Xenharmonic Wiki

Sensipent family: Difference between revisions

← Older edit
VisualWikitext
m →Sensi: update a link
Tags: Mobile edit Mobile web edit
 
(61 intermediate revisions by 11 users not shown)
Line 5: Line 5:
| ja =  
| ja =  
}}
}}
= Sensipent =
{{Technical data page}}
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.


Subgroup: 2.3.5
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]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)
 
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
 
== Sensipent ==
{{Main| Sensipent }}
 
[[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]] ([[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


[[POTE generator]]: 162/125 = 443.058
==== 2.3.5.11.17.23 subgroup ====
[[Subgroup]]: 2.3.5.11.17.23


{{Val list|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375


= Sensi =
{{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}


{{main| Sensi }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
{{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."
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185


== 7-limit ==
[[Badness]] (Sintel): 0.555


Subgroup: 2.3.5.7
==== 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 }}
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
 
[[Optimal tuning]]s:
* [[CTE]]: 2/1 = 1\1, ~22/17 = 443.183
* [[CEE]]: 2/1 = 1\1, ~22/17 = 443.115
 
[[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 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 ===
[[Subgroup]]: 2.3.5.7


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


[[Mapping]]: [{{val| 1 6 8 11 }}, {{val| 0 -7 -9 -13 }}]
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}


Mapping generators: ~2, ~14/9
: mapping generators: ~2, ~9/7


{{Multival|legend=1| 7 9 13 -2 1 5 }}
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
* [[POTE]]: ~2 = 1200.000, ~9/7 = 443.383


[[POTE generator]]: ~9/7 = 443.383
[[Minimax tuning]]:
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


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


[[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]]: 0.025622


==== 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 }}]
 
Gencom: [2 9/7; 91/90 126/125 169/168]


POTE generator: ~9/7 = 443.322
: mapping generators: ~2, ~9/7


{{Val list|legend=1| 19, 27, 46, 111de, 157de }}
Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016


== Sensor ==
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}


=== Sensor ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: [{{val| 1 6 8 11 -6 }}, {{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


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


Badness: 0.0379
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}


=== 13-limit ===
Badness: 0.037942


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


Comma list: 91/90, 126/125, 169/168, 385/384
Comma list: 91/90, 126/125, 169/168, 385/384


Mapping: [{{val| 1 6 8 11 -6 10 }}, {{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


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


Badness: 0.0256
Badness: 0.025575


== Sensis ==
==== 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:
* CTE: ~2 = 1200.000, ~9/7 = 443.3775
* POTE: ~2 = 1200.000, ~9/7 = 443.365
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
Badness: 0.022908
=== Sensus ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 100/99, 245/243
Comma list: 126/125, 176/175, 245/243


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


POTE generator: ~9/7 = 443.962
: mapping generators: ~2, ~9/7


{{Val list|legend=1| 19, 27e, 73ee }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.4783
* POTE: ~2 = 1200.000, ~9/7 = 443.626


Badness: 0.0287
{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}


=== 13-limit ===
Badness: 0.029486


==== 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: 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


Mapping: [{{val| 1 6 8 11 6 10 }}, {{val| 0 -7 -9 -13 -4 -10 }}]
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168


POTE generator: ~9/7 = 443.945
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}


{{Val list|legend=1| 19, 27e, 46e, 73ee }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.5050
* POTE: ~2 = 1200.000, ~9/7 = 443.551


Badness: 0.0200
{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}


== Sensus ==
Badness: 0.016238


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


Comma list: 126/125, 176/175, 245/243
Comma list: 56/55, 100/99, 245/243


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


POTE generator: ~9/7 = 443.626
: mapping generators: ~2, ~9/7


{{Val list|legend=1| 19e, 27e, 46, 119c, 165c }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.1886
* POTE: ~2 = 1200.000, ~9/7 = 443.962


Badness: 0.0295
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}


=== 13-limit ===
Badness: 0.028680


==== 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: 56/55, 78/77, 91/90, 100/99
 
Mapping: [{{val| 1 6 8 11 23 10 }}, {{val| 0 -7 -9 -13 -31 -10 }}]


POTE generator: ~9/7 = 443.559
Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}


{{Val list|legend=1| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.2863
* POTE: ~2 = 1200.000, ~9/7 = 443.945


Badness: 0.0208
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}


== Sensa ==
Badness: 0.020017


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


Comma list: 55/54, 77/75, 99/98
Comma list: 55/54, 77/75, 99/98


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


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


{{Val list|legend=1| 19e, 27, 46ee }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.7814
* POTE: ~2 = 1200.000, ~9/7 = 443.518


Badness: 0.0368
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}


=== 13-limit ===
Badness: 0.036835


==== 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: 55/54, 66/65, 77/75, 143/140


Mapping: [{{val| 1 6 8 11 11 10 }}, {{val| 0 -7 -9 -13 -12 -11 }}]
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.7877
* POTE: ~2 = 1200.000, ~9/7 = 443.506
 
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
 
Badness: 0.023258
 
=== Bisensi ===
Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).
 
