Sensipent family: Difference between revisions

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

{{Main| Sensipent }}

[[Subgroup]]: 2.3.5

[[Comma list]]: 78732/78125

{{Mapping|legend=1| 1 -1 -1 | 0 7 9 }}

: mapping generators: ~2, ~162/125

* [[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}}

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

[[Badness]] (Sintel): 0.826

=== 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.

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.  

* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth|Breedsmic temperaments]] (+2401/2400)

* ''[[Browser]]'' → [[Canopic clan #Browser|Canopic clan]] (+16875/16807)

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

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.  

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.

[[#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.

{{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 [[S-expression|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 {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.

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

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

Subgroup: 2.3.5.11

Comma list: 8019/8000, 16384/16335

Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}

: mapping generators: ~2, ~128/99

* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}

Badness (Sintel): 0.728

==== 2.3.5.11.17 subgroup ====

Comma list: 256/255, 1089/1088, 1377/1375

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}

: mapping generators: ~2, ~22/17

* WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

Badness (Sintel): 0.639

==== 2.3.5.11.17.23 subgroup ====

Subgroup: 2.3.5.11.17.23

Comma list: 256/255, 576/575, 1089/1088, 1377/1375

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}

* WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

Badness (Sintel): 0.555

==== 2.3.5.11.17.23.31 subgroup ====

Subgroup: 2.3.5.11.17.23.31

Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375

Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}

* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}

{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}

Badness (Sintel): 0.490

== Sensi ==

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.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 245/243

{{Mapping|legend=1| 1-1 -1 -2 | 0 7 9 13 }}

[[Optimal tuning]]s:  

* [[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 }}

* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}

* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}

: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[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 tradeoff: ~9/7 = [435.084, 445.628]

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

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

[[Badness]] (Sintel): 0.648

==== 2.3.5.7.13 subgroup (sensation) ====

Subgroup: 2.3.5.7.13

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

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

Optimal tunings:  

* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}

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

Badness (Sintel): 0.484

Subgroup: 2.3.5.7.11

 

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

 

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

 

Optimal tunings:  

* WE: ~2 = 1200.0367{{c}}, ~9/7 = 443.3074{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.2947{{c}}

 

 

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

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

 

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

 

Optimal tunings:  

* WE: ~2 = 1200.3171{{c}}, ~9/7 = 443.4382{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3290{{c}}

 

 

Badness (Sintel): 1.06

 

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

 

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

 

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

 

Optimal tunings:  

* WE: ~2 = 1200.1572{{c}}, ~9/7 = 443.4230{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3666{{c}}

 

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

 

Badness (Sintel): 1.17

 

=== Sensus ===

Subgroup: 2.3.5.7.11

 

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

 

Mapping: {{mapping| 1 -1 -1 -2 -8| 0 7 9 13 31 }}

 

Optimal tunings:  

* WE: ~2 = 1199.0709{{c}}, ~9/7 = 443.2830{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5664{{c}}

 

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

 

Badness (Sintel): 0.975

 

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

 

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

 

Mapping: {{mapping| 1 -1 -1 -2 -8 0 | 0 7 9 13 31 10 }}

 

Optimal tunings:  

* WE: ~2 = 1199.6887{{c}}, ~9/7 = 443.4441{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5400{{c}}

 

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

 

Badness (Sintel): 0.859

 

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

 

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

 

Mapping: {{mapping| 1 -1 -1 -2 -8 0 -7 | 0 7 9 13 31 10 30 }}

 

Optimal tunings:  

* WE: ~2 = 1199.7033{{c}}, ~9/7 = 443.4418{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.5345{{c}}

 

 

Badness (Sintel): 0.827

 

Subgroup: 2.3.5.7.11

 

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

 

Mapping: {{mapping| 1 -1 -1 -2 2| 0 7 9 13 4 }}

 

Optimal tunings:  

* WE: ~2 = 1196.8330{{c}}, ~9/7 = 443.7907{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6554{{c}}

 

 

Badness (Sintel): 0.948

 

==== 13-limit ====

 

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

 

Mapping: {{mapping| 1 -1 -1 -2 2 0 | 0 7 9 13 4 10 }}

 

Optimal tunings:  

* WE: ~2 = 1197.4337{{c}}, ~9/7 = 442.9960{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.6925{{c}}

 

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

 

Badness (Sintel): 0.827

 

=== Sensa ===

Subgroup: 2.3.5.7.11

 

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

 

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

 

Optimal tunings:  

* WE: ~2 = 1201.0322{{c}}, ~9/7 = 443.8994{{c}}

* 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}}

 

* 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 }}

 

 

=== 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.)

 

 

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

 

* 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 }}

 

 

==== 13-limit ====

 

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

 

Mapping: {{mapping| 2 -2 -2 -4 1 0 | 0 7 9 13 8 10 }}

 

* 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)).

 

 

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

 

 

: 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 }}

 

 

==== 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 }}

 

* WE: ~2 = 1200.6518{{c}}, ~25/22 = 221.6764{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~25/22 = 221.5908{{c}}

 

 

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}}

* [[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 }}

 

* WE: ~2 = 1199.4073{{c}}, ~128/99 = 443.0552{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.2784{{c}}

 

 

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 }}

 

 

[[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 ===

 

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, c⁴P4/14)).

 

[[Subgroup]]: 2.3.5.7

 

 

{{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

 

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, 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).

 

 

[[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 }}

 

 

=== 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

 

* WE: ~2 = 1200.6094{{c}}, ~11/8 = 547.9095{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6413{{c}}

 

 

Badness (Sintel): 1.40

 

=== 13-limit ===

 

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

 

Mapping: {{mapping| 1 -8 -10 6 3 11 | 0 21 27 -7 1 -16}}

 

* 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}}

 

* 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 }}

 

 

=== 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 }}

 

* WE: ~2 = 1200.7181{{c}}, ~11/8 = 547.9418{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 547.6175{{c}}

 

 

Badness (Sintel): 1.16

 

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

 

 

[[Comma list]]: 1728/1715, 78732/78125

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Category:Sensipent family| ]] <!-- main article -->