Xenharmonic Wiki

Sensipent family: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
m recat
Move the opening note to a reasonable place. Don't shock the readers with stuff like 31/24 or 40/31 without proper context
 
(95 intermediate revisions by 13 users not shown)
Line 1: Line 1:
__FORCETOC__
{{Technical data page}}


<span style="display: block; text-align: right;">[[de:Sensi]]</span>
[[Regular temperament|Temperaments]] of the '''sensipent family''' [[tempering out|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.


=Sensipent=
== Sensipent ==
[[Comma|Comma]]: 78732/78125 = |2 9 -7&gt;
{{Main| Sensipent }}


[[POTE_tuning|POTE generator]]: 162/125 = 443.058 cents
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.


[[Map|Map]]: [&lt;1 6 8|, &lt;0 -7 -9|]
[[Subgroup]]: 2.3.5


[[EDO|EDO]]s: [[8edo|8]], [[19edo|19]], [[46edo|46]], [[65edo|65]], [[539edo|539]], [[604edo|604c]], [[669edo|669c]], [[734edo|734c]], [[799edo|799c]], [[864edo|864c]], [[929edo|929c]]
[[Comma list]]: 78732/78125


=Sensi=
{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}
{{main|Sensi}}
: mapping generators: ~2, ~162/125
Sensi tempers out 686/675, 245/243 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 [[13-limit|13-limit]] sensi tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo|46edo]] is an excellent sensi tuning, and MOS of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."


[[Comma|Commas]]: 126/125, 245/243
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9429{{c}}, ~162/125 = 443.0364{{c}}
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.0507{{c}}
: error map: {{val| 0.000 -0.600 +1.143 }}


7-limit minimax
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}


[|1 0 0 0&gt;, |1/13 0 0 7/13&gt;, |5/13 0 0 9/13&gt;, |0 0 0 1&gt;]
[[Badness]] (Sintel): 0.826


[[Eigenmonzo|Eigenmonzos]]: 2, 7
=== 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.


9-limit minimax
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.


[|1 0 0 0&gt;, |2/5 14/5 -7/5 0&gt;,
Temperaments discussed elsewhere include:
|4/5 18/5 -9/5 0&gt;, |3/5 26/5 -13/5 0&gt;<nowiki>]</nowiki>
* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)
* ''[[Browser]]'' → [[Canopic clan #Browser|Canopic clan]] (+16875/16807)


[[Eigenmonzo|Eigenmonzos]]: 2, 9/5
Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.


[[POTE_tuning|POTE generator]]: ~9/7 = 443.383
==== Subgroup extensions ====
The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. 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.  


Algebraic generator: The [[Algebraic_number|real root]] of x^5+x^4-4x^2+x-1, at 443.3783 cents.
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.


[[Map|Map]]: [&lt;1 6 8 11|, &lt;0 -7 -9 -13|]
[[#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.


Wedgie: &lt;&lt;7 9 13 -2 1 5||
=== Sensible ===
{{See also| Sensipent #Sensible interval table }}


[[generator|Generators]]: 2, 14/9
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.


[[EDO|EDO]]s: [[19edo|19]], [[27edo|27]], [[46edo|46]], [[157edo|157d]], [[203edo|203cd]], [[249edo|249cdd]], [[295edo|295ccdd]]
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) }}.


[[Badness|Badness]]: 0.0256
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]].


