Sensipent family: Difference between revisions

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Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125, seven make harmonic 6 and nine make harmonic 10.

Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125. Two generators make 5/3, seven make harmonic 6 and nine make harmonic 10. Its [[ploidacot]] is beta-heptacot ([[pergen]] (P8, ccP5/7)) and its color name is Sepguti.

The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These all use the same ~~nominal~~ generator as sensipent.

The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.

Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.

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== Sensipent ==

[[Subgroup]]: 2.3.5

[[Comma list]]: 78732/78125

: mapping generators: ~2, ~125~~/81~~

: mapping generators: ~2, ~162/125

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

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.058

{{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

=== 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

: mapping generators: ~2, ~31/24

{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}

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

[[Badness]] (Sintel): 0.243

=== Sendai ===

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

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

[[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.

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

: mapping generators: ~2, ~128/99

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

[[Badness]] (Sintel): 0.728

==== 2.3.5.11.17 subgroup ====

[[Subgroup]]: 2.3.5.11.17

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

: mapping generators: ~2, ~22/17

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

[[Badness]] (Sintel): 0.639

==== 2.3.5.11.17.23 subgroup ====

[[Subgroup]]: 2.3.5.11.17.23

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

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

[[Badness]] (Sintel): 0.555

==== 2.3.5.11.17.23.31 subgroup ====

[[Subgroup]]: 2.3.5.11.17.23.31

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

[[Optimal tuning]]s:

* [[CTE]]: 2/1 = 1\1, ~22/17 = 443.183

* [[CEE]]: 2/1 = 1\1, ~22/17 = 443.115

[[Badness]] (Sintel): 0.490

== Sensi ==

~~{{See also| Sensamagic clan #Sensi }}~~

Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the 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 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 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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[[Comma list]]: 126/125, 245/243

~~: mapping generators: ~2, ~14/9~~

~~{{Multival|legend=1|~~ 7 ~~9 13 -2 1 5 }}~~

: mapping generators: ~2, ~9/7

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~9/7 = 443.3166

[[Optimal tuning]]s:

* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166

* [[POTE]]: ~2 = 1200.000, ~9/7 = 443.383

[[Minimax tuning]]:

* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.7

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7

* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}

: [[Eigenmonzo basis|~~eigenmonzo (~~unchanged-interval) basis]]: 2.9/5

: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[Tuning ranges of regular temperaments|Tuning ranges]]:

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

~~Sval mapping~~: {{mapping| 1 ~~6 8 11 10 | 0~~ -7 -9 -~~13 -10 }}~~

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

~~Gencom mapping: {{mapping| 1 6 8 11~~ 0 10 | 0 -7 -9 -13 ~~0 -~~10 }}

: ~~gencom~~: [2 ~~14/~~9~~; 91/90 126/125 169~~/~~168]~~

: mapping generators: ~2, ~9/7

Optimal tuning (CTE): ~2 = ~~1\1~~, ~9/7 = 443.4016

Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016

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Comma list: 126/125, 245/243, 385/384

Mapping: {{mapping| 1 ~~6 8 11~~ -6 | 0 -7 -9 -13 15 }}

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

: mapping generators: ~2, ~9/7

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.2987

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.2987

* POTE: ~2 = 1200.000, ~9/7 = 443.294

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Comma list: 91/90, 126/125, 169/168, 385/384

Mapping: {{mapping| 1 ~~6 8 11~~ -~~6 10~~ | 0 -7 -9 -13 15 -10 }}

Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.3658

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.3658

* POTE: ~2 = 1200.000, ~9/7 = 443.321

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Comma list: 91/90, 126/125, 154/153, 169/168, 256/255

Mapping: {{mapping| 1 ~~6 8 11~~ -6 10 -6 | 0 -7 -9 -13 15 -10 16 }}

Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.3775

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.3775

* POTE: ~2 = 1200.000, ~9/7 = 443.365

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Comma list: 126/125, 176/175, 245/243

Mapping: {{mapping| 1 6 8 ~~11 23~~ | 0 -7 -9 -13 -31 }}

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

: mapping generators: ~2, ~9/7

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.4783

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.4783

* POTE: ~2 = 1200.000, ~9/7 = 443.626

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Comma list: 91/90, 126/125, 169/168, 352/351

