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Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. 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.


Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. This temperament family includes sensi, sensei, bison, and heinz.
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:  
Temperaments discussed elsewhere include:  
* ''[[subpental]]'', {3136/3125, 19683/19600} → [[Hemimean clan #Subpental]]
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[catafourth]]'', {2401/2400, 78732/78125} → [[Breedsmic temperaments #Catafourth]]
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)
 
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.


== Sensipent ==
== Sensipent ==
Subgroup: 2.3.5
{{Main| Sensipent }}
 
[[Subgroup]]: 2.3.5


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


[[Mapping]]: [{{val| 1 -1 -1 }}, {{val| 0 7 9 }}]
{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
 
: mapping generators: ~2, ~162/125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.058
 
{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
 
[[Badness]]:
* Smith: 0.035220
* Dirichlet: 0.826
 
=== 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]].
 
[[Subgroup]]: 2.3.5.31
 
[[Comma list]]: 961/960, 2511/2500
 
{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
 
: mapping generators: ~2, ~31/24
 
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
 
[[Badness]] (Sintel): 0.243
 
=== Sendai ===
{{ See also | Sensipent#Sendai interval table }}
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).
 
[[Subgroup]]: 2.3.5.23.29.31
 
[[Comma list]]: 465/464, 576/575, 621/620, 900/899
 
{{Mapping|legend=1| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
 
{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989
 
[[Badness]] (Sintel): 0.283
 
=== 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 [[961/960|S31]] × [[1024/1023|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 {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]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
 
[[Subgroup]]: 2.3.5.11
 
[[Comma list]]: 8019/8000, 16384/16335
 
{{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
 
: mapping generators: ~2, ~128/99
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115
 
[[Badness]] (Sintel): 0.728
 
==== 2.3.5.11.17 subgroup ====
 
[[Subgroup]]: 2.3.5.11.17
 
[[Comma list]]: 256/255, 1089/1088, 1377/1375
 
{{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
: mapping generators: ~2, ~22/17
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.188
 
[[Badness]] (Sintel): 0.639
 
==== 2.3.5.11.17.23 subgroup ====
[[Subgroup]]: 2.3.5.11.17.23
 
[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375
 
{{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185
 
[[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
 
{{Mapping|legend=1| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}


[[POTE generator]]: 162/125 = 443.058
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}


{{Val list|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
[[Optimal tuning]]s:
* [[CTE]]: 2/1 = 1\1, ~22/17 = 443.183
* [[CEE]]: 2/1 = 1\1, ~22/17 = 443.115


[[Badness]]: 0.035220
[[Badness]] (Sintel): 0.490


== Sensi ==
== Sensi ==
{{main| Sensi }}
{{Main| Sensi }}
{{see also| Sensamagic clan #Sensi }}


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."
Sensi tempers out [[245/243]], [[686/675]] 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 scale]]s of size 8, 11, 19 and 27 are available.  


=== Septimal sensi ===
=== Septimal sensi ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 245/243
[[Comma list]]: 126/125, 245/243


[[Mapping]]: [{{val| 1 -1 -1 -2 }}, {{val| 0 7 9 13 }}]
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
 
Mapping generators: ~2, ~9/7


{{Multival|legend=1| 7 9 13 -2 1 5 }}
: mapping generators: ~2, ~9/7


[[POTE generator]]: ~9/7 = 443.383
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
* [[POTE]]: ~2 = 1200.000, ~9/7 = 443.383


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}  
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}  
: [[Eigenmonzo]]s (unchanged intervals): 2, 7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 9/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges of regular temperaments|Tuning ranges]]:  
[[Tuning ranges of regular temperaments|Tuning ranges]]:  
Line 55: Line 165:
* 7-odd-limit [[diamond tradeoff]]: ~9/7 = [442.179, 445.628]
* 7-odd-limit [[diamond tradeoff]]: ~9/7 = [442.179, 445.628]
* 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]
* 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]
* 7-odd-limit diamond monotone and tradeoff: ~9/7 = [442.179, 445.628]
* 9-odd-limit diamond monotone and tradeoff: ~9/7 = [442.105, 444.444]