==Sensor==
Subgroup: 2.3.5.11
[[Comma|Comma]]s: 126/125, 245/243, 385/384


[[POTE_tuning|POTE generator]]: ~9/7 = 443.294
Comma list: 8019/8000, 16384/16335


[[Map|Map]]: [&lt;1 6 8 11 -6|, &lt;0 -7 -9 -13 15|]
Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}


[[EDO|EDO]]s: 19, 27, 46, 111d, 157d, 268cdd
: mapping generators: ~2, ~128/99


[[Badness|Badness]]: 0.0379
Optimal tunings:  
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}


===13-limit===
{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
[[Comma|Comma]]s: 91/90, 126/125, 169/168, 385/384


[[POTE_tuning|POTE generator]]: ~9/7 = 443.321
Badness (Sintel): 0.728


[[Map|Map]]: [&lt;1 6 8 11 -6 10|, &lt;0 -7 -9 -13 15 -10|]
==== 2.3.5.11.17 subgroup ====
Subgroup: 2.3.5.11.17


[[EDO|EDO]]s: [[19edo|19]], [[27edo|27]], [[46edo|46]], [[111edo|111df]], [[157edo|157df]]
Comma list: 256/255, 1089/1088, 1377/1375


[[Badness|Badness]]: 0.0256
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}


==Sensis==
: mapping generators: ~2, ~22/17
[[Comma|Comma]]s: 56/55, 100/99, 245/243


[[POTE_tuning|POTE generator]]: 443.962
Optimal tunings:  
* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}


[[Map|Map]]: [&lt;1 6 8 11 6|, &lt;0 -7 -9 -13 -4|]
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


[[EDO|EDO]]s: [[19edo|19]], [[27edo|27e]], [[73edo|73ee]]
Badness (Sintel): 0.639


[[Badness|Badness]]: 0.0287
==== 2.3.5.11.17.23 subgroup ====
Subgroup: 2.3.5.11.17.23


===13-limit===
Comma list: 256/255, 576/575, 1089/1088, 1377/1375
[[Comma|Comma]]s: 56/55, 78/77, 91/90, 100/99


[[POTE_tuning|POTE generator]]: 443.945
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}


[[Map|Map]]: [&lt;1 6 8 11 6 10|, &lt;0 -7 -9 -13 -4 -10|]
Optimal tunings:  
* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}


[[EDO|EDO]]s: 19, 27e, 46e, 73ee
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


[[Badness|Badness]]: 0.0200
Badness (Sintel): 0.555


==Sensus==
==== 2.3.5.11.17.23.31 subgroup ====
[[Comma|Comma]]s: 126/125, 176/175, 245/243
Subgroup: 2.3.5.11.17.23.31


[[POTE_generator|POTE generator]]: ~9/7 = 443.626
Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375


[[Map|Map]]: [&lt;1 6 8 11 23|, &lt;0 -7 -9 -13 -31|]
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}


[[EDO|EDO]]s: 19e, 27e, 46, 119c, 165c
Optimal tunings:  
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}


[[Badness|Badness]]: 0.0295
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}


===13-limit===
Badness (Sintel): 0.490
[[Comma|Comma]]s: 91/90, 126/125, 169/168, 352/351


[[POTE_generator|POTE generator]]: ~9/7 = 443.559
== Sensi ==
{{Main| Sensi }}


[[Map|Map]]: [&lt;1 6 8 11 23 10|, &lt;0 -7 -9 -13 -31 -10|]
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.


[[EDO|EDO]]s: 19e, 27e, 46, 165cf, 211bccf, 257bccff, [[303edo|303bccdff]]
=== Septimal sensi ===
[[Subgroup]]: 2.3.5.7


[[Badness|Badness]]: 0.0208
[[Comma list]]: 126/125, 245/243


See [[Chords_of_sensus|Chords of sensus]] for a listing of chords.
{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}


==Sensa==
[[Optimal tuning]]s:
Commas: 55/54, 77/75, 99/98
* [[WE]]: ~2 = 1199.7081{{c}}, ~9/7 = 443.2748{{c}}
: [[error map]]: {{val| -0.292 +1.261 +3.452 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3493{{c}}
: error map: {{val| 0.000 +1.490 +3.830 -5.285 }}


POTE generator: ~9/7 = 443.518
[[Minimax tuning]]:
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


Map: [&lt;1 6 8 11 11|, &lt;0 -7 -9 -13 -12|]
[[Tuning ranges of regular temperaments|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]


EDOs: 19e, 27, 46ee
[[Algebraic generator]]: The real root of ''x''<sup>5</sup> + ''x''<sup>4</sup> - 4''x''<sup>2</sup> + ''x'' - 1, at 443.3783 cents.


