Sensipent family: Difference between revisions

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→Sensi: removing for now because 1) it's definitely not a "play on words", though possibly it may be a contraction of those two words; 2) this could be covered better on the Temperament names page; 3) this fact is contentious and under discussion here: https://en.xen.wiki/w/Talk:Temperament_names
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{{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.


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)
* ''[[browser]]'', {16875/16807, 78732/78125} → [[Mirkwai clan #Browser]]
 
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 }}


[[POTE generator]]: 162/125 = 443.058
: 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 }}
{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}


[[Badness]]: 0.035220
[[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 }}
 
{{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 ==
== 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.  
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 56: 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.


{{Optimal ET sequence|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 }}]


Gencom: [2 9/7; 91/90 126/125 169/168]
: mapping generators: ~2, ~9/7


POTE generator: ~9/7 = 443.322
Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016


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


=== Sensor ===
=== Sensor ===
Line 85: 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 ET sequence|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
 
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}


Badness: 0.037942
Badness: 0.037942
Line 98: 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 ET sequence|legend=1| 19, 27, 46, 111df, 157df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}


Badness: 0.025575
Badness: 0.025575
Line 111: 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 ET sequence|legend=1| 19, 27, 46, 157df, 203cdff, 249cddff }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}


Badness: 0.022908
Badness: 0.022908


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


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


POTE generator: ~9/7 = 443.962
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.4783
* POTE: ~2 = 1200.000, ~9/7 = 443.626


{{Optimal ET sequence|legend=1| 8d, 19, 27e, 73ee }}
{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}


Badness: 0.028680
Badness: 0.029486


==== 13-limit ====
==== 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 }}


Mapping: [{{val| 1 -1 -1 -2 2 0 }}, {{val| 0 7 9 13 4 10 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.5075
* POTE: ~2 = 1200.000, ~9/7 = 443.559


POTE generator: ~9/7 = 443.945
{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}


{{Optimal ET sequence|legend=1| 19, 27e, 46e, 73ee }}
Badness: 0.020789


Badness: 0.020017
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


=== Sensus ===
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Subgroup: 2.3.5.7.11


Comma list: 126/125, 176/175, 245/243
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}


Mapping: [{{val| 1 -1 -1 -2 -8 }}, {{val| 0 7 9 13 31 }}]
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.5050
* POTE: ~2 = 1200.000, ~9/7 = 443.551


POTE generator: ~9/7 = 443.626
{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}


{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c, 165c }}
Badness: 0.016238


Badness: 0.029486
=== Sensis ===
Subgroup: 2.3.5.7.11


==== 13-limit ====
Comma list: 56/55, 100/99, 245/243
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 126/125, 169/168, 352/351
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}


Mapping: [{{val| 1 -1 -1 -2 -8 0 }}, {{val| 0 7 9 13 31 10 }}]
: mapping generators: ~2, ~9/7


POTE generator: ~9/7 = 443.559
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.1886
* POTE: ~2 = 1200.000, ~9/7 = 443.962


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


Badness: 0.020789
Badness: 0.028680


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


Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Comma list: 56/55, 78/77, 91/90, 100/99


Mapping: [{{val| 1 -1 -1 -2 -8 0 -7 }}, {{val| 0 7 9 13 31 10 30 }}]
Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}


POTE generator: ~9/7 = 443.551
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.2863
* POTE: ~2 = 1200.000, ~9/7 = 443.945


{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}


Badness: 0.016238
Badness: 0.020017


=== Sensa ===
=== Sensa ===
Line 189: Line 316:
Comma list: 55/54, 77/75, 99/98
Comma list: 55/54, 77/75, 99/98


Mapping: [{{val| 1 -1 -1 -2 -1 }}, {{val| 0 7 9 13 12 }}]
Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
 
: mapping generators: ~2, ~9/7


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


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


Badness: 0.036835
Badness: 0.036835
Line 202: Line 333:
Comma list: 55/54, 66/65, 77/75, 143/140
Comma list: 55/54, 66/65, 77/75, 143/140


