Sensipent family: Difference between revisions

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{{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.  
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.  


Line 18: Line 19:


== Sensipent ==
== Sensipent ==
{{Main| Sensipent }}
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


Line 26: Line 29:
: mapping generators: ~2, ~162/125
: 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


Badness (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]].
=== 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
[[Subgroup]]: 2.3.5.31


[[Comma list]]: 78732/78125, 961/960
[[Comma list]]: 961/960, 2511/2500


{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
{{Mapping|legend=1| 1 -1 -1 2 | 0 7 9 8}}
Line 47: Line 50:
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~31/24 = 443.050
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 443.050


Badness (Dirichlet): 0.243
[[Badness]] (Sintel): 0.243


=== Sendai ===
=== Sendai ===
{{ See also | User:VIxen/Table_of_sensipent_intervals }}
{{ See also | Sensipent#Sendai interval table }}
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 (and where an international music competition is held).
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
[[Subgroup]]: 2.3.5.23.29.31


[[Comma list]]: 465/464, 576/575, 621/620, 729/725
[[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 }}
{{Mapping|legend=1| 1 -1 -1 6 -4 2| 0 7 9 -4 24 8 }}
Line 63: Line 66:
{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}
{{Optimal ET sequence|legend=1| 19, 46j, 65, 149, 363j }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~31/24 = 442.989
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~31/24 = 442.989


Badness (Dirichlet): 0.283
[[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 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.
{{ 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 -1 -1 9 | 0 7 9 -15 }}
{{Mapping|legend=1| 1 -1 -1 9 | 0 7 9 -15 }}
Line 80: Line 88:
{{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 -1 -1 9 10 | 0 7 9 -15 -16 }}
{{Mapping|legend=1| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
Line 97: Line 104:
{{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


==== 2.3.5.11.17.23 ====
==== 2.3.5.11.17.23 subgroup ====
[[Subgroup]]: 2.3.5.11.17.23
[[Subgroup]]: 2.3.5.11.17.23


[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]]
[[Comma list]]: 256/255, 576/575, 1089/1088, 1377/1375


{{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
{{Mapping|legend=1| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
Line 110: Line 117:
{{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 = 443.185
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000, ~22/17 = 443.185


Badness (Dirichlet): 0.555
[[Badness]] (Sintel): 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]],) [[529/528|S23]], [[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]].


==== 2.3.5.11.17.23.31 subgroup ====
[[Subgroup]]: 2.3.5.11.17.23.31
[[Subgroup]]: 2.3.5.11.17.23.31


[[Comma list]]: [[8019/8000]], [[16384/16335]], [[256/255]], [[576/575]], [[961/960]]
[[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 }}
{{Mapping|legend=1| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
Line 125: Line 130:
{{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 = 443.183
[[Optimal tuning]]s:
* [[CTE]]: 2/1 = 1\1, ~22/17 = 443.183
* [[CEE]]: 2/1 = 1\1, ~22/17 = 443.115


Badness (Dirichlet): 0.490
[[Badness]] (Sintel): 0.490


== Sensi ==
== Sensi ==
Line 142: Line 149:


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


[[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]]:  
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: mapping generators: ~2, ~9/7
: 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 188: Line 193:


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{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
* 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 205: Line 208:


Mapping: {{mapping| 1 -1 -1 -2 9 0 | 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 224: Line 225:


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 239: Line 240:


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{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
* 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 256: Line 255:


Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 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 275: Line 272:


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 290: Line 287:


: mapping generators: ~2, ~9/7
: mapping generators: ~2, ~9/7
{{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
* 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 307: Line 302:


Mapping: {{mapping| 1 -1 -1 -2 2 0 | 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 328: Line 321:


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 343: Line 336:


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 364: Line 357:


Optimal tunings:  
Optimal tunings:  
* CTE: ~99/70 = 1\2, ~9/7 = 443.3688 (~11/10 = 156.6312)
* CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
* POTE: ~99/70 = 1\2, ~9/7 = 443.308 (~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 383: Line 376:


Optimal tunings:  
Optimal tunings:  
* CTE: ~55/39 = 1\2, ~9/7 = 443.4416, ~11/10 = 156.5584
* CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
* POTE: ~55/39 = 1\2, ~9/7 = 443.275, ~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 402: Line 395:


Optimal tunings:  
Optimal tunings:  
* CTE: ~17/12 = 1\2, ~9/7 = 443.4466 (~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 420: Line 413:


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 435: Line 428:


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 451: Line 444:
: mapping generators: ~2, ~162/125
: mapping generators: ~2, ~162/125


{{Multival|legend=1| 7 9 32 -2 31 49 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 442.755
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 468: Line 459:
: mapping generators: ~2, ~162/125
: mapping generators: ~2, ~162/125


{{Multival|legend=1| 7 9 -33 -2 -72 -102 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~162/125 = 443.289
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~162/125 = 443.289


{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
{{Optimal ET sequence|legend=1| 46, 111, 157, 268cd }}
Line 485: Line 474:
: mapping generators: ~2, ~128/99
: 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 500: Line 489:
: mapping generators: ~2, ~84/65
: mapping generators: ~2, ~84/65


Optimal tuning (POTE): ~2 = 1\1, ~84/65 = 443.270
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 515: Line 504:
: mapping generators: ~2, ~22/17
: 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 534: Line 523:
:: mapping generators: ~567/400, ~35/32
:: mapping generators: ~567/400, ~35/32


{{Multival|legend=1| 14 18 -20 -4 -71 -97 }}
[[Optimal tuning]] ([[POTE]]): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)
 
[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~162/125 = 443.075 (~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 553: Line 540:
:: mapping generators: ~567/400, ~35/32
:: mapping generators: ~567/400, ~35/32


Optimal tuning (POTE): ~99/70 = 1\2, ~162/125 = 443.117 (~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 570: Line 557:
:: mapping generators: ~55/39, ~35/32
:: mapping generators: ~55/39, ~35/32


Optimal tuning (POTE): ~55/39 = 1\2, ~162/125 = 443.096 (~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 587: Line 574:
: mapping generators: ~2, ~56/45
: mapping generators: ~2, ~56/45


{{Multival|legend=1| 14 18 45 -4 32 54 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~56/45 = 378.467
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~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 602: 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 615: 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 632: Line 617:
: mapping generators: ~2, ~48/35
: 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 649: Line 632:
: mapping generators: ~2, ~11/8
: mapping generators: ~2, ~11/8


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 547.631
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 662: Line 645:
{{Mapping|legend=1| 1 -8 -10 6 3 11 | 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 675: Line 658:
{{Mapping|legend=1| 1 -8 -10 6 3 11 5 | 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 688: Line 671:
{{Mapping|legend=1| 1 -8 -10 6 3 11 5 12 | 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 705: Line 688:
: mapping generators: ~63/50, ~36/35
: mapping generators: ~63/50, ~36/35


{{Multival|legend=1| 21 27 12 -6 -40 -48 }}
[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000, ~36/35 = 43.147
 
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~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 720: 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 735: Line 716:
: mapping generators: ~49/39, ~36/35
: mapping generators: ~49/39, ~36/35


Optimal tuning (POTE): ~49/39 = 1\3, ~36/35 = 43.288
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 748: 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 761: 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 768: 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]]

Latest revision as of 00:30, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

Temperaments of the sensipent family temper out the sensipent comma, 78732/78125, also known as medium semicomma. 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 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.

Temperaments discussed elsewhere include:

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

Sensipent

Main article: Sensipent

Subgroup: 2.3.5

Comma list: 78732/78125

Mapping: [⟨1 -1 -1], ⟨0 7 9]]

mapping generators: ~2, ~162/125

Optimal tuning (POTE): ~2 = 1200.000, ~162/125 = 443.058

Optimal ET sequence: 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c

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 539 is the one with prime 31 by interpreting the generator accurately as 31/24~40/31, tempering out 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: [⟨1 -1 -1 2], ⟨0 7 9 8]]

mapping generators: ~2, ~31/24

Optimal ET sequence: 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c

Optimal tuning (CTE): ~2 = 1200.000, ~31/24 = 443.050

Badness (Sintel): 0.243

Sendai

See also: Sensipent § Sendai interval table

Sendai is an accurate extension of (2.3.5.31) sensipent with primes 23 and 29 found by VIxen. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).

