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add canonical extension of sensipent to prime 31 which preserves accuracy and the "pent" can still refer to being primarily 5-limit (as we only add a relevant & accurate large prime). also add extension to 2.3.5.11.17.31 that really deserves documentation and explain the commonness and strange indirect accuracy of this 5-limit structure in terms of higher-limit harmony.
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m →Sensible: nowrap for visual sanity
 
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{{interwiki
{{Technical data page}}
| de = Sensi
| en = Sensipent family
| es =
| ja =
}}
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, seven make harmonic 6 and nine make harmonic 10.


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 all use the same nominal generator as sensipent.  
[[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 ===
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.  
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.  
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Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.


== Sensipent ==
=== Sensible ===
[[Subgroup]]: 2.3.5
{{See also| Sensipent #Sensible interval table }}


[[Comma list]]: 78732/78125
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.


{{Mapping|legend=1| 1 6 8 | 0 -7 -9 }}
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) }}.


: mapping generators: ~2, ~125/81
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]].


[[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~162/125 = 443.058
Subgroup: 2.3.5.11


{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
Comma list: 8019/8000, 16384/16335


[[Badness]]: 0.035220
Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}


Badness (Dirichlet): 0.826
: mapping generators: ~2, ~128/99


=== 2.3.5.31 ===
Optimal tunings:
Fascinatingly, essentially the only extension that preserves the occurrence of sensi's 5-limit structure in such large edos as [[539edo|539]] is the one to prime 31 by interpreting the generator accurately as [[40/31]]~[[31/24]] by tempering [[961/960|S31 = 961/960]], so that the large [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated. 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]].
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}


[[Subgroup]]: 2.3.5.31
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}


[[Comma list]]: 78732/78125, 961/960
Badness (Sintel): 0.728


=== Sensible ===
==== 2.3.5.11.17 subgroup ====
An extension of sensipent  to prime 11 of dubious canonicity (but significantly higher accuracy than [[sensi]]) interprets the generator as [[165/128]]~[[128/99]] by tempering [[8019/8000|S9/S10]] 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 is equivalent to tempering the [[semiporwellisma]] which as it is equal to [[961/960|S31]] * [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list), which by the equivalence of the aforementioned [[lopsided comma]] also 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 to prime 17 is reasonably accurate through tempering [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
Subgroup: 2.3.5.11.17


[[Subgroup]]: 2.3.5.11
Comma list: 256/255, 1089/1088, 1377/1375


[[Comma list]]: [[8019/8000]], [[16384/16335]]
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}


{{Mapping|legend=1| 1 6 8 -6 | 0 -7 -9 15 }}
: mapping generators: ~2, ~22/17


{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~128/99 = 443.115
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


Badness (Dirichlet): 0.728
Badness (Sintel): 0.639


=== 2.3.5.11.17 ===
==== 2.3.5.11.17.23 subgroup ====
The aforementioned extension to prime 17 through tempering [[1089/1088|S33]] is equivalent to the one by tempering [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
Subgroup: 2.3.5.11.17.23


[[Subgroup]]: 2.3.5.11.17
Comma list: 256/255, 576/575, 1089/1088, 1377/1375


[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}


{{Mapping|legend=1| 1 6 8 -6 -6 | 0 -7 -9 15 16 }}
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=1| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~22/17 = 443.188
Badness (Sintel): 0.555


Badness (Dirichlet): 0.639
==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31


=== 2.3.5.11.17.31 ===
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony without the need for interpreting it as high-damage ~[[9/7]] or ~[[13/10]] intervals. Its S-expression-based comma list is {([[256/255|S16]],) [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495|496/495 = S31 * S32]] and [[528/527|528/527 = S32 * S33]] as well as [[16337/16335|16337/16335 = S31/S33]] = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>.


