Sensipent family: Difference between revisions

Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.  

* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)

* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)

Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.

 

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 [[S-expression|S31⋅S32²]] (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 {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, 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]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.

 

The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 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 {([[8019/8000|S9/S10]], [[256/255|S16]],) [[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]])/([[33/31]])². A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].

 

 

 

Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}

 

 

* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}

 

 

 

 

 

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}

 

: mapping generators: ~2, ~22/17

 

* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}

 

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

 

 

 

 

 

* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}

 

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

 

 

 

 

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}

 

 

 

Badness (Sintel): 0.490

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

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

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

Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}

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

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

Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}

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

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

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

Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}

Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).  

Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}

Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}

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

Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}

Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}

Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}

Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}

Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}

Bison has a 1/2-octave period and the generator can be taken as ~162/125 or its semi-octave complement, ~35/32. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].

Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}

Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}

Mapping: {{mapping| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}

Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}

Mapping: {{mapping| 1 -8 -10 6 3 | 0 21 27 -7 1}}

Mapping: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}

Mapping: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}

Mapping: {{mapping| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}

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

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

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

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

== Other subgroup extensions ==

The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.  

For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].

Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}

=== Sendai ===

Sendai is an accurate extension of 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-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}

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

[[Category:Rank 2]]