Subgroup: 2.3.5.23.29.31

Comma list: 465/464, 576/575, 621/620, 900/899

Mapping: [⟨1 -1 -1 6 -4 2], ⟨0 7 9 -4 24 8]]

Optimal ET sequence: 19, 46j, 65, 149, 363j

Optimal tuning (CTE): ~2 = 1200.000, ~31/24 = 442.989

Badness (Sintel): 0.283

Sensible

See also: Sensipent § Sensible interval table

Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than sensi. It interprets the generator as 165/128~128/99 by tempering out 8019/8000 so that 11/8 is reached as (10/9)3. This extension is very strong as supported by the optimal ET sequence going very far and as supported by another observation that it also tempers out the semiporwellisma, which is equal to S31 × S322 (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a lopsided comma, implies that this temperament equates (33/32)2 with 16/15 as well as that a natural extension 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 S33 so that a slightly sharp ~22/17 is equated with the generator.

The aforementioned extension with prime 17 through tempering out S33 is equivalent to the one by tempering out 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 {(S16, S9/S10,) S23, S24, S31, S32, S33} implying also tempering out 496/495 = S31 × S32 and 528/527 = S32 × S33 as well as 16337/16335 = S31/S33 = (34/30)/(33/31)2 = (17/15)/(33/31)2. A notable patent val tuning not appearing in the optimal ET sequence is 157edo.

Subgroup: 2.3.5.11

Comma list: 8019/8000, 16384/16335

Mapping: [⟨1 -1 -1 9], ⟨0 7 9 -15]]

mapping generators: ~2, ~128/99

Optimal ET sequence: 19, 46, 65, 176, 241, 306

Optimal tuning (CTE): ~2 = 1200.000, ~128/99 = 443.115

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: [⟨1 -1 -1 9 10], ⟨0 7 9 -15 -16]]

mapping generators: ~2, ~22/17

Optimal ET sequence: 19, 46, 65, 111, 176g

Optimal tuning (CTE): ~2 = 1200.000, ~22/17 = 443.188

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: [⟨1 -1 -1 9 10 6], ⟨0 7 9 -15 -16 -4]]

Optimal ET sequence: 19, 46, 65, 111, 176g

Optimal tuning (CTE): ~2 = 1200.000, ~22/17 = 443.185

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: [⟨1 -1 -1 9 10 6 2], ⟨0 7 9 -15 -16 -4 8]]

Optimal ET sequence: 19, 46, 65, 111, 176g

Optimal tunings:

  • CTE: 2/1 = 1\1, ~22/17 = 443.183
  • CEE: 2/1 = 1\1, ~22/17 = 443.115

Badness (Sintel): 0.490

Sensi

Main article: Sensi

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

Septimal sensi

Subgroup: 2.3.5.7

Comma list: 126/125, 245/243

Mapping: [⟨1-1 -1 -2], ⟨0 7 9 13]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.3166
  • POTE: ~2 = 1200.000, ~9/7 = 443.383

Minimax tuning:

unchanged-interval (eigenmonzo) basis: 2.7
unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

  • 7-odd-limit diamond monotone: ~9/7 = [442.105, 450.000] (7\19 to 3\8)
  • 9-odd-limit diamond monotone: ~9/7 = [442.105, 444.444] (7\19 to 10\27)
  • 7-odd-limit diamond tradeoff: ~9/7 = [442.179, 445.628]
  • 9-odd-limit diamond tradeoff: ~9/7 = [435.084, 445.628]

Algebraic generator: The real root of x5 + x4 - 4x2 + x - 1, at 443.3783 cents.

Optimal ET sequence: 19, 27, 46

Badness: 0.025622

2.3.5.7.13 subgroup (sensation)

Subgroup: 2.3.5.7.13

Comma list: 91/90, 126/125, 169/168

Mapping: [⟨1 -1 -1 -2 0], ⟨0 7 9 13 10]]

mapping generators: ~2, ~9/7

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

Optimal ET sequence: 19, 27, 46, 111df

Sensor

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/243, 385/384

Mapping: [⟨1 -1 -1 -2 9], ⟨0 7 9 13 -15]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.2987
  • POTE: ~2 = 1200.000, ~9/7 = 443.294

Optimal ET sequence: 19, 27, 46, 111d

Badness: 0.037942

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 385/384

Mapping: [⟨1 -1 -1 -2 9 0], ⟨0 7 9 13 -15 10]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.3658
  • POTE: ~2 = 1200.000, ~9/7 = 443.321

