Sensipent family: Difference between revisions

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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]]

Latest revision as of 12:59, 21 May 2026

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

Temperaments of the sensipent family temper out the sensipent comma, 78732/78125, also known as medium semicomma.

Sensipent

Main article: 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: [⟨1 -1 -1], ⟨0 7 9]]

mapping generators: ~2, ~162/125

Optimal tunings:

  • WE: ~2 = 1199.9429 ¢, ~162/125 = 443.0364 ¢
error map: ⟨-0.057 -0.643 +1.071]
  • CWE: ~2 = 1200.0000 ¢, ~162/125 = 443.0507 ¢
error map: ⟨0.000 -0.600 +1.143]

Optimal ET sequence: 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.

Temperaments discussed elsewhere include:

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

Sensible

See also: Sensipent § Sensible interval table

Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than sensi. It interprets the generator as 165/128~128/99 by tempering out 8019/8000 so that 11/8 is reached as (10/9)3. This extension is very strong as supported by the optimal ET sequence going very far and as supported by another observation that it also tempers out the semiporwellisma, which is equal to S31⋅S322 (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a lopsided comma, implies that this temperament equates (33/32)2 with 16/15 as well as that a natural extension to prime 31 exists through {961/960 (S31), 1024/1023 (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 implies tempering out 256/255 (S16), as 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 {(S9/S10, S16,) S23, S24, S31, S32, S33} implying also tempering out 496/495 (S31⋅S32) and 528/527 (S32⋅S33) as well as 16337/16335 (S31/S33) = (17/15)/(33/31)2. A notable patent val tuning not appearing in the optimal ET sequence is 157edo.

Subgroup: 2.3.5.11

Comma list: 8019/8000, 16384/16335

Subgroup-val mapping: [⟨1 -1 -1 9], ⟨0 7 9 -15]]

mapping generators: ~2, ~128/99

Optimal tunings:

  • WE: ~2 = 1199.6725 ¢, ~128/99 = 443.0183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~128/99 = 443.1341 ¢

Optimal ET sequence: 19, 46, 65, 176, 241, 306

Badness (Sintel): 0.728

2.3.5.11.17 subgroup

Subgroup: 2.3.5.11.17

Comma list: 256/255, 1089/1088, 1377/1375

Subgroup-val mapping: [⟨1 -1 -1 9 10], ⟨0 7 9 -15 -16]]

mapping generators: ~2, ~22/17

Optimal tunings:

  • WE: ~2 = 1199.5016 ¢, ~22/17 = 443.0038 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/17 = 443.1878 ¢

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

Badness (Sintel): 0.639

2.3.5.11.17.23 subgroup

Subgroup: 2.3.5.11.17.23

Comma list: 256/255, 576/575, 1089/1088, 1377/1375

Subgroup-val mapping: [⟨1 -1 -1 9 10 6], ⟨0 7 9 -15 -16 -4]]

Optimal tunings:

  • WE: ~2 = 1199.6207 ¢, ~22/17 = 443.0400 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/17 = 443.1808 ¢

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

Badness (Sintel): 0.555

2.3.5.11.17.23.31 subgroup

Subgroup: 2.3.5.11.17.23.31

Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375

Subgroup-val mapping: [⟨1 -1 -1 9 10 6 2], ⟨0 7 9 -15 -16 -4 8]]

Optimal tunings:

  • WE: ~2 = 1199.6623 ¢, ~22/17 = 443.0616 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/17 = 443.1858 ¢

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

Badness (Sintel): 0.490

Sensi

Main article: Sensi

Sensi tempers out 245/243, 686/675 and 4375/4374 in addition to 126/125, and can be described as the 19 & 27 temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. 46edo is an excellent sensi tuning, and mos scales of size 8, 11, 19 and 27 are available.

