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Sensipent family: Difference between revisions

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Move the opening note to a reasonable place. Don't shock the readers with stuff like 31/24 or 40/31 without proper context
 
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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


Line 21: Line 20:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 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 28: Line 26:


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


Line 34: Line 33:
Temperaments discussed elsewhere include:  
Temperaments discussed elsewhere include:  
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Mirkwai clan #Browser|Mirkwai clan]] (+16875/16807)
* ''[[Browser]]'' → [[Canopic clan #Browser|Canopic clan]] (+16875/16807)


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


=== 2.3.5.31 subgroup ===
==== Subgroup extensions ====
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 [[#Other 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
This extension can be laid onto 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.


Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
[[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, considered immediately below.  
 
Optimal tunings:
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~31/24 = 442.989{{c}} -->
 
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
 
Badness (Sintel): 0.283


=== Sensible ===
=== Sensible ===
{{See also| Sensipent #Sensible interval table }}
{{See also| Sensipent #Sensible interval table }}


Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[961/960|S31]] × [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {S31, S32}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088|S33]] so that a slightly sharp ~[[22/17]] is equated with the generator.
Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[S-expression|S31⋅S32<sup>2</sup>]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.


The aforementioned extension with prime 17 through tempering out [[1089/1088|S33]] is equivalent to the one by tempering out [[256/255|S16]] = [[256/255]] = ([[22/17]])/([[165/128]]).
The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.


Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list is {([[256/255|S16]], [[8019/8000|S9/S10]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering out [[496/495]] = S31 × S32 and [[528/527]] = S32 × S33 as well as [[16337/16335]] = S31/S33 = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].
Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]]. A notable non-patent val is [[84edo|84g]].


Subgroup: 2.3.5.11
Subgroup: 2.3.5.11
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* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~128/99 = 443.115{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
Line 116: Line 81:
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.188{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Line 132: Line 96:
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.185{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Line 148: Line 111:
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.183{{c}}
* CEE: ~2 = 1200.000{{c}}, ~22/17 = 443.115{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
Line 158: Line 119:
{{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 201: Line 158:


Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4016{{c}} -->


{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
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Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* 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 242: Line 192:
* 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 259: Line 207:
* 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 272: Line 218:


Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* 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 295: Line 237:
* 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 312: Line 252:
* 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 325: Line 263:


Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* 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 348: Line 282:
* 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 361: Line 293:


Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
Mapping: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* 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 307:
Comma list: 55/54, 66/65, 77/75, 143/140
Comma list: 55/54, 66/65, 77/75, 143/140


Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}


Optimal tunings:  
Optimal tunings:  
* 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 318:


=== 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 324:
Comma list: 121/120, 126/125, 245/243
Comma list: 121/120, 126/125, 245/243


Mapping:
Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
* common form: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
 
:: mapping generators: ~99/70, ~9/7
: mapping generators: ~99/70, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 | 0 -7 -9 -13 -8 }}
:: mapping generators: ~99/70, ~11/10


Optimal tunings:  
Optimal tunings:  
* 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 341:
Comma list: 91/90, 121/120, 126/125, 169/168
Comma list: 91/90, 121/120, 126/125, 169/168


Mapping:
Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}
* common form: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10}}
:: mapping generators: ~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 356:
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168


Mapping:
Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}
* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}
:: mapping generators: ~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 367:


=== 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 469: Line 380:
* 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 486: Line 395:
* 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 406:


{{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 412:
* [[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 423:


{{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 429:
* [[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 539: Line 440:


Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}
: mapping generators: ~2, ~128/99


Optimal tunings:  
Optimal tunings:  
* 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 557: Line 455:


Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}
Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}
: mapping generators: ~2, ~84/65


Optimal tunings:  
Optimal tunings:  
* 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 575: Line 470:


Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}
Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}
: mapping generators: ~2, ~22/17


Optimal tunings:  
Optimal tunings:  
* 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 480:


== 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 486:
[[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:
Line 605: Line 494:
* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})
* [[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 }}
: error map: {{val| 0.000 -0.446 +1.341 +0.446 }}
<!-- * [[POTE]]: ~567/400 = 600.000{{c}}, ~162/125 = 443.075{{c}} (~35/32 = 156.925{{c}}) -->


{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
Line 616: Line 504:
Comma list: 441/440, 6144/6125, 8019/8000
Comma list: 441/440, 6144/6125, 8019/8000


Mapping:
Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
* common form: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
:: mapping generators: ~567/400, ~162/125
* mingen form: {{mapping| 2 5 7 3 3 | 0 -7 -9 10 15 }}
:: mapping generators: ~567/400, ~35/32


Optimal tunings:
Optimal tunings:
* WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{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}})
* CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})
<!-- * POTE: ~99/70 = 600.000{{c}}, ~162/125 = 443.117{{c}} (~35/32 = 156.883{{c}}) -->


{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}
Line 636: Line 519:
Comma list: 351/350, 364/363, 441/440, 10985/10976
Comma list: 351/350, 364/363, 441/440, 10985/10976


Mapping:
Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
* common form: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
:: mapping generators: ~55/39, ~162/125
* mingen form: {{mapping| 2 5 7 3 3 4 | 0 -7 -9 10 15 13 }}
:: mapping generators: ~55/39, ~35/32


Optimal tunings:
Optimal tunings:
* WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{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}})
* CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})
<!-- * POTE: ~55/39 = 600.000{{c}}, ~162/125 = 443.096{{c}} (~35/32 = 156.904{{c}}) -->


{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}
Line 652: Line 530:


== 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 659: Line 537:


{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
: mapping generators: ~2, ~45/28
: mapping generators: ~2, ~45/28


Line 667: Line 544:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}
: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~45/28 = 821.533{{c}} -->


{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}
{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}
Line 683: Line 559:
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~45/28 = 821.560{{c}} -->


{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}
Line 699: Line 574:
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~45/28 = 821.563{{c}} -->


{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}
Line 706: Line 580:


== 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 713: Line 587:


{{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


Line 721: Line 594:
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~48/35 = 546.815{{c}} -->


{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
Line 739: Line 611:
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.631{{c}} -->


{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}
Line 755: Line 626:
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.629{{c}} -->


{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}
Line 771: Line 641:
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.635{{c}} -->


{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}
Line 787: Line 656:
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~11/8 = 547.614{{c}} -->


{{Optimal ET sequence|legend=0| 46, 103h, 149h }}
{{Optimal ET sequence|legend=0| 46, 103h, 149h }}
Line 801: Line 669:


{{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


Line 809: Line 676:
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}
<!-- * [[POTE]]: ~63/50 = 400.000{{c}}, ~36/35 = 43.147{{c}} -->


{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
Line 825: Line 691:
* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}
<!-- * POTE: ~63/50 = 400.000{{c}}, ~36/35 = 43.292{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}
Line 843: Line 708:
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}
<!-- * POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.288{{c}} -->


{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
Line 859: Line 723:
* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}
<!-- * POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.276{{c}} -->


{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
Line 875: Line 738:
* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}
<!-- * POTE: ~49/39 = 400.000{{c}}, ~36/35 = 43.292{{c}} -->


{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}


Badness (Sintel): 1.12
Badness (Sintel): 1.12
== Other subgroup extensions ==
=== Sensipent (2.3.5.31 subgroup) ===
Subgroup: 2.3.5.31
Comma list: 961/960, 2511/2500
Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}
Optimal tunings:
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
Badness (Sintel): 0.243
=== Sendai ===
{{See also| Sensipent #Sendai interval table }}
Sendai is an accurate extension of sensipent with primes [[23/16|23]] and [[29/16|29]] found by [[User:VIxen|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence.
Subgroup: 2.3.5.23.29.31
Comma list: 465/464, 576/575, 621/620, 900/899
Subgroup-val mapping: {{mapping| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
Optimal tunings:
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
Badness (Sintel): 0.283


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Sensipent family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Sensipent_family"