Optimal ET sequence: 19, 27, 46, 111df

Badness: 0.025575

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 154/153, 169/168, 256/255

Mapping: [⟨1 -1 -1 -2 9 0 10], ⟨0 7 9 13 -15 10 -16]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.3775
  • POTE: ~2 = 1200.000, ~9/7 = 443.365

Optimal ET sequence: 19, 27, 46

Badness: 0.022908

Sensus

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 245/243

Mapping: [⟨1 -1 -1 -2 -8], ⟨0 7 9 13 31]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.4783
  • POTE: ~2 = 1200.000, ~9/7 = 443.626

Optimal ET sequence: 19e, 27e, 46, 119c

Badness: 0.029486

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 352/351

Mapping: [⟨1 -1 -1 -2 -8 0], ⟨0 7 9 13 31 10]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.5075
  • POTE: ~2 = 1200.000, ~9/7 = 443.559

Optimal ET sequence: 19e, 27e, 46

Badness: 0.020789

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 126/125, 136/135, 154/153, 169/168

Mapping: [⟨1 -1 -1 -2 -8 0 -7], ⟨0 7 9 13 31 10 30]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.5050
  • POTE: ~2 = 1200.000, ~9/7 = 443.551

Optimal ET sequence: 19eg, 27eg, 46

Badness: 0.016238

Sensis

Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 245/243

Mapping: [⟨1 -1 -1 -2 2], ⟨0 7 9 13 4]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.1886
  • POTE: ~2 = 1200.000, ~9/7 = 443.962

Optimal ET sequence: 8d, 19, 27e

Badness: 0.028680

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90, 100/99

Mapping: [⟨1 -1 -1 -2 2 0], ⟨0 7 9 13 4 10]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.2863
  • POTE: ~2 = 1200.000, ~9/7 = 443.945

Optimal ET sequence: 8d, 19, 27e

Badness: 0.020017

Sensa

Subgroup: 2.3.5.7.11

Comma list: 55/54, 77/75, 99/98

Mapping: [⟨1 -1 -1 -2 -1], ⟨0 7 9 13 12]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.7814
  • POTE: ~2 = 1200.000, ~9/7 = 443.518

Optimal ET sequence: 8d, 19e, 27

Badness: 0.036835

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 55/54, 66/65, 77/75, 143/140

Mapping: [⟨1 -1 -1 -2 -1 0], ⟨0 7 9 13 12 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~9/7 = 443.7877
  • POTE: ~2 = 1200.000, ~9/7 = 443.506

Optimal ET sequence: 8d, 19e, 27

Badness: 0.023258

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

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

Mapping:

  • common form: [⟨2 -2 -2 -4 1], ⟨0 7 9 13 8]]
mapping generators: ~99/70, ~9/7
  • mingen form: [⟨2 5 7 9 9], ⟨0 -7 -9 -13 -8]]
mapping generators: ~99/70, ~11/10

Optimal tunings:

  • CTE: ~99/70 = 600.000, ~9/7 = 443.3688 (~11/10 = 156.6312)
  • POTE: ~99/70 = 600.000, ~9/7 = 443.308 (~11/10 = 156.692)

Optimal ET sequence: 8d, …, 38d, 46

Badness: 0.041723

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 126/125, 169/168

Mapping:

  • common form: [⟨2 -2 -2 -4 1 0], ⟨0 7 9 13 8 10]]
mapping generators: ~99/70, ~9/7
  • mingen form: [⟨2 5 7 9 9 10], ⟨0 -7 -9 -13 -8 -10]]
mapping generators: ~99/70, ~11/10

Optimal tunings:

  • CTE: ~55/39 = 600.000, ~9/7 = 443.4416, ~11/10 = 156.5584
  • POTE: ~55/39 = 600.000, ~9/7 = 443.275, ~11/10 = 156.725

Optimal ET sequence: 8d, …, 38df, 46

Badness: 0.026339

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 126/125, 154/153, 169/168

Mapping:

  • common form: [⟨2 -2 -2 -4 1 0 3], ⟨0 7 9 13 8 10 7]]
mapping generators: ~99/70, ~9/7
  • mingen form: [⟨2 5 7 9 9 10 10], ⟨0 -7 -9 -13 -8 -10 -7]]
mapping generators: ~99/70, ~11/10

Optimal tunings:

