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{{interwiki
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| de = Sensi
| en = Sensipent family
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}}
Temperaments of the '''sensipent family''' temper out the [[sensipent comma]], 78732/78125, also known as medium semicomma. The head of this family is sensipent i.e. the 5-limit version of [[sensi]], generated by the naiadic interval of tempered 162/125, seven make harmonic 6 and nine make harmonic 10.


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 all use the same nominal generator as sensipent.  
[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.  


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.
== Sensipent ==
 
{{Main| Sensipent }}
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.


'''IMPORTANT:''' Please be careful, when reading the mappings, to observe the orientation of the generator; most if not all mappings here use the generator that is closer to the octave than the unison, but show its tuning in terms of the generator that is closer to the unison than the octave.
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.  


== Sensipent ==
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 78732/78125
[[Comma list]]: 78732/78125


{{Mapping|legend=1| 1 6 8 | 0 -7 -9 }}
{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
: mapping generators: ~2, ~162/125


: mapping generators: ~2, ~125/81
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
[[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~162/125 = 443.058
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
: error map: {{val| 0.000 -0.600 +1.143 }}
 
[[Badness]]: 0.035220
 
Badness (Dirichlet): 0.826
 
=== 2.3.5.31 ===
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 to prime 31 by interpreting the generator accurately as [[40/31]]~[[31/24]] by tempering [[961/960|S31 = 961/960]], so that the large [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of slight accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].


[[Subgroup]]: 2.3.5.31
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}


[[Comma list]]: 78732/78125, 961/960
[[Badness]] (Sintel): 0.826


{{Mapping|legend=1| 1 6 8 10 | 0 -7 -9 -8}}
=== 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.


{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
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.


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/28 = 443.050
==== 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.  


Badness (Dirichlet): 0.243
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.


=== Sendai ===
[[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.
{{ See also | User:VIxen/Table_of_sensipent_intervals }}
Sendai is an accurate extension of (2.3.5.31) [[#Sensipent|sensipent]] to 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.
 
[[Subgroup]]: 2.3.5.23.29.31
 
[[Comma list]]: 729/725, 3888/3875, 2511/2500
 
{{Mapping|legend=1| 1 6 8 2 20 10 | 0 -7 -9 4 -24 -8 }}
 
{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 442.989
 
Badness (Dirichlet): 0.283
 
=== Sensible ===
An extension of sensipent  to prime 11 of dubious canonicity (but significantly higher accuracy than [[sensi]]) interprets the generator as [[165/128]]~[[128/99]] by tempering [[8019/8000|S9/S10]] 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 is equivalent to tempering the [[semiporwellisma]] which is equal to [[961/960|S31]] * [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). The equivalence of the aforementioned [[lopsided comma]] also 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 to prime 17 is reasonably accurate through tempering [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
 
[[Subgroup]]: 2.3.5.11
 
[[Comma list]]: [[8019/8000]], [[16384/16335]]
 
{{Mapping|legend=1| 1 6 8 -6 | 0 -7 -9 15 }}
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~128/99 = 443.115
 
Badness (Dirichlet): 0.728
 
==== 2.3.5.11.17 ====
The aforementioned extension to prime 17 through tempering [[1089/1088|S33]] is equivalent to the one by tempering [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
 
[[Subgroup]]: 2.3.5.11.17
 
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]
 
{{Mapping|legend=1| 1 6 8 -6 -6 | 0 -7 -9 15 16 }}
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~22/17 = 443.188
 
Badness (Dirichlet): 0.639
 
==== 2.3.5.11.17.23 ====
[[Subgroup]]: 2.3.5.11.17.23
 
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]]
 
{{Mapping|legend=1| 1 6 8 -6 -6 2 | 0 -7 -9 15 16 4 }}
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.185
 
Badness (Dirichlet): 0.555
 
==== 2.3.5.11.17.23.31 ====
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony without the need for interpreting it as high-damage [[~]][[9/7]] or [[~]][[13/10]] intervals. Its S-expression-based comma list is {([[256/255|S16]],) [[8019/8000|S9/S10]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[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]]: 2.3.5.11.17.23.31
 
[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]], [[961/960]]
 
{{Mapping|legend=1| 1 6 8 -6 -6 2 10 | 0 -7 -9 15 16 4 -8 }}
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
 
[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.183
 
Badness (Dirichlet): 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 ===
Line 135: Line 48:
[[Comma list]]: 126/125, 245/243
[[Comma list]]: 126/125, 245/243