Septimal sensi

Subgroup: 2.3.5.7

Comma list: 126/125, 245/243

Mapping: [⟨1-1 -1 -2], ⟨0 7 9 13]]

Optimal tunings:

  • WE: ~2 = 1199.7081 ¢, ~9/7 = 443.2748 ¢
error map: ⟨-0.292 +1.261 +3.452 -5.669]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3493 ¢
error map: ⟨0.000 +1.490 +3.830 -5.285]

Minimax tuning:

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

Tuning ranges:

  • 7-odd-limit diamond monotone: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
  • 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
  • 7-odd-limit diamond tradeoff: ~9/7 = [442.179, 445.628]
  • 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]

Algebraic generator: The real root of x5 + x4 - 4x2 + x - 1, at 443.3783 cents.

Optimal ET sequence: 19, 27, 46

Badness (Sintel): 0.648

2.3.5.7.13 subgroup (sensation)

Subgroup: 2.3.5.7.13

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

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

Optimal tunings:

  • WE: ~2 = 1200.3138 ¢, ~9/7 = 443.4379 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3581 ¢

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

Badness (Sintel): 0.484

Sensor

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1200.0367 ¢, ~9/7 = 443.3074 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.2947 ¢

Optimal ET sequence: 19, 27, 46, 111d

Badness (Sintel): 1.25

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1200.3171 ¢, ~9/7 = 443.4382 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3290 ¢

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

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -1 -1 -2 9 0 10], ⟨0 7 9 13 -15 10 -16]]

Optimal tunings:

  • WE: ~2 = 1200.1572 ¢, ~9/7 = 443.4230 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3666 ¢

Optimal ET sequence: 19, 27, 46

Badness (Sintel): 1.17

Sensus

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 -1 -2 -8], ⟨0 7 9 13 31]]

Optimal tunings:

  • WE: ~2 = 1199.0709 ¢, ~9/7 = 443.2830 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.5664 ¢

Optimal ET sequence: 19e, 27e, 46, 119c

Badness (Sintel): 0.975

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 352/351

Mapping: [⟨1 -1 -1 -2 -8 0], ⟨0 7 9 13 31 10]]

Optimal tunings:

  • WE: ~2 = 1199.6887 ¢, ~9/7 = 443.4441 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.5400 ¢

Optimal ET sequence: 19e, 27e, 46

Badness (Sintel): 0.859

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 136/135, 154/153, 169/168

Mapping: [⟨1 -1 -1 -2 -8 0 -7], ⟨0 7 9 13 31 10 30]]

Optimal tunings:

  • WE: ~2 = 1199.7033 ¢, ~9/7 = 443.4418 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.5345 ¢

Optimal ET sequence: 19eg, 27eg, 46

Badness (Sintel): 0.827

Sensis

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1196.8330 ¢, ~9/7 = 443.7907 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.6554 ¢

Optimal ET sequence: 8d, 19, 27e

Badness (Sintel): 0.948

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1197.4337 ¢, ~9/7 = 442.9960 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.6925 ¢

Optimal ET sequence: 8d, 19, 27e

Badness (Sintel): 0.827

Sensa

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1201.0322 ¢, ~9/7 = 443.8994 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.6392 ¢

Optimal ET sequence: 8d, 19e, 27

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 -1 -2 -1 0], ⟨0 7 9 13 12 10]]

Optimal tunings:

  • WE: ~2 = 1201.1279 ¢, ~9/7 = 443.9232 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.6386 ¢

Optimal ET sequence: 8d, 19e, 27

Badness (Sintel): 0.961

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

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

Mapping: [⟨2 -2 -2 -4 1], ⟨0 7 9 13 8]]

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

Optimal tunings:

  • WE: ~99/70 = 600.1183 ¢, ~9/7 = 443.3956 ¢ (~11/10 = 156.7227 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~9/7 = 443.3348 ¢ (~11/10 = 156.6652 ¢)

Optimal ET sequence: 8d, …, 38d, 46

Badness (Sintel): 1.38

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -2 -2 -4 1 0], ⟨0 7 9 13 8 10]]

Optimal tunings:

  • WE: ~55/39 = 600.1183 ¢, ~9/7 = 443.5071 ¢ (~11/10 = 156.8074 ¢)
  • CWE: ~55/39 = 600.0000 ¢, ~9/7 = 443.3459 ¢ (~11/10 = 156.6541 ¢)

Optimal ET sequence: 8d, …, 38df, 46

Badness (Sintel): 1.09

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 126/125, 154/153, 169/168

Mapping: [⟨2 -2 -2 -4 1 0 3], ⟨0 7 9 13 8 10 7]]

Optimal tunings:

  • WE: ~17/12 = 600.2912 ¢, ~9/7 = 443.4993 ¢ (~11/10 = 156.7919 ¢)
  • CWE: ~17/12 = 600.0000 ¢, ~9/7 = 443.3456 ¢ (~11/10 = 156.6544 ¢)

Optimal ET sequence: 8d, …, 38df, 46

Badness (Sintel): 0.960

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

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

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

mapping generators: ~2, ~25/22

Optimal tunings:

  • WE: ~2 = 1199.9253 ¢, ~25/22 = 221.5916 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/22 = 221.6014 ¢

Optimal ET sequence: 27e, 38d, 65

Badness (Sintel): 1.61

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 -1 -2 -3 0], ⟨0 14 18 26 35 20]]

Optimal tunings:

  • WE: ~2 = 1200.6518 ¢, ~25/22 = 221.6764 ¢
  • CWE: ~2 = 1200.0000 ¢, ~25/22 = 221.5908 ¢

Optimal ET sequence: 27e, 38df, 65f

Badness (Sintel): 1.36

Sensei

Subgroup: 2.3.5.7

Comma list: 225/224, 78732/78125

Mapping: [⟨1 -1 -1 -9], ⟨0 7 9 32]]

Optimal tunings:

  • WE: ~2 = 1200.6422 ¢, ~162/125 = 442.9920 ¢
error map: ⟨+0.642 -1.653 -0.028 +1.139]
  • CWE: ~2 = 1200.0000 ¢, ~162/125 = 442.7842 ¢
error map: ⟨0.000 -2.466 -1.256 +0.267]

Optimal ET sequence: 19, 65d, 84, 103, 187, 290b

Badness (Sintel): 1.50

Warrior

Subgroup: 2.3.5.7

Comma list: 5120/5103, 78732/78125

Mapping: [⟨1 -1 -1 15], ⟨0 7 9 -33]]

Optimal tunings:

  • WE: ~2 = 1199.2419 ¢, ~162/125 = 443.0087 ¢
error map: ⟨-0.758 -0.136 +1.523 +0.516]
  • CWE: ~2 = 1200.0000 ¢, ~162/125 = 443.2918 ¢
error map: ⟨0.000 +1.088 +3.313 +2.544]

Optimal ET sequence: 19d, 46, 111, 157, 268cd

Badness (Sintel): 2.99

11-limit

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

  • WE: ~2 = 1199.4073 ¢, ~128/99 = 443.0552 ¢
  • CWE: ~2 = 1200.0000 ¢, ~128/99 = 443.2784 ¢

Optimal ET sequence: 19d, 46, 65d, 111, 268cd

Badness (Sintel): 1.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 1331/1323

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

Optimal tunings:

  • WE: ~2 = 1199.4202 ¢, ~84/65 = 443.0554 ¢
  • CWE: ~2 = 1200.0000 ¢, ~84/65 = 443.2755 ¢

Optimal ET sequence: 19df, 46, 65d, 111, 268cd

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 -1 -1 15 9 17 10], ⟨0 7 9 -33 -15 -36 -16]]

Optimal tunings:

  • WE: ~2 = 1199.4084 ¢, ~22/17 = 443.0513 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/17 = 443.2764 ¢

