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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 27: Line 25:
[[Badness]] (Sintel): 0.826
[[Badness]] (Sintel): 0.826


=== Overview to extensions ===
==== Subgroup extensions ====
[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai|Sendai]] is perhaps more accurate 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.
 
=== 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.  


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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 ====
Fascinatingly, essentially the only simple and accurate extension that preserves the occurrence of sensipent's tempered [[5-limit]] structure in such large edos as [[539edo|539]] is the one with prime 31 by interpreting the generator accurately as [[31/24]]~[[40/31]], tempering out [[961/960|S31 = 961/960]], so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].
The 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. 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 tempered 5-limit structure.  
 
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 ===
{{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]] is equivalent to the one by tempering out [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
 
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list is {([[256/255|S16]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering out [[496/495]] = S31 × S32 and [[528/527]] = S32 × S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
 
Subgroup: 2.3.5.11
 
Comma list: 8019/8000, 16384/16335
 
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}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~128/99 = 443.115{{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}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.188{{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}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.185{{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 }}
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 31/24~40/31 such that 24:31:40 is 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|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy.  
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{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 }}
 
Badness (Sintel): 0.490


== Sensi ==
== Sensi ==
{{Main| Sensi }}
{{Main| Sensi }}


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


=== Septimal sensi ===
=== Septimal sensi ===
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{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
: mapping generators: ~2, ~9/7


[[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 200: Line 87:
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 }}
 
: 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 218: Line 102:
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 }}
 
: mapping generators: ~2, ~9/7


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


{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}
Line 237: Line 117:
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:  
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3658{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.321{{c}} -->


{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
Line 254: Line 132:
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:  
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3775{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.365{{c}} -->


{{Optimal ET sequence|legend=0| 19, 27, 46 }}
{{Optimal ET sequence|legend=0| 19, 27, 46 }}
Line 271: Line 147:
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 }}
 
: mapping generators: ~2, ~9/7


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


{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}
Line 290: Line 162:
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:  
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5075{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.559{{c}} -->


{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}
Line 307: Line 177:
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:  
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5050{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.551{{c}} -->


{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}
Line 324: Line 192:
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 }}
 
: mapping generators: ~2, ~9/7


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


{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
Line 343: Line 207:
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:  
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2863{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.945{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
Line 360: Line 222:
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 }}
 
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.7814{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.518{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
Line 379: Line 237:
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:  
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.7877{{c}}
* POTE: ~2 = 1200.000{{c}}, ~9/7 = 443.506{{c}} -->


{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
Line 392: Line 248:


=== 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)). (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 398: Line 254:
Comma list: 121/120, 126/125, 245/243
Comma list: 121/120, 126/125, 245/243


Mapping:
{{Mapping|legend=0| 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:  
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})
<!-- * CTE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3688{{c}} (~11/10 = 156.6312{{c}})
* POTE: ~99/70 = 600.000{{c}}, ~9/7 = 443.308{{c}} (~11/10 = 156.692{{c}}) -->


{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}
Line 419: Line 271:
Comma list: 91/90, 121/120, 126/125, 169/168
Comma list: 91/90, 121/120, 126/125, 169/168


Mapping:
{{Mapping|legend=0| 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: ~55/39, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
:: mapping generators: ~55/39, ~11/10


Optimal tunings:  
Optimal tunings:  
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})
<!-- * CTE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.4416{{c}} (~11/10 = 156.5584{{c}})
* POTE: ~55/39 = 600.000{{c}}, ~9/7 = 443.275{{c}} (~11/10 = 156.725{{c}}) -->


{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
Line 440: Line 286:
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|legend=0| 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: ~17/12, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
:: mapping generators: ~17/12, ~11/10


Optimal tunings:  
Optimal tunings:  
* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{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}})
* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})
<!-- * CTE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.4466{{c}} (~11/10 = 156.5534{{c}}) -->


{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
Line 456: Line 297:


=== 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 462: Line 303:
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 469: Line 310:
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5981{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.605{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}
{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}
Line 481: Line 320:
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:  
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6333{{c}}
* POTE: ~2 = 1200.000{{c}}, ~25/22 = 221.556{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}
{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}
Line 499: Line 336:


{{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:  
Line 507: Line 342:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 442.755{{c}} -->


{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}
Line 519: Line 353:


{{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:  
Line 527: Line 359:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 443.289{{c}} -->


{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}
Line 538: Line 369:
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 }}
 
: mapping generators: ~2, ~128/99


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~128/99 = 443.274{{c}} -->


{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}
{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}
Line 556: Line 384:
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 }}
 
: mapping generators: ~2, ~84/65


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~84/65 = 443.270{{c}} -->


{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}
Line 574: Line 399:
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 }}
 
: mapping generators: ~2, ~22/17


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


{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}
Line 588: Line 410:


== 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)).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 594: Line 416:
[[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 612: Line 434:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping:
{{Mapping|legend=0| 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 630: Line 449:
Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 364/363, 441/440, 10985/10976


Mapping:
{{Mapping|legend=0| 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 650: Line 466:
[[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 666: Line 484:
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|legend=0| 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 680: Line 499:
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|legend=0| 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 696: Line 516:
[[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 712: Line 534:
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


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 728: Line 551:
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:  
* 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 742: Line 566:
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:  
* 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 756: Line 581:
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:  
* 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 773: Line 599:


{{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 788: Line 616:
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:  
* 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 802: Line 631:
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


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 818: Line 648:
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:  
* 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 832: Line 663:
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:  
* 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
 
== Subgroup extensions ==
=== Sensipent (2.3.5.31) ===
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
 
=== 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]].
 
Subgroup: 2.3.5.11
 
Comma list: 6912/6875, 8019/8000
 
Subgroup-val mapping: {{mapping| 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
Subgroup-val mapping: {{mapping| 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
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
=== 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
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:Sensipent family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Rank 2]]
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