{{Mapping|legend=1| 1 6 8 11 | 0 -7 -9 -13 }}
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}
 
: mapping generators: ~2, ~14/9
 
{{Multival|legend=1| 7 9 13 -2 1 5 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
* [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}  
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}  
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges of regular temperaments|Tuning ranges]]:  
[[Tuning ranges of regular temperaments|Tuning ranges]]:  
Line 161: Line 72:
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}


[[Badness]]: 0.025622
[[Badness]] (Sintel): 0.648


==== 2.3.5.7.13 subgroup (sensation) ====
==== 2.3.5.7.13 subgroup (sensation) ====
Line 168: Line 79:
Comma list: 91/90, 126/125, 169/168
Comma list: 91/90, 126/125, 169/168


Sval mapping: {{mapping| 1 6 8 11 10 | 0 -7 -9 -13 -10 }}
{{Mapping|legend=0| 1 -1 -1 -2 0| 0 7 9 13 10 }}


Gencom mapping: {{mapping| 1 6 8 11 0 10 | 0 -7 -9 -13 0 -10 }}
Optimal tunings:  
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}


: gencom: [2 14/9; 91/90 126/125 169/168]
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
Badness (Sintel): 0.484
 
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}


=== Sensor ===
=== Sensor ===
Line 183: Line 94:
Comma list: 126/125, 245/243, 385/384
Comma list: 126/125, 245/243, 385/384


Mapping: {{mapping| 1 6 8 11 -6 | 0 -7 -9 -13 15 }}
{{Mapping|legend=0| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
 
{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2987
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.294
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}


{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}


Badness: 0.037942
Badness (Sintel): 1.25


==== 13-limit ====
==== 13-limit ====
Line 200: 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 6 8 11 -6 10 | 0 -7 -9 -13 15 -10 }}
{{Mapping|legend=0| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}
 
{{Multival|legend=1| 7 9 13 -15 10 -2 1 -48 -10 5 -66 -10 -87 -20 90 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.3658
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.321
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


Badness: 0.025575
Badness (Sintel): 1.06


==== 17-limit ====
==== 17-limit ====
Line 217: 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 6 8 11 -6 10 -6 | 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:  
* CTE: ~2 = 1\1, ~9/7 = 443.3775
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.365
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}


{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=0| 19, 27, 46 }}


Badness: 0.022908
Badness (Sintel): 1.17


=== Sensus ===
=== Sensus ===
Line 232: Line 139:
Comma list: 126/125, 176/175, 245/243
Comma list: 126/125, 176/175, 245/243


Mapping: {{mapping| 1 6 8 11 23 | 0 -7 -9 -13 -31 }}
{{Mapping|legend=0| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
 
{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.4783
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.626
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}


{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}


Badness: 0.029486
Badness (Sintel): 0.975


==== 13-limit ====
==== 13-limit ====
Line 249: 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 6 8 11 23 10 | 0 -7 -9 -13 -31 -10 }}
{{Mapping|legend=0| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}
 
{{Multival|legend=1| 7 9 13 31 10 -2 1 25 -10 5 41 -10 42 -20 -80 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.5075
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.559
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}


{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}


Badness: 0.020789
Badness (Sintel): 0.859


==== 17-limit ====
==== 17-limit ====
Line 266: 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 6 8 11 23 10 23 | 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:  
* CTE: ~2 = 1\1, ~9/7 = 443.5050
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.551
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}


{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}


Badness: 0.016238
Badness (Sintel): 0.827


=== Sensis ===
=== Sensis ===
Line 281: Line 184:
Comma list: 56/55, 100/99, 245/243
Comma list: 56/55, 100/99, 245/243


Mapping: {{mapping| 1 6 8 11 6 | 0 -7 -9 -13 -4 }}
{{Mapping|legend=0| 1 -1 -1 -2 2| 0 7 9 13 4 }}
 
{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.1886
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.962
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}


Badness: 0.028680
Badness (Sintel): 0.948


==== 13-limit ====
==== 13-limit ====
Line 298: 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 6 8 11 6 10 | 0 -7 -9 -13 -4 -10 }}
{{Mapping|legend=0| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}
 
{{Multival|legend=1| 7 9 13 4 10 -2 1 -18 -10 5 -22 -10 -34 -20 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2863
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.945
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}


Badness: 0.020017
Badness (Sintel): 0.827


=== Sensa ===
=== Sensa ===
Line 315: Line 214:
Comma list: 55/54, 77/75, 99/98
Comma list: 55/54, 77/75, 99/98