[[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]]: 0.025622
[[Badness]]: 0.025622


==== Sensation ====
==== 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 -1 -1 -2 0 }}, {{val| 0 7 9 13 10 }}]
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}


Gencom mapping: [{{val| 1 -1 -1 -2 0 0 }}, {{val| 0 7 9 13 0 10 }}]
: mapping generators: ~2, ~9/7


Gencom: [2 9/7; 91/90 126/125 169/168]
Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016


POTE generator: ~9/7 = 443.322
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
 
Optimal GPV sequence: {{Val list| 19, 27, 46, 111de, 157de }}


=== Sensor ===
=== Sensor ===
Line 84: Line 190:
Comma list: 126/125, 245/243, 385/384
Comma list: 126/125, 245/243, 385/384


Mapping: [{{val| 1 -1 -1 -2 9 }}, {{val| 0 7 9 13 -15 }}]
Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}


POTE generator: ~9/7 = 443.294
: mapping generators: ~2, ~9/7


Optimal GPV sequence: {{Val list| 19, 27, 46, 111d, 157d, 268cdd }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.2987
* POTE: ~2 = 1200.000, ~9/7 = 443.294
 
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}


Badness: 0.037942
Badness: 0.037942
Line 97: Line 207:
Comma list: 91/90, 126/125, 169/168, 385/384
Comma list: 91/90, 126/125, 169/168, 385/384


Mapping: [{{val| 1 -1 -1 -2 9 0 }}, {{val| 0 7 9 13 -15 10 }}]
Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}


POTE generator: ~9/7 = 443.321
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.3658
* POTE: ~2 = 1200.000, ~9/7 = 443.321


Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}


Badness: 0.025575
Badness: 0.025575
Line 110: Line 222:
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255


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


POTE generator: ~9/7 = 443.365
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.3775
* POTE: ~2 = 1200.000, ~9/7 = 443.365


Optimal GPV sequence: {{Val list| 19, 27, 46, 157df, 203cdff, 249cddff }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}


Badness: 0.022908
Badness: 0.022908
=== Sensus ===
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 245/243
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
: mapping generators: ~2, ~9/7
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.4783
* POTE: ~2 = 1200.000, ~9/7 = 443.626
{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
Badness: 0.029486
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 126/125, 169/168, 352/351
Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.5075
* POTE: ~2 = 1200.000, ~9/7 = 443.559
{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
Badness: 0.020789
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.5050
* POTE: ~2 = 1200.000, ~9/7 = 443.551
{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
Badness: 0.016238


=== Sensis ===
=== Sensis ===
Line 123: Line 284:
Comma list: 56/55, 100/99, 245/243
Comma list: 56/55, 100/99, 245/243


Mapping: [{{val| 1 -1 -1 -2 2 }}, {{val| 0 7 9 13 4 }}]
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
 
: mapping generators: ~2, ~9/7


POTE generator: ~9/7 = 443.962
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.1886
* POTE: ~2 = 1200.000, ~9/7 = 443.962


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


Badness: 0.028680
Badness: 0.028680
Line 136: Line 301:
Comma list: 56/55, 78/77, 91/90, 100/99
Comma list: 56/55, 78/77, 91/90, 100/99


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


POTE generator: ~9/7 = 443.945
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.2863
* POTE: ~2 = 1200.000, ~9/7 = 443.945


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


Badness: 0.020017
Badness: 0.020017


=== Sensus ===
=== Sensa ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 245/243
Comma list: 55/54, 77/75, 99/98
 
Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}


Mapping: [{{val| 1 -1 -1 -2 -8 }}, {{val| 0 7 9 13 31 }}]
: mapping generators: ~2, ~9/7


