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{{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]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.9429{c}}, ~9/7 = 443.0364{{c}}
* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.0507{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
: error map: {{val| 0.000 -0.600 +1.143 }}
: error map: {{val| 0.000 -0.600 +1.143 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 443.058{{c}} -->


{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}
Line 37: Line 35:


Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
=== 2.3.5.31 subgroup ===
Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one with prime 31 by interpreting the generator accurately as [[31/24]]~[[40/31]], tempering out [[961/960|S31 = 961/960]], so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
Subgroup: 2.3.5.31
Comma list: 961/960, 2511/2500
Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
: mapping generators: ~2, ~31/24
Optimal tunings:
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~31/24 = 443.050{{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 (2.3.5.31) [[#Sensipent|sensipent]] with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).
Subgroup: 2.3.5.23.29.31
Comma list: 465/464, 576/575, 621/620, 900/899
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}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~31/24 = 442.989{{c}} -->
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
Badness (Sintel): 0.283


=== Sensible ===
=== Sensible ===
{{See also| Sensipent #Sensible interval table }}
{{See also| Sensipent #Sensible interval table }}


Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[961/960|S31]] × [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
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|S33]] is equivalent to the one by tempering out [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
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 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]].
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]].


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11
Line 98: Line 56:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~128/99 = 443.115{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
Line 116: Line 73:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.188{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Line 132: Line 88:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.185{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Line 148: Line 103:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.183{{c}}
* CEE: ~2 = 1200.000{{c}}, ~22/17 = 443.115{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Line 158: Line 111:
{{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 166: Line 119:


{{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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.7081{c}}, ~9/7 = 443.2748{{c}}
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3166{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~9/7 = 443.383{{c}} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
Line 201: Line 150:


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:  
Optimal tunings:  
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4016{{c}} -->


{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
Line 219: 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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2987{{c}}
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.294{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}


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


Badness (Smith): 0.037942
Badness (Sintel): 1.25


==== 13-limit ====
==== 13-limit ====
Line 238: Line 182:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3658{{c}}
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.321{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}


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


Badness (Smith): 0.025575
Badness (Sintel): 1.06


==== 17-limit ====
==== 17-limit ====
Line 253: Line 197:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3775{{c}}
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.365{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}


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


Badness (Smith): 0.022908
Badness (Sintel): 1.17


=== Sensus ===
=== Sensus ===
Line 266: 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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4783{{c}}
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.626{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}


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


Badness (Smith): 0.029486
Badness (Sintel): 0.975


==== 13-limit ====
==== 13-limit ====
Line 285: Line 227:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5075{{c}}
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.559{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}


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


Badness (Smith): 0.020789
Badness (Sintel): 0.859


==== 17-limit ====
==== 17-limit ====
Line 300: Line 242:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5050{{c}}
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}


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


Badness (Smith): 0.016238
Badness (Sintel): 0.827


=== Sensis ===
=== Sensis ===
Line 313: 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


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.1886{{c}}
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}


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


Badness (Smith): 0.028680
Badness (Sintel): 0.948


==== 13-limit ====
==== 13-limit ====
Line 332: Line 272:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2863{{c}}
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.945{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}


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


Badness (Smith): 0.020017
Badness (Sintel): 0.827


=== Sensa ===
=== Sensa ===
Line 345: 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 = 1200.0000{{c}}, ~9/7 = 443.7814{{c}}
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.518{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}


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


Badness (Smith): 0.036835
Badness (Sintel): 1.22


==== 13-limit ====
==== 13-limit ====
Line 361: 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 = 1200.0000{{c}}, ~9/7 = 443.7877{{c}}
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.506{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}


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


Badness (Smith): 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 378: 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 = 600.0000{{c}}, ~9/7 = 443.3688{{c}} (~11/10 = 156.6312{{c}})
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* POTE: ~99/70 = 600.000{{c}}, ~9/7 = 443.308{{c}} (~11/10 = 156.692{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})


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


Badness (Smith): 0.041723
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 397: 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 = 600.0000{{c}}, ~9/7 = 443.4416{{c}} (~11/10 = 156.5584{{c}})
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* POTE: ~55/39 = 600.000{{c}}, ~9/7 = 443.275{{c}} (~11/10 = 156.725{{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 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Badness (Smith): 0.026339
Badness (Sintel): 1.09


==== 17-limit ====
==== 17-limit ====
Line 416: 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 = 600.0000{{c}}, ~9/7 = 443.4466{{c}} (~11/10 = 156.5534{{c}})
* 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=0| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Badness (Smith): 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 441: Line 370:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5981{{c}}
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.605{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}


