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+sensation (2.3.5.7.13 sensi, copied from chromatic pairs)
Move the opening note to a reasonable place. Don't shock the readers with stuff like 31/24 or 40/31 without proper context
 
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= Sensipent =


Subgroup: 2.3.5
[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.
 
== Sensipent ==
{{Main| Sensipent }}
 
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.
 
[[Subgroup]]: 2.3.5


[[Comma list]]: 78732/78125
[[Comma list]]: 78732/78125


[[Mapping]]: [{{val| 1 6 8 }}, {{val| 0 -7 -9 }}]
{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
: mapping generators: ~2, ~162/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
: error map: {{val| 0.000 -0.600 +1.143 }}
 
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}
 
[[Badness]] (Sintel): 0.826
 
=== Overview to extensions ===
==== Full 7-limit extensions ====
The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.
 
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.
 
Temperaments discussed elsewhere include:
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Canopic clan #Browser|Canopic clan]] (+16875/16807)
 
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
 
==== 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. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Other subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.
 
This extension can be laid onto 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 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|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, considered immediately below.
 
=== Sensible ===
{{See also| Sensipent #Sensible interval table }}
 
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]])<sup>3</sup>. 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⋅S32<sup>2</sup>]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> 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 {{nowrap| 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]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]]. A notable non-patent val is [[84edo|84g]].
 
Subgroup: 2.3.5.11
 
Comma list: 8019/8000, 16384/16335
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}
 
: mapping generators: ~2, ~128/99
 
Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
 
Badness (Sintel): 0.728
 
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17
 
Comma list: 256/255, 1089/1088, 1377/1375
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
: mapping generators: ~2, ~22/17
 
Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}


[[POTE generator]]: 162/125 = 443.058
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


{{Val list|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
Badness (Sintel): 0.639


= Sensi =
==== 2.3.5.11.17.23 subgroup ====
Subgroup: 2.3.5.11.17.23


{{main| Sensi }}
Comma list: 256/255, 576/575, 1089/1088, 1377/1375


Sensi tempers out [[686/675]], [[245/243]] and [[4375/4374]] in addition to [[126/125]], and can be described as the 19&amp;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 of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}


== 7-limit ==
Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


[[Comma list]]: 126/125, 245/243
Badness (Sintel): 0.555
 
==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31
 
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
 
Optimal tunings:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.490
 
== Sensi ==
{{Main| Sensi }}
 
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.


[[Mapping]]: [{{val| 1 6 8 11 }}, {{val| 0 -7 -9 -13 }}]
=== Septimal sensi ===
[[Subgroup]]: 2.3.5.7


Mapping generators: ~2, ~14/9
[[Comma list]]: 126/125, 245/243


{{Multival|legend=1| 7 9 13 -2 1 5 }}
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}


[[POTE generator]]: ~9/7 = 443.383
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}  
: [{{monzo| 1 0 0 0 }}, {{monzo| 1/13 0 0 7/13 }}, {{monzo| 5/13 0 0 9/13 }}, {{monzo| 0 0 0 1 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
: [[Eigenmonzo]]s: 2, 7
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
* [[9-odd-limit]]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5
: [{{monzo| 1 0 0 0 }}, {{monzo| 2/5 14/5 -7/5 0 }}, {{monzo| 4/5 18/5 -9/5 0 }}, {{monzo| 3/5 26/5 -13/5 0 }}]
 
: [[Eigenmonzo]]s: 2, 9/5
[[Tuning ranges of regular temperaments|Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
* 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
* 7-odd-limit [[diamond tradeoff]]: ~9/7 = [442.179, 445.628]
* 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]


[[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.
[[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.