POTE generator: ~9/7 = 443.626
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.7814
* POTE: ~2 = 1200.000, ~9/7 = 443.518


Optimal GPV sequence: {{Val list| 19e, 27e, 46, 119c, 165c }}
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}


Badness: 0.029486
Badness: 0.036835


==== 13-limit ====
==== 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: 55/54, 66/65, 77/75, 143/140
 
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}


Mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.7877
* POTE: ~2 = 1200.000, ~9/7 = 443.506


POTE generator: ~9/7 = 443.559
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}


Optimal GPV sequence: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
Badness: 0.023258


Badness: 0.020789
=== Bisensi ===
Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).  


==== 17-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13.17


Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Comma list: 121/120, 126/125, 245/243


Mapping: [{{val| 1 -1 -1 -2 -8 0 -7 }}, {{val| 0 7 9 13 31 10 30 }}]
Mapping:  
* common form: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
:: mapping generators: ~99/70, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
:: mapping generators: ~99/70, ~11/10


POTE generator: ~9/7 = 443.551
Optimal tunings:
* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
* POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~11/10 = 156.692)


Optimal GPV sequence: {{Val list| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}


Badness: 0.016238
Badness: 0.041723


=== Sensa ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 55/54, 77/75, 99/98
Comma list: 91/90, 121/120, 126/125, 169/168


Mapping: [{{val| 1 -1 -1 -2 -1 }}, {{val| 0 7 9 13 12 }}]
Mapping:  
* common form: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10}}
:: mapping generators: ~99/70, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
:: mapping generators: ~99/70, ~11/10


POTE generator: ~9/7 = 443.518
Optimal tunings:
* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
* POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~11/10 = 156.725


Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}


Badness: 0.036835
Badness: 0.026339


==== 13-limit ====
==== 17-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13.17


Comma list: 55/54, 66/65, 77/75, 143/140
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168


Mapping: [{{val| 1 -1 -1 -2 -1 0 }}, {{val| 0 7 9 13 12 11 }}]
Mapping:  
* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}
:: mapping generators: ~99/70, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
:: mapping generators: ~99/70, ~11/10


POTE generator: ~9/7 = 443.506
Optimal tunings:
* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)


Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}


Badness: 0.023258
Badness: 0.0188


=== Hemisensi ===
=== Hemisensi ===
Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-tetradecacot (pergen (P8, ccP5/14)).
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 126/125, 243/242, 245/242
Comma list: 126/125, 243/242, 245/242


Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]
Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}
 
: mapping generators: ~2, ~25/22


POTE generator: ~25/22 = 221.605
Optimal tunings:
* CTE: ~2 = 1200.000, ~25/22 = 221.5981
* POTE: ~2 = 1200.000, ~25/22 = 221.605


Optimal GPV sequence: {{Val list| 27e, 65, 157de, 222cde }}
{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}


Badness: 0.048714
Badness: 0.048714
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
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:
* CTE: ~2 = 1200.000, ~25/22 = 221.6333
* POTE: ~2 = 1200.000, ~25/22 = 221.556
{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
Badness: 0.033016


== Sensei ==
== Sensei ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 78732/78125
[[Comma list]]: 225/224, 78732/78125


[[Mapping]]: [{{val| 1 -1 -1 -9 }}, {{val| 0 7 9 32 }}]
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}


{{Multival|legend=1| 7 9 32 -2 31 49 }}
: mapping generators: ~2, ~162/125


[[POTE generator]]: ~125/81 = 757.245
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755


{{Val list|legend=1| 19, 84, 103, 187, 290b }}
{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}


[[Badness]]: 0.059218
[[Badness]]: 0.059218


== Warrior ==
== Warrior ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 5120/5103, 78732/78125
[[Comma list]]: 5120/5103, 78732/78125


[[Mapping]]: [{{val|1 -1 -1 15}}, {{val|0 7 9 -33}}]
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}