Mapping: {{mapping| 1 6 8 ~~11 23 10~~ | 0 -7 -9 -13 -31 -10 }}

Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.5075

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.5075

* POTE: ~2 = 1200.000, ~9/7 = 443.559

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Comma list: 91/90, 126/125, 136/135, 154/153, 169/168

Mapping: {{mapping| 1 6 8 ~~11 23 10 23~~ | 0 -7 -9 -13 -31 -10 -30 }}

Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.5050

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.5050

* POTE: ~2 = 1200.000, ~9/7 = 443.551

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Comma list: 56/55, 100/99, 245/243

Mapping: {{mapping| 1 ~~6 8 11 6~~ | 0 -7 -9 -13 -4 }}

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

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.1886

: mapping generators: ~2, ~9/7

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.1886

* POTE: ~2 = 1200.000, ~9/7 = 443.962

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Comma list: 56/55, 78/77, 91/90, 100/99

Mapping: {{mapping| 1 ~~6 8 11 6 10~~ | 0 -7 -9 -13 -4 -10 }}

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

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.2863

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.2863

* POTE: ~2 = 1200.000, ~9/7 = 443.945

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Comma list: 55/54, 77/75, 99/98

Mapping: {{mapping| 1 ~~6 8 11 11~~ | 0 -7 -9 -13 -12 }}

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

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.7814

: mapping generators: ~2, ~9/7

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.7814

* POTE: ~2 = 1200.000, ~9/7 = 443.518

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

Mapping: {{mapping| 1 ~~6 8 11 11 11~~ | 0 -7 -9 -13 -12 -11 }}

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

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~9/7 = 443.7877

Optimal tunings:

* CTE: ~2 = 1200.000, ~9/7 = 443.7877

* POTE: ~2 = 1200.000, ~9/7 = 443.506

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=== Bisensi ===

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

Subgroup: 2.3.5.7.11

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

Mapping: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}

Mapping:

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

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

:: 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 ~~tuning~~ (~~CTE~~): ~99/70 = ~~1\2~~, ~11/10 = 156.~~6312~~

Optimal tunings:

* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)

* POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~11/10 = 156.692)

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

Mapping: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}

Mapping:

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

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

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

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

Optimal ~~tuning (~~CTE): ~55/39 = ~~1\2~~, ~11/10 = 156.5584

Optimal tunings:

* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584

* POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~11/10 = 156.725

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

Mapping: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}

Mapping:

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

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

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

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

Optimal ~~tuning (~~CTE): ~17/12 = ~~1\2~~, ~11/10 = 156.5534

Optimal tunings:

* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)

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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)).

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 245/242

Mapping: {{mapping| 1 ~~13 17 24 32~~ | 0 -14 -18 -26 -35 }}

Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}

: mapping generators: ~2, ~44/25

: mapping generators: ~2, ~25/22

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~25/22 = 221.5981

Optimal tunings:

* CTE: ~2 = 1200.000, ~25/22 = 221.5981

* POTE: ~2 = 1200.000, ~25/22 = 221.605

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Comma list: 91/90, 126/125, 169/168, 243/242

Mapping: {{mapping| 1 ~~13 17 24 32 30~~ | 0 -14 -18 -26 -35 ~~-30~~ }}

Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}

Optimal ~~tuning (~~CTE): ~2 = ~~1\1~~, ~25/22 = 221.6333

Optimal tunings:

* CTE: ~2 = 1200.000, ~25/22 = 221.6333

* POTE: ~2 = 1200.000, ~25/22 = 221.556

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[[Comma list]]: 225/224, 78732/78125

~~{{Multival|legend=1| 7 9 32 -~~2 ~~31 49 }}~~

: mapping generators: ~2, ~162/125

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~162/125 = 442.755

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755

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[[Comma list]]: 5120/5103, 78732/78125

~~{{Multival|legend=1| 7 9 -33 -~~2 ~~-72 -102 }}~~

: mapping generators: ~2, ~162/125

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~162/125 = 443.289

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289

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Comma list: 176/175, 1331/1323, 5120/5103