Badness: 0.0368
{{Optimal ET sequence|legend=1| 19, 27, 46 }}


===13-limit===
[[Badness]] (Sintel): 0.648
Commas: 55/54, 66/65, 77/75, 143/140


POTE generator: ~9/7 = 443.506
==== 2.3.5.7.13 subgroup (sensation) ====
Subgroup: 2.3.5.7.13


Map: [&lt;1 6 8 11 11 10|, &lt;0 -7 -9 -13 -12 -11|]
Comma list: 91/90, 126/125, 169/168


EDOs: 19e, 27, 46ee
Mapping: {{mapping| 1 -1 -1 -2 0| 0 7 9 13 10 }}


Badness: 0.0233
Optimal tunings:  
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}


=Hemisensi=
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
Commas: 126/125, 243/242, 245/242


POTE generator: ~25/22 = 221.605
Badness (Sintel): 0.484


Map: [&lt;1 13 17 24 32|, &lt;0 -14 -18 -26 -35|]
=== Sensor ===
Subgroup: 2.3.5.7.11


EDOs: 27e, 65, 157de, 222cde
Comma list: 126/125, 245/243, 385/384


Badness: 0.0487
Mapping: {{mapping| 1 -1 -1 -2 9 | 0 7 9 13 -15 }}


=Sensei=
Optimal tunings:
Commas: 225/224, 78732/78125
* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}


POTE generator: ~125/81 = 757.245
{{Optimal ET sequence|legend=0| 19, 27, 46, 111d }}


Map: [&lt;1 6 8 23|, &lt;0 -7 -9 -32|]
Badness (Sintel): 1.25


Wedgie: &lt;&lt;7 9 32 -2 31 49||
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


EDOs: 19, 84, 103, 187, 290b
Comma list: 91/90, 126/125, 169/168, 385/384


Badness: 0.0592
Mapping: {{mapping| 1 -1 -1 -2 9 0 | 0 7 9 13 -15 10 }}


=Bison=
Optimal tunings:
Commas: 6144/6125, 78732/78125
* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}


POTE generator: ~35/32 = 156.925
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}


Map: [&lt;2 5 7 3|, &lt;0 -7 -9 10|]
Badness (Sintel): 1.06


Wedgie: &lt;&lt;14 18 -20 -4 -71 -97||
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


EDOs: 8, 38, 46, 84, 130
Comma list: 91/90, 126/125, 154/153, 169/168, 256/255


Badness: 0.0704
Mapping: {{mapping| 1 -1 -1 -2 9 0 10 | 0 7 9 13 -15 10 -16 }}


==11-limit==
Optimal tunings:
Commas: 441/440, 6144/6125, 8019/8000
* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}


POTE generator: ~35/32 = 156.883
{{Optimal ET sequence|legend=0| 19, 27, 46 }}


Map: [&lt;2 5 7 3 3|, &lt;0 -7 -9 10 15|]
Badness (Sintel): 1.17


EDOs: 46, 84, 130, 306, 436ce
=== Sensus ===
Subgroup: 2.3.5.7.11


Badness: 0.0371
Comma list: 126/125, 176/175, 245/243


==13-limit==
Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}
Commas: 351/350, 364/363, 441/440, 10985/10976


POTE generator: ~35/32 = 156.904
Optimal tunings:  
* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}


Map: [&lt;2 5 7 3 3 4|, &lt;0 -7 -9 10 15 13|]
{{Optimal ET sequence|legend=0| 19e, 27e, 46, 119c }}


EDOs: 46, 84, 130, 566ce, 596cef
Badness (Sintel): 0.975


Badness: 0.0235
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


=Heinz=
Comma list: 91/90, 126/125, 169/168, 352/351
[[Comma|Comma]]s: 78732/78125, 1029/1024


