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. 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.

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== Sensipent ==

[[Subgroup]]: 2.3.5

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: 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

~~Badness (Dirichlet): 0.826~~

=== 2.3.5.31 subgroup ===

Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one with prime 31 by interpreting the generator accurately as [[31/24]]~[[40/31]], tempering out [[961/960|S31 = 961/960]], so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].

=== 2.3.5.31 ===

Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one to prime 31 by interpreting the generator accurately as [[40/31]]~[[31~~/24~~]] by tempering [[961/960|S31 = 961/960]], so that the ~~large~~ [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].

[[Subgroup]]: 2.3.5.31

[[Comma list]]: ~~78732~~/~~78125~~, ~~961~~/~~960~~

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

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

[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~31/24 = 443.050

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

Badness (~~Dirichlet~~): 0.243

[[Badness]] (Sintel): 0.243

=== Sendai ===

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

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

[[Subgroup]]: 2.3.5.23.29.31

[[Comma list]]: 465/464, 576/575, 621/620, ~~729~~/~~725~~

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

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~31/24 = 442.989

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

Badness (~~Dirichlet~~): 0.283

[[Badness]] (Sintel): 0.283

=== Sensible ===

An extension of sensipent to prime 11 of dubious canonicity (but significantly higher accuracy than [[sensi]]) interprets the generator as [[165/128]]~[[128/99]] by tempering [[8019/8000~~|S9/S10~~]] so that [[11/8]] is reached as ([[10/9]])3. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation: that it ~~is equivalent to tempering~~ the [[semiporwellisma]] which is equal to [[961/960|S31]] * [[1024/1023|S32]]2 (thus forming the S-expression-based comma list). The ~~equivalence~~ of the ~~aforementioned~~ [[lopsided comma]] ~~also~~ implies that this temperament equates ([[33/32]])2 with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension to prime 17 is reasonably accurate through tempering [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.

Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])3. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[961/960|S31]] × [[1024/1023|S32]]2 (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])2 with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.

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

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

[[Subgroup]]: 2.3.5.11

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

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

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~128/99 = 443.115

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

Badness (~~Dirichlet~~): 0.728

[[Badness]] (Sintel): 0.728

==== 2.3.5.11.17 ====

==== 2.3.5.11.17 subgroup ====

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

[[Subgroup]]: 2.3.5.11.17

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

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

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~22/17 = 443.188

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

Badness (~~Dirichlet~~): 0.639

[[Badness]] (Sintel): 0.639

==== 2.3.5.11.17.23 ====

==== 2.3.5.11.17.23 subgroup ====

[[Subgroup]]: 2.3.5.11.17.23

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

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

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[[Optimal tuning]] ([[CTE]]): ~2 = ~~1\1~~, ~22/17 = 443.185

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

Badness (~~Dirichlet~~): 0.555

[[Badness]] (Sintel): 0.555

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

Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony without the need for interpreting it as high-damage [[~]][[9/7]] or [[~]][[13/10]] intervals. Its S-expression-based comma list is {([[256/255|S16]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495]] = S31 * S32 and [[528/527]] = S32 * S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])2 = ([[17/15]])/([[33/31]])2. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].

==== 2.3.5.11.17.23.31 subgroup ====

[[Subgroup]]: 2.3.5.11.17.23.31

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

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

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[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~22/17 = 443.~~183~~

[[Optimal tuning]]s:

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

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

Badness (~~Dirichlet~~): 0.490

[[Badness]] (Sintel): 0.490

== Sensi ==

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: mapping generators: ~2, ~9/7

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

[[Optimal tuning]]s:

* [[CTE]]: ~2 = ~~1\1~~, ~9/7 = 443.3166

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

* [[POTE]]: ~2 = ~~1\1~~, ~9/7 = 443.383

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

~~{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}~~

Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.2987

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.294

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

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

~~{{Multival|legend=1| 7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90 }}~~

Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.3658

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.321

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

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.3775

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.365

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

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: mapping generators: ~2, ~9/7

~~{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}~~

Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.4783

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.626

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

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

~~{{Multival|legend=1| 7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80 }}~~

Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.5075

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.559

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

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.5050

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.551

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

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: mapping generators: ~2, ~9/7

~~{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}~~

Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.1886

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.962

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

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

~~{{Multival|legend=1| 7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20 }}~~

Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.2863

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.945

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

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.7814

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.518

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

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~9/7 = 443.7877

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

* POTE: ~2 = ~~1\1~~, ~9/7 = 443.506

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

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Optimal tunings:

* CTE: ~99/70 = ~~1\2~~, ~9/7 = 443.3688 (~11/10 = 156.6312)

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

* POTE: ~99/70 = ~~1\2~~, ~9/7 = 443.308 (~11/10 = 156.692)

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

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Optimal tunings:

* CTE: ~55/39 = ~~1\2~~, ~9/7 = 443.4416, ~11/10 = 156.5584

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

* POTE: ~55/39 = ~~1\2~~, ~9/7 = 443.275, ~11/10 = 156.725

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

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Optimal tunings:

* CTE: ~17/12 = ~~1\2~~, ~9/7 = 443.4466 (~11/10 = 156.5534)

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

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~25/22 = 221.5981

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

* POTE: ~2 = ~~1\1~~, ~25/22 = 221.605

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

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Optimal tunings:

* CTE: ~2 = ~~1\1~~, ~25/22 = 221.6333

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

* POTE: ~2 = ~~1\1~~, ~25/22 = 221.556

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

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: mapping generators: ~2, ~162/125

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

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

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

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: mapping generators: ~2, ~162/125

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

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

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

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

~~{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}~~

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

[[Optimal tuning]] ([[POTE]]): ~567/400 = ~~1\2~~, ~162/125 = 443.075 (~35/32 = 156.925)

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

Optimal tuning (POTE): ~99/70 = ~~1\2~~, ~162/125 = 443.117 (~35/32 = 156.883)

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

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

Optimal tuning (POTE): ~55/39 = ~~1\2~~, ~162/125 = 443.096 (~35/32 = 156.904)

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

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: mapping generators: ~2, ~56/45

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

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

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

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

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

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

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

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

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

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

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{{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

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

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

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

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

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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

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

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

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

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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