Mapping: {{mapping| 1 ~~6 8~~ -18 -6 | 0 -7 -9 33 15 }}

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

: mapping generators: ~2, ~128/99

Optimal tuning (POTE): ~2 = ~~1\1~~, ~128/99 = 443.274

Optimal tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274

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Comma list: 176/175, 351/350, 847/845, 1331/1323

Mapping: {{mapping| 1 ~~6 8~~ -18 -~~6 -19~~ | 0 -7 -9 33 15 36 }}

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

: mapping generators: ~2, ~84/65

Optimal tuning (POTE): ~2 = ~~1\1~~, ~84/65 = 443.270

Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270

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Comma list: 176/175, 256/255, 351/350, 442/441, 715/714

Mapping: {{mapping| 1 ~~6 8~~ -18 -~~6 -19 -6~~ | 0 -7 -9 33 15 36 16 }}

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

: mapping generators: ~2, ~22/17

Optimal tuning (POTE): ~2 = ~~1\1~~, ~22/17 = 443.270

Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270

{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}

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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]].

[[Subgroup]]: 2.3.5.7

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

[[Mapping]]:

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

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

:: 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]] ([[POTE]]): ~567/400 = ~~1\2~~, ~35/32 = 156.925

[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)

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

Mapping: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}

Mapping:

* 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 tuning (POTE): ~99/70 = ~~1\2~~, ~35/32 = 156.883

Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)

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

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

Mapping:

* 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 tuning (POTE): ~55/39 = ~~1\2~~, ~35/32 = 156.904

Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)

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== Subpental ==

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

[[Subgroup]]: 2.3.5.7

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~~{{Multival|legend=1| 14 18~~ 45 ~~-4 32 54 }}~~

: mapping generators: ~2, ~56/45

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~56/45 = 378.467

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~56/45 = 378.467

{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}

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Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~56/45 = 378.440

Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.440

{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}

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Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~56/45 = 378.437

Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.437

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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).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 78732/78125

~~: mapping generators: ~2, ~35/24~~

~~{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}~~

: mapping generators: ~2, ~48/35

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~48/35 = 546.815

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815

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Comma list: 385/384, 441/440, 78732/78125

~~Mapping:~~ {{~~mapping~~| 1 ~~13 17~~ -~~1 4~~ | 0 -21 -27 7 -1 }}

: mapping generators: ~2, ~11/8

Optimal tuning (POTE): ~2 = ~~1\1~~, ~11/8 = 547.631

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631

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Line 643:

Comma list: 351/350, 385/384, 441/440, 847/845

~~Mapping:~~ {{~~mapping~~| 1 ~~13 17~~ -~~1 4~~ -5 | 0 -21 -27 7 -1 16 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~11/8 = 547.629

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.629

Line 484:

Line 656:

Comma list: 273/272, 351/350, 385/384, 441/440, 847/845

~~Mapping:~~ {{~~mapping~~| 1 ~~13 17~~ -~~1 4~~ -5 3 | 0 -21 -27 7 -1 16 2 }}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~11/8 = 547.635

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.635

Line 497:

Line 669:

Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968

~~Mapping:~~ {{~~mapping~~| 1 ~~13 17~~ -~~1 4~~ -5 3 -5 | 0 -21 -27 7 -1 16 2 17 }}

{{Mapping|legend=1| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17}}

Optimal tuning (POTE): ~2 = ~~1\1~~, ~11/8 = 547.614

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614

Line 506:

Line 678:

== Trisensory ==

Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).