[[POTE_generator|POTE generator]]: ~48/35 = 546.815
Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}


[[Map|Map]]: [&lt;1 13 17 -1|, &lt;0 -21 -27 7|]
Optimal tunings:  
* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}


[[EDO|EDO]]s: [[46edo|46]], [[103edo|103]], [[149edo|149]], [[699edo|699bd]]
{{Optimal ET sequence|legend=0| 19e, 27e, 46 }}


[[Badness|Badness]]: 0.1154
Badness (Sintel): 0.859


==11-limit==
==== 17-limit ====
[[Comma|Comma]]s: 385/384, 441/440, 88208/87500
Subgroup: 2.3.5.7.11.13.17


[[POTE_generator|POTE generator]]: ~11/8 = 547.631
Comma list: 91/90, 126/125, 136/135, 154/153, 169/168


[[Map|Map]]: [&lt;1 13 17 -1 4|, &lt;0 -21 -27 7 -1|]
Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}


[[EDO|EDO]]s: 46, 103, 149
Optimal tunings:  
* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}


[[Badness|Badness]]: 0.0424
{{Optimal ET sequence|legend=0| 19eg, 27eg, 46 }}


==13-limit==
Badness (Sintel): 0.827
[[Comma|Comma]]s: 351/350, 385/384, 441/440, 847/845


[[POTE_generator|POTE generator]]: ~11/8 = 547.629
=== Sensis ===
Subgroup: 2.3.5.7.11


[[Map|Map]]: [&lt;1 13 17 -1 4 -5|, &lt;0 -21 -27 7 -1 16|]
Comma list: 56/55, 100/99, 245/243


[[EDO|EDO]]s: 46, 57, 103, 149, 252e, 401bde
Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}


[[Badness|Badness]]: 0.0258
Optimal tunings:  
* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}


==17-limit==
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}
[[Comma|Comma]]s: 273/272, 351/350, 385/384, 441/440, 847/845


[[POTE_generator|POTE generator]]: ~11/8 = 547.635
Badness (Sintel): 0.948


[[Map|Map]]: [&lt;1 13 17 -1 4 -5 3|, &lt;0 -21 -27 7 -1 16 2|]
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


[[EDO|EDO]]s: 46, 103, 149, [[252edo|252ef]], [[401edo|401bdef]]
Comma list: 56/55, 78/77, 91/90, 100/99


[[Badness|Badness]]: 0.0185
Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}


==19-limit==
Optimal tunings:
[[Comma|Comma]]s: 171/170, 209/208, 351/350, 385/384, 441/440, 969/968
* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}


[[POTE_generator|POTE generator]]: ~11/8 = 547.614
{{Optimal ET sequence|legend=0| 8d, 19, 27e }}


[[Map|Map]]: [&lt;1 13 17 -1 4 -5 3 -5|, &lt;0 -21 -27 7 -1 16 2 17|]
Badness (Sintel): 0.827


[[EDO|EDO]]s: 46, 103h, 149h, 252efh
=== Sensa ===
Subgroup: 2.3.5.7.11


[[Badness|Badness]]: 0.0190
Comma list: 55/54, 77/75, 99/98


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


[[Category:Theory]]
Optimal tunings:
[[Category:Temperament family]]
* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}
[[Category:Sensipent]]
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6392{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
 
Badness (Sintel): 1.22
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 55/54, 66/65, 77/75, 143/140
 
Mapping: {{mapping| 1 -1 -1 -2 -1 0 | 0 7 9 13 12 10}}
 
Optimal tunings:
* WE: ~2 = 1201.1279{{c}}, ~9/7 = 443.9232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6386{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 19e, 27 }}
 
Badness (Sintel): 0.961
 
=== Bisensi ===
Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). (As 84edo is even, one might be interested to know its [[17-limit]] val is 84def, corresponding to a strong flat tendency.)
 