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


POTE generator: ~9/7 = 443.506
Optimal tunings:
* CTE: ~2 = 1200.000, ~9/7 = 443.7877
* POTE: ~2 = 1200.000, ~9/7 = 443.506


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


Badness: 0.023258
Badness: 0.023258


=== Hemisensi ===
=== 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
Subgroup: 2.3.5.7.11


Comma list: 126/125, 243/242, 245/242
Comma list: 121/120, 126/125, 245/243


Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]
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: ~25/22 = 221.605
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| 27e, 38d, 65, 157de, 222cde }}
{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}


Badness: 0.048714
Badness: 0.041723


==== 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, 243/242
Comma list: 91/90, 121/120, 126/125, 169/168
 
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
 
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 ET sequence|legend=1| 8d, …, 38df, 46 }}
 
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


Mapping: [{{val| 1 -1 -1 -2 -3 0 }}, {{val| 0 14 18 26 35 30 }}]
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)


POTE generator: ~25/22 = 221.556
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}


{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
Badness: 0.0188


Badness: 0.033016
=== Hemisensi ===
Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-tetradecacot (pergen (P8, ccP5/14)).  


=== Bisensi ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 126/125, 245/243
Comma list: 126/125, 243/242, 245/242
 
Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}


Mapping: [{{val| 2 5 7 9 9 }}, {{val| 0 -7 -9 -13 -8 }}]
: mapping generators: ~2, ~25/22


POTE generator: ~11/10 = 156.692
Optimal tunings:
* CTE: ~2 = 1200.000, ~25/22 = 221.5981
* POTE: ~2 = 1200.000, ~25/22 = 221.605


{{Optimal ET sequence|legend=1| 8d, …, 38d, 46, 176dde, 222cdde, 268cddee }}
{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}


Badness: 0.041723
Badness: 0.048714


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


Comma list: 91/90, 121/120, 126/125, 169/168
Comma list: 91/90, 126/125, 169/168, 243/242


Mapping: [{{val| 2 5 7 9 9 10 }}, {{val| 0 -7 -9 -13 -8 -10 }}]
Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}


POTE generator: ~11/10 = 156.725
Optimal tunings:
* CTE: ~2 = 1200.000, ~25/22 = 221.6333
* POTE: ~2 = 1200.000, ~25/22 = 221.556


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}


Badness: 0.026339
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


{{Optimal ET sequence|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


{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
Line 297: 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 tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
Line 310: 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 tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
Line 323: 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}}
 
: mapping generators: ~2, ~22/17


POTE generator: ~22/17 = 443.270
Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
Line 332: Line 511:


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


{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)
 
[[POTE generator]]: ~35/32 = 156.925


{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
Line 351: 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 ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
Line 364: 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 ET sequence|legend=1| 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


{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
Line 392: 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}}


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


{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
Line 405: 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 ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
Line 418: 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 ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
Line 431: 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 ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
Line 440: Line 678:


== 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


{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
Line 459: 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 ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 472: 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 ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 485: 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 ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Line 498: 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 ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Line 507: Line 749:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[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]]

Latest revision as of 00:30, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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.

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:

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

Sensipent

Main article: Sensipent

Subgroup: 2.3.5

Comma list: 78732/78125

Mapping: [⟨1 -1 -1], ⟨0 7 9]]

mapping generators: ~2, ~162/125

Optimal tuning (POTE): ~2 = 1200.000, ~162/125 = 443.058

Optimal ET sequence: 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 539 is the one with prime 31 by interpreting the generator accurately as 31/24~40/31, tempering out 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: [⟨1 -1 -1 2], ⟨0 7 9 8]]

mapping generators: ~2, ~31/24

Optimal ET sequence: 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 with primes 23 and 29 found by 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: [⟨1 -1 -1 6 -4 2], ⟨0 7 9 -4 24 8]]

Optimal ET sequence: 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)3. This extension is very strong as supported by the optimal ET sequence going very far and as supported by another observation that it also tempers out the semiporwellisma, which is equal to S31 × S322 (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a lopsided comma, implies that this temperament equates (33/32)2 with 16/15 as well as that a natural extension to prime 31 exists through {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 S33 so that a slightly sharp ~22/17 is equated with the generator.

