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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.  
{{Technical data page}}
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. 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.  


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


Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.  
Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.  
Line 18: Line 19:


== Sensipent ==
== Sensipent ==
{{Main| 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, ~125/81
: mapping generators: ~2, ~162/125


[[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~162/125 = 443.058
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.058


{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}


[[Badness]]: 0.035220
[[Badness]]:
* Smith: 0.035220
* Dirichlet: 0.826
 
=== 2.3.5.31 subgroup ===
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]].
 
[[Subgroup]]: 2.3.5.31
 
[[Comma list]]: 961/960, 2511/2500
 
{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
 
: mapping generators: ~2, ~31/24
 
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050
 
[[Badness]] (Sintel): 0.243


Badness (Dirichlet): 0.826
=== 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).


=== 2.3.5.31 ===
[[Subgroup]]: 2.3.5.23.29.31
Fascinatingly, essentially the only extension that preserves the occurrence of sensi's 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. 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
[[Comma list]]: 465/464, 576/575, 621/620, 900/899
 
{{Mapping|legend=1| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
 
{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989


[[Comma list]]: 78732/78125, 961/960
[[Badness]] (Sintel): 0.283


=== Sensible ===
=== 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 as it is equal to [[961/960|S31]] * [[1024/1023|S32]]<sup>2</sup> (thus forming the S-expression-based comma list), which by 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.
{{ 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
[[Subgroup]]: 2.3.5.11


[[Comma list]]: [[8019/8000]], [[16384/16335]]
[[Comma list]]: 8019/8000, 16384/16335


{{Mapping|legend=1| 1 6 8 -6 | 0 -7 -9 15 }}
{{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
 
: mapping generators: ~2, ~128/99


{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 176, 241, 306 }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~128/99 = 443.115
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~128/99 = 443.115


Badness (Dirichlet): 0.728
[[Badness]] (Sintel): 0.728


=== 2.3.5.11.17 ===
==== 2.3.5.11.17 subgroup ====
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
[[Subgroup]]: 2.3.5.11.17


[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]]
[[Comma list]]: 256/255, 1089/1088, 1377/1375


{{Mapping|legend=1| 1 6 8 -6 -6 | 0 -7 -9 15 16 }}
{{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
 
: mapping generators: ~2, ~22/17


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~22/17 = 443.188
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.188


Badness (Dirichlet): 0.639
[[Badness]] (Sintel): 0.639


=== 2.3.5.11.17.31 ===
==== 2.3.5.11.17.23 subgroup ====
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]],) [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} implying also tempering [[496/495|496/495 = S31 * S32]] and [[528/527|528/527 = S32 * S33]] as well as [[16337/16335|16337/16335 = S31/S33]] = ([[17/15|34/30]])/([[33/31]])<sup>2</sup> = ([[17/15]])/([[33/31]])<sup>2</sup>.
[[Subgroup]]: 2.3.5.11.17.23


[[Subgroup]]: 2.3.5.11.17.31
[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375


[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[961/960]]
{{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}


{{Mapping|legend=1| 1 6 8 -6 -6 10 | 0 -7 -9 15 16 -8 }}
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}


{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g, 287cg }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~31/24 = 443.185
[[Badness]] (Sintel): 0.555


Badness (Dirichlet): 0.519
==== 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=1| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
 
{{Optimal ET sequence|legend=1| 19, 46, 65, 111, 176g }}
 
[[Optimal tuning]]s:
* [[CTE]]: 2/1 = 1\1, ~22/17 = 443.183
* [[CEE]]: 2/1 = 1\1, ~22/17 = 443.115
 
[[Badness]] (Sintel): 0.490


== Sensi ==
== Sensi ==
Line 96: Line 146:
[[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 }}
: mapping generators: ~2, ~9/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~9/7 = 443.3166
* [[CTE]]: ~2 = 1200.000, ~9/7 = 443.3166
* [[POTE]]: ~2 = 1\1, ~9/7 = 443.383
* [[POTE]]: ~2 = 1200.000, ~9/7 = 443.383


[[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 129: Line 177:
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: {{mapping| 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 }}


: gencom: [2 14/9; 91/90 126/125 169/168]
: mapping generators: ~2, ~9/7


Optimal tuning (CTE): ~2 = 1\1, ~9/7 = 443.4016
Optimal tuning (CTE): ~2 = 1200.000, ~9/7 = 443.4016


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
Line 144: Line 190:
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: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}


{{Multival|legend=1| 7 9 13 -15 -2 1 -48 5 -66 -87 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.2987
* CTE: ~2 = 1200.000, ~9/7 = 443.2987
* POTE: ~2 = 1\1, ~9/7 = 443.294
* POTE: ~2 = 1200.000, ~9/7 = 443.294


{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111d }}
Line 161: Line 207:
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: {{mapping| 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
* CTE: ~2 = 1200.000, ~9/7 = 443.3658
* POTE: ~2 = 1\1, ~9/7 = 443.321
* POTE: ~2 = 1200.000, ~9/7 = 443.321