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 126/125, 245/243
 
Mapping: {{mapping| 2 -2 -2 -4 1 | 0 7 9 13 8 }}
 
: mapping generators: ~99/70, ~9/7
 
Optimal tunings:
* 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}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38d, 46 }}
 
Badness (Sintel): 1.38
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 121/120, 126/125, 169/168
 
Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}
 
Optimal tunings:
* 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}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
 
Badness (Sintel): 1.09
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 121/120, 126/125, 154/153, 169/168
 
Mapping: {{mapping| 2 -2 -2 -4 1 0 3 | 0 7 9 13 8 10 7 }}
 
Optimal tunings:
* WE: ~17/12 = 600.2912{{c}}, ~9/7 = 443.4993{{c}} (~11/10 = 156.7919{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~9/7 = 443.3456{{c}} (~11/10 = 156.6544{{c}})
 
{{Optimal ET sequence|legend=0| 8d, …, 38df, 46 }}
 
Badness (Sintel): 0.960
 
=== Hemisensi ===
Hemisensi splits the ~9/7 generator in two, each for ~25/22. Its ploidacot is beta-14-cot (pergen (P8, ccP5/14)).
 
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 243/242, 245/242
 
Mapping: {{mapping| 1 -1 -1 -2 -3 | 0 14 18 26 35 }}
 
: mapping generators: ~2, ~25/22
 
Optimal tunings:
* WE: ~2 = 1199.9253{{c}}, ~25/22 = 221.5916{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.6014{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 38d, 65 }}
 
Badness (Sintel): 1.61
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 91/90, 126/125, 169/168, 243/242
 
Mapping: {{mapping| 1 -1 -1 -2 -3 0 | 0 14 18 26 35 20 }}
 
Optimal tunings:
* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 38df, 65f }}
 
Badness (Sintel): 1.36
 
== Sensei ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 78732/78125
 
{{Mapping|legend=1| 1 -1 -1 -9 | 0 7 9 32 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6422{{c}}, ~162/125 = 442.9920{{c}}
: [[error map]]: {{val| +0.642 -1.653 -0.028 +1.139 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 442.7842{{c}}
: error map: {{val| 0.000 -2.466 -1.256 +0.267 }}
 
{{Optimal ET sequence|legend=0| 19, 65d, 84, 103, 187, 290b }}
 
[[Badness]] (Sintel): 1.50
 
== Warrior ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 5120/5103, 78732/78125
 
{{Mapping|legend=1| 1 -1 -1 15 | 0 7 9 -33 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2419{{c}}, ~162/125 = 443.0087{{c}}
: [[error map]]: {{val| -0.758 -0.136 +1.523 +0.516 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~162/125 = 443.2918{{c}}
: error map: {{val| 0.000 +1.088 +3.313 +2.544 }}
 
{{Optimal ET sequence|legend=1| 19d, 46, 111, 157, 268cd }}
 
[[Badness]] (Sintel): 2.99
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 1331/1323, 5120/5103
 
Mapping: {{mapping| 1 -1 -1 15 9 | 0 7 9 -33 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}
 
{{Optimal ET sequence|legend=0| 19d, 46, 65d, 111, 268cd }}
 
Badness (Sintel): 1.53
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 847/845, 1331/1323
 
Mapping: {{mapping| 1 -1 -1 15 9 17 | 0 7 9 -33 -15 -36 }}
 
Optimal tunings:
* WE: ~2 = 1199.4202{{c}}, ~84/65 = 443.0554{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~84/65 = 443.2755{{c}}
 
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cd }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 256/255, 351/350, 442/441, 715/714
 
Mapping: {{mapping| 1 -1 -1 15 9 17 10 | 0 7 9 -33 -15 -36 -16 }}
 
Optimal tunings:
* WE: ~2 = 1199.4084{{c}}, ~22/17 = 443.0513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.2764{{c}}
 