[[Subgroup]]: 2.3.5.11.17.31
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}


[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[]]
Optimal tunings:  
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}


{{Mapping|legend=1| 1 6 8 -6 -6 | 0 -7 -9 15 16 }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g, 287cg }}
Badness (Sintel): 0.490
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.185
 
Badness (Dirichlet): 0.519


== Sensi ==
== Sensi ==
{{Main| 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.  
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 ===
Line 96: Line 118:
[[Comma list]]: 126/125, 245/243
[[Comma list]]: 126/125, 245/243


{{Mapping|legend=1| 1 6 8 11 | 0 -7 -9 -13 }}
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
 
: mapping generators: ~2, ~14/9
 
{{Multival|legend=1| 7 9 13 -2 1 5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
* [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
: [[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 basis|eigenmonzo (unchanged-interval) basis]]: 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 basis|eigenmonzo (unchanged-interval) basis]]: 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 122: Line 142:
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}


[[Badness]]: 0.025622
[[Badness]] (Sintel): 0.648


==== 2.3.5.7.13 subgroup (sensation) ====
==== 2.3.5.7.13 subgroup (sensation) ====
Line 129: Line 149:
Comma list: 91/90, 126/125, 169/168
Comma list: 91/90, 126/125, 169/168


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


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


: gencom: [2 14/9; 91/90 126/125 169/168]
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
Badness (Sintel): 0.484
 
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}


=== Sensor ===
=== Sensor ===
Line 144: Line 164:
Comma list: 126/125, 245/243, 385/384
Comma list: 126/125, 245/243, 385/384


Mapping: {{mapping| 1 6 8 11 -6 | 0 -7 -9 -13 15 }}
Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
 
{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2987
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.294
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}


{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}


Badness: 0.037942
Badness (Sintel): 1.25


==== 13-limit ====
==== 13-limit ====
Line 161: Line 179:
Comma list: 91/90, 126/125, 169/168, 385/384
Comma list: 91/90, 126/125, 169/168, 385/384


Mapping: {{mapping| 1 6 8 11 -6 10 | 0 -7 -9 -13 15 -10 }}
Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
 
{{Multival|legend=1| 7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.3658
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.321
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


Badness: 0.025575
Badness (Sintel): 1.06


==== 17-limit ====
==== 17-limit ====
Line 178: Line 194:
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: {{mapping| 1 6 8 11 -6 10 -6 | 0 -7 -9 -13 15 -10 16 }}
Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.3775
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.365
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}


{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=0| 19, 27, 46 }}


Badness: 0.022908
Badness (Sintel): 1.17


=== Sensus ===
=== Sensus ===
Line 193: Line 209:
Comma list: 126/125, 176/175, 245/243
Comma list: 126/125, 176/175, 245/243


Mapping: {{mapping| 1 6 8 11 23 | 0 -7 -9 -13 -31 }}
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
 
{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.4783
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.626
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}


{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}


Badness: 0.029486
Badness (Sintel): 0.975


==== 13-limit ====
==== 13-limit ====
Line 210: Line 224:
Comma list: 91/90, 126/125, 169/168, 352/351
Comma list: 91/90, 126/125, 169/168, 352/351


Mapping: {{mapping| 1 6 8 11 23 10 | 0 -7 -9 -13 -31 -10 }}
Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
 
{{Multival|legend=1| 7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.5075
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.559
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}


{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}


Badness: 0.020789
Badness (Sintel): 0.859


==== 17-limit ====
==== 17-limit ====
Line 227: Line 239:
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168


Mapping: {{mapping| 1 6 8 11 23 10 23 | 0 -7 -9 -13 -31 -10 -30 }}
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.5050
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.551
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}


{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}


Badness: 0.016238
Badness (Sintel): 0.827


=== Sensis ===
=== Sensis ===
Line 242: Line 254:
Comma list: 56/55, 100/99, 245/243
Comma list: 56/55, 100/99, 245/243


Mapping: {{mapping| 1 6 8 11 6 | 0 -7 -9 -13 -4 }}
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
 
{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.1886
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.962
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}


Badness: 0.028680
Badness (Sintel): 0.948


==== 13-limit ====
==== 13-limit ====
Line 259: Line 269:
Comma list: 56/55, 78/77, 91/90, 100/99
Comma list: 56/55, 78/77, 91/90, 100/99


Mapping: {{mapping| 1 6 8 11 6 10 | 0 -7 -9 -13 -4 -10 }}
Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
 
