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Move the opening note to a reasonable place. Don't shock the readers with stuff like 31/24 or 40/31 without proper context
 
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{{interwiki
{{Technical data page}}
| de = Sensi
| en = Sensipent family
| es =
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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.
[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.  
 
== Sensipent ==
{{Main| Sensipent }}
 
The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125. Two generators make 5/3, seven make harmonic 6 and nine make harmonic 10. Its [[ploidacot]] is beta-heptacot ([[pergen]] (P8, ccP5/7)) and its color name is Sepguti.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 78732/78125
 
{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
: mapping generators: ~2, ~162/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
: error map: {{val| 0.000 -0.600 +1.143 }}
 
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}
 
[[Badness]] (Sintel): 0.826
 
=== Overview to extensions ===
==== Full 7-limit extensions ====
The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.
 
Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.  


Temperaments discussed elsewhere include:  
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]]'' → [[Canopic clan #Browser|Canopic clan]] (+16875/16807)
* ''[[browser]]'', {16875/16807, 78732/78125} → [[Mirkwai clan #Browser]]
 
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
 
==== Subgroup extensions ====
The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Other subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.
 
This extension can be laid onto many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 31/24~40/31 such that 24:31:40 is a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.
 
[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, considered immediately below.
 
=== Sensible ===
{{See also| Sensipent #Sensible interval table }}
 
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[S-expression|S31⋅S32<sup>2</sup>]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.
 
The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.
 
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]]. A notable non-patent val is [[84edo|84g]].
 
Subgroup: 2.3.5.11
 
Comma list: 8019/8000, 16384/16335
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}
 
: mapping generators: ~2, ~128/99
 
Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
 
Badness (Sintel): 0.728
 
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17
 
Comma list: 256/255, 1089/1088, 1377/1375
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
: mapping generators: ~2, ~22/17
 
Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
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


== Sensipent ==
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
Subgroup: 2.3.5
 
Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.555
 
==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31


[[Comma list]]: 78732/78125
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375


[[Mapping]]: [{{val| 1 -1 -1 }}, {{val| 0 7 9 }}]
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}


[[POTE generator]]: 162/125 = 443.058
Optimal tunings:  
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}


{{Val list|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


[[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 {{nowrap| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.  


=== 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 }}


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


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~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 145:
* 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]] (Sintel): 0.648


==== 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]
Optimal tunings:  
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}


POTE generator: ~9/7 = 443.322
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


Optimal GPV sequence: {{Val list| 19, 27, 46, 111de, 157de }}
Badness (Sintel): 0.484


=== Sensor ===
=== Sensor ===
Line 85: Line 172:
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
Optimal tunings:  
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}


Optimal GPV sequence: {{Val list| 19, 27, 46, 111d, 157d, 268cdd }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}


Badness: 0.037942
Badness (Sintel): 1.25


==== 13-limit ====
==== 13-limit ====
Line 98: Line 187:
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:  
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}


Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


Badness: 0.025575
Badness (Sintel): 1.06


==== 17-limit ====
==== 17-limit ====
Line 111: Line 202:
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:  
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}


Optimal GPV sequence: {{Val list| 19, 27, 46, 157df, 203cdff, 249cddff }}
{{Optimal ET sequence|legend=0| 19, 27, 46 }}


Badness: 0.022908
Badness (Sintel): 1.17


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


POTE generator: ~9/7 = 443.962
Optimal tunings:  
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}


Optimal GPV sequence: {{Val list| 8d, 19, 27e, 73ee }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}


Badness: 0.028680
Badness (Sintel): 0.975


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


POTE generator: ~9/7 = 443.945
Optimal tunings:  
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}


Optimal GPV sequence: {{Val list| 19, 27e, 46e, 73ee }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}


Badness: 0.020017
Badness (Sintel): 0.859


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


Comma list: 126/125, 176/175, 245/243
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168


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


POTE generator: ~9/7 = 443.626
Optimal tunings:  
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}


Optimal GPV sequence: {{Val list| 19e, 27e, 46, 119c, 165c }}
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}


Badness: 0.029486
Badness (Sintel): 0.827


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


Comma list: 91/90, 126/125, 169/168, 352/351
Comma list: 56/55, 100/99, 245/243


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


POTE generator: ~9/7 = 443.559
Optimal tunings:  
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}


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


Badness: 0.020789
Badness (Sintel): 0.948


==== 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:  
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}


Optimal GPV sequence: {{Val list| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}


Badness: 0.016238
Badness (Sintel): 0.827


=== Sensa ===
=== Sensa ===
Line 189: Line 292:
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 }}


POTE generator: ~9/7 = 443.518
Optimal tunings:  
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}


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


Badness: 0.036835
Badness (Sintel): 1.22


==== 13-limit ====
==== 13-limit ====
Line 202: Line 307:
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 10}}
 
Optimal tunings:
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}


POTE generator: ~9/7 = 443.506
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}


Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}
Badness (Sintel): 0.961


Badness: 0.023258
=== Bisensi ===
Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). (As 84edo is even, one might be interested to know its [[17-limit]] val is 84def, corresponding to a strong flat tendency.)