Optimal ET sequence: 19df, 46, 65d, 111, 268cdg

Badness (Sintel): 0.922

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

Comma list: 6144/6125, 78732/78125

Mapping: [⟨2 -2 -2 13], ⟨0 7 9 -10]]

mapping generators: ~567/400, ~162/125

Optimal tunings:

  • WE: ~567/400 = 599.9413 ¢, ~162/125 = 443.0320 ¢ (~35/32 = 156.9093 ¢)
error map: ⟨-0.117 -0.613 +1.092 +0.091]
  • CWE: ~567/400 = 1200.0000 ¢, ~162/125 = 443.0728 ¢ (~35/32 = 156.9272 ¢)
error map: ⟨0.000 -0.446 +1.341 +0.446]

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

Badness (Sintel): 1.78

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -2 -2 13 18], ⟨0 7 9 -10 -15]]

Optimal tunings:

  • WE: ~99/70 = 599.8776 ¢, ~162/125 = 443.0265 ¢ (~35/32 = 156.8511 ¢)
  • CWE: ~99/70 = 600.0000 ¢, ~162/125 = 443.1166 ¢ (~35/32 = 156.8834 ¢)

Optimal ET sequence: 38e, 46, 84, 130, 306, 436ce

Badness (Sintel): 1.23

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -2 -2 13 18 17], ⟨0 7 9 -10 -15 -13]]

Optimal tunings:

  • WE: ~55/39 = 599.9161 ¢, ~162/125 = 443.0343 ¢ (~35/32 = 156.8817 ¢)
  • CWE: ~55/39 = 600.0000 ¢, ~162/125 = 443.0973 ¢ (~35/32 = 156.9027 ¢)

Optimal ET sequence: 38e, 46, 84, 130, 566ce, 596cef

Badness (Sintel): 0.971

Subpental

Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c4P4/14)).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 19683/19600

Mapping: [⟨1 -8 -10 -28], ⟨0 14 18 45]]

mapping generators: ~2, ~45/28

Optimal tunings:

  • WE: ~2 = 1199.9261 ¢, ~45/28 = 821.4823 ¢
error map: ⟨-0.074 -0.611 +1.107 -0.052]
  • CWE: ~2 = 1200.0000 ¢, ~45/28 = 821.5303 ¢
error map: ⟨0.000 -0.531 +1.231 +0.036]

Optimal ET sequence: 19, …, 111, 130

Badness (Sintel): 1.37

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 8019/8000

Mapping: [⟨1 -8 -10 -28 24], ⟨0 14 18 45 -30]]

Optimal tunings:

  • WE: ~2 = 1199.6571 ¢, ~45/28 = 821.3249 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/28 = 821.5560 ¢

Optimal ET sequence: 19, 111, 130, 241, 371ce, 501cde

Badness (Sintel): 1.50

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 3136/3125

Mapping: [⟨1 -8 -10 -28 24 -23], ⟨0 14 18 45 -30 39]]

Optimal tunings:

  • WE: ~2 = 1199.6819 ¢, ~45/28 = 821.3451 ¢
  • CWE: ~2 = 1200.0000 ¢, ~45/28 = 821.5591 ¢

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

Badness (Sintel): 0.989

Heinz

Heinz splits the sensipent generator ~324/125 in three. Its ploidacot is theta-21-cot (pergen (P8, c9P5/21)). A notable tuning of heinz not shown below for those who like 19edo's representation of the 5-limit is 57edo (57 = 103 - 46).

Subgroup: 2.3.5.7

Comma list: 1029/1024, 78732/78125

Mapping: [⟨1 -8 -10 6], ⟨0 21 27 -7]]

mapping generators: ~2, ~48/35

Optimal tunings:

  • WE: ~2 = 1200.4250 ¢, ~48/35 = 547.8379 ¢
error map: ⟨+0.425 -0.758 +1.061 -1.141]
  • CWE: ~2 = 1200.0000 ¢, ~48/35 = 547.6528 ¢
error map: ⟨0.000 -1.247 +0.311 -2.395]