{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
: mapping generators: ~2, ~162/125


[[POTE generator]]: ~162/125 = 443.289
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289


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


[[Badness]]: 0.118239
[[Badness]]: 0.118239
Line 257: Line 470:
Comma list: 176/175, 1331/1323, 5120/5103
Comma list: 176/175, 1331/1323, 5120/5103


Mapping: [{{val|1 -1 -1 15 9}}, {{val|0 7 9 -33 -15}}]
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15}}


POTE generator: ~128/99 = 443.274
: mapping generators: ~2, ~128/99


Optimal GPV sequence: {{Val list| 46, 65d, 111, 268cd, 379cdd}}
Optimal tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274
 
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}


Badness: 0.046383
Badness: 0.046383
Line 270: Line 485:
Comma list: 176/175, 351/350, 847/845, 1331/1323
Comma list: 176/175, 351/350, 847/845, 1331/1323


Mapping: [{{val|1 -1 -1 15 9 17}}, {{val|0 7 9 -33 -15 -36}}]
Mapping: {{mapping| 1 -1 -1 15 9 17| 0 7 9 -33 -15 -36}}


POTE generator: ~84/65 = 443.270
: mapping generators: ~2, ~84/65


Optimal GPV sequence: {{Val list| 46, 65d, 111, 268cd, 379cddf}}
Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270
 
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}


Badness: 0.028735
Badness: 0.028735
Line 283: Line 500:
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714


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


POTE generator: ~22/17 = 443.270
: mapping generators: ~2, ~22/17


Optimal GPV sequence: {{Val list| 46, 65d, 111, 268cdg, 379cddfg }}
Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270
 
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}


Badness: 0.018105
Badness: 0.018105


== Bison ==
== Bison ==
Subgroup: 2.3.5.7
Bison has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 78732/78125
[[Comma list]]: 6144/6125, 78732/78125


[[Mapping]]: [{{val| 2 5 7 3 }}, {{val| 0 -7 -9 10 }}]
[[Mapping]]:  
 
* common form: {{mapping| 2 -2 -2 13 | 0 7 9 -10}}
{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
:: mapping generators: ~567/400, ~162/125
* mingen form: {{mapping| 2 5 7 3 | 0 -7 -9 10 }}
:: mapping generators: ~567/400, ~35/32


[[POTE generator]]: ~35/32 = 156.925
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)


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


[[Badness]]: 0.070375
[[Badness]]: 0.070375
Line 311: Line 534:
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:  
* common form: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15}}
:: mapping generators: ~567/400, ~162/125
* mingen form: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}
:: mapping generators: ~567/400, ~35/32


POTE generator: ~35/32 = 156.883
Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)


Optimal GPV sequence: {{Val list| 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}


Badness: 0.037132
Badness: 0.037132
Line 324: Line 551:
Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 364/363, 441/440, 10985/10976


Mapping: [{{val| 2 5 7 3 3 4 }}, {{val| 0 -7 -9 10 15 13 }}]
Mapping:  
* common form: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
:: mapping generators: ~55/39, ~162/125
* mingen form: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
:: mapping generators: ~55/39, ~35/32


POTE generator: ~35/32 = 156.904
Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)


Optimal GPV sequence: {{Val list| 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}


Badness: 0.023504
Badness: 0.023504
== Subpental ==
Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c<sup>4</sup>P4/14)).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3136/3125, 19683/19600
{{Mapping|legend=1| 1 6 8 17 | 0 -14 -18 -45 }}
: mapping generators: ~2, ~56/45
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~56/45 = 378.467
{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
[[Badness]]: 0.054303
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 8019/8000
Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}
Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.440
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
Badness: 0.045352
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 3136/3125
Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}
Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.437
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
Badness: 0.023940


== Heinz ==
== Heinz ==
Subgroup: 2.3.5.7
Heinz splits the generator ~18/7 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


