Sensipent family: Difference between revisions

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: mapping generators: ~2, ~162/125

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9429{c}}, ~9/7 = 443.0364{{c}}

* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}

: [[error map]]: {{val| -0.057 -0.643 +1.071 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.0507{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}

: error map: {{val| 0.000 -0.600 +1.143 }}

~~~~

{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}

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

~~=== 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]].

~~Subgroup: 2.3.5.31~~

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

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

~~: mapping generators: ~2, ~31/24~~

~~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 (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~~

~~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]] is equivalent to the one by~~ tempering out [[256/255~~|S16~~]] ~~= [[~~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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* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}

~~~~

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* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}

~~~~

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* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}

~~~~

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* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}

~~<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.183{{c}}~~

* CEE: ~2 = 1200.000{{c}}, ~22/17 = 443.115{{c}} -->

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Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the 19 &~~amp;~~ 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.

Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.

=== Septimal sensi ===

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~~: mapping generators: ~2, ~9/7~~

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.7081{c}}, ~9/7 = 443.2748{{c}}

* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}

: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}

: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}

~~<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3166{{c}}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~9/7 = 443.383{{c}} -->

[[Minimax tuning]]:

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

~~: mapping generators: ~2, ~9/7~~

Optimal tunings:

* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}

~~~~

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Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}

~~: mapping generators: ~2, ~9/7~~

Optimal tunings:

* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2987{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.294{{c}} -->

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* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3658{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.321{{c}} -->

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* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3775{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.365{{c}} -->

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Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}

~~: mapping generators: ~2, ~9/7~~

Optimal tunings:

* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4783{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.626{{c}} -->

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* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5075{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.559{{c}} -->

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* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5050{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.551{{c}} -->

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Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}

~~: mapping generators: ~2, ~9/7~~

Optimal tunings:

* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.1886{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.962{{c}} -->

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* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2863{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.945{{c}} -->

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

~~: mapping generators: ~2, ~9/7~~

Optimal tunings:

* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.7814{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.518{{c}} -->

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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:

* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.7877{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.506{{c}} -->

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=== Bisensi ===

Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).

Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).

Subgroup: 2.3.5.7.11

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Comma list: 121/120, 126/125, 245/243

Mapping:

Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}

* common form: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}

:: mapping generators: ~99/70, ~9/7

: mapping generators: ~99/70, ~9/7

* mingen form: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}

~~:: mapping generators: ~99/70, ~11/10~~

Optimal tunings:

* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})

* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})

~~<!-- * CTE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3688{{c}} (~11/10 = 156.6312{{c}})~~

* POTE: ~99/70 = 600.000{{c}}, ~9/7 = 443.308{{c}} (~11/10 = 156.692{{c}}) -->

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Comma list: 91/90, 121/120, 126/125, 169/168

Mapping:

Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}

* common form: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10}}

~~:: mapping generators: ~55/39, ~9/7~~

* mingen form: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}

~~:: mapping generators: ~55/39, ~11/10~~

Optimal tunings:

* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})

* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})

~~<!-- * CTE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.4416{{c}} (~11/10 = 156.5584{{c}})~~

* POTE: ~55/39 = 600.000{{c}}, ~9/7 = 443.275{{c}} (~11/10 = 156.725{{c}}) -->

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Comma list: 91/90, 121/120, 126/125, 154/153, 169/168

Mapping:

Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}

* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}

~~:: mapping generators: ~17/12, ~9/7~~

* mingen form: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}

~~:: mapping generators: ~17/12, ~11/10~~

Optimal tunings:

* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})

* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})

~~~~

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=== Hemisensi ===

Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-~~tetradecacot~~ (pergen (P8, ccP5/14)).

Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-14-cot (pergen (P8, ccP5/14)).

Subgroup: 2.3.5.7.11

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* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5981{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.605{{c}} -->

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* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}

~~<!-- * CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6333{{c}}~~

* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.556{{c}} -->

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~~: mapping generators: ~2, ~162/125~~

[[Optimal tuning]]s:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}

: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}

~~~~

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~~: mapping generators: ~2, ~162/125~~

[[Optimal tuning]]s:

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}

: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}

~~~~

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Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}

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

Optimal tunings:

* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}

~~~~

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Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}

~~: mapping generators: ~2, ~84/65~~

Optimal tunings:

* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}

~~~~

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Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}

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

Optimal tunings:

* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}

~~~~

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== Bison ==

Bison has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].

Bison has a 1/2-octave period and the generator can be taken as ~162/125 or its semi-octave complement, ~35/32. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].

[[Subgroup]]: 2.3.5.7

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[[Comma list]]: 6144/6125, 78732/78125

~~[[Mapping]]:~~

* common form: {{~~mapping~~| 2 -2 -2 13 | 0 7 9 -10}}

: mapping generators: ~567/400, ~162/125

:: mapping generators: ~567/400, ~162/125

* mingen form: {{mapping| 2 5 7 3 | 0 -7 -9 10 }}

~~:: mapping generators: ~567/400, ~35/32~~

[[Optimal tuning]]s:

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* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})

: error map: {{val| 0.000 -0.446 +1.341 +0.446 }}

~~~~

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Comma list: 441/440, 6144/6125, 8019/8000

Mapping:

Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}

* common form: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}

~~:: mapping generators: ~567/400, ~162/125~~

* mingen form: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}

~~:: mapping generators: ~567/400, ~35/32~~

Optimal tunings:

* WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{c}})

* CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})

~~~~

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Comma list: 351/350, 364/363, 441/440, 10985/10976

Mapping:

Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}

* common form: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}

~~:: mapping generators: ~55/39, ~162/125~~

* mingen form: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}

~~:: mapping generators: ~55/39, ~35/32~~

Optimal tunings:

* WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{c}})

* CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})

~~~~

{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}

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== Subpental ==

Subpental splits the generator ~14/9 in two. Its ploidacot is theta-~~tetradecacot~~ (pergen (P8, c4P4/14)).

Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c4P4/14)).

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~45/28

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}

: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}

~~~~

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* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}

~~~~

{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}

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* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}

~~~~

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== Heinz ==

Heinz splits the generator ~18/7 in three. Its ploidacot is theta-21-cot (pergen (P8, c9P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).

Heinz splits the sensipent generator ~324/125 in three. Its ploidacot is theta-21-cot (pergen (P8, c9P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~2, ~48/35

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Line 586:

* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}

: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}

~~~~

Line 739:

Line 603:

* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}

~~~~

Line 755:

Line 618:

* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}

~~~~

Line 771:

Line 633:

* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}

~~~~

Line 787:

Line 648:

* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}

~~~~

Line 801:

Line 661:

: mapping generators: ~63/50, ~36/35

Line 809:

Line 668:

* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}

: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}

~~~~

Line 825:

Line 683:

* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}

* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}

~~~~

Line 843:

Line 700:

* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}

* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}

~~~~

Line 859:

Line 715:

* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}

* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}

~~~~

Line 875:

Line 730:

* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}

* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}

~~~~

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 13: / Line 13: @@
 {{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
 : mapping generators: ~2, ~162/125
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.9429{c}}, ~9/7 = 443.0364{{c}}
+* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
 : [[error map]]: {{val| -0.057 -0.643 +1.071 }}
-* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.0507{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
 : error map: {{val| 0.000 -0.600 +1.143 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 443.058{{c}} -->
 {{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}
@@ Line 37: / Line 35: @@
 Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
-=== 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]].
-Subgroup: 2.3.5.31
-Comma list: 961/960, 2511/2500
-Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
-: mapping generators: ~2, ~31/24
-Optimal tunings:
-* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
-* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~31/24 = 443.050{{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 (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
-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}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~31/24 = 442.989{{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]] is equivalent to the one by tempering out [[256/255|S16]] = [[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 98: / Line 56: @@
 * WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~128/99 = 443.115{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
@@ Line 116: / Line 73: @@
 * WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.188{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
@@ Line 132: / Line 88: @@
 * WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.185{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
@@ Line 148: / Line 103: @@
 * WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
-<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.183{{c}}
-* CEE: ~2 = 1200.000{{c}}, ~22/17 = 443.115{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
@@ Line 158: / Line 111: @@
 {{Main| Sensi }}
-Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the 19 &amp; 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.
+Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.
 === Septimal sensi ===
@@ Line 166: / Line 119: @@
 {{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
-: mapping generators: ~2, ~9/7
 [[Optimal tuning]]s:
-* [[WE]]: ~2 = 1199.7081{c}}, ~9/7 = 443.2748{{c}}
+* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
 : [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
 : error map: {{val| 0.000 +1.490 +3.830 -5.285 }}
-<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3166{{c}}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~9/7 = 443.383{{c}} -->
 [[Minimax tuning]]:
@@ Line 201: / Line 150: @@
 Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
-: mapping generators: ~2, ~9/7
 Optimal tunings:
 * WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4016{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
@@ Line 219: / Line 165: @@
 Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
-: mapping generators: ~2, ~9/7
 Optimal tunings:
 * WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2987{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.294{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}
@@ Line 242: / Line 184: @@
 * WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3658{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.321{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
@@ Line 259: / Line 199: @@
 * WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3775{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.365{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 27, 46 }}
@@ Line 272: / Line 210: @@
 Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
-: mapping generators: ~2, ~9/7
 Optimal tunings:
 * WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4783{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.626{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}
@@ Line 295: / Line 229: @@
 * WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5075{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.559{{c}} -->
 {{Optimal ET sequence|legend=0| 19e, 27e, 46 }}
@@ Line 312: / Line 244: @@
 * WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5050{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.551{{c}} -->
 {{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}
@@ Line 325: / Line 255: @@
 Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
-: mapping generators: ~2, ~9/7
 Optimal tunings:
 * WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.1886{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.962{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 19, 27e }}
@@ Line 348: / Line 274: @@
 * WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2863{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.945{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 19, 27e }}
@@ Line 361: / Line 285: @@
 Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
-: mapping generators: ~2, ~9/7
 Optimal tunings:
 * WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.7814{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.518{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
@@ Line 379: / 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:
 * WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.7877{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.506{{c}} -->
 {{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
@@ Line 392: / Line 310: @@
 === Bisensi ===
-Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).
+Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).
 Subgroup: 2.3.5.7.11
@@ Line 398: / Line 316: @@
 Comma list: 121/120, 126/125, 245/243
-Mapping:
+Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
-* common form: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
-:: mapping generators: ~99/70, ~9/7
+: mapping generators: ~99/70, ~9/7
-* mingen form: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
-:: mapping generators: ~99/70, ~11/10
 Optimal tunings:
 * WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
 * CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})
-<!-- * CTE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3688{{c}} (~11/10 = 156.6312{{c}})
-* POTE: ~99/70 = 600.000{{c}}, ~9/7 = 443.308{{c}} (~11/10 = 156.692{{c}}) -->
 {{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}
@@ Line 419: / Line 333: @@
 Comma list: 91/90, 121/120, 126/125, 169/168
-Mapping:
+Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}
-* common form: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10}}
-:: mapping generators: ~55/39, ~9/7
-* mingen form: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
-:: mapping generators: ~55/39, ~11/10
 Optimal tunings:
 * WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
 * CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})
-<!-- * CTE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.4416{{c}} (~11/10 = 156.5584{{c}})
-* POTE: ~55/39 = 600.000{{c}}, ~9/7 = 443.275{{c}} (~11/10 = 156.725{{c}}) -->
 {{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
@@ Line 440: / Line 348: @@
 Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
-Mapping:
+Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}
-* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}
-:: mapping generators: ~17/12, ~9/7
-* mingen form: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
-:: mapping generators: ~17/12, ~11/10
 Optimal tunings:
 * WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})
 * CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})
-<!-- * CTE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.4466{{c}} (~11/10 = 156.5534{{c}}) -->
 {{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
@@ Line 456: / Line 359: @@
 === Hemisensi ===
-Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-tetradecacot (pergen (P8, ccP5/14)).
+Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-14-cot (pergen (P8, ccP5/14)).