=== Hemisensi ===
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: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}


Mapping: [{{val| 1 -1 -1 -2 -3 }}, {{val| 0 14 18 26 35 }}]
: mapping generators: ~99/70, ~9/7


POTE generator: ~25/22 = 221.605
Optimal tunings:  
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})


Optimal GPV sequence: {{Val list| 27e, 38d, 65, 157de, 222cde }}
{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}


Badness: 0.048714
Badness (Sintel): 1.38


==== 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: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}
 
Optimal tunings:
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
 
Badness (Sintel): 1.09
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
 
Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}


Mapping: [{{val| 1 -1 -1 -2 -3 0 }}, {{val| 0 14 18 26 35 30 }}]
Optimal tunings:  
* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})


POTE generator: ~25/22 = 221.556
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Optimal GPV sequence: {{Val list| 27e, 38df, 65f }}
Badness (Sintel): 0.960


Badness: 0.033016
=== Hemisensi ===
Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-14-cot (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:  
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}


Optimal GPV sequence: {{Val list| 8d, …, 38d, 46, 176dde, 222cdde, 268cddee }}
{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}


Badness: 0.041723
Badness (Sintel): 1.61


==== 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:  
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}


Optimal GPV sequence: {{Val list| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}


Badness: 0.026339
Badness (Sintel): 1.36


== 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 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6422{{c}}, ~162/125 = 442.9920{{c}}
: [[error map]]: {{val| +0.642 -1.653 -0.028 +1.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}


[[POTE generator]]: ~125/81 = 757.245
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}


{{Val list|legend=1| 19, 84, 103, 187, 290b }}
[[Badness]] (Sintel): 1.50
 
[[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 }}


[[POTE generator]]: ~162/125 = 443.289
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2419{{c}}, ~162/125 = 443.0087{{c}}
: [[error map]]: {{val| -0.758 -0.136 +1.523 +0.516 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}


{{Val list|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}


[[Badness]]: 0.118239
[[Badness]] (Sintel): 2.99


=== 11-limit ===
=== 11-limit ===
Line 297: Line 439:
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
Optimal tunings:  
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}


Optimal GPV sequence: {{Val list| 46, 65d, 111, 268cd, 379cdd}}
{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}


Badness: 0.046383
Badness (Sintel): 1.53


=== 13-limit ===
=== 13-limit ===
Line 310: Line 454:
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
Optimal tunings:  
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}


Optimal GPV sequence: {{Val list| 46, 65d, 111, 268cd, 379cddf}}
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}


Badness: 0.028735
Badness (Sintel): 1.19


=== 17-limit ===
=== 17-limit ===
Line 323: Line 469:
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
Optimal tunings:  
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}


Optimal GPV sequence: {{Val list| 46, 65d, 111, 268cdg, 379cddfg }}
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}


Badness: 0.018105
Badness (Sintel): 0.922


== Bison ==
== Bison ==
Subgroup: 2.3.5.7
Bison has a 1/2-octave period and the generator can be taken as [[~]][[162/125]] or its semi-octave complement, ~[[35/32]]. Its [[ploidacot]] is diploid delta-heptacot ([[pergen]] (P8/2, ccP5/7)).
 
[[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|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
: mapping generators: ~567/400, ~162/125


{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 599.9413{{c}}, ~162/125 = 443.0320{{c}} (~35/32 = 156.9093{{c}})
: [[error map]]: {{val| -0.117 -0.613 +1.092 +0.091 }}
* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})
: error map: {{val| 0.000 -0.446 +1.341 +0.446 }}


[[POTE generator]]: ~35/32 = 156.925
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}


{{Val list|legend=1| 8, 38, 46, 84, 130 }}
[[Badness]] (Sintel): 1.78
 
[[Badness]]: 0.070375


=== 11-limit ===
=== 11-limit ===
Line 351: Line 504:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping: [{{val| 2 5 7 3 3 }}, {{val| 0 -7 -9 10 15 }}]
Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}


POTE generator: ~35/32 = 156.883
Optimal tunings:
* WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})


Optimal GPV sequence: {{Val list| 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}


Badness: 0.037132
Badness (Sintel): 1.23


=== 13-limit ===
=== 13-limit ===
Line 364: Line 519:
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: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
 
Optimal tunings:
* WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})
 
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}
 
Badness (Sintel): 0.971
 
== Subpental ==
Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c<sup>4</sup>P4/14)).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 19683/19600
 