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 126/125, 245/243
 
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
 
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 ET sequence|legend=1| 8d, …, 38d, 46 }}
 
Badness: 0.041723
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 126/125, 169/168


POTE generator: ~9/7 = 443.506
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


{{Val list|legend=1| 19e, 27, 46ee }}
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


Badness: 0.0233
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}


== Hemisensi ==
Badness: 0.026339
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
 
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
 
Optimal tunings:
* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)
 
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
 
Badness: 0.0188
 
=== 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
Line 186: Line 408:
Comma list: 126/125, 243/242, 245/242
Comma list: 126/125, 243/242, 245/242


Mapping: [{{val| 1 13 17 24 32 }}, {{val| 0 -14 -18 -26 -35 }}]
Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}


POTE generator: ~25/22 = 221.605
: mapping generators: ~2, ~25/22


{{Val list|legend=1| 27e, 65, 157de, 222cde }}
Optimal tunings:
* CTE: ~2 = 1200.000, ~25/22 = 221.5981
* POTE: ~2 = 1200.000, ~25/22 = 221.605


Badness: 0.0487
{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}


= Sensei =
Badness: 0.048714


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


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


[[Mapping]]: [{{val| 1 6 8 23 }}, {{val| 0 -7 -9 -32 }}]
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
 
: mapping generators: ~2, ~162/125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755
 
{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
 
[[Badness]]: 0.059218
 
== Warrior ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 78732/78125
 
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}
 
: mapping generators: ~2, ~162/125
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289
 
{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
 
[[Badness]]: 0.118239
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1331/1323, 5120/5103
 
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15}}
 
: mapping generators: ~2, ~128/99
 
Optimal tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274
 
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
 
Badness: 0.046383
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 847/845, 1331/1323
 
Mapping: {{mapping| 1 -1 -1 15 9 17| 0 7 9 -33 -15 -36}}


{{Multival|legend=1| 7 9 32 -2 31 49 }}
: mapping generators: ~2, ~84/65


[[POTE generator]]: ~125/81 = 757.245
Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270


{{Val list|legend=1| 19, 84, 103, 187, 290b }}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}


[[Badness]]: 0.0592
Badness: 0.028735


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


Subgroup: 2.3.5.7
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714


[[Comma list]]: 6144/6125, 78732/78125
Mapping: {{mapping| 1 -1 -1 15 9 17 10| 0 7 9 -33 -15 -36 -16}}
 
: mapping generators: ~2, ~22/17
 
Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270
 
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
 
Badness: 0.018105
 
== Bison ==
Bison has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].


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


{{Val list|legend=1| 8, 38, 46, 84, 130 }}
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)


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


== 11-limit ==
[[Badness]]: 0.070375


=== 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:  
* 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
 
Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)
 
{{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
 
Badness: 0.037132


POTE generator: ~35/32 = 156.883
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 46, 84, 130, 306, 436ce }}
Comma list: 351/350, 364/363, 441/440, 10985/10976


Badness: 0.0371
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


== 13-limit ==
Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)


{{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}
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
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 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}


POTE generator: ~35/32 = 156.904
Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.437


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


Badness: 0.0235
Badness: 0.023940


= Heinz =
== Heinz ==
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
[[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 }}


[[POTE generator]]: ~48/35 = 546.815
: mapping generators: ~2, ~48/35


{{Val list|legend=1| 46, 103, 149, 699bd }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815


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


== 11-limit ==
[[Badness]]: 0.115385


=== 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|legend=1| 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 tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631


Badness: 0.0424
{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}


== 13-limit ==
Badness: 0.042412


=== 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|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


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


Badness: 0.0258
Badness: 0.025779
 
== 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|legend=1| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}
 
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.635
 
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
 
Badness: 0.018479


POTE generator: ~11/8 = 547.635
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


{{Val list|legend=1| 46, 103, 149, 252ef, 401bdef }}
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968


Badness: 0.0185
{{Mapping|legend=1| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17}}


== 19-limit ==
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614


{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
Badness: 0.019005
== 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]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
[[Badness]]: 0.089740
=== 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 tuning (POTE): ~63/50 = 400.000, ~36/35 = 43.292
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Badness: 0.058413
=== 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 tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Badness: 0.034829
=== 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 tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.276
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Badness: 0.024120
=== 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: [{{val| 1 13 17 -1 4 -5 3 -5 }}, {{val| 0 -21 -27 7 -1 16 2 17 }}]
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}


POTE generator: ~11/8 = 547.614
Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.292


{{Val list|legend=1| 46, 103h, 149h, 252efh }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}


Badness: 0.0190
Badness: 0.018466


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