{{Multival|legend=1| 7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2863
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.945
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}


Badness: 0.020017
Badness (Sintel): 0.827


=== Sensa ===
=== Sensa ===
Line 276: Line 284:
Comma list: 55/54, 77/75, 99/98
Comma list: 55/54, 77/75, 99/98


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7814
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.518
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}


Badness: 0.036835
Badness (Sintel): 1.22


==== 13-limit ====
==== 13-limit ====
Line 291: Line 299:
Comma list: 55/54, 66/65, 77/75, 143/140
Comma list: 55/54, 66/65, 77/75, 143/140


Mapping: {{mapping| 1 6 8 11 11 11 | 0 -7 -9 -13 -12 -11 }}
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7877
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.506
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}


Badness: 0.023258
Badness (Sintel): 0.961


=== Bisensi ===
=== 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)).
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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


Mapping: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}


: mapping generators: ~99/70, ~11/10
: mapping generators: ~99/70, ~9/7


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 1\2, ~11/10 = 156.6312
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* POTE: ~99/70 = 1\2, ~11/10 = 156.692
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})


{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}


Badness: 0.041723
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 323: Line 333:
Comma list: 91/90, 121/120, 126/125, 169/168
Comma list: 91/90, 121/120, 126/125, 169/168


Mapping: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 1\2, ~11/10 = 156.5584
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* POTE: ~55/39 = 1\2, ~11/10 = 156.725
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Badness: 0.026339
Badness (Sintel): 1.09


==== 17-limit ====
==== 17-limit ====
Line 338: Line 348:
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168


Mapping: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~11/10 = 156.5534
* 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}})


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Badness: 0.0188
Badness (Sintel): 0.960


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


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


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


: mapping generators: ~2, ~44/25
: mapping generators: ~2, ~25/22


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.5981
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* POTE: ~2 = 1\1, ~25/22 = 221.605
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}


{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}


Badness: 0.048714
Badness (Sintel): 1.61


==== 13-limit ====
==== 13-limit ====
Line 369: Line 382:
Comma list: 91/90, 126/125, 169/168, 243/242
Comma list: 91/90, 126/125, 169/168, 243/242


Mapping: {{mapping| 1 13 17 24 32 30 | 0 -14 -18 -26 -35 -30 }}
Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.6333
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* POTE: ~2 = 1\1, ~25/22 = 221.556
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}


{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}


Badness: 0.033016
Badness (Sintel): 1.36


== Sensei ==
== Sensei ==
Line 384: Line 397:
[[Comma list]]: 225/224, 78732/78125
[[Comma list]]: 225/224, 78732/78125


{{Mapping|legend=1| 1 6 8 23 | 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}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 442.755
: [[error map]]: {{val| +0.642 -1.653 -0.028 +1.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}


{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}


[[Badness]]: 0.059218
[[Badness]] (Sintel): 1.50


== Warrior ==
== Warrior ==
Line 399: Line 414:
[[Comma list]]: 5120/5103, 78732/78125
[[Comma list]]: 5120/5103, 78732/78125


{{Mapping|legend=1| 1 6 8 -18 | 0 -7 -9 33 }}
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}


{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}


{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
[[Badness]] (Sintel): 2.99
 
[[Badness]]: 0.118239


=== 11-limit ===
=== 11-limit ===
Line 414: Line 431:
Comma list: 176/175, 1331/1323, 5120/5103
Comma list: 176/175, 1331/1323, 5120/5103


Mapping: {{mapping| 1 6 8 -18 -6 | 0 -7 -9 33 15 }}
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 427: Line 446:
Comma list: 176/175, 351/350, 847/845, 1331/1323
Comma list: 176/175, 351/350, 847/845, 1331/1323


Mapping: {{mapping| 1 6 8 -18 -6 -19 | 0 -7 -9 33 15 36 }}
Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 440: Line 461:
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: {{mapping| 1 6 8 -18 -6 -19 -6 | 0 -7 -9 33 15 36 16 }}
Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}


Badness: 0.018105
Badness (Sintel): 0.922


== Bison ==
== Bison ==
Bison has a 1/2-octave period and the generator can be taken as ~162/125 or its semi-octave complement, ~35/32. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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