Mapping: {{mapping| 1 6 8 11 11 | 0 -7 -9 -13 -12 }}
{{Mapping|legend=0| 1 -1 -1 -2 -1| 0 7 9 13 12 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7814
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.518
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}


Badness: 0.036835
Badness (Sintel): 1.22


==== 13-limit ====
==== 13-limit ====
Line 330: 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 6 8 11 11 11 | 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:  
* CTE: ~2 = 1\1, ~9/7 = 443.7877
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* POTE: ~2 = 1\1, ~9/7 = 443.506
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}


Badness: 0.023258
Badness (Sintel): 0.961


=== 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)). (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


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


Mapping: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
{{Mapping|legend=0| 2 -2 -2 -4 1 | 0 7 9 13 8 }}


: mapping generators: ~99/70, ~11/10
: mapping generators: ~99/70, ~9/7


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 1\2, ~11/10 = 156.6312
* WE: ~99/70 = 600.1183{{c}}, ~9/7 = 443.3956{{c}} (~11/10 = 156.7227{{c}})
* POTE: ~99/70 = 1\2, ~11/10 = 156.692
* CWE: ~99/70 = 600.0000{{c}}, ~9/7 = 443.3348{{c}} (~11/10 = 156.6652{{c}})


{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}


Badness: 0.041723
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 362: 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 5 7 9 9 10 | 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:  
* CTE: ~55/39 = 1\2, ~11/10 = 156.5584
* WE: ~55/39 = 600.1183{{c}}, ~9/7 = 443.5071{{c}} (~11/10 = 156.8074{{c}})
* POTE: ~55/39 = 1\2, ~11/10 = 156.725
* CWE: ~55/39 = 600.0000{{c}}, ~9/7 = 443.3459{{c}} (~11/10 = 156.6541{{c}})


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Badness: 0.026339
Badness (Sintel): 1.09


==== 17-limit ====
==== 17-limit ====
Line 377: 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 5 7 9 9 10 10 | 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:  
* CTE: ~17/12 = 1\2, ~11/10 = 156.5534
* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}


Badness: 0.0188
Badness (Sintel): 0.960


=== Hemisensi ===
=== 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
Subgroup: 2.3.5.7.11


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


Mapping: {{mapping| 1 13 17 24 32 | 0 -14 -18 -26 -35 }}
{{Mapping|legend=0| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}


: mapping generators: ~2, ~44/25
: mapping generators: ~2, ~25/22


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.5981
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* POTE: ~2 = 1\1, ~25/22 = 221.605
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}


{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}


Badness: 0.048714
Badness (Sintel): 1.61


==== 13-limit ====
==== 13-limit ====
Line 408: 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 13 17 24 32 30 | 0 -14 -18 -26 -35 -30 }}
{{Mapping|legend=0| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.6333
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* POTE: ~2 = 1\1, ~25/22 = 221.556
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}


{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}


Badness: 0.033016
Badness (Sintel): 1.36


== Sensei ==
== Sensei ==
Line 423: Line 327:
[[Comma list]]: 225/224, 78732/78125
[[Comma list]]: 225/224, 78732/78125


{{Mapping|legend=1| 1 6 8 23 | 0 -7 -9 -32 }}
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}


{{Multival|legend=1| 7 9 32 -2 31 49 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6422{{c}}, ~162/125 = 442.9920{{c}}
: [[error map]]: {{val| +0.642 -1.653 -0.028 +1.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 442.755
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}


{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
[[Badness]] (Sintel): 1.50
 
[[Badness]]: 0.059218


== Warrior ==
== Warrior ==
Line 438: Line 344:
[[Comma list]]: 5120/5103, 78732/78125
[[Comma list]]: 5120/5103, 78732/78125


{{Mapping|legend=1| 1 6 8 -18 | 0 -7 -9 33 }}
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}


{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.2419{{c}}, ~162/125 = 443.0087{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289
: [[error map]]: {{val| -0.758 -0.136 +1.523 +0.516 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}


{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}


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


=== 11-limit ===
=== 11-limit ===
Line 453: Line 361:
Comma list: 176/175, 1331/1323, 5120/5103
Comma list: 176/175, 1331/1323, 5120/5103


Mapping: {{mapping| 1 6 8 -18 -6 | 0 -7 -9 33 15 }}
{{Mapping|legend=0| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~128/99 = 443.274
Optimal tunings:
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}


Badness: 0.046383
Badness (Sintel): 1.53


=== 13-limit ===
=== 13-limit ===
Line 466: 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 6 8 -18 -6 -19 | 0 -7 -9 33 15 36 }}
{{Mapping|legend=0| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}


Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
Optimal tunings:
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}


Badness: 0.028735
Badness (Sintel): 1.19


=== 17-limit ===
=== 17-limit ===
Line 479: 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 6 8 -18 -6 -19 -6 | 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 tuning (POTE): ~2 = 1\1, ~22/17 = 443.270
Optimal tunings:
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}


Badness: 0.018105
Badness (Sintel): 0.922


== Bison ==
== Bison ==
Bison has a 1/2-octave period and the generator can be taken as [[~]][[162/125]] or its semi-octave complement, ~[[35/32]]. Its [[ploidacot]] is diploid delta-heptacot ([[pergen]] (P8/2, ccP5/7)).
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 78732/78125
[[Comma list]]: 6144/6125, 78732/78125


{{Mapping|legend=1| 2 5 7 3 | 0 -7 -9 10 }}
{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
: mapping generators: ~567/400, ~162/125


: mapping generators: ~567/400, ~35/32
[[Optimal tuning]]s:
 
* [[WE]]: ~567/400 = 599.9413{{c}}, ~162/125 = 443.0320{{c}} (~35/32 = 156.9093{{c}})
{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
: [[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}})
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~35/32 = 156.925
: 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]]: 0.070375
[[Badness]] (Sintel): 1.78


=== 11-limit ===
=== 11-limit ===
Line 509: Line 426:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}
{{Mapping|legend=0| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}


Optimal tuning (POTE): ~99/70 = 1\2, ~35/32 = 156.883
Optimal tunings:
* 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=1| 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}


Badness: 0.037132
Badness (Sintel): 1.23


=== 13-limit ===
=== 13-limit ===
Line 522: 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 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
{{Mapping|legend=0| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}


Optimal tuning (POTE): ~55/39 = 1\2, ~35/32 = 156.904
Optimal tunings:
* 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=1| 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}


Badness: 0.023504
Badness (Sintel): 0.971


== Subpental ==
== 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, c<sup>4</sup>P4/14)).
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[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


{{Multival|legend=1| 14 18 45 -4 32 54 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.9261{{c}}, ~45/28 = 821.4823{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 378.467
: [[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]]: 0.054303
[[Badness]] (Sintel): 1.37


=== 11-limit ===
=== 11-limit ===
Line 550: Line 476:
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 tuning (POTE): ~2 = 1\1, ~56/45 = 378.440
Optimal tunings:
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}


{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}


Badness: 0.045352
Badness (Sintel): 1.50


=== 13-limit ===
=== 13-limit ===
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 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 tuning (POTE): ~2 = 1\1, ~56/45 = 378.437
Optimal tunings:
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}


{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}


Badness: 0.023940
Badness (Sintel): 0.989


== Heinz ==
== Heinz ==
A notable tuning of heinz not shown below for those who like [[19edo]]'s representation of the [[5-limit]] is [[57edo]] (= 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 578: Line 508:
[[Comma list]]: 1029/1024, 78732/78125
[[Comma list]]: 1029/1024, 78732/78125


{{Mapping|legend=1| 1 13 17 -1 | 0 -21 -27 7 }}
{{Mapping|legend=1| 1 -8 -10 6 | 0 21 27 -7 }}
: mapping generators: ~2, ~48/35


: mapping generators: ~2, ~35/24
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
: [[error map]]: {{val| +0.425 -0.758 +1.061 -1.141 }}
 
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 546.815
: 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]]: 0.115385
[[Badness]] (Sintel): 2.92


=== 11-limit ===
=== 11-limit ===
Line 595: Line 526:
Comma list: 385/384, 441/440, 78732/78125
Comma list: 385/384, 441/440, 78732/78125


Mapping: {{mapping| 1 13 17 -1 4 | 0 -21 -27 7 -1 }}
{{Mapping|legend=0| 1 -8 -10 6 3 | 0 21 27 -7 1}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
: mapping generators: ~2, ~11/8


{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
Optimal tunings:
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}


Badness: 0.042412
{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}
 
Badness (Sintel): 1.40


=== 13-limit ===
=== 13-limit ===
Line 608: 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 13 17 -1 4 -5 | 0 -21 -27 7 -1 16 }}
{{Mapping|legend=0| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.629
Optimal tunings:
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}


{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}


Badness: 0.025779
Badness (Sintel): 1.07


=== 17-limit ===
=== 17-limit ===
Line 621: 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 13 17 -1 4 -5 3 | 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 tuning (POTE): ~2 = 1\1, ~11/8 = 547.635
Optimal tunings:
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}