  • CTE: ~17/12 = 600.000, ~9/7 = 443.4466 (~11/10 = 156.5534)

Optimal ET sequence: 8d, …, 38df, 46

Badness: 0.0188

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

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

Mapping: [⟨1 -1 -1 -2 -3], ⟨0 14 18 26 35]]

mapping generators: ~2, ~25/22

Optimal tunings:

  • CTE: ~2 = 1200.000, ~25/22 = 221.5981
  • POTE: ~2 = 1200.000, ~25/22 = 221.605

Optimal ET sequence: 27e, 38d, 65

Badness: 0.048714

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 169/168, 243/242

Mapping: [⟨1 -1 -1 -2 -3 0], ⟨0 14 18 26 35 20]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~25/22 = 221.6333
  • POTE: ~2 = 1200.000, ~25/22 = 221.556

Optimal ET sequence: 27e, 38df, 65f

Badness: 0.033016

Sensei

Subgroup: 2.3.5.7

Comma list: 225/224, 78732/78125

Mapping: [⟨1 -1 -1 -9], ⟨0 7 9 32]]

mapping generators: ~2, ~162/125

Optimal tuning (POTE): ~2 = 1200.000, ~162/125 = 442.755

Optimal ET sequence: 19, 65d, 84, 103, 187, 290b

Badness: 0.059218

Warrior

Subgroup: 2.3.5.7

Comma list: 5120/5103, 78732/78125

Mapping: [⟨1 -1 -1 15], ⟨0 7 9 -33]]

mapping generators: ~2, ~162/125

Optimal tuning (POTE): ~2 = 1200.000, ~162/125 = 443.289

Optimal ET sequence: 46, 111, 157, 268cd

Badness: 0.118239

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323, 5120/5103

Mapping: [⟨1 -1 -1 15 9], ⟨0 7 9 -33 -15]]

mapping generators: ~2, ~128/99

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

Optimal ET sequence: 46, 65d, 111, 268cd, 379cdd

Badness: 0.046383

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 847/845, 1331/1323

Mapping: [⟨1 -1 -1 15 9 17], ⟨0 7 9 -33 -15 -36]]

mapping generators: ~2, ~84/65

Optimal tuning (POTE): ~2 = 1200.000, ~84/65 = 443.270

Optimal ET sequence: 46, 65d, 111, 268cd, 379cddf

Badness: 0.028735

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 442/441, 715/714

Mapping: [⟨1 -1 -1 15 9 17 10], ⟨0 7 9 -33 -15 -36 -16]]

mapping generators: ~2, ~22/17

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

Optimal ET sequence: 46, 65d, 111, 268cdg, 379cddfg

Badness: 0.018105

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

Comma list: 6144/6125, 78732/78125

Mapping:

  • common form: [⟨2 -2 -2 13], ⟨0 7 9 -10]]
mapping generators: ~567/400, ~162/125
  • mingen form: [⟨2 5 7 3], ⟨0 -7 -9 10]]
mapping generators: ~567/400, ~35/32

Optimal tuning (POTE): ~567/400 = 600.000, ~162/125 = 443.075 (~35/32 = 156.925)

Optimal ET sequence: 8, 38, 46, 84, 130

Badness: 0.070375

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 6144/6125, 8019/8000

Mapping:

  • common form: [⟨2 -2 -2 13 18], ⟨0 7 9 -10 -15]]
mapping generators: ~567/400, ~162/125
  • mingen form: [⟨2 5 7 3 3], ⟨0 -7 -9 10 15]]
mapping generators: ~567/400, ~35/32

Optimal tuning (POTE): ~99/70 = 600.000, ~162/125 = 443.117 (~35/32 = 156.883)

Optimal ET sequence: 46, 84, 130, 306, 436ce

Badness: 0.037132

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 441/440, 10985/10976

Mapping:

  • common form: [⟨2 -2 -2 13 18 17], ⟨0 7 9 -10 -15 -13]]
mapping generators: ~55/39, ~162/125
  • mingen form: [⟨2 5 7 3 3 4], ⟨0 -7 -9 10 15 13]]
mapping generators: ~55/39, ~35/32

Optimal tuning (POTE): ~55/39 = 600.000, ~162/125 = 443.096 (~35/32 = 156.904)

