3) is the midpoint of 5/4 and 4/3, which it is the difference of. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Other subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.
-This extension can be applied to many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 31/24~40/31 such that 24:31:40 is a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.
-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
 ==== Subgroup extensions ====
-[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai|Sendai]] is perhaps more accurate but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one might consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and [[111edo]] are good general tunings, and as is their sum, [[176edo]]. All of these have in common that they support sensible by patent val.
+[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai|Sendai]] is perhaps more accurate but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and [[111edo]] are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap| 65 + 111 = 176g }} is almost patent val.
 === Full 7-limit extensions ===
@@ Line 57: / Line 38: @@
 Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
+==== Subgroup extensions ====
+The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.
+This extension can be applied to many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 31/24~40/31 such that 24:31:40 is a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.
+[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy.
 == Sensi ==
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 Badness (Sintel): 1.12
-== Other subgroup extensions ==
+== Subgroup extensions ==
-=== Sendai ===
+=== Sensipent (2.3.5.31) ===
-{{See also| Sensipent #Sendai interval table }}
+Subgroup: 2.3.5.31
-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). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence.
+Comma list: 961/960, 2511/2500
-Subgroup: 2.3.5.23.29.31
-Comma list: 465/464, 576/575, 621/620, 900/899
+{{Mapping|legend=2| 1 -1 -1 2 | 0 7 9 8 }}
-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}}
+* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
-{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
+{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
-Badness (Sintel): 0.283
+Badness (Sintel): 0.243
-=== Sensible ===
+=== Sensible (2.3.5.11) ===
 {{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 [[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.
+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 for 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 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 {{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]]. A notable non-patent val is [[84edo|84g]].
@@ Line 716: / Line 700: @@
 Subgroup: 2.3.5.11
-Comma list: 8019/8000, 16384/16335
+Comma list: 6912/6875, 8019/8000
-Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}
-: mapping generators: ~2, ~128/99
+{{Mapping|legend=2| 1 -1 -1 9 | 0 7 9 -15 }}
 Optimal tunings:
@@ Line 735: / Line 717: @@
 Comma list: 256/255, 1089/1088, 1377/1375
-Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
+{{Mapping|legend=2| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
-: mapping generators: ~2, ~22/17
 Optimal tunings:
@@ Line 752: / Line 732: @@
 Comma list: 256/255, 576/575, 1089/1088, 1377/1375
-Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
+{{Mapping|legend=2| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
 Optimal tunings:
@@ Line 767: / Line 747: @@
 Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
-Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
+{{Mapping|legend=2| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
 Optimal tunings:
@@ Line 777: / Line 757: @@
 Badness (Sintel): 0.490
+=== Sendai (2.3.5.23) ===
+{{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). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence. It is ideal in tuning for ambiguating the generator's 2.3.5.31 interpretation as 40/31~31/24, being between these (and hence also a good tuning of 2.3.5.31 sensipent at the cost of some 5-limit accuracy).
+{{Todo|inline=1|complete section|comment=Add the data for the intermediate subgroups. }}
+==== 2.3.5.23.29.31 subgroup ====
+Subgroup: 2.3.5.23.29.31
+Comma list: 465/464, 576/575, 621/620, 900/899
+{{Mapping|legend=2| 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:Sensipent family| ]] <!-- main article -->
 [[Category:Temperament families]]
-[[Category:Sensipent family| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Rank 2]]