The aforementioned extension with prime 17 through tempering out S33 is equivalent to the one by tempering out 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 {(S16, S9/S10,) S23, S24, S31, S32, S33} implying also tempering out 496/495 = S31 × S32 and 528/527 = S32 × S33 as well as 16337/16335 = S31/S33 = (34/30)/(33/31)2 = (17/15)/(33/31)2. 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: [⟨1 -1 -1 9], ⟨0 7 9 -15]]

mapping generators: ~2, ~128/99

Optimal ET sequence: 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: [⟨1 -1 -1 9 10], ⟨0 7 9 -15 -16]]

mapping generators: ~2, ~22/17

Optimal ET sequence: 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: [⟨1 -1 -1 9 10 6], ⟨0 7 9 -15 -16 -4]]

Optimal ET sequence: 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: [⟨1 -1 -1 9 10 6 2], ⟨0 7 9 -15 -16 -4 8]]

Optimal ET sequence: 19, 46, 65, 111, 176g

Optimal tunings:

  • CTE: 2/1 = 1\1, ~22/17 = 443.183
  • CEE: 2/1 = 1\1, ~22/17 = 443.115

Badness (Sintel): 0.490

Sensi

Main article: Sensi

Sensi tempers out 245/243, 686/675 and 4375/4374 in addition to 126/125, and can be described as the 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 scales of size 8, 11, 19 and 27 are available.

Septimal sensi

Subgroup: 2.3.5.7

Comma list: 126/125, 245/243

Mapping: [⟨1-1 -1 -2], ⟨0 7 9 13]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.3166
  • POTE: ~2 = 1200.000, ~9/7 = 443.383

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.7
unchanged-interval (eigenmonzo) basis: 2.9/5

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 x5 + x4 - 4x2 + x - 1, at 443.3783 cents.

Optimal ET sequence: 19, 27, 46

Badness: 0.025622

2.3.5.7.13 subgroup (sensation)

Subgroup: 2.3.5.7.13

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

Mapping: [⟨1 -1 -1 -2 0], ⟨0 7 9 13 10]]

mapping generators: ~2, ~9/7

Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016

Optimal ET sequence: 19, 27, 46, 111df

Sensor

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -1 -2 9], ⟨0 7 9 13 -15]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.2987
  • POTE: ~2 = 1200.000, ~9/7 = 443.294

Optimal ET sequence: 19, 27, 46, 111d

Badness: 0.037942

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 -1 -2 9 0], ⟨0 7 9 13 -15 10]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.3658
  • POTE: ~2 = 1200.000, ~9/7 = 443.321

Optimal ET sequence: 19, 27, 46, 111df

Badness: 0.025575

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168, 256/255

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: 19, 27, 46

Badness: 0.022908

Sensus

Subgroup: 2.3.5.7.11

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

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: 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: [⟨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: 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: [⟨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: 19eg, 27eg, 46

Badness: 0.016238

Sensis

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -1 -2 2], ⟨0 7 9 13 4]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.1886
  • POTE: ~2 = 1200.000, ~9/7 = 443.962

Optimal ET sequence: 8d, 19, 27e

Badness: 0.028680

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90, 100/99

Mapping: [⟨1 -1 -1 -2 2 0], ⟨0 7 9 13 4 10]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.2863
  • POTE: ~2 = 1200.000, ~9/7 = 443.945

Optimal ET sequence: 8d, 19, 27e

Badness: 0.020017

Sensa

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -1 -2 -1], ⟨0 7 9 13 12]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.7814
  • POTE: ~2 = 1200.000, ~9/7 = 443.518

Optimal ET sequence: 8d, 19e, 27

Badness: 0.036835

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 77/75, 143/140

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: 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: [⟨2 -2 -2 -4 1], ⟨0 7 9 13 8]]
mapping generators: ~99/70, ~9/7
  • mingen form: [⟨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: 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

Mapping:

  • common form: [⟨2 -2 -2 -4 1 0], ⟨0 7 9 13 8 10]]
mapping generators: ~99/70, ~9/7
  • mingen form: [⟨2 5 7 9 9 10], ⟨0 -7 -9 -13 -8 -10]]
mapping generators: ~99/70, ~11/10

