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Changed 1\1 to 1200.000. Good point about significant digits. But optimal tunings routinely use slightly stretched/compressed octaves. "1200.000" unlike "1\1" makes it clear that the octave is exactly just. Granted, the reader could click on the link for POTE or CTE, read through several paragraphs, and find out that the octave is just, but why not communicate that fact directly? Likewise, 1\2 and 1\3 are changed to cents. (P.S. thanks for correcting mincents to mingen and other helpful changes)
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{{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.  
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 ==
== Sensipent ==
{{Main| Sensipent }}
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


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{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
{{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
[[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}}
{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050


Badness (Dirichlet): 0.243
[[Badness]] (Sintel): 0.243


=== Sendai ===
=== 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
[[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 }}
{{Mapping|legend=1| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989


Badness (Dirichlet): 0.283
[[Badness]] (Sintel): 0.283


=== Sensible ===
=== 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
[[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 }}
{{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~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
[[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 }}
{{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~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
[[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 }}
{{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
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[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~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
[[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 }}
{{Mapping|legend=1| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
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{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~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 ==
== Sensi ==
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: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 7 9 13 -2 1 5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}  
* [[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 }}
* [[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]]:  
[[Tuning ranges of regular temperaments|Tuning ranges]]:  
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: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}


Optimal tunings:  
Optimal tunings:  
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Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
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:  
Optimal tunings:  
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: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}


Optimal tunings:  
Optimal tunings:  
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Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
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:  
Optimal tunings:  
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: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}


Optimal tunings:  
Optimal tunings:  
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Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
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:  
Optimal tunings:  
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: mapping generators: ~2, ~162/125
: 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 = 1200.000, ~162/125 = 442.755
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: mapping generators: ~2, ~162/125
: 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 = 1200.000, ~162/125 = 443.289
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* mingen form: {{mapping| 2 5 7 3 | 0 -7 -9 10 }}
* mingen form: {{mapping| 2 5 7 3 | 0 -7 -9 10 }}
:: mapping generators: ~567/400, ~35/32
:: 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 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)
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: mapping generators: ~2, ~56/45
: 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 = 1200.000, ~56/45 = 378.467
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: mapping generators: ~2, ~48/35
: 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 = 1200.000, ~48/35 = 546.815
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: mapping generators: ~63/50, ~36/35
: 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 = 400.000, ~36/35 = 43.147
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Sensipent| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Sensipent_family"