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

Badness (Sintel): 2.92

11-limit

Subgroup: 2.3.5.7.11

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

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

mapping generators: ~2, ~11/8

Optimal tunings:

  • WE: ~2 = 1200.6094 ¢, ~11/8 = 547.9095 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 547.6413 ¢

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

Badness (Sintel): 1.40

13-limit

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:

  • WE: ~2 = 1200.6343 ¢, ~11/8 = 547.9182 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 547.6345 ¢

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

Badness (Sintel): 1.07

17-limit

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:

  • WE: ~2 = 1200.5351 ¢, ~11/8 = 547.8790 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 547.6388 ¢

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

Badness (Sintel): 0.941

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968

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

Optimal tunings:

  • WE: ~2 = 1200.7181 ¢, ~11/8 = 547.9418 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/8 = 547.6175 ¢

Optimal ET sequence: 46, 103h, 149h

Badness (Sintel): 1.16

Trisensory

Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).

Subgroup: 2.3.5.7

Comma list: 1728/1715, 78732/78125

Mapping: [⟨3 4 6 8], ⟨0 7 9 4]]

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

Optimal tunings:

  • WE: ~63/50 = 399.8117 ¢, ~36/35 = 43.1270 ¢
error map: ⟨-0.565 -0.819 +0.700 +2.176]
  • CWE: ~63/50 = 400.0000 ¢, ~36/35 = 43.0852 ¢
error map: ⟨0.000 -0.359 +1.453 +3.515]

Optimal ET sequence: 27, 57, 84, 111, 195d, 306d

Badness (Sintel): 2.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 78732/78125

Mapping: [⟨3 4 6 8 8], ⟨0 7 9 4 22]]

Optimal tunings:

  • WE: ~63/50 = 399.7341 ¢, ~36/35 = 43.2633 ¢
  • CWE: ~63/50 = 400.0000 ¢, ~36/35 = 43.2290 ¢

Optimal ET sequence: 27e, 84e, 111, 360ccdde

Badness (Sintel): 1.93

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 9295/9261

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

mapping generators: ~49/39, ~36/35

Optimal tunings:

  • WE: ~49/39 = 399.7403 ¢, ~36/35 = 43.2602 ¢
  • CWE: ~49/39 = 400.0000 ¢, ~36/35 = 43.2415 ¢

Optimal ET sequence: 27e, 84e, 111, 360ccddef

Badness (Sintel): 1.44

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 351/350, 442/441, 540/539, 715/714

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

Optimal tunings:

  • WE: ~49/39 = 399.7422 ¢, ~36/35 = 43.2480 ¢
  • CWE: ~49/39 = 400.0000 ¢, ~36/35 = 43.2305 ¢

Optimal ET sequence: 27eg, 84e, 111

Badness (Sintel): 1.23

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

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

Optimal tunings:

  • WE: ~49/39 = 399.7059 ¢, ~36/35 = 43.2600 ¢
  • CWE: ~49/39 = 400.0000 ¢, ~36/35 = 43.2433 ¢

Optimal ET sequence: 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 (S31), 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: [⟨1 -1 -1 2], ⟨0 7 9 8]]

Optimal tunings:

  • WE: ~2 = 1200.0154 ¢, ~31/24 = 443.0514 ¢
  • CWE: ~2 = 1200.0000 ¢, ~31/24 = 443.0474 ¢

Optimal ET sequence: 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 and 29 found by VIxen. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).

Subgroup: 2.3.5.23.29.31

Comma list: 465/464, 576/575, 621/620, 900/899

Subgroup-val mapping: [⟨1 -1 -1 6 -4 2], ⟨0 7 9 -4 24 8]]

Optimal tunings:

  • WE: ~2 = 1200.0782 ¢, ~31/24 = 443.0005 ¢
  • CWE: ~2 = 1200.0000 ¢, ~31/24 = 442.9762 ¢

Optimal ET sequence: 19, 46j, 65, 149, 363j

Badness (Sintel): 0.283

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