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=== Overview to extensions ===
=== Overview to extensions ===
==== Full 7-limit extensions ====
The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.  
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.  


Line 32: Line 33:
Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)
* ''[[Browser]]'' → [[Canopic clan #Browser|Canopic clan]] (+16875/16807)


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 ===
==== Subgroup extensions ====
The generator 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.  
The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts, which is important as it is the difference between 5/4 and 4/3 (whose product is 5/3 at 2 gens). This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions_2|#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's fine-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]].
This extension can be applied to many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 40/31~31/24 such that 24:31:40 is tuned as a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.


Subgroup: 2.3.5.31
[[Sensible]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap| 65 + 111 = 176g }} is almost patent val.
 
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
 
=== Sensible ===
{{See also| Sensipent #Sensible interval table }}
 
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[961/960|S31]] × [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
 
The aforementioned extension with prime 17 through tempering out [[1089/1088|S33]] 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]].
 
Subgroup: 2.3.5.11
 
Comma list: 8019/8000, 16384/16335
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}
 
: mapping generators: ~2, ~128/99
 
Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
 
Badness (Sintel): 0.728
 
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17
 
Comma list: 256/255, 1089/1088, 1377/1375
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
: mapping generators: ~2, ~22/17
 
Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.639
 
==== 2.3.5.11.17.23 subgroup ====
Subgroup: 2.3.5.11.17.23
 
Comma list: 256/255, 576/575, 1089/1088, 1377/1375
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
 
Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.555
 
==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31
 
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
 
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
 
Optimal tunings:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
 
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
 
Badness (Sintel): 0.490


== Sensi ==
== Sensi ==
Line 159: Line 57:


[[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}}
Line 187: Line 85:
Comma list: 91/90, 126/125, 169/168
Comma list: 91/90, 126/125, 169/168


Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
{{Mapping|legend=0| 1 -1 -1 -2 0| 0 7 9 13 10 }}


Optimal tunings:  
Optimal tunings:  
Line 202: Line 100:
Comma list: 126/125, 245/243, 385/384
Comma list: 126/125, 245/243, 385/384


Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
{{Mapping|legend=0| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 217: Line 115:
Comma list: 91/90, 126/125, 169/168, 385/384
Comma list: 91/90, 126/125, 169/168, 385/384


Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
{{Mapping|legend=0| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}


Optimal tunings:  
Optimal tunings:  
Line 232: Line 130:
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255


Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}
{{Mapping|legend=0| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}


Optimal tunings:  
Optimal tunings:  
Line 247: Line 145:
Comma list: 126/125, 176/175, 245/243
Comma list: 126/125, 176/175, 245/243


Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
{{Mapping|legend=0| 1 -1 -1 -2 -8| 0 7 9 13 31 }}


Optimal tunings:  
Optimal tunings:  
Line 262: Line 160:
Comma list: 91/90, 126/125, 169/168, 352/351
Comma list: 91/90, 126/125, 169/168, 352/351


Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
{{Mapping|legend=0| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}


Optimal tunings:  
Optimal tunings:  
Line 277: Line 175:
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168


Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}
{{Mapping|legend=0| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}


Optimal tunings:  
Optimal tunings:  
Line 292: Line 190:
Comma list: 56/55, 100/99, 245/243
Comma list: 56/55, 100/99, 245/243


Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
{{Mapping|legend=0| 1 -1 -1 -2 2| 0 7 9 13 4 }}


Optimal tunings:  
Optimal tunings:  
Line 307: Line 205:
Comma list: 56/55, 78/77, 91/90, 100/99
Comma list: 56/55, 78/77, 91/90, 100/99


Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
{{Mapping|legend=0| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}


Optimal tunings:  
Optimal tunings:  
Line 322: Line 220:
Comma list: 55/54, 77/75, 99/98
Comma list: 55/54, 77/75, 99/98


Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
{{Mapping|legend=0| 1 -1 -1 -2 -1| 0 7 9 13 12 }}


Optimal tunings:  
Optimal tunings:  
Line 337: Line 235:
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|legend=0| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}


Optimal tunings:  
Optimal tunings:  
Line 348: Line 246:


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


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 354: Line 252:
Comma list: 121/120, 126/125, 245/243
Comma list: 121/120, 126/125, 245/243


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


: mapping generators: ~99/70, ~9/7
: mapping generators: ~99/70, ~9/7
Line 371: Line 269:
Comma list: 91/90, 121/120, 126/125, 169/168
Comma list: 91/90, 121/120, 126/125, 169/168


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


Optimal tunings:  
Optimal tunings:  
Line 386: Line 284:
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168


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


Optimal tunings:  
Optimal tunings:  
Line 403: Line 301:
Comma list: 126/125, 243/242, 245/242
Comma list: 126/125, 243/242, 245/242


Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}
{{Mapping|legend=0| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}


: mapping generators: ~2, ~25/22
: mapping generators: ~2, ~25/22
Line 420: Line 318:
Comma list: 91/90, 126/125, 169/168, 243/242
Comma list: 91/90, 126/125, 169/168, 243/242


Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}
{{Mapping|legend=0| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}


Optimal tunings:  
Optimal tunings:  
Line 469: Line 367:
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|legend=0| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}


