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[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.  
[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.  
Note that optimal ET sequences here often omit some [[val]] of [[84edo]].


== Sensipent ==
== Sensipent ==
Line 28: Line 26:


=== 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.  


Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.  
Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~189/125; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~90/49; five make the original. All are considered below.  
 
Temperaments discussed elsewhere include:
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)
 
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
 
=== 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 [[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 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]].
 
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 }}
==== Subgroup extensions ====
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.


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


Optimal tunings:
[[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.
* 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 151: Line 79:
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 166: Line 94:
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 181: Line 109:
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 196: Line 124:
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 211: Line 139:
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 226: Line 154:
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 241: Line 169:
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 256: Line 184:
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 271: Line 199:
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 286: Line 214:
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 301: Line 229:
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 10}}
{{Mapping|legend=0| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}


Optimal tunings:  
Optimal tunings:  
Line 312: Line 240:


=== 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 318: Line 246:
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 335: Line 263:
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 350: Line 278:
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 367: Line 295:
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 384: Line 312:
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 433: Line 361:
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 448: Line 376:
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 463: Line 391:
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 498: Line 426:
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 513: Line 441:
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 548: Line 476:
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 563: Line 491:
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 598: Line 526:
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 615: Line 543:
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 630: Line 558:
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 645: Line 573:
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 680: Line 608:
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 695: Line 623:
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 712: Line 640:
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 727: Line 655:
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 737: Line 665:
Badness (Sintel): 1.12
Badness (Sintel): 1.12


== Other subgroup extensions ==
== Catafourth ==
=== Sensipent (2.3.5.31 subgroup) ===
Catafourth tempers out 2401/2400, the [[breedsma]], and may be described as the 27 & 130 temperament. [[130edo]] itself may be recommended as a tuning. [[157edo]] also makes for a viable tuning.
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.  
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 78732/78125
 
{{Mapping|legend=1| 1 -15 -19 -12 | 0 28 36 25 }}
: mapping generators: ~2, ~189/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9278{{c}}, ~189/125 = 710.7220{{c}}
: [[error map]]: {{val| -0.072 -0.656 +1.050 +0.091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/125 = 710.7626{{c}}
: error map: {{val| 0.000 -0.603 +1.139 +0.238 }}
 
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
 
[[Badness]] (Sintel): 2.01
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 78408/78125
 
{{Mapping|legend=0| 1 -15 -19 -12 -38 | 0 28 36 25 70 }}
 
Optimal tunings:
* WE: ~2 = 1200.0219{{c}}, ~189/125 = 710.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/125 = 710.7487{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363, 493e }}
 
Badness (Sintel): 1.22
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 10985/10976
 
{{Mapping|legend=0| 1 -15 -19 -12 -38 -4 | 0 28 36 25 70 13 }}
 
Optimal tunings:
* WE: ~2 = 1200.1023{{c}}, ~98/65 = 710.8043{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~98/65 = 710.7459{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363 }}
 
Badness (Sintel): 0.896
 
== Browser ==
Named by [[Xenllium]] in 2022, browser may be described as the {{nowrap| 103 & 111 }} temperament.
 
This can also be considered a [[non-over-1 temperament]], with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 78732/78125
 
{{Mapping|legend=1| 1 -29 -37 -47 | 0 35 45 57 }}
: mapping generators: ~2, ~90/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9313{{c}}, ~90/49 = 1048.5414{{c}}
: [[error map]]: {{val| -0.069 -1.013 +0.592 +1.264 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~90/49 = 1048.5998{{c}}
: error map: {{val| 0.000 -0.962 +0.677 +1.362 }}
 
{{Optimal ET sequence|legend=1| 103, 111, 214 }}
 
[[Badness]] (Sintel): 4.57
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 78732/78125
 
{{Mapping|legend=0| 1 -29 -37 -47 -28 | 0 35 45 57 36 }}
 
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5981{{c}}
 
{{Optimal ET sequence|legend=0| 103, 214 }}
 
Badness (Sintel): 1.91
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 847/845, 1375/1372
 
{{Mapping|legend=0| 1 -29 -37 -47 -28 -33 | 0 35 45 57 36 42 }}
 
Optimal tunings:
* WE: ~2 = 1200.1344{{c}}, ~11/6 = 1048.7124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.5984{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
 
{{Mapping|legend=0| 1 -29 -37 -47 -28 -33 -23 | 0 35 45 57 36 42 31 }}
 
Optimal tunings:
* WE: ~2 = 1199.9191{{c}}, ~11/6 = 1048.5324{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6014{{c}}
 
{{Optimal ET sequence|legend=0| 103, 111, 214 }}
 
Badness (Sintel): 1.04


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]].
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714
{{Mapping|legend=0| 1 -29 -37 -47 -28 -33 -23 -91 | 0 35 45 57 36 42 31 109 }}
Optimal tunings:
* WE: ~2 = 1199.9145{{c}}, ~11/6 = 1048.5290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1048.6021{{c}}
{{Optimal ET sequence|legend=0| 103h, 111, 214 }}
Badness (Sintel): 1.07
== Subgroup extensions ==
=== Sensipent (2.3.5.31) ===
Subgroup: 2.3.5.31
Subgroup: 2.3.5.31


Comma list: 961/960, 2511/2500
Comma list: 961/960, 2511/2500


Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
{{Mapping|legend=2| 1 -1 -1 2 | 0 7 9 8 }}


Optimal tunings:  
Optimal tunings:  
Line 757: Line 813:
Badness (Sintel): 0.243
Badness (Sintel): 0.243


=== Sendai ===
=== 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 }}
{{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).
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
Subgroup: 2.3.5.23.29.31


Comma list: 465/464, 576/575, 621/620, 900/899  
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 }}
{{Mapping|legend=2| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}


Optimal tunings:  
Optimal tunings:  
Line 776: Line 905:
Badness (Sintel): 0.283
Badness (Sintel): 0.283


[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 09:06, 11 October 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

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.

