Sensipent family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings, per community consensus (1/)
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 443.058{{c}}
* [[WE]]: ~2 = 1199.9429{c}}, ~9/7 = 443.0364{{c}}
: [[error map]]: {{val| -0.057 -0.643 +1.071 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.0507{{c}}
: error map: {{val| 0.000 -0.600 +1.143 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~162/125 = 443.058{{c}} -->


{{Optimal ET sequence|legend=1| 8, 19, 46, 65, 539, 604c, 669c, 734c, 799c, 864c, 929c }}
{{Optimal ET sequence|legend=1| 8, 11c, 19, 46, 65, 539, 604c, 669c }}


[[Badness]] (Sintel): 0.826
[[Badness]] (Sintel): 0.826
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Comma list: 961/960, 2511/2500
Comma list: 961/960, 2511/2500


Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8}}
Subgroup-val mapping: {{mapping| 1 -1 -1 2 | 0 7 9 8 }}


: mapping generators: ~2, ~31/24
: mapping generators: ~2, ~31/24


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~31/24 = 443.050{{c}}
* WE: ~2 = 1200.0154{{c}}, ~31/24 = 443.0514{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 443.0474{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~31/24 = 443.050{{c}} -->


{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
{{Optimal ET sequence|legend=0| 8, 11c, 19, 46, 65, 344, 409, 474, 539, 604c }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~31/24 = 442.989{{c}}
* WE: ~2 = 1200.0782{{c}}, ~31/24 = 443.0005{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~31/24 = 442.9762{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~31/24 = 442.989{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
{{Optimal ET sequence|legend=0| 19, 46j, 65, 149, 363j }}
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~128/99 = 443.115{{c}}
* WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~128/99 = 443.115{{c}} -->


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


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


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
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Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}


Optimal tunings:
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.183{{c}}
* WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
* CEE: ~2 = 1200.000{{c}}, ~22/17 = 443.115{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~22/17 = 443.183{{c}}
* CEE: ~2 = 1200.000{{c}}, ~22/17 = 443.115{{c}} -->


{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
{{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3166{{c}}
* [[WE]]: ~2 = 1199.7081{c}}, ~9/7 = 443.2748{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~9/7 = 443.383{{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 }}
<!-- * [[CTE]]: ~2 = 1200.0000{{c}}, ~9/7 = 443.3166{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~9/7 = 443.383{{c}} -->


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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{{Optimal ET sequence|legend=1| 19, 27, 46 }}
{{Optimal ET sequence|legend=1| 19, 27, 46 }}


[[Badness]] (Smith): 0.025622
[[Badness]] (Sintel): 0.648


==== 2.3.5.7.13 subgroup (sensation) ====
==== 2.3.5.7.13 subgroup (sensation) ====
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Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4016{{c}}
* WE: ~2 = 1200.3138{{c}}, ~9/7 = 443.4379{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/7 = 443.3581{{c}}
<!-- * CTE: ~2 = 1200.0000{{c}}, ~9/7 = 443.4016{{c}} -->


{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
{{Optimal ET sequence|legend=0| 19, 27, 46, 111df }}
Badness (Sintel): 0.484


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


Badness: 0.037942
Badness (Smith): 0.037942


==== 13-limit ====
==== 13-limit ====

Revision as of 16:52, 6 October 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.

Sensipent

Main article: Sensipent

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.

Subgroup: 2.3.5

Comma list: 78732/78125

Mapping: [⟨1 -1 -1], ⟨0 7 9]]

mapping generators: ~2, ~162/125

Optimal tunings:

  • WE: ~2 = 1199.9429{c}}, ~9/7 = 443.0364 ¢
error map: ⟨-0.057 -0.643 +1.071]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.0507 ¢
error map: ⟨0.000 -0.600 +1.143]

Optimal ET sequence: 8, 11c, 19, 46, 65, 539, 604c, 669c

Badness (Sintel): 0.826

Overview to 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.

Temperaments discussed elsewhere include:

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

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

Subgroup-val mapping: [⟨1 -1 -1 2], ⟨0 7 9 8]]

mapping generators: ~2, ~31/24

Optimal tunings:

  • WE: ~2 = 1200.0154 ¢, ~31/24 = 443.0514 ¢
  • CWE: ~2 = 1200.0000 ¢, ~31/24 = 443.0474 ¢

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

Badness (Sintel): 0.243

Sendai

See also: Sensipent § Sendai interval table

Sendai is an accurate extension of (2.3.5.31) sensipent 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

Subgroup-val mapping: [⟨1 -1 -1 6 -4 2], ⟨0 7 9 -4 24 8]]

Optimal tunings:

  • WE: ~2 = 1200.0782 ¢, ~31/24 = 443.0005 ¢
  • CWE: ~2 = 1200.0000 ¢, ~31/24 = 442.9762 ¢

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

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

Subgroup-val mapping: [⟨1 -1 -1 9], ⟨0 7 9 -15]]

mapping generators: ~2, ~128/99

Optimal tunings:

  • WE: ~2 = 1199.6725 ¢, ~128/99 = 443.0183 ¢
  • CWE: ~2 = 1200.0000 ¢, ~128/99 = 443.1341 ¢

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

Badness (Sintel): 0.728

2.3.5.11.17 subgroup

Subgroup: 2.3.5.11.17

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

Subgroup-val mapping: [⟨1 -1 -1 9 10], ⟨0 7 9 -15 -16]]

mapping generators: ~2, ~22/17

Optimal tunings:

  • WE: ~2 = 1199.5016 ¢, ~22/17 = 443.0038 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/17 = 443.1878 ¢

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

Optimal tunings:

  • WE: ~2 = 1199.6207 ¢, ~22/17 = 443.0400 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/17 = 443.1808 ¢

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

Optimal tunings:

  • WE: ~2 = 1199.6623 ¢, ~22/17 = 443.0616 ¢
  • CWE: ~2 = 1200.0000 ¢, ~22/17 = 443.1858 ¢

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

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:

  • WE: ~2 = 1199.7081{c}}, ~9/7 = 443.2748 ¢
error map: ⟨-0.292 +1.261 +3.452 -5.669]
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3493 ¢
error map: ⟨0.000 +1.490 +3.830 -5.285]

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 (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: [⟨1 -1 -1 -2 0], ⟨0 7 9 13 10]]

mapping generators: ~2, ~9/7

Optimal tunings:

  • WE: ~2 = 1200.3138 ¢, ~9/7 = 443.4379 ¢
  • CWE: ~2 = 1200.0000 ¢, ~9/7 = 443.3581 ¢

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

Badness (Sintel): 0.484

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 = 1200.0000 ¢, ~9/7 = 443.2987 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.294 ¢

Optimal ET sequence: 19, 27, 46, 111d

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.3658 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.321 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.3775 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.365 ¢

Optimal ET sequence: 19, 27, 46

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.4783 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.626 ¢

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

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.5075 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.559 ¢

Optimal ET sequence: 19e, 27e, 46

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.5050 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.551 ¢

Optimal ET sequence: 19eg, 27eg, 46

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.1886 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.962 ¢

Optimal ET sequence: 8d, 19, 27e

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.2863 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.945 ¢

Optimal ET sequence: 8d, 19, 27e

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.7814 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.518 ¢

Optimal ET sequence: 8d, 19e, 27

Badness (Smith): 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 = 1200.0000 ¢, ~9/7 = 443.7877 ¢
  • POTE: ~2 = 1200.000 ¢, ~9/7 = 443.506 ¢

Optimal ET sequence: 8d, 19e, 27

Badness (Smith): 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 = 600.0000 ¢, ~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 (Smith): 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 = 600.0000 ¢, ~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 (Smith): 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 = 600.0000 ¢, ~9/7 = 443.4466 ¢ (~11/10 = 156.5534 ¢)

Optimal ET sequence: 8d, …, 38df, 46

Badness (Smith): 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 = 1200.0000 ¢, ~25/22 = 221.5981 ¢
  • POTE: ~2 = 1200.000 ¢, ~25/22 = 221.605 ¢

Optimal ET sequence: 27e, 38d, 65

Badness (Smith): 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 = 1200.0000 ¢, ~25/22 = 221.6333 ¢
  • POTE: ~2 = 1200.000 ¢, ~25/22 = 221.556 ¢

Optimal ET sequence: 27e, 38df, 65f

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~162/125 = 442.755 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~162/125 = 443.289 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~128/99 = 443.274 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~84/65 = 443.270 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~22/17 = 443.270 ¢

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

Badness (Smith): 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 tunings:

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

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

Badness (Smith): 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 tunings:

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

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

Badness (Smith): 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 tunings:

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

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~56/45 = 378.467 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~56/45 = 378.440 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~56/45 = 378.437 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~48/35 = 546.815 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~11/8 = 547.631 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~11/8 = 547.629 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~11/8 = 547.635 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~2 = 1200.000 ¢, ~11/8 = 547.614 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~63/50 = 400.000 ¢, ~36/35 = 43.147 ¢

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

Badness (Smith): 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 tunings:

  • POTE: ~63/50 = 400.000 ¢, ~36/35 = 43.292 ¢

Optimal ET sequence: 27e, 84e, 111

Badness (Smith): 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 tunings:

  • POTE: ~49/39 = 400.000 ¢, ~36/35 = 43.288 ¢

Optimal ET sequence: 27e, 84e, 111

Badness (Smith): 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 tunings:

  • POTE: ~49/39 = 400.000 ¢, ~36/35 = 43.276 ¢

Optimal ET sequence: 27eg, 84e, 111

Badness (Smith): 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 tunings:

  • POTE: ~49/39 = 400.000 ¢, ~36/35 = 43.292 ¢

Optimal ET sequence: 27eg, 84e, 111

Badness (Smith): 0.018466

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