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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 =
 }}
+{{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.
@@ Line 18: / Line 19: @@
 == Sensipent ==
+{{Main| Sensipent }}
 [[Subgroup]]: 2.3.5
@@ Line 26: / Line 29: @@
 : 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
-Badness (Dirichlet): 0.826
+=== 2.3.5.31 subgroup ===
+Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one with prime 31 by interpreting the generator accurately as [[31/24]]~[[40/31]], tempering out [[961/960|S31 = 961/960]], so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
-=== 2.3.5.31 ===
-Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one to prime 31 by interpreting the generator accurately as [[40/31]]~[[31/24]] by tempering [[961/960|S31 = 961/960]], so that the large [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
 [[Subgroup]]: 2.3.5.31
-[[Comma list]]: 78732/78125, 961/960
+[[Comma list]]: 961/960, 2511/2500
 {{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
@@ Line 47: / Line 50: @@
 {{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~31/24 = 443.050
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
-Badness (Dirichlet): 0.243
+[[Badness]] (Sintel): 0.243
 === Sendai ===
-{{ See also | User:VIxen/Table_of_sensipent_intervals }}
+{{ See also | Sensipent#Sendai interval table }}
-Sendai is an accurate extension of (2.3.5.31) [[#Sensipent|sensipent]] to primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).
+Sendai is an accurate extension of (2.3.5.31) [[#Sensipent|sensipent]] with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).
 [[Subgroup]]: 2.3.5.23.29.31
-[[Comma list]]: 465/464, 576/575, 621/620, 729/725
+[[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 }}
@@ Line 63: / Line 66: @@
 {{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~31/24 = 442.989
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989
-Badness (Dirichlet): 0.283
+[[Badness]] (Sintel): 0.283
 === Sensible ===
-An extension of sensipent  to prime 11 of dubious canonicity (but significantly higher accuracy than [[sensi]]) interprets the generator as [[165/128]]~[[128/99]] by tempering [[8019/8000|S9/S10]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation: that it is equivalent to tempering the [[semiporwellisma]] which is equal to [[961/960|S31]] * [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). The equivalence of the aforementioned [[lopsided comma]] also implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension to prime 17 is reasonably accurate through tempering [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
+{{ 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]]
+[[Comma list]]: 8019/8000, 16384/16335
 {{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
@@ Line 80: / Line 88: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~128/99 = 443.115
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115
-Badness (Dirichlet): 0.728
+[[Badness]] (Sintel): 0.728
-==== 2.3.5.11.17 ====
+==== 2.3.5.11.17 subgroup ====
-The aforementioned extension to prime 17 through tempering [[1089/1088|S33]] is equivalent to the one by tempering [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
 [[Subgroup]]: 2.3.5.11.17
-[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]
+[[Comma list]]: 256/255, 1089/1088, 1377/1375
 {{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
@@ Line 97: / Line 104: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~22/17 = 443.188
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.188
-Badness (Dirichlet): 0.639
+[[Badness]] (Sintel): 0.639
-==== 2.3.5.11.17.23 ====
+==== 2.3.5.11.17.23 subgroup ====
 [[Subgroup]]: 2.3.5.11.17.23
-[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]]
+[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375
 {{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
@@ Line 110: / Line 117: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~22/17 = 443.185
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185
-Badness (Dirichlet): 0.555
+[[Badness]] (Sintel): 0.555
-==== 2.3.5.11.17.23.31 ====
-Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony without the need for interpreting it as high-damage [[~]][[9/7]] or [[~]][[13/10]] intervals. Its S-expression-based comma list is {([[256/255|S16]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495]] = S31 * S32 and [[528/527]] = S32 * S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
+==== 2.3.5.11.17.23.31 subgroup ====
 [[Subgroup]]: 2.3.5.11.17.23.31
-[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]], [[961/960]]
+[[Comma list]]: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
 {{Mapping|legend=1| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
@@ Line 125: / Line 130: @@
 {{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
-[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~22/17 = 443.183
+[[Optimal tuning]]s:
+* [[CTE]]: 2/1 = 1\1, ~22/17 = 443.183
+* [[CEE]]: 2/1 = 1\1, ~22/17 = 443.115
-Badness (Dirichlet): 0.490
+[[Badness]] (Sintel): 0.490
 == Sensi ==
@@ Line 142: / Line 149: @@
 : mapping generators: ~2, ~9/7
-{{Multival|legend=1| 7 9 13 -2 1 5 }}
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
+* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
-* [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
+* [[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 176: / Line 181: @@
 : 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 188: / Line 193: @@
 : mapping generators: ~2, ~9/7
-{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.2987
+* CTE: ~2 = 1200.000, ~9/7 = 443.2987
-* POTE: ~2 = 1\1, ~9/7 = 443.294
+* POTE: ~2 = 1200.000, ~9/7 = 443.294
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
@@ Line 205: / Line 208: @@
 Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
-{{Multival|legend=1| 7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90 }}
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.3658
+* CTE: ~2 = 1200.000, ~9/7 = 443.3658
-* POTE: ~2 = 1\1, ~9/7 = 443.321
+* POTE: ~2 = 1200.000, ~9/7 = 443.321
 {{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
@@ Line 224: / Line 225: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.3775
+* CTE: ~2 = 1200.000, ~9/7 = 443.3775
-* POTE: ~2 = 1\1, ~9/7 = 443.365
+* POTE: ~2 = 1200.000, ~9/7 = 443.365
 {{Optimal ET sequence|legend=1| 19, 27, 46 }}
@@ Line 239: / Line 240: @@
 : mapping generators: ~2, ~9/7
-{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.4783
