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


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.
== Sensipent ==
 
{{Main| Sensipent }}
Temperaments discussed elsewhere include:
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)


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


== Sensipent ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 23: Line 13:


{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
: mapping generators: ~2, ~162/125
: mapping generators: ~2, ~162/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.058
[[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, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}


[[Badness]]:
[[Badness]] (Sintel): 0.826
* Smith: 0.035220
* Dirichlet: 0.826


=== 2.3.5.31 subgroup ===
=== Overview to extensions ===
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]].
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.  


[[Subgroup]]: 2.3.5.31
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.  


[[Comma list]]: 961/960, 2511/2500
Temperaments discussed elsewhere include:
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)


{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.


: mapping generators: ~2, ~31/24
=== Sensible ===
{{See also| Sensipent #Sensible interval table }}


{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
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.


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
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) }}.


[[Badness]] (Dirichlet): 0.243
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]].


=== Sendai ===
Subgroup: 2.3.5.11
{{ See also | User:VIxen/Table of sensipent intervals }}
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: 8019/8000, 16384/16335


[[Comma list]]: 465/464, 576/575, 621/620, 729/725
Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}
 
{{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]] (Dirichlet): 0.283
 
=== Sensible ===
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
: mapping generators: ~2, ~128/99


{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}


[[Badness]] (Dirichlet): 0.728
Badness (Sintel): 0.728


==== 2.3.5.11.17 subgroup ====
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17


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


[[Comma list]]: 256/255, 1089/1088, 1377/1375
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
{{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}


: mapping generators: ~2, ~22/17
: mapping generators: ~2, ~22/17


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
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 = 1200.000, ~22/17 = 443.188
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


[[Badness]] (Dirichlet): 0.639
Badness (Sintel): 0.639


==== 2.3.5.11.17.23 subgroup ====
==== 2.3.5.11.17.23 subgroup ====
[[Subgroup]]: 2.3.5.11.17.23
Subgroup: 2.3.5.11.17.23


[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375
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 }}
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


[[Badness]] (Dirichlet): 0.555
Badness (Sintel): 0.555


==== 2.3.5.11.17.23.31 subgroup ====
==== 2.3.5.11.17.23.31 subgroup ====
[[Subgroup]]: 2.3.5.11.17.23.31
Subgroup: 2.3.5.11.17.23.31


[[Comma list]]: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
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 }}
Subgroup-val mapping: {{mapping| 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 tunings:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}


[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.183
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


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


== 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 ===
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{{Mapping|legend=1| 1-1 -1 -2 | 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 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
* [[POTE]]: ~2 = 1200.000, ~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 166: 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 175: Line 151:
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}


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


Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
Badness (Sintel): 0.484


=== Sensor ===
=== Sensor ===
Line 187: Line 165:


Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.2987
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* POTE: ~2 = 1200.000, ~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 206: Line 180:


Mapping: {{mapping| 1 -1 -1 -2 9 0 | 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 = 1200.000, ~9/7 = 443.3658
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* POTE: ~2 = 1200.000, ~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 225: Line 197:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.3775
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* POTE: ~2 = 1200.000, ~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 238: Line 210:


Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.4783
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* POTE: ~2 = 1200.000, ~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 257: Line 225:


Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 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 = 1200.000, ~9/7 = 443.5075
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* POTE: ~2 = 1200.000, ~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 276: Line 242:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.5050
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* POTE: ~2 = 1200.000, ~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 289: Line 255:


Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
: mapping generators: ~2, ~9/7
{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.1886
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* POTE: ~2 = 1200.000, ~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 308: Line 270:


Mapping: {{mapping| 1 -1 -1 -2 2 0 | 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 = 1200.000, ~9/7 = 443.2863
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* POTE: ~2 = 1200.000, ~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 325: Line 285:


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~9/7 = 443.7814
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* POTE: ~2 = 1200.000, ~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 341: 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 -1 -1 -2 -1 0 | 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 = 1200.000, ~9/7 = 443.7877
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* POTE: ~2 = 1200.000, ~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. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).  
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
Line 358: Line 316:
Comma list: 121/120, 126/125, 245/243
Comma list: 121/120, 126/125, 245/243


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


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~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 377: Line 333:
Comma list: 91/90, 121/120, 126/125, 169/168
Comma list: 91/90, 121/120, 126/125, 169/168