{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=1| 19, 27, 46, 111df }}
Line 178: Line 222:
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: {{mapping| 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
* CTE: ~2 = 1200.000, ~9/7 = 443.3775
* POTE: ~2 = 1\1, ~9/7 = 443.365
* POTE: ~2 = 1200.000, ~9/7 = 443.365


{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}
Line 193: Line 237:
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: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}


{{Multival|legend=1| 7 9 13 31 -2 1 25 5 41 42 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.4783
* CTE: ~2 = 1200.000, ~9/7 = 443.4783
* POTE: ~2 = 1\1, ~9/7 = 443.626
* POTE: ~2 = 1200.000, ~9/7 = 443.626


{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
{{Optimal ET sequence|legend=1| 19e, 27e, 46, 119c }}
Line 210: Line 254:
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: {{mapping| 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
* CTE: ~2 = 1200.000, ~9/7 = 443.5075
* POTE: ~2 = 1\1, ~9/7 = 443.559
* POTE: ~2 = 1200.000, ~9/7 = 443.559


{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
{{Optimal ET sequence|legend=1| 19e, 27e, 46 }}
Line 227: Line 269:
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: {{mapping| 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
* CTE: ~2 = 1200.000, ~9/7 = 443.5050
* POTE: ~2 = 1\1, ~9/7 = 443.551
* POTE: ~2 = 1200.000, ~9/7 = 443.551


{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
{{Optimal ET sequence|legend=1| 19eg, 27eg, 46 }}
Line 242: Line 284:
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: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}


{{Multival|legend=1| 7 9 13 4 -2 1 -18 5 -22 -34 }}
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.1886
* CTE: ~2 = 1200.000, ~9/7 = 443.1886
* POTE: ~2 = 1\1, ~9/7 = 443.962
* POTE: ~2 = 1200.000, ~9/7 = 443.962


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
Line 259: Line 301:
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: {{mapping| 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
* CTE: ~2 = 1200.000, ~9/7 = 443.2863
* POTE: ~2 = 1\1, ~9/7 = 443.945
* POTE: ~2 = 1200.000, ~9/7 = 443.945


{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
{{Optimal ET sequence|legend=1| 8d, 19, 27e }}
Line 276: Line 316:
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: {{mapping| 1 -1 -1 -2 -1| 0 7 9 13 12 }}
 
: mapping generators: ~2, ~9/7


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7814
* CTE: ~2 = 1200.000, ~9/7 = 443.7814
* POTE: ~2 = 1\1, ~9/7 = 443.518
* POTE: ~2 = 1200.000, ~9/7 = 443.518


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
Line 291: Line 333:
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: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 11}}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~9/7 = 443.7877
* CTE: ~2 = 1200.000, ~9/7 = 443.7877
* POTE: ~2 = 1\1, ~9/7 = 443.506
* POTE: ~2 = 1200.000, ~9/7 = 443.506


{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
{{Optimal ET sequence|legend=1| 8d, 19e, 27 }}
Line 302: Line 344:


=== Bisensi ===
=== Bisensi ===
Bisensi has a 1/2-octave period. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).
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:
 
* common form: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
: mapping generators: ~99/70, ~11/10
:: 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:  
* CTE: ~99/70 = 1\2, ~11/10 = 156.6312
* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
* POTE: ~99/70 = 1\2, ~11/10 = 156.692
* POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~11/10 = 156.692)


{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
{{Optimal ET sequence|legend=1| 8d, …, 38d, 46 }}
Line 323: Line 369:
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:
* common form: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10}}
:: mapping generators: ~99/70, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 10 | 0 -7 -9 -13 -8 -10 }}
:: mapping generators: ~99/70, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 1\2, ~11/10 = 156.5584
* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
* POTE: ~55/39 = 1\2, ~11/10 = 156.725
* POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~11/10 = 156.725


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
Line 338: Line 388:
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:
* common form: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7}}
:: mapping generators: ~99/70, ~9/7
* mingen form: {{mapping| 2 5 7 9 9 10 10 | 0 -7 -9 -13 -8 -10 -7 }}
:: mapping generators: ~99/70, ~11/10


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~11/10 = 156.5534
* CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)


{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
{{Optimal ET sequence|legend=1| 8d, …, 38df, 46 }}
Line 348: Line 402:


=== Hemisensi ===
=== Hemisensi ===
Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-tetradecacot (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: {{mapping| 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
* CTE: ~2 = 1200.000, ~25/22 = 221.5981
* POTE: ~2 = 1\1, ~25/22 = 221.605
* POTE: ~2 = 1200.000, ~25/22 = 221.605


{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
{{Optimal ET sequence|legend=1| 27e, 38d, 65 }}
Line 369: Line 425:
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: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~25/22 = 221.6333
* CTE: ~2 = 1200.000, ~25/22 = 221.6333
* POTE: ~2 = 1\1, ~25/22 = 221.556
* POTE: ~2 = 1200.000, ~25/22 = 221.556