+* CTE: ~2 = 1200.000, ~9/7 = 443.4783
-* POTE: ~2 = 1\1, ~9/7 = 443.626
+* POTE: ~2 = 1200.000, ~9/7 = 443.626
 {{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
@@ Line 256: / Line 255: @@
 Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
-{{Multival|legend=1| 7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80 }}
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.5075
+* CTE: ~2 = 1200.000, ~9/7 = 443.5075
-* POTE: ~2 = 1\1, ~9/7 = 443.559
+* POTE: ~2 = 1200.000, ~9/7 = 443.559
 {{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
@@ Line 275: / Line 272: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.5050
+* CTE: ~2 = 1200.000, ~9/7 = 443.5050
-* POTE: ~2 = 1\1, ~9/7 = 443.551
+* POTE: ~2 = 1200.000, ~9/7 = 443.551
 {{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
@@ Line 290: / Line 287: @@
 : mapping generators: ~2, ~9/7
-{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.1886
+* CTE: ~2 = 1200.000, ~9/7 = 443.1886
-* POTE: ~2 = 1\1, ~9/7 = 443.962
+* POTE: ~2 = 1200.000, ~9/7 = 443.962
 {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
@@ Line 307: / Line 302: @@
 Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
-{{Multival|legend=1| 7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20 }}
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.2863
+* CTE: ~2 = 1200.000, ~9/7 = 443.2863
-* POTE: ~2 = 1\1, ~9/7 = 443.945
+* POTE: ~2 = 1200.000, ~9/7 = 443.945
 {{Optimal ET sequence|legend=1| 8d, 19, 27e }}
@@ Line 328: / Line 321: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.7814
+* CTE: ~2 = 1200.000, ~9/7 = 443.7814
-* POTE: ~2 = 1\1, ~9/7 = 443.518
+* POTE: ~2 = 1200.000, ~9/7 = 443.518
 {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
@@ Line 343: / Line 336: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~9/7 = 443.7877
+* CTE: ~2 = 1200.000, ~9/7 = 443.7877
-* POTE: ~2 = 1\1, ~9/7 = 443.506
+* POTE: ~2 = 1200.000, ~9/7 = 443.506
 {{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
@@ Line 364: / Line 357: @@
 Optimal tunings:
-* CTE: ~99/70 = 1\2, ~9/7 = 443.3688 (~11/10 = 156.6312)
+* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
-* POTE: ~99/70 = 1\2, ~9/7 = 443.308 (~11/10 = 156.692)
+* POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~11/10 = 156.692)
 {{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
@@ Line 383: / Line 376: @@
 Optimal tunings:
-* CTE: ~55/39 = 1\2, ~9/7 = 443.4416, ~11/10 = 156.5584
+* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
-* POTE: ~55/39 = 1\2, ~9/7 = 443.275, ~11/10 = 156.725
+* POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~11/10 = 156.725
 {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
@@ Line 402: / Line 395: @@
 Optimal tunings:
-* CTE: ~17/12 = 1\2, ~9/7 = 443.4466 (~11/10 = 156.5534)
+* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)
 {{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
@@ Line 420: / Line 413: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~25/22 = 221.5981
+* CTE: ~2 = 1200.000, ~25/22 = 221.5981
-* POTE: ~2 = 1\1, ~25/22 = 221.605
+* POTE: ~2 = 1200.000, ~25/22 = 221.605
 {{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
@@ Line 435: / Line 428: @@
 Optimal tunings:
-* CTE: ~2 = 1\1, ~25/22 = 221.6333
+* CTE: ~2 = 1200.000, ~25/22 = 221.6333
-* POTE: ~2 = 1\1, ~25/22 = 221.556
+* POTE: ~2 = 1200.000, ~25/22 = 221.556
 {{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
@@ Line 451: / Line 444: @@
 : mapping generators: ~2, ~162/125
-{{Multival|legend=1| 7 9 32 -2 31 49 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 442.755
 {{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
@@ Line 468: / Line 459: @@
 : mapping generators: ~2, ~162/125
-{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289
 {{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
@@ Line 485: / Line 474: @@
 : 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 500: / Line 489: @@
 : 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 515: / Line 504: @@
 : 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 534: / Line 523: @@
 :: mapping generators: ~567/400, ~35/32
-{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
+[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)
-[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~162/125 = 443.075 (~35/32 = 156.925)
 {{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
@@ Line 553: / Line 540: @@
 :: mapping generators: ~567/400, ~35/32
-Optimal tuning (POTE): ~99/70 = 1\2, ~162/125 = 443.117 (~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 570: / Line 557: @@
 :: mapping generators: ~55/39, ~35/32
-Optimal tuning (POTE): ~55/39 = 1\2, ~162/125 = 443.096 (~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 587: / Line 574: @@
 : mapping generators: ~2, ~56/45
-{{Multival|legend=1| 14 18 45 -4 32 54 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~56/45 = 378.467
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 378.467
 {{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
@@ Line 602: / 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 615: / 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 632: / Line 617: @@
 : mapping generators: ~2, ~48/35
-{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 546.815
 {{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
@@ Line 649: / Line 632: @@
 : 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 662: / Line 645: @@
 {{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 675: / Line 658: @@
 {{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 688: / Line 671: @@
 {{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 705: / Line 688: @@
 : mapping generators: ~63/50, ~36/35
-{{Multival|legend=1| 21 27 12 -6 -40 -48 }}
+[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147
-[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~36/35 = 43.147
 {{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
@@ Line 720: / 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 735: / Line 716: @@
 : mapping generators: ~49/39, ~36/35
-Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
+Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288
 {{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
@@ Line 748: / 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 761: / 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 768: / Line 749: @@
 [[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
 [[Category:Sensipent family| ]] <!-- main article -->
-[[Category:Sensipent| ]] <!-- key article -->
 [[Category:Rank 2]]