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


[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~35/32 = 156.925
[[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 }}


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


[[Badness]]: 0.070375
[[Badness]] (Sintel): 1.78


=== 11-limit ===
=== 11-limit ===
Line 470: Line 496:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


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


Optimal tuning (POTE): ~99/70 = 1\2, ~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 ET sequence|legend=1| 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 483: Line 511:
Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 364/363, 441/440, 10985/10976


Mapping: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~35/32 = 156.904
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=1| 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}


Badness: 0.023504
Badness (Sintel): 0.971


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


[[Comma list]]: 3136/3125, 19683/19600
[[Comma list]]: 3136/3125, 19683/19600


{{Mapping|legend=1| 1 6 8 17 | 0 -14 -18 -45 }}
{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
: mapping generators: ~2, ~45/28


{{Multival|legend=1| 14 18 45 -4 32 54 }}
[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 378.467
{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}


{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
[[Badness]] (Sintel): 1.37
 
[[Badness]]: 0.054303


=== 11-limit ===
=== 11-limit ===
Line 511: Line 546:
Comma list: 540/539, 3136/3125, 8019/8000
Comma list: 540/539, 3136/3125, 8019/8000


Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}
Mapping: {{mapping| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.440
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=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}


Badness: 0.045352
Badness (Sintel): 1.50


=== 13-limit ===
=== 13-limit ===
Line 524: Line 561:
Comma list: 351/350, 540/539, 676/675, 3136/3125
Comma list: 351/350, 540/539, 676/675, 3136/3125


Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}
Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.437
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=1| 19, 111, 130, 241, 371ce }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}


Badness: 0.023940
Badness (Sintel): 0.989


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 539: Line 578:
[[Comma list]]: 1029/1024, 78732/78125
[[Comma list]]: 1029/1024, 78732/78125


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


: mapping generators: ~2, ~35/24
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
: [[error map]]: {{val| +0.425 -0.758 +1.061 -1.141 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 546.815
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}


{{Optimal ET sequence|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 556: Line 596:
Comma list: 385/384, 441/440, 78732/78125
Comma list: 385/384, 441/440, 78732/78125


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


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
: mapping generators: ~2, ~11/8
 
Optimal tunings:
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}


{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}


Badness: 0.042412
Badness (Sintel): 1.40


=== 13-limit ===
=== 13-limit ===
Line 569: Line 613:
Comma list: 351/350, 385/384, 441/440, 847/845
Comma list: 351/350, 385/384, 441/440, 847/845


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


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 582: Line 628:
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: {{mapping| 1 13 17 -1 4 -5 3 | 0 -21 -27 7 -1 16 2 }}
Mapping: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 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 595: Line 643:
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: {{mapping| 1 13 17 -1 4 -5 3 -5 | 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 }}


Optimal tuning (POTE): ~2 = 1\1, ~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 ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=0| 46, 103h, 149h }}


Badness: 0.019005
Badness (Sintel): 1.16


== Trisensory ==
== Trisensory ==
Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 609: Line 661:


{{Mapping|legend=1| 3 4 6 8 | 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}}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~36/35 = 43.147
: [[error map]]: {{val| -0.565 -0.819 +0.700 +2.176 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}


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


[[Badness]]: 0.089740
[[Badness]] (Sintel): 2.27


=== 11-limit ===
=== 11-limit ===
Line 625: Line 680:
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~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 ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}


Badness: 0.058413
Badness (Sintel): 1.93


=== 13-limit ===
=== 13-limit ===
Line 638: Line 695:
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
: mapping generators: ~49/39, ~36/35


{{Optimal ET sequence|legend=1| 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 651: Line 712:
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}


Optimal tuning (POTE): ~49/39 = 1\3, ~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 ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}


Badness: 0.024120
Badness (Sintel): 1.23


=== 19-limit ===
=== 19-limit ===
Line 664: Line 727:
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 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 tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.292
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) ===
The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.
 
For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little 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
 
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).
 
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 }}
 
Optimal tunings:
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}


Badness: 0.018466
Badness (Sintel): 0.283


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