{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}


Badness: 0.018479
Badness (Sintel): 0.941


=== 19-limit ===
=== 19-limit ===
Line 634: 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 13 17 -1 4 -5 3 -5 | 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 tuning (POTE): ~2 = 1\1, ~11/8 = 547.614
Optimal tunings:
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}


{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=0| 46, 103h, 149h }}


Badness: 0.019005
Badness (Sintel): 1.16


== Trisensory ==
== Trisensory ==
Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 648: Line 591:


{{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


{{Multival|legend=1| 21 27 12 -6 -40 -48 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~63/50 = 399.8117{{c}}, ~36/35 = 43.1270{{c}}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~36/35 = 43.147
: [[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]]: 0.089740
[[Badness]] (Sintel): 2.27


=== 11-limit ===
=== 11-limit ===
Line 662: 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 tuning (POTE): ~63/50 = 1\3, ~36/35 = 43.292
Optimal tunings:
* 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=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}


Badness: 0.058413
Badness (Sintel): 1.93


=== 13-limit ===
=== 13-limit ===
Line 675: 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 }}


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
: mapping generators: ~49/39, ~36/35


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Optimal tunings:
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}


Badness: 0.034829
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
 
Badness (Sintel): 1.44


=== 17-limit ===
=== 17-limit ===
Line 688: 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 tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.276
Optimal tunings:
* 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=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}


Badness: 0.024120
Badness (Sintel): 1.23


=== 19-limit ===
=== 19-limit ===
Line 701: 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:
* 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 }}
 
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|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
 
=== 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
 
Comma list: 961/960, 2511/2500
 
{{Mapping|legend=2| 1 -1 -1 2 | 0 7 9 8 }}
 
Optimal tunings:
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
 
{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.292
Badness (Sintel): 0.243


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
=== Sensible (2.3.5.11) ===
{{See also| Sensipent #Sensible interval table }}


Badness: 0.018466
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[S-expression|S31⋅S32<sup>2</sup>]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension for prime 31 exists through {[[961/960]] ({{S|31}}), [[1024/1023]] ({{S|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.


The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{S|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]]. A notable non-patent val is [[84edo|84g]].
Finally, despite not appearing in the final optimal ET sequence, [[241edo]] is a good tuning for it via the val {{nowrap| 65 + 176g {{=}} 241g. }} (Specifically, in the sequence of edos 111, 176, 241, 65 we have increasingly good 5-limit, an essentially-just 11/8 in 111 that gets slightly sharper, a 23/16 that gets slightly flatter and becomes essentially-just in 65, and a 31 that is essentially-just in 241, leaving only the sharp 17 as encouraging a tuning closer to 111 than 65, and with more unusual complex subgroup harmony extensions (or simpler but less accurate ones) available in most of these tunings.)
Subgroup: 2.3.5.11
Comma list: 6912/6875, 8019/8000
{{Mapping|legend=2| 1 -1 -1 9 | 0 7 9 -15 }}
Optimal tunings:
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
Badness (Sintel): 0.728
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17
Comma list: 256/255, 1089/1088, 1377/1375
{{Mapping|legend=2| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
Optimal tunings:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Badness (Sintel): 0.639
==== 2.3.5.11.17.23 subgroup ====
Subgroup: 2.3.5.11.17.23
Comma list: 256/255, 576/575, 1089/1088, 1377/1375
{{Mapping|legend=2| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
Optimal tunings:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Badness (Sintel): 0.555
==== 2.3.5.11.17.23.31 subgroup ====
Subgroup: 2.3.5.11.17.23.31
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
{{Mapping|legend=2| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
Optimal tunings:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Badness (Sintel): 0.490
=== Sendai (2.3.5.23) ===
{{See also| Sensipent #Sendai interval table }}
Sendai is an accurate extension of sensipent with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence. It is ideal in tuning for ambiguating the generator's 2.3.5.31 interpretation as 40/31~31/24, being between these (and hence also a good tuning of 2.3.5.31 sensipent at the cost of some 5-limit accuracy).
{{Todo|inline=1|complete section|comment=Add the data for the intermediate subgroups. }}
==== 2.3.5.23.29.31 subgroup ====
Subgroup: 2.3.5.23.29.31
Comma list: 465/464, 576/575, 621/620, 900/899
{{Mapping|legend=2| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
Optimal tunings:
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
Badness (Sintel): 0.283
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Sensipent| ]] <!-- key article -->
[[Category:Rank 2]]
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