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


Badness (Smith): 0.048714
Badness (Sintel): 1.61


==== 13-limit ====
==== 13-limit ====
Line 456: Line 385:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6333{{c}}
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.556{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}


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


Badness (Smith): 0.033016
Badness (Sintel): 1.36


== Sensei ==
== Sensei ==
Line 469: Line 398:


{{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:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 442.755{{c}}
* [[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 }}


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


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


== Warrior ==
== Warrior ==
Line 485: Line 415:


{{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:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 443.289{{c}}
* [[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 ET sequence|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}


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


=== 11-limit ===
=== 11-limit ===
Line 500: 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:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~128/99 = 443.274{{c}}
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}


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


Badness (Smith): 0.046383
Badness (Sintel): 1.53


=== 13-limit ===
=== 13-limit ===
Line 516: 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:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~84/65 = 443.270{{c}}
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}


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


Badness (Smith): 0.028735
Badness (Sintel): 1.19


=== 17-limit ===
=== 17-limit ===
Line 532: 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:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~22/17 = 443.270{{c}}
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}


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


Badness (Smith): 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 550: 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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:
* [[POTE]]: ~567/400 = 600.000{{c}}, ~162/125 = 443.075{{c}} (~35/32 = 156.925{{c}})
* [[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]] (Smith): 0.070375
[[Badness]] (Sintel): 1.78


=== 11-limit ===
=== 11-limit ===
Line 568: 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 tunings:  
Optimal tunings:
* POTE: ~99/70 = 600.000{{c}}, ~162/125 = 443.117{{c}} (~35/32 = 156.883{{c}})
* 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=0| 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}


Badness (Smith): 0.037132
Badness (Sintel): 1.23


=== 13-limit ===
=== 13-limit ===
Line 586: 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 tunings:  
Optimal tunings:
* POTE: ~55/39 = 600.000{{c}}, ~162/125 = 443.096{{c}} (~35/32 = 156.904{{c}})
* 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| 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}


Badness (Smith): 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 606: 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:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~56/45 = 378.467{{c}}
* [[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, 929c, 1059c, 1189bc, 1319bc }}
{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}


[[Badness]] (Smith): 0.054303
[[Badness]] (Sintel): 1.37


=== 11-limit ===
=== 11-limit ===
Line 622: 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 tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~56/45 = 378.440{{c}}
* 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, 872cde }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}


Badness (Smith): 0.045352
Badness (Sintel): 1.50


=== 13-limit ===
=== 13-limit ===
Line 636: 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 tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~56/45 = 378.437{{c}}
* 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 }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}


Badness (Smith): 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 652: 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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~48/35 = 546.815{{c}}
* [[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 673: Line 601:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.631{{c}}
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}


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


Badness (Smith): 0.042412
Badness (Sintel): 1.40


=== 13-limit ===
=== 13-limit ===
Line 687: Line 616:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.629{{c}}
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}


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


Badness (Smith): 0.025779
Badness (Sintel): 1.07


=== 17-limit ===
=== 17-limit ===
Line 701: Line 631:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.635{{c}}
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}


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


Badness (Smith): 0.018479
Badness (Sintel): 0.941


=== 19-limit ===
=== 19-limit ===
Line 715: Line 646:


Optimal tunings:  
Optimal tunings:  
* POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.614{{c}}
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}


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


Badness (Smith): 0.019005
Badness (Sintel): 1.16


== Trisensory ==
== Trisensory ==
Line 729: 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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~63/50 = 400.000{{c}}, ~36/35 = 43.147{{c}}
* [[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 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 747: Line 681:


Optimal tunings:  
Optimal tunings:  
* POTE: ~63/50 = 400.000{{c}}, ~36/35 = 43.292{{c}}
* 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=0| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}


Badness (Smith): 0.058413
Badness (Sintel): 1.93


=== 13-limit ===
=== 13-limit ===
Line 763: Line 698:


Optimal tunings:  
Optimal tunings:  
* POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.288{{c}}
* 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=0| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}


Badness (Smith): 0.034829
Badness (Sintel): 1.44


=== 17-limit ===
=== 17-limit ===
Line 777: Line 713:


Optimal tunings:  
Optimal tunings:  
* POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.276{{c}}
* 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=0| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}


Badness (Smith): 0.024120
Badness (Sintel): 1.23


=== 19-limit ===
=== 19-limit ===
Line 791: Line 728:


Optimal tunings:  
Optimal tunings:  
* POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.292{{c}}
* 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 }}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}


Badness (Smith): 0.018466
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=0| 19, 46j, 65, 149, 363j }}
 
Badness (Sintel): 0.283


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