 Subgroup: 2.3.5.7.11
@@ Line 469: / Line 372: @@
 * WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5981{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.605{{c}} -->
 {{Optimal ET sequence|legend=0| 27e, 38d, 65 }}
@@ Line 486: / Line 387: @@
 * WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}
-<!-- * CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6333{{c}}
-* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.556{{c}} -->
 {{Optimal ET sequence|legend=0| 27e, 38df, 65f }}
@@ Line 499: / Line 398: @@
 {{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
-: mapping generators: ~2, ~162/125
 [[Optimal tuning]]s:
@@ Line 507: / Line 404: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
 : error map: {{val| 0.000 -2.466 -1.256 +0.267 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 442.755{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}
@@ Line 519: / Line 415: @@
 {{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}
-: mapping generators: ~2, ~162/125
 [[Optimal tuning]]s:
@@ Line 527: / Line 421: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
 : error map: {{val| 0.000 +1.088 +3.313 +2.544 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 443.289{{c}} -->
 {{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}
@@ Line 539: / Line 432: @@
 Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}
-: mapping generators: ~2, ~128/99
 Optimal tunings:
 * WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~128/99 = 443.274{{c}} -->
 {{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}
@@ Line 557: / Line 447: @@
 Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}
-: mapping generators: ~2, ~84/65
 Optimal tunings:
 * WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~84/65 = 443.270{{c}} -->
 {{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}
@@ Line 575: / Line 462: @@
 Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}
-: mapping generators: ~2, ~22/17
 Optimal tunings:
 * WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~22/17 = 443.270{{c}} -->
 {{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}
@@ Line 588: / Line 472: @@
 == Bison ==
-Bison has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].
+Bison has a 1/2-octave period and the generator can be taken as ~162/125 or its semi-octave complement, ~35/32. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].
 [[Subgroup]]: 2.3.5.7
@@ Line 594: / Line 478: @@
 [[Comma list]]: 6144/6125, 78732/78125
-[[Mapping]]:
+{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
-* common form: {{mapping| 2 -2 -2 13 | 0 7 9 -10}}
+: mapping generators: ~567/400, ~162/125
-:: mapping generators: ~567/400, ~162/125
-* mingen form: {{mapping| 2 5 7 3 | 0 -7 -9 10 }}
-:: mapping generators: ~567/400, ~35/32
 [[Optimal tuning]]s:
@@ Line 605: / Line 486: @@
 * [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})
 : error map: {{val| 0.000 -0.446 +1.341 +0.446 }}
-<!-- * [[POTE]]: ~567/400 = 600.000{{c}}, ~162/125 = 443.075{{c}} (~35/32 = 156.925{{c}}) -->
 {{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
@@ Line 616: / Line 496: @@
 Comma list: 441/440, 6144/6125, 8019/8000
-Mapping:
+Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
-* common form: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
-:: mapping generators: ~567/400, ~162/125
-* mingen form: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}
-:: mapping generators: ~567/400, ~35/32
 Optimal tunings:
 * WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{c}})
 * CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})
-<!-- * POTE: ~99/70 = 600.000{{c}}, ~162/125 = 443.117{{c}} (~35/32 = 156.883{{c}}) -->
 {{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}
@@ Line 636: / Line 511: @@
 Comma list: 351/350, 364/363, 441/440, 10985/10976
-Mapping:
+Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
-* common form: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
-:: mapping generators: ~55/39, ~162/125
-* mingen form: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
-:: mapping generators: ~55/39, ~35/32
 Optimal tunings:
 * WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{c}})
 * CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})
-<!-- * POTE: ~55/39 = 600.000{{c}}, ~162/125 = 443.096{{c}} (~35/32 = 156.904{{c}}) -->
 {{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}
@@ Line 652: / Line 522: @@
 == Subpental ==
-Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c<sup>4</sup>P4/14)).
+Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c<sup>4</sup>P4/14)).
 [[Subgroup]]: 2.3.5.7
@@ Line 659: / Line 529: @@
 {{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
 : mapping generators: ~2, ~45/28
@@ Line 667: / Line 536: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
 : error map: {{val| 0.000 -0.531 +1.231 +0.036 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~45/28 = 821.533{{c}} -->
 {{Optimal ET sequence|legend=1| 19, …, 111, 130 }}
@@ Line 683: / Line 551: @@
 * WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~45/28 = 821.560{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}
@@ Line 699: / Line 566: @@
 * WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~45/28 = 821.563{{c}} -->
 {{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}
@@ Line 706: / Line 572: @@
 == Heinz ==
-Heinz splits the generator ~18/7 in three. Its ploidacot is theta-21-cot (pergen (P8, c<sup>9</sup>P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).
+Heinz splits the sensipent generator ~324/125 in three. Its ploidacot is theta-21-cot (pergen (P8, c<sup>9</sup>P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).
 [[Subgroup]]: 2.3.5.7
@@ Line 713: / Line 579: @@
 {{Mapping|legend=1| 1 -8 -10 6 | 0 21 27 -7 }}
 : mapping generators: ~2, ~48/35
@@ Line 721: / Line 586: @@
 * [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
 : error map: {{val| 0.000 -1.247 +0.311 -2.395 }}
-<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~48/35 = 546.815{{c}} -->
 {{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
@@ Line 739: / Line 603: @@
 * WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.631{{c}} -->
 {{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}
@@ Line 755: / Line 618: @@
 * WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.629{{c}} -->
 {{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}
@@ Line 771: / Line 633: @@
 * WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.635{{c}} -->
 {{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}
@@ Line 787: / Line 648: @@
 * WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
 * CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}
-<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.614{{c}} -->
 {{Optimal ET sequence|legend=0| 46, 103h, 149h }}
@@ Line 801: / Line 661: @@
 {{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
 : mapping generators: ~63/50, ~36/35
@@ Line 809: / Line 668: @@
 * [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
 : error map: {{val| 0.000 -0.359 +1.453 +3.515 }}
-<!-- * [[POTE]]: ~63/50 = 400.000{{c}}, ~36/35 = 43.147{{c}} -->
 {{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
@@ Line 825: / Line 683: @@
 * WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
 * CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}
-<!-- * POTE: ~63/50 = 400.000{{c}}, ~36/35 = 43.292{{c}} -->
 {{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}
@@ Line 843: / Line 700: @@
 * WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
 * CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}
-<!-- * POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.288{{c}} -->
 {{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
@@ Line 859: / Line 715: @@
 * WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
 * CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}
-<!-- * POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.276{{c}} -->
 {{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
@@ Line 875: / Line 730: @@
 * WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
 * CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}
-<!-- * POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.292{{c}} -->
 {{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
 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]]