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


[[Mapping]]: [{{val| 1 -8 -10 6 }}, {{val| 0 21 27 -7 }}]
{{Mapping|legend=1| 1 -8 -10 6| 0 21 27 -7 }}


{{Multival|legend=1|21 27 -7 -6 -70 -92}}
: mapping generators: ~2, ~48/35


[[POTE generator]]: ~48/35 = 546.815
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815


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


[[Badness]]: 0.115385
[[Badness]]: 0.115385
Line 352: Line 628:
Comma list: 385/384, 441/440, 78732/78125
Comma list: 385/384, 441/440, 78732/78125


Mapping: [{{val| 1 -8 -10 6 3 }}, {{val| 0 21 27 -7 1 }}]
{{Mapping|legend=1| 1 -8 -10 6 3 | 0 21 27 -7 1}}
 
: mapping generators: ~2, ~11/8


POTE generator: ~11/8 = 547.631
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631


Optimal GPV sequence: {{Val list| 46, 103, 149, 252e, 401bdee }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}


Badness: 0.042412
Badness: 0.042412
Line 365: Line 643:
Comma list: 351/350, 385/384, 441/440, 847/845
Comma list: 351/350, 385/384, 441/440, 847/845


Mapping: [{{val| 1 -8 -10 6 3 11 }}, {{val| 0 21 27 -7 1 -16 }}]
{{Mapping|legend=1| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}


POTE generator: ~11/8 = 547.629
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.629


Optimal GPV sequence: {{Val list| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}


Badness: 0.025779
Badness: 0.025779
Line 378: Line 656:
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 -8 -10 6 3 11 5 }}, {{val| 0 21 27 -7 1 -16 -2 }}]
{{Mapping|legend=1| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}


POTE generator: ~11/8 = 547.635
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.635


Optimal GPV sequence: {{Val list| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}


Badness: 0.018479
Badness: 0.018479
Line 391: Line 669:
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968


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


POTE generator: ~11/8 = 547.614
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614


Optimal GPV sequence: {{Val list| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}


Badness: 0.019005
Badness: 0.019005


== Trisensory ==
== Trisensory ==
Subgroup: 2.3.5.7
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
[[Comma list]]: 1728/1715, 78732/78125


[[Mapping]]: [{{val|3 4 6 8}}, {{val|0 7 9 4}}]
{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}


{{Multival|legend=1|21 27 12 -6 -40 -48}}
: mapping generators: ~63/50, ~36/35


[[POTE generator]]: ~36/35 = 43.147
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147


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


[[Badness]]: 0.089740
[[Badness]]: 0.089740
Line 419: Line 699:
Comma list: 176/175, 540/539, 78732/78125
Comma list: 176/175, 540/539, 78732/78125


Mapping: [{{val|3 4 6 8 8}}, {{val|0 7 9 4 22}}]
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}


POTE generator: ~36/35 = 43.292
Optimal tuning (POTE): ~63/50 = 400.000, ~36/35 = 43.292


Optimal GPV sequence: {{Val list| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}


Badness: 0.058413
Badness: 0.058413
Line 432: Line 712:
Comma list: 176/175, 351/350, 540/539, 9295/9261
Comma list: 176/175, 351/350, 540/539, 9295/9261


Mapping: [{{val|3 4 6 8 8 11}}, {{val|0 7 9 4 22 1}}]
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
 
: mapping generators: ~49/39, ~36/35


POTE generator: ~36/35 = 43.288
Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288


Optimal GPV sequence: {{Val list| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}


Badness: 0.034829
Badness: 0.034829
Line 445: Line 727:
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714


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


POTE generator: ~36/35 = 43.276
Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.276


Optimal GPV sequence: {{Val list| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}


Badness: 0.024120
Badness: 0.024120
Line 458: Line 740:
Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475
Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475


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


POTE generator: ~36/35 = 43.292
Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.292


Optimal GPV sequence: {{Val list| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}


Badness: 0.018466
Badness: 0.018466


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