{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
{{Optimal ET sequence|legend=1| 27e, 38df, 65f }}
Line 384: Line 440:
[[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 }}
: mapping generators: ~2, ~162/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 442.755
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755


{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
{{Optimal ET sequence|legend=1| 19, 65d, 84, 103, 187, 290b }}
Line 399: Line 455:
[[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 }}
: mapping generators: ~2, ~162/125


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289


{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
Line 414: Line 470:
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: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15}}
 
: mapping generators: ~2, ~128/99


Optimal tuning (POTE): ~2 = 1\1, ~128/99 = 443.274
Optimal tuning (POTE): ~2 = 1200.000, ~128/99 = 443.274


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cdd}}
Line 427: Line 485:
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: {{mapping| 1 -1 -1 15 9 17| 0 7 9 -33 -15 -36}}


Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
: mapping generators: ~2, ~84/65
 
Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cd, 379cddf}}
Line 440: Line 500:
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: {{mapping| 1 -1 -1 15 9 17 10| 0 7 9 -33 -15 -36 -16}}
 
: mapping generators: ~2, ~22/17


Optimal tuning (POTE): ~2 = 1\1, ~22/17 = 443.270
Optimal tuning (POTE): ~2 = 1200.000, ~22/17 = 443.270


{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
{{Optimal ET sequence|legend=1| 46, 65d, 111, 268cdg, 379cddfg }}
Line 449: Line 511:


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


: mapping generators: ~567/400, ~35/32
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)
 
{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
 
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~35/32 = 156.925


{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
Line 470: Line 534:
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:
* 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 tuning (POTE): ~99/70 = 1\2, ~35/32 = 156.883
Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)


{{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
{{Optimal ET sequence|legend=1| 46, 84, 130, 306, 436ce }}
Line 483: Line 551:
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:
* 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 tuning (POTE): ~55/39 = 1\2, ~35/32 = 156.904
Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)


{{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}
{{Optimal ET sequence|legend=1| 46, 84, 130, 566ce, 596cef }}
Line 492: Line 564:


== Subpental ==
== Subpental ==
Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c<sup>4</sup>P4/14)).
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 498: Line 572:
{{Mapping|legend=1| 1 6 8 17 | 0 -14 -18 -45 }}
{{Mapping|legend=1| 1 6 8 17 | 0 -14 -18 -45 }}


{{Multival|legend=1| 14 18 45 -4 32 54 }}
: mapping generators: ~2, ~56/45


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~56/45 = 378.467
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~56/45 = 378.467


{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
{{Optimal ET sequence|legend=1| 19, 111, 130, 929c, 1059c, 1189bc, 1319bc }}
Line 513: Line 587:
Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}
Mapping: {{mapping| 1 6 8 17 -6 | 0 -14 -18 -45 30 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.440
Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.440


{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce, 501cde, 872cde }}
Line 526: Line 600:
Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}
Mapping: {{mapping| 1 6 8 17 -6 16 | 0 -14 -18 -45 30 -39 }}


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 378.437
Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.437


{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
{{Optimal ET sequence|legend=1| 19, 111, 130, 241, 371ce }}
Line 533: Line 607:


== 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 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).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 539: Line 613:
[[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, ~35/24
: mapping generators: ~2, ~48/35


{{Multival|legend=1| 21 27 -7 -6 -70 -92 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~48/35 = 546.815
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 546.815


{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
Line 556: Line 628:
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=1| 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 tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631


{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252e, 401bdee }}
Line 569: Line 643:
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=1| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.629
Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.629


{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
Line 582: Line 656:
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=1| 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 tuning (POTE): ~2 = 1200.000, ~11/8 = 547.635


{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
{{Optimal ET sequence|legend=1| 46, 103, 149, 252ef, 401bdeef }}
Line 595: Line 669:
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=1| 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 tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614


{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
{{Optimal ET sequence|legend=1| 46, 103h, 149h, 252efhh }}
Line 604: Line 678:


== 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 610: Line 686:
{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}
{{Mapping|legend=1| 3 4 6 8 | 0 7 9 4 }}


{{Multival|legend=1| 21 27 12 -6 -40 -48 }}
: mapping generators: ~63/50, ~36/35


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~36/35 = 43.147
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147


{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
Line 625: Line 701:
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}


Optimal tuning (POTE): ~63/50 = 1\3, ~36/35 = 43.292
Optimal tuning (POTE): ~63/50 = 400.000, ~36/35 = 43.292


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 638: Line 714:
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
Mapping: {{mapping| 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 tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288


{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27e, 84e, 111 }}
Line 651: Line 729:
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
Mapping: {{mapping| 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 tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.276


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Line 664: Line 742:
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}
Mapping: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.292
Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.292


{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
{{Optimal ET sequence|legend=1| 27eg, 84e, 111 }}
Line 671: Line 749:


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