[[Subgroup]]: 2.3.5.7

Line 512:

Line 686:

~~{{Multival|legend=1| 21 27 12 -6 -40 -48 }}~~

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

[[Optimal tuning]] ([[POTE]]): ~63/50 = ~~1\3~~, ~36/35 = 43.147

[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147

Line 527:

Line 701:

Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}

Optimal tuning (POTE): ~63/50 = ~~1\3~~, ~36/35 = 43.292

Optimal tuning (POTE): ~63/50 = 400.000, ~36/35 = 43.292

Line 540:

Line 714:

Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}

Optimal tuning (POTE): ~49/39 = ~~1\3~~, ~36/35 = 43.288

: mapping generators: ~49/39, ~36/35

Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288

Line 553:

Line 729:

Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}

Optimal tuning (POTE): ~49/39 = ~~1\3~~, ~36/35 = 43.276

Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.276

Line 566:

Line 742:

Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}

Optimal tuning (POTE): ~49/39 = ~~1\3~~, ~36/35 = 43.292

Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.292

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[[Category:Temperament families]]

[[Category:Pages with mostly numerical content]]

[[Category:Sensipent family| ]]

~~[[Category:Sensipent| ]] ~~

[[Category:Rank 2]]

@@ Line 5: / Line 5: @@
 | ja =
 }}
-Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125, seven make harmonic 6 and nine make harmonic 10.
+{{Technical data page}}
+Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125. Two generators make 5/3, seven make harmonic 6 and nine make harmonic 10. Its [[ploidacot]] is beta-heptacot ([[pergen]] (P8, ccP5/7)) and its color name is Sepguti.
-The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These all use the same nominal generator as sensipent.
+The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.
 Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.
@@ Line 18: / Line 19: @@
 == Sensipent ==
+{{Main| Sensipent }}
 [[Subgroup]]: 2.3.5
 [[Comma list]]: 78732/78125
-{{Mapping|legend=1| 1 6 8 | 0 -7 -9 }}
+{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
-: mapping generators: ~2, ~125/81
+: mapping generators: ~2, ~162/125
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, 162/125 = 443.058
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.058
 {{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
+=== 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
+{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
+: mapping generators: ~2, ~31/24
+{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
+[[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
+{{Mapping|legend=1| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
+{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989
+[[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.
+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
+{{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
+: mapping generators: ~2, ~128/99
+{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115
+[[Badness]] (Sintel): 0.728
+==== 2.3.5.11.17 subgroup ====
+[[Subgroup]]: 2.3.5.11.17
+[[Comma list]]: 256/255, 1089/1088, 1377/1375
+{{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
+: mapping generators: ~2, ~22/17
+{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.188
+[[Badness]] (Sintel): 0.639
+==== 2.3.5.11.17.23 subgroup ====
+[[Subgroup]]: 2.3.5.11.17.23
+[[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 }}
+{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185
+[[Badness]] (Sintel): 0.555
+==== 2.3.5.11.17.23.31 subgroup ====
+[[Subgroup]]: 2.3.5.11.17.23.31
+[[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 }}
+{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
+[[Optimal tuning]]s:
+* [[CTE]]: 2/1 = 1\1, ~22/17 = 443.183
+* [[CEE]]: 2/1 = 1\1, ~22/17 = 443.115
+[[Badness]] (Sintel): 0.490
 == Sensi ==
 {{Main| Sensi }}
-{{See also| Sensamagic clan #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 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 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.
 === Septimal sensi ===
@@ Line 43: / Line 146: @@
 [[Comma list]]: 126/125, 245/243
-{{Mapping|legend=1| 1 6 8 11 | 0 -7 -9 -13 }}
+{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
-: mapping generators: ~2, ~14/9
-{{Multival|legend=1| 7 9 13 -2 1 5 }}
+: mapping generators: ~2, ~9/7
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~9/7 = 443.3166
+[[Optimal tuning]]s:
+* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
+* [[POTE]]: ~2 = 1200.000, ~9/7 = 443.383
 [[Minimax tuning]]:
 * [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
 * [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
-: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
+: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
 [[Tuning ranges of regular temperaments|Tuning ranges]]:
@@ Line 74: / Line 177: @@
 Comma list: 91/90, 126/125, 169/168