Optimal tunings:  
Optimal tunings:  
Line 484: Line 382:
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|legend=0| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}


Optimal tunings:  
Optimal tunings:  
Line 499: Line 397:
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|legend=0| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}


Optimal tunings:  
Optimal tunings:  
Line 510: Line 408:


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


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 534: Line 432:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
{{Mapping|legend=0| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}


Optimal tunings:
Optimal tunings:
Line 549: Line 447:
Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 364/363, 441/440, 10985/10976


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


Optimal tunings:
Optimal tunings:
Line 584: Line 482:
Comma list: 540/539, 3136/3125, 8019/8000
Comma list: 540/539, 3136/3125, 8019/8000


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


Optimal tunings:  
Optimal tunings:  
Line 599: Line 497:
Comma list: 351/350, 540/539, 676/675, 3136/3125
Comma list: 351/350, 540/539, 676/675, 3136/3125


Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}
{{Mapping|legend=0| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}


Optimal tunings:  
Optimal tunings:  
Line 634: Line 532:
Comma list: 385/384, 441/440, 78732/78125
Comma list: 385/384, 441/440, 78732/78125


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


: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8
Line 651: Line 549:
Comma list: 351/350, 385/384, 441/440, 847/845
Comma list: 351/350, 385/384, 441/440, 847/845


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


Optimal tunings:  
Optimal tunings:  
Line 666: Line 564:
Comma list: 273/272, 351/350, 385/384, 441/440, 847/845
Comma list: 273/272, 351/350, 385/384, 441/440, 847/845


Mapping: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}
{{Mapping|legend=0| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}


Optimal tunings:  
Optimal tunings:  
Line 681: Line 579:
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968


Mapping: {{mapping| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}
{{Mapping|legend=0| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}


Optimal tunings:  
Optimal tunings:  
Line 716: Line 614:
Comma list: 176/175, 540/539, 78732/78125
Comma list: 176/175, 540/539, 78732/78125


Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
{{Mapping|legend=0| 3 4 6 8 8 | 0 7 9 4 22 }}


Optimal tunings:  
Optimal tunings:  
Line 731: Line 629:
Comma list: 176/175, 351/350, 540/539, 9295/9261
Comma list: 176/175, 351/350, 540/539, 9295/9261


Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
{{Mapping|legend=0| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}


: mapping generators: ~49/39, ~36/35
: mapping generators: ~49/39, ~36/35
Line 748: Line 646:
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714


Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
{{Mapping|legend=0| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}


Optimal tunings:  
Optimal tunings:  
Line 763: Line 661:
Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475
Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475


Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}
{{Mapping|legend=0| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}


Optimal tunings:  
Optimal tunings:  
Line 773: Line 671:
Badness (Sintel): 1.12
Badness (Sintel): 1.12


== Subgroup extensions ==
=== Sensipent (2.3.5.31) ===
Subgroup: 2.3.5.31
Comma list: 961/960, 2511/2500
{{Mapping|legend=2| 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
=== Sensible (2.3.5.11) ===
{{See also| Sensipent #Sensible interval table }}
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[S-expression|S31⋅S32<sup>2</sup>]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension for prime 31 exists through {[[961/960]] ({{S|31}}), [[1024/1023]] ({{S|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.
The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{S|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]]. A notable non-patent val is [[84edo|84g]].
Finally, despite not appearing in the final optimal ET sequence, [[241edo]] is a good tuning for it via the val {{nowrap| 65 + 176g {{=}} 241g. }} (Specifically, in the sequence of edos 111, 176, 241, 65 we have increasingly good 5-limit, an essentially-just 11/8 in 111 that gets slightly sharper, a 23/16 that gets slightly flatter and becomes essentially-just in 65, and a 31 that is essentially-just in 241, leaving only the sharp 17 as encouraging a tuning closer to 111 than 65, and with more unusual complex subgroup harmony extensions (or simpler but less accurate ones) available in most of these tunings.)
Subgroup: 2.3.5.11
Comma list: 6912/6875, 8019/8000
{{Mapping|legend=2| 1 -1 -1 9 | 0 7 9 -15 }}
Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
Badness (Sintel): 0.728
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17
Comma list: 256/255, 1089/1088, 1377/1375
{{Mapping|legend=2| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Badness (Sintel): 0.639
==== 2.3.5.11.17.23 subgroup ====
Subgroup: 2.3.5.11.17.23
Comma list: 256/255, 576/575, 1089/1088, 1377/1375
{{Mapping|legend=2| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Badness (Sintel): 0.555
==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
{{Mapping|legend=2| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
Optimal tunings:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Badness (Sintel): 0.490
=== Sendai (2.3.5.23) ===
{{See also| Sensipent #Sendai interval table }}
Sendai is an accurate extension of sensipent with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence. It is ideal in tuning for ambiguating the generator's 2.3.5.31 interpretation as 40/31~31/24, being between these (and hence also a good tuning of 2.3.5.31 sensipent at the cost of some 5-limit accuracy).
{{Todo|inline=1|complete section|comment=Add the data for the intermediate subgroups. }}
==== 2.3.5.23.29.31 subgroup ====
Subgroup: 2.3.5.23.29.31
Comma list: 465/464, 576/575, 621/620, 900/899
{{Mapping|legend=2| 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
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