{{Val list|legend=1| 19, 27, 46, 157d, 203cd, 249cdd, 295ccdd }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
 
[[Badness]]: 0.0256


=== Sensation ===
[[Badness]] (Sintel): 0.648


==== 2.3.5.7.13 subgroup (sensation) ====
Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7.13


Comma list: 91/90, 126/125, 169/168
Comma list: 91/90, 126/125, 169/168


Sval mapping: [{{val| 1 6 8 11 10 }}, {{val| 0 -7 -9 -13 -10 }}]
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}


Gencom mapping: [{{val| 1 6 8 11 0 10 }}, {{val| 0 -7 -9 -13 0 -10 }}]
Optimal tunings:  
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}


Gencom: [2 9/7; 91/90 126/125 169/168]
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


POTE generator: ~9/7 = 443.322
Badness (Sintel): 0.484


{{Val list|legend=1| 19, 27, 46, 111de, 157de }}
=== Sensor ===
Subgroup: 2.3.5.7.11


== Sensor ==
Comma list: 126/125, 245/243, 385/384


Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
Optimal tunings:
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}
Badness (Sintel): 1.25
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 126/125, 169/168, 385/384
Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
Optimal tunings:
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
Badness (Sintel): 1.06
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}
Optimal tunings:
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}
{{Optimal ET sequence|legend=0| 19, 27, 46 }}
Badness (Sintel): 1.17
=== Sensus ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 245/243, 385/384
Comma list: 126/125, 176/175, 245/243
 
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
 
Optimal tunings:
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}


Mapping: [{{val| 1 6 8 11 -6 }}, {{val| 0 -7 -9 -13 15 }}]
Badness (Sintel): 0.975


POTE generator: ~9/7 = 443.294
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 19, 27, 46, 111d, 157d, 268cdd }}
Comma list: 91/90, 126/125, 169/168, 352/351


Badness: 0.0379
Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}


=== 13-limit ===
Optimal tunings:
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}
 
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}


Subgroup: 2.3.5.7.11.13
Badness (Sintel): 0.859


Comma list: 91/90, 126/125, 169/168, 385/384
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Mapping: [{{val| 1 6 8 11 -6 10 }}, {{val| 0 -7 -9 -13 15 -10 }}]
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168


POTE generator: ~9/7 = 443.321
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}


{{Val list|legend=1| 19, 27, 46, 111df, 157df }}
Optimal tunings:
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}


Badness: 0.0256
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}


== Sensis ==
Badness (Sintel): 0.827


=== Sensis ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 100/99, 245/243
Comma list: 56/55, 100/99, 245/243


Mapping: [{{val| 1 6 8 11 6 }}, {{val| 0 -7 -9 -13 -4 }}]
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
 
Optimal tunings:
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}


POTE generator: ~9/7 = 443.962
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}


{{Val list|legend=1| 19, 27e, 73ee }}
Badness (Sintel): 0.948


Badness: 0.0287
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=== 13-limit ===
Comma list: 56/55, 78/77, 91/90, 100/99
 
Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
 
Optimal tunings:
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
 
Badness (Sintel): 0.827
 
=== Sensa ===
Subgroup: 2.3.5.7.11
 
Comma list: 55/54, 77/75, 99/98
 
Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
 
Optimal tunings:
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
 
Badness (Sintel): 1.22


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 78/77, 91/90, 100/99
Comma list: 55/54, 66/65, 77/75, 143/140


Mapping: [{{val| 1 6 8 11 6 10 }}, {{val| 0 -7 -9 -13 -4 -10 }}]
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}


POTE generator: ~9/7 = 443.945
Optimal tunings:  
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}


{{Val list|legend=1| 19, 27e, 46e, 73ee }}
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}


Badness: 0.0200
Badness (Sintel): 0.961


== Sensus ==
=== Bisensi ===
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.)


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 245/243
Comma list: 121/120, 126/125, 245/243


Mapping: [{{val| 1 6 8 11 23 }}, {{val| 0 -7 -9 -13 -31 }}]
Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}


POTE generator: ~9/7 = 443.626
: mapping generators: ~99/70, ~9/7


{{Val list|legend=1| 19e, 27e, 46, 119c, 165c }}
Optimal tunings:
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})


Badness: 0.0295
{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}


=== 13-limit ===
Badness (Sintel): 1.38


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 126/125, 169/168, 352/351
Comma list: 91/90, 121/120, 126/125, 169/168
 
Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
 
Badness (Sintel): 1.09
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168


Mapping: [{{val| 1 6 8 11 23 10 }}, {{val| 0 -7 -9 -13 -31 -10 }}]
Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}


POTE generator: ~9/7 = 443.559
Optimal tunings:
* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})


{{Val list|legend=1| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Badness: 0.0208
Badness (Sintel): 0.960


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 55/54, 77/75, 99/98
Comma list: 126/125, 243/242, 245/242