{{Optimal ET sequence|legend=0| 19df, 46, 65d, 111, 268cdg }}
 
Badness (Sintel): 0.922
 
== Bison ==
Bison has a 1/2-octave period and the generator can be taken as [[~]][[162/125]] or its semi-octave complement, ~[[35/32]]. Its [[ploidacot]] is diploid delta-heptacot ([[pergen]] (P8/2, ccP5/7)).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 6144/6125, 78732/78125
 
{{Mapping|legend=1| 2 -2 -2 13 | 0 7 9 -10 }}
: mapping generators: ~567/400, ~162/125
 
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 599.9413{{c}}, ~162/125 = 443.0320{{c}} (~35/32 = 156.9093{{c}})
: [[error map]]: {{val| -0.117 -0.613 +1.092 +0.091 }}
* [[CWE]]: ~567/400 = 1200.0000{{c}}, ~162/125 = 443.0728{{c}} (~35/32 = 156.9272{{c}})
: error map: {{val| 0.000 -0.446 +1.341 +0.446 }}
 
{{Optimal ET sequence|legend=1| 8, 38, 46, 84, 130 }}
 
[[Badness]] (Sintel): 1.78
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 6144/6125, 8019/8000
 
Mapping: {{mapping| 2 -2 -2 13 18 | 0 7 9 -10 -15 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8776{{c}}, ~162/125 = 443.0265{{c}} (~35/32 = 156.8511{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~162/125 = 443.1166{{c}} (~35/32 = 156.8834{{c}})
 
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 306, 436ce }}
 
Badness (Sintel): 1.23
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 364/363, 441/440, 10985/10976
 
Mapping: {{mapping| 2 -2 -2 13 18 17 | 0 7 9 -10 -15 -13 }}
 
Optimal tunings:
* WE: ~55/39 = 599.9161{{c}}, ~162/125 = 443.0343{{c}} (~35/32 = 156.8817{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~162/125 = 443.0973{{c}} (~35/32 = 156.9027{{c}})
 
{{Optimal ET sequence|legend=0| 38e, 46, 84, 130, 566ce, 596cef }}
 
Badness (Sintel): 0.971
 
== Subpental ==
Subpental splits the generator of sensipent plus an octave, ~324/125, in two, each for ~45/28 of about 821.5 cents. Alternatively, the generator may be taken to be its octave complement, ~56/45, of about 378.5 cents. Its ploidacot is theta-14-cot (pergen (P8, c<sup>4</sup>P4/14)).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 3136/3125, 19683/19600
 
{{Mapping|legend=1| 1 -8 -10 -28 | 0 14 18 45 }}
: mapping generators: ~2, ~45/28
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9261{{c}}, ~45/28 = 821.4823{{c}}
: [[error map]]: {{val| -0.074 -0.611 +1.107 -0.052 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~45/28 = 821.5303{{c}}
: error map: {{val| 0.000 -0.531 +1.231 +0.036 }}
 
{{Optimal ET sequence|legend=1| 19, …, 111, 130 }}
 
[[Badness]] (Sintel): 1.37
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 3136/3125, 8019/8000
 
Mapping: {{mapping| 1 -8 -10 -28 24 | 0 14 18 45 -30 }}
 
Optimal tunings:
* WE: ~2 = 1199.6571{{c}}, ~45/28 = 821.3249{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5560{{c}}
 
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce, 501cde }}
 
Badness (Sintel): 1.50
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 540/539, 676/675, 3136/3125
 
Mapping: {{mapping| 1 -8 -10 -28 24 -23 | 0 14 18 45 -30 39 }}
 
Optimal tunings:
* WE: ~2 = 1199.6819{{c}}, ~45/28 = 821.3451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/28 = 821.5591{{c}}
 
{{Optimal ET sequence|legend=0| 19, 111, 130, 241, 371ce }}
 
Badness (Sintel): 0.989
 
== Heinz ==
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
 
[[Comma list]]: 1029/1024, 78732/78125
 
{{Mapping|legend=1| 1 -8 -10 6 | 0 21 27 -7 }}
: mapping generators: ~2, ~48/35
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4250{{c}}, ~48/35 = 547.8379{{c}}
: [[error map]]: {{val| +0.425 -0.758 +1.061 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 547.6528{{c}}
: error map: {{val| 0.000 -1.247 +0.311 -2.395 }}
 
