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Sensipent family: Difference between revisions

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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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[[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 ===
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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 ===
{{ See also | Sensipent#Sensible interval table }}
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[961/960|S31]] × [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[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]]).
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]].
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
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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 subgroup ====
==== 2.3.5.11.17 subgroup ====
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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 subgroup ====
==== 2.3.5.11.17.23 subgroup ====
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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 subgroup ====
==== 2.3.5.11.17.23.31 subgroup ====
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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:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Sensipent_family"