-Sval mapping: {{mapping| 1 6 8 11 10 | 0 -7 -9 -13 -10 }}
+Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
-Gencom mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 }}
-: gencom: [2 14/9; 91/90 126/125 169/168]
+: mapping generators: ~2, ~9/7
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
+Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
@@ Line 89: / Line 190: @@
 Comma list: 126/125, 245/243, 385/384
-Mapping: {{mapping| 1 6 8 11 -6 | 0 -7 -9 -13 15 }}
+Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
+: mapping generators: ~2, ~9/7
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.2987
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.2987
+* POTE: ~2 = 1200.000, ~9/7 = 443.294
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
@@ Line 102: / Line 207: @@
 Comma list: 91/90, 126/125, 169/168, 385/384
-Mapping: {{mapping| 1 6 8 11 -6 10 | 0 -7 -9 -13 15 -10 }}
+Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.3658
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.3658
+* POTE: ~2 = 1200.000, ~9/7 = 443.321
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
@@ Line 115: / Line 222: @@
 Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
-Mapping: {{mapping| 1 6 8 11 -6 10 -6 | 0 -7 -9 -13 15 -10 16 }}
+Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.3775
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.3775
+* POTE: ~2 = 1200.000, ~9/7 = 443.365
 {{Optimal ET sequence|legend=1| 19, 27, 46 }}
@@ Line 128: / Line 237: @@
 Comma list: 126/125, 176/175, 245/243
-Mapping: {{mapping| 1 6 8 11 23 | 0 -7 -9 -13 -31 }}
+Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
+: mapping generators: ~2, ~9/7
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4783
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.4783
+* POTE: ~2 = 1200.000, ~9/7 = 443.626
 {{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
@@ Line 141: / Line 254: @@
 Comma list: 91/90, 126/125, 169/168, 352/351
-Mapping: {{mapping| 1 6 8 11 23 10 | 0 -7 -9 -13 -31 -10 }}
+Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.5075
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.5075
+* POTE: ~2 = 1200.000, ~9/7 = 443.559
 {{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
@@ Line 154: / Line 269: @@
 Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
-Mapping: {{mapping| 1 6 8 11 23 10 23 | 0 -7 -9 -13 -31 -10 -30 }}
+Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.5050
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.5050
+* POTE: ~2 = 1200.000, ~9/7 = 443.551
 {{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
@@ Line 167: / Line 284: @@
 Comma list: 56/55, 100/99, 245/243
-Mapping: {{mapping| 1 6 8 11 6 | 0 -7 -9 -13 -4 }}
+Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.1886
+: mapping generators: ~2, ~9/7
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.1886
+* POTE: ~2 = 1200.000, ~9/7 = 443.962
 {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
@@ Line 180: / Line 301: @@
 Comma list: 56/55, 78/77, 91/90, 100/99
-Mapping: {{mapping| 1 6 8 11 6 10 | 0 -7 -9 -13 -4 -10 }}
+Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.2863
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.2863
+* POTE: ~2 = 1200.000, ~9/7 = 443.945
 {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
@@ Line 193: / Line 316: @@
 Comma list: 55/54, 77/75, 99/98
-Mapping: {{mapping| 1 6 8 11 11 | 0 -7 -9 -13 -12 }}
+Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.7814
+: mapping generators: ~2, ~9/7
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.7814
+* POTE: ~2 = 1200.000, ~9/7 = 443.518
 {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
@@ Line 206: / Line 333: @@
 Comma list: 55/54, 66/65, 77/75, 143/140
-Mapping: {{mapping| 1 6 8 11 11 11 | 0 -7 -9 -13 -12 -11 }}
+Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}
-Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.7877
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~9/7 = 443.7877
+* POTE: ~2 = 1200.000, ~9/7 = 443.506
 {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
@@ Line 215: / Line 344: @@
 === Bisensi ===
+Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).
 Subgroup: 2.3.5.7.11
 Comma list: 121/120, 126/125, 245/243
-Mapping: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
+Mapping:
+* common form: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
-: mapping generators: ~99/70, ~11/10
+:: 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 tuning (CTE): ~99/70 = 1\2, ~11/10 = 156.6312
+Optimal tunings:
+* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
+* POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~11/10 = 156.692)
 {{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
@@ Line 234: / Line 369: @@
 Comma list: 91/90, 121/120, 126/125, 169/168
-Mapping: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
+Mapping:
+* common form: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10}}
+:: mapping generators: ~99/70, ~9/7