{{Optimal ET sequence|legend=1| 46, 103, 149, 699bdd }}
 
[[Badness]] (Sintel): 2.92
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 441/440, 78732/78125
 
Mapping: {{mapping| 1 -8 -10 6 3 | 0 21 27 -7 1}}
 
: mapping generators: ~2, ~11/8
 
Optimal tunings:
* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103, 149, 252e, 401bdee }}
 
Badness (Sintel): 1.40
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 351/350, 385/384, 441/440, 847/845
 
Mapping: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}
 
Optimal tunings:
* WE: ~2 = 1200.6343{{c}}, ~11/8 = 547.9182{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6345{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef, 401bdeef }}
 
Badness (Sintel): 1.07
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 273/272, 351/350, 385/384, 441/440, 847/845
 
Mapping: {{mapping| 1 -8 -10 6 3 11 5 | 0 21 27 -7 1 -16 -2}}
 
Optimal tunings:
* WE: ~2 = 1200.5351{{c}}, ~11/8 = 547.8790{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6388{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103, 149, 252ef }}
 
Badness (Sintel): 0.941
 
=== 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: {{mapping| 1 -8 -10 6 3 11 5 12 | 0 21 27 -7 1 -16 -2 -17 }}
 
Optimal tunings:
* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}
 
{{Optimal ET sequence|legend=0| 46, 103h, 149h }}
 
Badness (Sintel): 1.16
 
== 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|legend=1| 3 4 6 8 | 0 7 9 4 }}
: mapping generators: ~63/50, ~36/35
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 399.8117{{c}}, ~36/35 = 43.1270{{c}}
: [[error map]]: {{val| -0.565 -0.819 +0.700 +2.176 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~36/35 = 43.0852{{c}}
: error map: {{val| 0.000 -0.359 +1.453 +3.515 }}
 
{{Optimal ET sequence|legend=1| 27, 57, 84, 111, 195d, 306d }}
 
[[Badness]] (Sintel): 2.27
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 540/539, 78732/78125
 
Mapping: {{mapping| 3 4 6 8 8 | 0 7 9 4 22 }}
 
Optimal tunings:
* WE: ~63/50 = 399.7341{{c}}, ~36/35 = 43.2633{{c}}
* CWE: ~63/50 = 400.0000{{c}}, ~36/35 = 43.2290{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccdde }}
 
Badness (Sintel): 1.93
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 351/350, 540/539, 9295/9261
 
Mapping: {{mapping| 3 4 6 8 8 11 | 0 7 9 4 22 1 }}
 
: mapping generators: ~49/39, ~36/35
 
Optimal tunings:
* WE: ~49/39 = 399.7403{{c}}, ~36/35 = 43.2602{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2415{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 84e, 111, 360ccddef }}
 
Badness (Sintel): 1.44
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 351/350, 442/441, 540/539, 715/714
 
Mapping: {{mapping| 3 4 6 8 8 11 10 | 0 7 9 4 22 1 21 }}
 
Optimal tunings:
* WE: ~49/39 = 399.7422{{c}}, ~36/35 = 43.2480{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2305{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
 
Badness (Sintel): 1.23
 
=== 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: {{mapping| 3 4 6 8 8 11 10 12 | 0 7 9 4 22 1 21 7 }}
 
Optimal tunings:
* WE: ~49/39 = 399.7059{{c}}, ~36/35 = 43.2600{{c}}
* CWE: ~49/39 = 400.0000{{c}}, ~36/35 = 43.2433{{c}}
 
{{Optimal ET sequence|legend=0| 27eg, 84e, 111 }}
 
Badness (Sintel): 1.12
 
== 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:Sensipent family| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Sensipent_family"