Sensipent family: Difference between revisions

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==== 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]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts, which is important as it is the difference between 5/4 and 4/3 (whose product is 5/3 at 2 gens). This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions_2|#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's fine-tempered 5-limit structure.  

This extension can be applied to many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 31/24~40/31 such that the 24:31:40 is a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.

[[Sensible]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap| 65 + 111 = 176g }} is almost patent val.

Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap| 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|legend=0| 1 -1 -1 -2 0| 0 7 9 13 10 }}

{{Mapping|legend=0| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}

{{Mapping|legend=0| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}

{{Mapping|legend=0| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}

{{Mapping|legend=0| 1 -1 -1 -2 -8| 0 7 9 13 31 }}

{{Mapping|legend=0| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}

{{Mapping|legend=0| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}

{{Mapping|legend=0| 1 -1 -1 -2 2| 0 7 9 13 4 }}

{{Mapping|legend=0| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}

{{Mapping|legend=0| 1 -1 -1 -2 -1| 0 7 9 13 12 }}

{{Mapping|legend=0| 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)). (As 84edo is even, one might be interested to know its [[17-limit]] val is 84def, corresponding to a strong flat tendency.)

{{Mapping|legend=0| 2 -2 -2 -4 1 | 0 7 9 13 8 }}

 

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

{{Mapping|legend=0| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}

{{Mapping|legend=0| 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-14-cot (pergen (P8, ccP5/14)).  

{{Mapping|legend=0| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}

{{Mapping|legend=0| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}

{{Mapping|legend=0| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}

{{Mapping|legend=0| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}

{{Mapping|legend=0| 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)).  

{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}

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

{{Mapping|legend=0| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}

{{Mapping|legend=0| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}

Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c⁴P4/14)).  

{{Mapping|legend=0| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}

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

Heinz splits the sensipent generator ~324/125 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).

{{Mapping|legend=0| 1 -8 -10 6 3 | 0 21 27 -7 1}}

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

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

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

{{Mapping|legend=0| 3 4 6 8 8 | 0 7 9 4 22 }}

{{Mapping|legend=0| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}

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

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

[[Category:Catalogs of rank-2 temperaments]]