Sensipent family: Difference between revisions

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]].

Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the 19 & 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.  

* [[WE]]: ~2 = 1199.7081{c}}, ~9/7 = 443.2748{{c}}

Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}

Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).  

Mapping:

* common form: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}

:: mapping generators: ~99/70, ~9/7

Mapping:

* common form: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10}}

Mapping:

* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}

Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-tetradecacot (pergen (P8, ccP5/14)).  

Bison has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].

* common form: {{mapping| 2 -2 -2 13 | 0 7 9 -10}}

:: mapping generators: ~567/400, ~162/125

Mapping:

* common form: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}

Mapping:

* common form: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}

Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c⁴P4/14)).  

Heinz splits the generator ~18/7 in three. Its ploidacot is theta-21-cot (pergen (P8, c⁹P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).