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.  

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

=== 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).

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

=== 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]]).

[[Subgroup]]: 2.3.5.11

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

==== 2.3.5.11.17 subgroup ====

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

==== 2.3.5.11.17.23 subgroup ====

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

==== 2.3.5.11.17.23.31 subgroup ====

{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}

== Sensi ==

: mapping generators: ~2, ~9/7

[[Optimal tuning]]s:  

[[Minimax tuning]]:  

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

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

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

: mapping generators: ~2, ~9/7

Optimal tunings:  

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

Optimal tunings:  

: mapping generators: ~2, ~9/7

Optimal tunings:  

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

Optimal tunings:  

: mapping generators: ~2, ~9/7

Optimal tunings:  

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

Optimal tunings:  

: mapping generators: ~2, ~162/125

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

: mapping generators: ~2, ~162/125

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

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

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

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

: mapping generators: ~2, ~56/45

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

: mapping generators: ~2, ~48/35

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

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

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

[[Category:Temperament families]]

[[Category:Sensipent family| ]] <!-- main article -->

[[Category:Rank 2]]