Sensipent family: Difference between revisions

: mapping generators: ~2, ~162/125

{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}

=== 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]].

: mapping generators: ~2, ~31/24

=== 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).

=== 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.

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]].

: mapping generators: ~2, ~128/99

==== 2.3.5.11.17 subgroup ====

: mapping generators: ~2, ~22/17

==== 2.3.5.11.17.23 subgroup ====

==== 2.3.5.11.17.23.31 subgroup ====

== Sensi ==

[[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]]:  

{{Optimal ET sequence|legend=1| 19, 27, 46 }}

==== 2.3.5.7.13 subgroup (sensation) ====

: mapping generators: ~2, ~9/7

=== Sensor ===

Optimal tunings:  

Badness: 0.037942

Optimal tunings:  

==== 17-limit ====

Optimal tunings:  

=== Sensus ===

Optimal tunings:  

==== 13-limit ====

Optimal tunings:  

==== 17-limit ====

Optimal tunings:  

=== Sensis ===

Optimal tunings:  

==== 13-limit ====

Optimal tunings:  

=== Sensa ===

Optimal tunings:  

==== 13-limit ====

Optimal tunings:  

=== Bisensi ===

Optimal tunings:  

==== 13-limit ====

Optimal tunings:  

==== 17-limit ====

Optimal tunings:  

=== Hemisensi ===

Optimal tunings:  

==== 13-limit ====

Optimal tunings:  

== Sensei ==

: mapping generators: ~2, ~162/125

== Warrior ==

: mapping generators: ~2, ~162/125

{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}

=== 11-limit ===

: mapping generators: ~2, ~128/99

=== 13-limit ===

: mapping generators: ~2, ~84/65

=== 17-limit ===

: mapping generators: ~2, ~22/17

== Bison ==

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

{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}

=== 11-limit ===

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

=== 13-limit ===

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

== Subpental ==

: mapping generators: ~2, ~56/45

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

=== 11-limit ===

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

=== 13-limit ===

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

== Heinz ==

: mapping generators: ~2, ~48/35

{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}

=== 11-limit ===

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

: mapping generators: ~2, ~11/8

=== 13-limit ===

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

=== 17-limit ===

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

=== 19-limit ===

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

== Trisensory ==

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

{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}

=== 11-limit ===

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

=== 13-limit ===

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

=== 17-limit ===

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

=== 19-limit ===

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

[[Category:Temperament families]]