+* mingen form: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
+:: mapping generators: ~99/70, ~11/10
-Optimal tuning (CTE): ~55/39 = 1\2, ~11/10 = 156.5584
+Optimal tunings:
+* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
+* POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~11/10 = 156.725
 {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
@@ Line 247: / Line 388: @@
 Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
-Mapping: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
+Mapping:
+* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}
+:: mapping generators: ~99/70, ~9/7
+* mingen form: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
+:: mapping generators: ~99/70, ~11/10
-Optimal tuning (CTE): ~17/12 = 1\2, ~11/10 = 156.5534
+Optimal tunings:
+* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)
 {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
@@ Line 256: / Line 402: @@
 === Hemisensi ===
+Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-tetradecacot (pergen (P8, ccP5/14)).
 Subgroup: 2.3.5.7.11
 Comma list: 126/125, 243/242, 245/242
-Mapping: {{mapping| 1 13 17 24 32 | 0 -14 -18 -26 -35 }}
+Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}
-: mapping generators: ~2, ~44/25
+: mapping generators: ~2, ~25/22
-Optimal tuning (CTE): ~2 = 1\1, ~25/22 = 221.5981
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~25/22 = 221.5981
+* POTE: ~2 = 1200.000, ~25/22 = 221.605
 {{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
@@ Line 275: / Line 425: @@
 Comma list: 91/90, 126/125, 169/168, 243/242
-Mapping: {{mapping| 1 13 17 24 32 30 | 0 -14 -18 -26 -35 -30 }}
+Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}
-Optimal tuning (CTE): ~2 = 1\1, ~25/22 = 221.6333
+Optimal tunings:
+* CTE: ~2 = 1200.000, ~25/22 = 221.6333
+* POTE: ~2 = 1200.000, ~25/22 = 221.556
 {{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
@@ Line 288: / Line 440: @@
 [[Comma list]]: 225/224, 78732/78125
-{{Mapping|legend=1| 1 6 8 23 | 0 -7 -9 -32 }}
+{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
-{{Multival|legend=1| 7 9 32 -2 31 49 }}
+: mapping generators: ~2, ~162/125
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 442.755
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755
 {{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
@@ Line 303: / Line 455: @@
 [[Comma list]]: 5120/5103, 78732/78125
-{{Mapping|legend=1| 1 6 8 -18 | 0 -7 -9 33 }}
+{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}
-{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
+: mapping generators: ~2, ~162/125
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289
 {{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
@@ Line 318: / Line 470: @@
 Comma list: 176/175, 1331/1323, 5120/5103
-Mapping: {{mapping| 1 6 8 -18 -6 | 0 -7 -9 33 15 }}
+Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15}}
+: mapping generators: ~2, ~128/99
-Optimal tuning (POTE): ~2 = 1\1, ~128/99 = 443.274
+Optimal tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274
 {{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
@@ Line 331: / Line 485: @@
 Comma list: 176/175, 351/350, 847/845, 1331/1323
-Mapping: {{mapping| 1 6 8 -18 -6 -19 | 0 -7 -9 33 15 36 }}
+Mapping: {{mapping| 1 -1 -1 15 9 17| 0 7 9 -33 -15 -36}}
+: mapping generators: ~2, ~84/65
-Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
+Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270
 {{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
@@ Line 344: / Line 500: @@
 Comma list: 176/175, 256/255, 351/350, 442/441, 715/714
-Mapping: {{mapping| 1 6 8 -18 -6 -19 -6 | 0 -7 -9 33 15 36 16 }}
+Mapping: {{mapping| 1 -1 -1 15 9 17 10| 0 7 9 -33 -15 -36 -16}}
+: mapping generators: ~2, ~22/17
-Optimal tuning (POTE): ~2 = 1\1, ~22/17 = 443.270
+Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270
 {{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
@@ Line 353: / Line 511: @@
 == 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]].
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 6144/6125, 78732/78125
-{{Mapping|legend=1| 2 5 7 3 | 0 -7 -9 10 }}
+[[Mapping]]:
+* common form: {{mapping| 2 -2 -2 13 | 0 7 9 -10}}
-: mapping generators: ~567/400, ~35/32
+:: mapping generators: ~567/400, ~162/125
+* mingen form: {{mapping| 2 5 7 3 | 0 -7 -9 10 }}
-{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
+:: mapping generators: ~567/400, ~35/32
-[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~35/32 = 156.925
+[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)
 {{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
@@ Line 374: / Line 534: @@
 Comma list: 441/440, 6144/6125, 8019/8000
-Mapping: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}
+Mapping:
+* 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 tuning (POTE): ~99/70 = 1\2, ~35/32 = 156.883
+Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)
 {{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
@@ Line 387: / Line 551: @@