Optimal ET sequence: 46, 84, 130, 566ce, 596cef

Badness: 0.023504

Subpental

Subpental splits the generator ~14/9 in two. Its ploidacot is theta-tetradecacot (pergen (P8, c4P4/14)).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 19683/19600

Mapping: [⟨1 6 8 17], ⟨0 -14 -18 -45]]

mapping generators: ~2, ~56/45

Optimal tuning (POTE): ~2 = 1200.000, ~56/45 = 378.467

Optimal ET sequence: 19, 111, 130, 929c, 1059c, 1189bc, 1319bc

Badness: 0.054303

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 8019/8000

Mapping: [⟨1 6 8 17 -6], ⟨0 -14 -18 -45 30]]

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

Optimal ET sequence: 19, 111, 130, 241, 371ce, 501cde, 872cde

Badness: 0.045352

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 676/675, 3136/3125

Mapping: [⟨1 6 8 17 -6 16], ⟨0 -14 -18 -45 30 -39]]

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

Optimal ET sequence: 19, 111, 130, 241, 371ce

Badness: 0.023940

Heinz

Heinz splits the generator ~18/7 in three. Its ploidacot is theta-21-cot (pergen (P8, c9P5/21)). A notable tuning of heinz not shown below for those who like 19edo's representation of the 5-limit is 57edo (57 = 103 - 46).

Subgroup: 2.3.5.7

Comma list: 1029/1024, 78732/78125

Mapping: [⟨1 -8 -10 6], ⟨0 21 27 -7]]

mapping generators: ~2, ~48/35

Optimal tuning (POTE): ~2 = 1200.000, ~48/35 = 546.815

Optimal ET sequence: 46, 103, 149, 699bdd

Badness: 0.115385

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 78732/78125

Mapping: [⟨1 -8 -10 6 3], ⟨0 21 27 -7 1]]

mapping generators: ~2, ~11/8

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.631

Optimal ET sequence: 46, 103, 149, 252e, 401bdee

Badness: 0.042412

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 385/384, 441/440, 847/845

Mapping: [⟨1 -8 -10 6 3 11], ⟨0 21 27 -7 1 -16]]

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

Optimal ET sequence: 46, 103, 149, 252ef, 401bdeef

Badness: 0.025779

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440, 847/845

Mapping: [⟨1 -8 -10 6 3 11 5], ⟨0 21 27 -7 1 -16 -2]]

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.635

Optimal ET sequence: 46, 103, 149, 252ef, 401bdeef

Badness: 0.018479

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968

Mapping: [⟨1 -8 -10 6 3 11 5 12], ⟨0 21 27 -7 1 -16 -2 -17]]

Optimal tuning (POTE): ~2 = 1200.000, ~11/8 = 547.614

Optimal ET sequence: 46, 103h, 149h, 252efhh

Badness: 0.019005

Trisensory

Trisensory has 1/3-octave period. Its ploidacot is triploid digamma-heptacot (pergen (P8/3, M6/21)).

Subgroup: 2.3.5.7

Comma list: 1728/1715, 78732/78125

Mapping: [⟨3 4 6 8], ⟨0 7 9 4]]

mapping generators: ~63/50, ~36/35

Optimal tuning (POTE): ~63/50 = 400.000, ~36/35 = 43.147

Optimal ET sequence: 27, 57, 84, 111, 195d, 306d

Badness: 0.089740

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 78732/78125

Mapping: [⟨3 4 6 8 8], ⟨0 7 9 4 22]]

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

Optimal ET sequence: 27e, 84e, 111

Badness: 0.058413

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 9295/9261

Mapping: [⟨3 4 6 8 8 11], ⟨0 7 9 4 22 1]]

mapping generators: ~49/39, ~36/35

Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.288

Optimal ET sequence: 27e, 84e, 111

Badness: 0.034829

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 351/350, 442/441, 540/539, 715/714

Mapping: [⟨3 4 6 8 8 11 10], ⟨0 7 9 4 22 1 21]]

Optimal tuning (POTE): ~49/39 = 400.000, ~36/35 = 43.276

Optimal ET sequence: 27eg, 84e, 111

Badness: 0.024120

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 324/323, 351/350, 400/399, 476/475

Mapping: [⟨3 4 6 8 8 11 10 12], ⟨0 7 9 4 22 1 21 7]]

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

Optimal ET sequence: 27eg, 84e, 111

Badness: 0.018466

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