Mapping:
Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}
* 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:  
Optimal tunings:  
* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~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 396: 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: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}
* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}
:: mapping generators: ~99/70, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
:: mapping generators: ~99/70, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~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-tetradecacot (pergen (P8, ccP5/14)).  
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
Line 421: Line 370:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~25/22 = 221.5981
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* POTE: ~2 = 1200.000, ~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 436: Line 385:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000, ~25/22 = 221.6333
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* POTE: ~2 = 1200.000, ~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 450: Line 399:
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}


: mapping generators: ~2, ~162/125
[[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 }}


{{Multival|legend=1| 7 9 32 -2 31 49 }}
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755
[[Badness]] (Sintel): 1.50
 
{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
 
[[Badness]]: 0.059218


== Warrior ==
== Warrior ==
Line 467: Line 416:
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}


: mapping generators: ~2, ~162/125
[[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 }}


{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289
[[Badness]] (Sintel): 2.99
 
{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
 
[[Badness]]: 0.118239


=== 11-limit ===
=== 11-limit ===
Line 482: Line 431:
Comma list: 176/175, 1331/1323, 5120/5103
Comma list: 176/175, 1331/1323, 5120/5103


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


: mapping generators: ~2, ~128/99
Optimal tunings:  
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}


Optimal tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274
{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
Badness (Sintel): 1.53
 
Badness: 0.046383


=== 13-limit ===
=== 13-limit ===
Line 497: 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 -1 -1 15 9 17| 0 7 9 -33 -15 -36}}
Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}


: mapping generators: ~2, ~84/65
Optimal tunings:  
 
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270
* 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 512: 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 -1 -1 15 9 17 10| 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
Optimal tunings:  
 
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270
* 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. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). Related page: [[Bison/Eliora's Approach]].  
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
Line 529: Line 478:
[[Comma list]]: 6144/6125, 78732/78125
[[Comma list]]: 6144/6125, 78732/78125


[[Mapping]]:
{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
* common form: {{mapping| 2 -2 -2 13 | 0 7 9 -10}}
: mapping generators: ~567/400, ~162/125
:: 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)
[[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 548: Line 496:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping:
Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
* common form: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15}}
:: mapping generators: ~567/400, ~162/125
* mingen form: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}
:: mapping generators: ~567/400, ~35/32


Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)
Optimal 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 565: Line 511:
Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 364/363, 441/440, 10985/10976


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


Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~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 ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c<sup>4</sup>P4/14)).  
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
Line 584: Line 528:
[[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


: mapping generators: ~2, ~56/45
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9261{{c}}, ~45/28 = 821.4823{{c}}
{{Multival|legend=1| 14 18 45 -4 32 54 }}
: [[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 = 1200.000, ~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 601: 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 = 1200.000, ~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 614: 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 = 1200.000, ~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 ==
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).
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 629: Line 578:
[[Comma list]]: 1029/1024, 78732/78125
[[Comma list]]: 1029/1024, 78732/78125


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


{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815
: [[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 }}


{{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 646: Line 596:
Comma list: 385/384, 441/440, 78732/78125
Comma list: 385/384, 441/440, 78732/78125


{{Mapping|legend=1| 1 -8 -10 6 3 | 0 21 27 -7 1}}
Mapping: {{mapping| 1 -8 -10 6 3 | 0 21 27 -7 1}}


: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631
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 661: Line 613:
Comma list: 351/350, 385/384, 441/440, 847/845
Comma list: 351/350, 385/384, 441/440, 847/845


{{Mapping|legend=1| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}
Mapping: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}


Optimal tuning (POTE): ~2 = 1200.000, ~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 674: 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|legend=1| 1 -8 -10 6 3 11 5 | 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 = 1200.000, ~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 687: 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|legend=1| 1 -8 -10 6 3 11 5 12 | 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 = 1200.000, ~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 ==
Line 703: 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
: 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 = 400.000, ~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 721: 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 = 400.000, ~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 736: Line 697:
: mapping generators: ~49/39, ~36/35
: mapping generators: ~49/39, ~36/35


Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288
Optimal tunings:
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}


Badness: 0.034829
Badness (Sintel): 1.44


=== 17-limit ===
=== 17-limit ===
Line 749: 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 = 400.000, ~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 762: 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 = 400.000, ~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"