 Comma list: 351/350, 364/363, 441/440, 10985/10976
-Mapping: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
+Mapping:
+* 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 tuning (POTE): ~55/39 = 1\2, ~35/32 = 156.904
+Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)
 {{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}
@@ Line 396: / Line 564: @@
 == Subpental ==
+Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c<sup>4</sup>P4/14)).
 [[Subgroup]]: 2.3.5.7
@@ Line 402: / Line 572: @@
 {{Mapping|legend=1| 1 6 8 17 | 0 -14 -18 -45 }}
-{{Multival|legend=1| 14 18 45 -4 32 54 }}
+: mapping generators: ~2, ~56/45
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 378.467
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~56/45 = 378.467
 {{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
@@ Line 417: / Line 587: @@
 Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}
-Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.440
+Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.440
 {{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
@@ Line 430: / Line 600: @@
 Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}
-Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.437
+Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.437
 {{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
@@ Line 437: / Line 607: @@
 == 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).
 [[Subgroup]]: 2.3.5.7
 [[Comma list]]: 1029/1024, 78732/78125
-{{Mapping|legend=1| 1 13 17 -1 | 0 -21 -27 7 }}
+{{Mapping|legend=1| 1 -8 -10 6| 0 21 27 -7 }}
-: mapping generators: ~2, ~35/24
-{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
+: mapping generators: ~2, ~48/35
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 546.815
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815
 {{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
@@ Line 458: / Line 628: @@
 Comma list: 385/384, 441/440, 78732/78125
-Mapping: {{mapping| 1 13 17 -1 4 | 0 -21 -27 7 -1 }}
+{{Mapping|legend=1| 1 -8 -10 6 3 | 0 21 27 -7 1}}
+: mapping generators: ~2, ~11/8
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631
 {{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
@@ Line 471: / Line 643: @@
 Comma list: 351/350, 385/384, 441/440, 847/845
-Mapping: {{mapping| 1 13 17 -1 4 -5 | 0 -21 -27 7 -1 16 }}
+{{Mapping|legend=1| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.629
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.629
 {{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
@@ Line 484: / Line 656: @@
 Comma list: 273/272, 351/350, 385/384, 441/440, 847/845
-Mapping: {{mapping| 1 13 17 -1 4 -5 3 | 0 -21 -27 7 -1 16 2 }}
+{{Mapping|legend=1| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.635
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.635
 {{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
@@ Line 497: / Line 669: @@
 Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
-Mapping: {{mapping| 1 13 17 -1 4 -5 3 -5 | 0 -21 -27 7 -1 16 2 17 }}
+{{Mapping|legend=1| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17}}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.614
+Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614
 {{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
@@ Line 506: / Line 678: @@
 == Trisensory ==
+Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).
 [[Subgroup]]: 2.3.5.7
@@ Line 512: / Line 686: @@
 {{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
-{{Multival|legend=1| 21 27 12 -6 -40 -48 }}
+: mapping generators: ~63/50, ~36/35
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~36/35 = 43.147
+[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147
 {{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
@@ Line 527: / Line 701: @@
 Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
-Optimal tuning (POTE): ~63/50 = 1\3, ~36/35 = 43.292
+Optimal tuning (POTE): ~63/50 = 400.000, ~36/35 = 43.292
 {{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
@@ Line 540: / Line 714: @@
 Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
-Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
+: mapping generators: ~49/39, ~36/35
+Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288
 {{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
@@ Line 553: / Line 729: @@
 Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
-Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.276
+Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.276
 {{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
@@ Line 566: / Line 742: @@
 Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}
-Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.292
+Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.292
 {{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
@@ Line 573: / Line 749: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Sensipent family| ]] <!-- main article -->
-[[Category:Sensipent| ]] <!-- key article -->
 [[Category:Rank 2]]