Mapping: [{{val| 1 6 8 11 11 }}, {{val| 0 -7 -9 -13 -12 }}]
Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}


POTE generator: ~9/7 = 443.518
: mapping generators: ~2, ~25/22


{{Val list|legend=1| 19e, 27, 46ee }}
Optimal tunings:
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}


Badness: 0.0368
{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}


=== 13-limit ===
Badness (Sintel): 1.61


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 66/65, 77/75, 143/140
Comma list: 91/90, 126/125, 169/168, 243/242
 
Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}
 
Badness (Sintel): 1.36
 
== Sensei ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 78732/78125
 
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6422{{c}}, ~162/125 = 442.9920{{c}}
: [[error map]]: {{val| +0.642 -1.653 -0.028 +1.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}
 
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}
 
[[Badness]] (Sintel): 1.50
 
== Warrior ==
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 6 8 11 11 10 }}, {{val| 0 -7 -9 -13 -12 -11 }}]
[[Comma list]]: 5120/5103, 78732/78125


POTE generator: ~9/7 = 443.506
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}


{{Val list|legend=1| 19e, 27, 46ee }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2419{{c}}, ~162/125 = 443.0087{{c}}
: [[error map]]: {{val| -0.758 -0.136 +1.523 +0.516 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}


Badness: 0.0233
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}


== Hemisensi ==
[[Badness]] (Sintel): 2.99


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 243/242, 245/242
Comma list: 176/175, 1331/1323, 5120/5103
 
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}


Mapping: [{{val| 1 13 17 24 32 }}, {{val| 0 -14 -18 -26 -35 }}]
{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}


POTE generator: ~25/22 = 221.605
Badness (Sintel): 1.53


{{Val list|legend=1| 27e, 65, 157de, 222cde }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.0487
Comma list: 176/175, 351/350, 847/845, 1331/1323


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


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}


[[Comma list]]: 225/224, 78732/78125
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}


[[Mapping]]: [{{val| 1 6 8 23 }}, {{val| 0 -7 -9 -32 }}]
Badness (Sintel): 1.19


{{Multival|legend=1| 7 9 32 -2 31 49 }}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


[[POTE generator]]: ~125/81 = 757.245
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714


{{Val list|legend=1| 19, 84, 103, 187, 290b }}
Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}


[[Badness]]: 0.0592
Optimal tunings:  
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}


= Bison =
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}


Subgroup: 2.3.5.7
Badness (Sintel): 0.922


[[Comma list]]: 6144/6125, 78732/78125
== Bison ==
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]]: [{{val| 2 5 7 3 }}, {{val| 0 -7 -9 10 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
[[Comma list]]: 6144/6125, 78732/78125


[[POTE generator]]: ~35/32 = 156.925
{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
: mapping generators: ~567/400, ~162/125


{{Val list|legend=1| 8, 38, 46, 84, 130 }}
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 599.9413{{c}}, ~162/125 = 443.0320{{c}} (~35/32 = 156.9093{{c}})
: [[error map]]: {{val| -0.117 -0.613 +1.092 +0.091 }}
* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})
: error map: {{val| 0.000 -0.446 +1.341 +0.446 }}


[[Badness]]: 0.0704
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}


== 11-limit ==
[[Badness]] (Sintel): 1.78


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping: [{{val| 2 5 7 3 3 }}, {{val| 0 -7 -9 10 15 }}]
Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})
 
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}
 
Badness (Sintel): 1.23
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 441/440, 10985/10976
 
Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
 
Optimal tunings:
* WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})
 
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}
 
Badness (Sintel): 0.971
 
== Subpental ==
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, c<sup>4</sup>P4/14)).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 19683/19600
 
{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
: mapping generators: ~2, ~45/28
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9261{{c}}, ~45/28 = 821.4823{{c}}
: [[error map]]: {{val| -0.074 -0.611 +1.107 -0.052 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}
 
{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}


POTE generator: ~35/32 = 156.883
[[Badness]] (Sintel): 1.37


{{Val list|legend=1| 46, 84, 130, 306, 436ce }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0371
Comma list: 540/539, 3136/3125, 8019/8000


== 13-limit ==
Mapping: {{mapping| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}


Optimal tunings:
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}
Badness (Sintel): 1.50
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 540/539, 676/675, 3136/3125