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 ET sequence: 8d, …, 38df, 46

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: [⟨2 -2 -2 -4 1 0 3], ⟨0 7 9 13 8 10 7]]
mapping generators: ~99/70, ~9/7
  • mingen form: [⟨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: 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

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

Mapping: [⟨1 -1 -1 -2 -3], ⟨0 14 18 26 35]]

mapping generators: ~2, ~25/22

Optimal tunings:

  • CTE: ~2 = 1200.000, ~25/22 = 221.5981
  • POTE: ~2 = 1200.000, ~25/22 = 221.605

Optimal ET sequence: 27e, 38d, 65

Badness: 0.048714

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 243/242

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: 27e, 38df, 65f

Badness: 0.033016

Sensei

Subgroup: 2.3.5.7

Comma list: 225/224, 78732/78125

Mapping: [⟨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: 19, 65d, 84, 103, 187, 290b

Badness: 0.059218

Warrior

Subgroup: 2.3.5.7

Comma list: 5120/5103, 78732/78125

Mapping: [⟨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: 46, 111, 157, 268cd

Badness: 0.118239

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323, 5120/5103

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: 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: [⟨1 -1 -1 15 9 17], ⟨0 7 9 -33 -15 -36]]

mapping generators: ~2, ~84/65

Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270

Optimal ET sequence: 46, 65d, 111, 268cd, 379cddf

Badness: 0.028735

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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: 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.

Subgroup: 2.3.5.7

Comma list: 6144/6125, 78732/78125

Mapping:

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

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

Optimal ET sequence: 8, 38, 46, 84, 130

Badness: 0.070375

11-limit

Subgroup: 2.3.5.7.11

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

Mapping:

  • common form: [⟨2 -2 -2 13 18], ⟨0 7 9 -10 -15]]
mapping generators: ~567/400, ~162/125
  • mingen form: [⟨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: 46, 84, 130, 306, 436ce

Badness: 0.037132

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 10985/10976

Mapping:

  • common form: [⟨2 -2 -2 13 18 17], ⟨0 7 9 -10 -15 -13]]
mapping generators: ~55/39, ~162/125
  • mingen form: [⟨2 5 7 3 3 4], ⟨0 -7 -9 10 15 13]]
mapping generators: ~55/39, ~35/32

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

Optimal ET sequence: 46, 84, 130, 566ce, 596cef

Badness: 0.023504

Subpental

Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c4P4/14)).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 19683/19600

Mapping: [⟨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: 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: [⟨1 6 8 17 -6], ⟨0 -14 -18 -45 30]]

Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.440

Optimal ET sequence: 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: [⟨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: 19, 111, 130, 241, 371ce

Badness: 0.023940

Heinz

Heinz splits the generator ~18/7 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).

Subgroup: 2.3.5.7

Comma list: 1029/1024, 78732/78125

Mapping: [⟨1 -8 -10 6], ⟨0 21 27 -7]]

mapping generators: ~2, ~48/35

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

Optimal ET sequence: 46, 103, 149, 699bdd

Badness: 0.115385

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -8 -10 6 3], ⟨0 21 27 -7 1]]

mapping generators: ~2, ~11/8

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

Optimal ET sequence: 46, 103, 149, 252e, 401bdee

Badness: 0.042412

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -8 -10 6 3 11], ⟨0 21 27 -7 1 -16]]

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

Optimal ET sequence: 46, 103, 149, 252ef, 401bdeef

Badness: 0.025779

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨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: 46, 103, 149, 252ef, 401bdeef

Badness: 0.018479

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: [⟨1 -8 -10 6 3 11 5 12], ⟨0 21 27 -7 1 -16 -2 -17]]

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

Optimal ET sequence: 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: [⟨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: 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: [⟨3 4 6 8 8], ⟨0 7 9 4 22]]

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

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 27eg, 84e, 111

Badness: 0.024120

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475

Mapping: [⟨3 4 6 8 8 11 10 12], ⟨0 7 9 4 22 1 21 7]]

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

Optimal ET sequence: 27eg, 84e, 111

Badness: 0.018466

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