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 ~189/125; four make the original. Finally, browser adds 16875/16807 with a generator of ~90/49; five make the original. All are considered below.

Subgroup extensions

The generator of sensipent can be accurately interpreted as 31/24~40/31, tempering out 961/960 (S31), 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. It is essentially the only simple and accurate extension that preserves sensipent's fine-tempered 5-limit structure.

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.

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 176g is almost patent val.

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)). (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

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)).

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

Catafourth

Catafourth tempers out 2401/2400, the breedsma, and may be described as the 27 & 130 temperament. 130edo itself may be recommended as a tuning. 157edo also makes for a viable tuning.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 78732/78125

Mapping: [⟨1 -15 -19 -12], ⟨0 28 36 25]]

mapping generators: ~2, ~189/125

Optimal tunings:

  • WE: ~2 = 1199.9278 ¢, ~189/125 = 710.7220 ¢
error map: ⟨-0.072 -0.656 +1.050 +0.091]
  • CWE: ~2 = 1200.0000 ¢, ~189/125 = 710.7626 ¢
error map: ⟨0.000 -0.603 +1.139 +0.238]

Optimal ET sequence: 27, 76, 103, 130

Badness (Sintel): 2.01

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 78408/78125

Mapping: [⟨1 -15 -19 -12 -38], ⟨0 28 36 25 70]]

Optimal tunings:

  • WE: ~2 = 1200.0219 ¢, ~189/125 = 710.7610 ¢
  • CWE: ~2 = 1200.0000 ¢, ~189/125 = 710.7487 ¢

Optimal ET sequence: 27e, 76e, 103, 130, 233, 363, 493e

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 10985/10976

Mapping: [⟨1 -15 -19 -12 -38 -4], ⟨0 28 36 25 70 13]]

Optimal tunings:

  • WE: ~2 = 1200.1023 ¢, ~98/65 = 710.8043 ¢
  • CWE: ~2 = 1200.0000 ¢, ~98/65 = 710.7459 ¢

Optimal ET sequence: 27e, 76e, 103, 130, 233, 363

Badness (Sintel): 0.896

Browser

Named by Xenllium in 2022, browser may be described as the 103 & 111 temperament.

This can also be considered a non-over-1 temperament, with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with mos scales of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8d-et, with a generator very slightly flat of 7\8.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 78732/78125

Mapping: [⟨1 -29 -37 -47], ⟨0 35 45 57]]

mapping generators: ~2, ~90/49

Optimal tunings:

  • WE: ~2 = 1199.9313 ¢, ~90/49 = 1048.5414 ¢
error map: ⟨-0.069 -1.013 +0.592 +1.264]
  • CWE: ~2 = 1200.0000 ¢, ~90/49 = 1048.5998 ¢
error map: ⟨0.000 -0.962 +0.677 +1.362]

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 4.57

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 78732/78125

Mapping: [⟨1 -29 -37 -47 -28], ⟨0 35 45 57 36]]

Optimal tunings:

  • WE: ~2 = 1200.1344 ¢, ~11/6 = 1048.7124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.5981 ¢

Optimal ET sequence: 103, 214

Badness (Sintel): 1.91

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 1375/1372

Mapping: [⟨1 -29 -37 -47 -28 -33], ⟨0 35 45 57 36 42]]

Optimal tunings:

  • WE: ~2 = 1200.1344 ¢, ~11/6 = 1048.7124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.5984 ¢

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 715/714, 847/845

Mapping: [⟨1 -29 -37 -47 -28 -33 -23], ⟨0 35 45 57 36 42 31]]

Optimal tunings:

  • WE: ~2 = 1199.9191 ¢, ~11/6 = 1048.5324 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.6014 ¢

Optimal ET sequence: 103, 111, 214

Badness (Sintel): 1.04

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714

Mapping: [⟨1 -29 -37 -47 -28 -33 -23 -91], ⟨0 35 45 57 36 42 31 109]]

Optimal tunings:

  • WE: ~2 = 1199.9145 ¢, ~11/6 = 1048.5290 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/6 = 1048.6021 ¢

Optimal ET sequence: 103h, 111, 214

Badness (Sintel): 1.07

Subgroup extensions

Sensipent (2.3.5.31)

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

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)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 for 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. A notable non-patent val is 84g.

Finally, despite not appearing in the final optimal ET sequence, 241edo is a good tuning for it via the val 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

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

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]]

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

Sendai (2.3.5.23)

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). 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: complete section

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

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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