Mapping: [{{val| 2 5 7 3 3 4 }}, {{val| 0 -7 -9 10 15 13 }}]
Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}


POTE generator: ~35/32 = 156.904
Optimal tunings:  
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}


{{Val list|legend=1| 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}


Badness: 0.0235
Badness (Sintel): 0.989


= Heinz =
== Heinz ==
Heinz splits the sensipent generator ~324/125 in three. Its ploidacot is theta-21-cot (pergen (P8, c<sup>9</sup>P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 78732/78125
[[Comma list]]: 1029/1024, 78732/78125


[[Mapping]]: [{{val| 1 13 17 -1 }}, {{val| 0 -21 -27 7 }}]
{{Mapping|legend=1| 1 -8 -10 6 | 0 21 27 -7 }}
 
: mapping generators: ~2, ~48/35
[[POTE generator]]: ~48/35 = 546.815


{{Val list|legend=1| 46, 103, 149, 699bd }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
: [[error map]]: {{val| +0.425 -0.758 +1.061 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}


[[Badness]]: 0.1154
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}


== 11-limit ==
[[Badness]] (Sintel): 2.92


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 13 17 -1 4 }}, {{val| 0 -21 -27 7 -1 }}]
Mapping: {{mapping| 1 -8 -10 6 3 | 0 21 27 -7 1}}


POTE generator: ~11/8 = 547.631
: mapping generators: ~2, ~11/8


{{Val list|legend=1| 46, 103, 149 }}
Optimal tunings:
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}


Badness: 0.0424
{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}


== 13-limit ==
Badness (Sintel): 1.40


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 1 13 17 -1 4 -5 }}, {{val| 0 -21 -27 7 -1 16 }}]
Mapping: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}


POTE generator: ~11/8 = 547.629
Optimal tunings:  
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}


{{Val list|legend=1| 46, 57, 103, 149, 252e, 401bde }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}


Badness: 0.0258
Badness (Sintel): 1.07
 
== 17-limit ==


=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


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


Mapping: [{{val| 1 13 17 -1 4 -5 3 }}, {{val| 0 -21 -27 7 -1 16 2 }}]
Mapping: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}


POTE generator: ~11/8 = 547.635
Optimal tunings:  
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}


{{Val list|legend=1| 46, 103, 149, 252ef, 401bdef }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}


Badness: 0.0185
Badness (Sintel): 0.941


== 19-limit ==
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
Mapping: {{mapping| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}
Optimal tunings:
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}
{{Optimal ET sequence|legend=0| 46, 103h, 149h }}
Badness (Sintel): 1.16
== Trisensory ==
Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 1728/1715, 78732/78125
{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
: mapping generators: ~63/50, ~36/35
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.8117{{c}}, ~36/35 = 43.1270{{c}}
: [[error map]]: {{val| -0.565 -0.819 +0.700 +2.176 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
[[Badness]] (Sintel): 2.27
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 78732/78125
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
Optimal tunings:
* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}
Badness (Sintel): 1.93
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 540/539, 9295/9261
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
: mapping generators: ~49/39, ~36/35
Optimal tunings:
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
Badness (Sintel): 1.44
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
Optimal tunings:
* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
Badness (Sintel): 1.23
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475
 
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}
 
Optimal tunings:
* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
 
Badness (Sintel): 1.12
 
== Other subgroup extensions ==
=== Sensipent (2.3.5.31 subgroup) ===
Subgroup: 2.3.5.31
 
Comma list: 961/960, 2511/2500
 
Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
 
Optimal tunings:
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
 
{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
 
Badness (Sintel): 0.243
 
=== Sendai ===
{{See also| Sensipent #Sendai interval table }}
 
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). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence.
 
Subgroup: 2.3.5.23.29.31
 
Comma list: 465/464, 576/575, 621/620, 900/899


Mapping: [{{val| 1 13 17 -1 4 -5 3 -5 }}, {{val| 0 -21 -27 7 -1 16 2 17 }}]
Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}


POTE generator: ~11/8 = 547.614
Optimal tunings:  
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}


{{Val list|legend=1| 46, 103h, 149h, 252efh }}
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}


Badness: 0.0190
Badness (Sintel): 0.283


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Sensipent]]
[[Category:Sensi]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Sensipent_family"