{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
: mapping generators: ~2, ~45/28
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9261{{c}}, ~45/28 = 821.4823{{c}}
: [[error map]]: {{val| -0.074 -0.611 +1.107 -0.052 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}
 
{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}
 
[[Badness]] (Sintel): 1.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 3136/3125, 8019/8000
 
Mapping: {{mapping| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}
 
Optimal tunings:
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}
 
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}


POTE generator: ~35/32 = 156.904
Badness (Sintel): 1.50


Optimal GPV sequence: {{Val list| 46, 84, 130, 566ce, 596cef }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Badness: 0.023504
Comma list: 351/350, 540/539, 676/675, 3136/3125
 
Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}
 
Optimal tunings:
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}
 
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}
 
Badness (Sintel): 0.989


== Heinz ==
== Heinz ==
Subgroup: 2.3.5.7
Heinz splits the sensipent generator ~324/125 in three. Its ploidacot is theta-21-cot (pergen (P8, c<sup>9</sup>P5/21)). A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (57 = 103 - 46).
 
[[Subgroup]]: 2.3.5.7


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


[[POTE generator]]: ~48/35 = 546.815
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
: [[error map]]: {{val| +0.425 -0.758 +1.061 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}


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


[[Badness]]: 0.115385
[[Badness]] (Sintel): 2.92


=== 11-limit ===
=== 11-limit ===
Line 392: Line 604:
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: {{mapping| 1 -8 -10 6 3 | 0 21 27 -7 1}}


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


Optimal GPV sequence: {{Val list| 46, 103, 149, 252e, 401bdee }}
Optimal tunings:  
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}


Badness: 0.042412
{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}
 
Badness (Sintel): 1.40


=== 13-limit ===
=== 13-limit ===
Line 405: Line 621:
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: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}


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


Optimal GPV sequence: {{Val list| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}


Badness: 0.025779
Badness (Sintel): 1.07


=== 17-limit ===
=== 17-limit ===
Line 418: Line 636:
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: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}


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


Optimal GPV sequence: {{Val list| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}


Badness: 0.018479
Badness (Sintel): 0.941


=== 19-limit ===
=== 19-limit ===
Line 431: Line 651:
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: {{mapping| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}


POTE generator: ~11/8 = 547.614
Optimal tunings:  
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}


Optimal GPV sequence: {{Val list| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=0| 46, 103h, 149h }}


Badness: 0.019005
Badness (Sintel): 1.16


== 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 }}
: mapping generators: ~63/50, ~36/35


{{Multival|legend=1|21 27 12 -6 -40 -48}}
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.8117{{c}}, ~36/35 = 43.1270{{c}}
: [[error map]]: {{val| -0.565 -0.819 +0.700 +2.176 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}


[[POTE generator]]: ~36/35 = 43.147
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}


{{Val list|legend=1| 27, 57, 84, 111, 195d, 306d }}
[[Badness]] (Sintel): 2.27
 
[[Badness]]: 0.089740


=== 11-limit ===
=== 11-limit ===
Line 459: Line 686:
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 tunings:  
* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}


Optimal GPV sequence: {{Val list| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}


Badness: 0.058413
Badness (Sintel): 1.93


=== 13-limit ===
=== 13-limit ===
Line 472: Line 701:
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 }}


POTE generator: ~36/35 = 43.288
: mapping generators: ~49/39, ~36/35


Optimal GPV sequence: {{Val list| 27e, 84e, 111 }}
Optimal tunings:  
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}


Badness: 0.034829
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
 
Badness (Sintel): 1.44


=== 17-limit ===
=== 17-limit ===
Line 485: Line 718:
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 tunings:  
* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}


Optimal GPV sequence: {{Val list| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}


Badness: 0.024120
Badness (Sintel): 1.23


=== 19-limit ===
=== 19-limit ===
Line 498: Line 733:
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 }}
 
Optimal tunings:
* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
 
Badness (Sintel): 1.12
 
== Other subgroup extensions ==
=== Sensipent (2.3.5.31 subgroup) ===
Subgroup: 2.3.5.31
 
Comma list: 961/960, 2511/2500
 
Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
 
Optimal tunings:
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
 
{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
 
Badness (Sintel): 0.243
 
=== Sendai ===
{{See also| Sensipent #Sendai interval table }}
 
Sendai is an accurate extension of sensipent with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence.
 
Subgroup: 2.3.5.23.29.31
 
Comma list: 465/464, 576/575, 621/620, 900/899
 
Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}


POTE generator: ~36/35 = 43.292
Optimal tunings:  
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}


Optimal GPV sequence: {{Val list| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}


Badness: 0.018466
Badness (Sintel): 0.283


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[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"