Sensipent family: Difference between revisions

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{{see also| Sensamagic clan #Sensi }}
{{see also| Sensamagic clan #Sensi }}


Sensi tempers out [[686/675]], [[245/243]] 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|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."
Sensi tempers out [[686/675]], [[245/243]] 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 of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."


=== Septimal sensi ===
=== Septimal sensi ===
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[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]
* [[7-odd-limit]]: ~9/7 = {{monzo| 2/13 0 0 1/13 }}  
: [{{monzo| 1 0 0 0 }}, {{monzo| 1/13 0 0 7/13 }}, {{monzo| 5/13 0 0 9/13 }}, {{monzo| 0 0 0 1 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 7
: [[Eigenmonzo]]s (unchanged intervals): 2, 7
* [[9-odd-limit]]
* [[9-odd-limit]]: ~9/7 = {{monzo| 1/5 2/5 -1/5 0 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 2/5 14/5 -7/5 0 }}, {{monzo| 4/5 18/5 -9/5 0 }}, {{monzo| 3/5 26/5 -13/5 0 }}]
: [[Eigenmonzo]]s (unchanged intervals): 2, 9/5
: [[Eigenmonzo]]s (unchanged intervals): 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 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]
* 7-odd-limit diamond monotone and tradeoff: ~9/7 = [442.179, 445.628]
* 9-odd-limit diamond monotone and tradeoff: ~9/7 = [442.105, 444.444]


[[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.
[[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.
Line 71: Line 77:
POTE generator: ~9/7 = 443.322
POTE generator: ~9/7 = 443.322


Vals: {{Val list| 19, 27, 46, 111de, 157de }}
Optimal GPV sequence: {{Val list| 19, 27, 46, 111de, 157de }}


=== Sensor ===
=== Sensor ===
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POTE generator: ~9/7 = 443.294
POTE generator: ~9/7 = 443.294


Vals: {{Val list| 19, 27, 46, 111d, 157d, 268cdd }}
Optimal GPV sequence: {{Val list| 19, 27, 46, 111d, 157d, 268cdd }}


Badness: 0.037942
Badness: 0.037942
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POTE generator: ~9/7 = 443.321
POTE generator: ~9/7 = 443.321


Vals: {{Val list| 19, 27, 46, 111df, 157df }}
Optimal GPV sequence: {{Val list| 19, 27, 46, 111df, 157df }}


Badness: 0.025575
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: [{{val| 1 -1 -1 -2 9 0 10 }}, {{val| 0 7 9 13 -15 10 -16 }}]
POTE generator: ~9/7 = 443.365
Optimal GPV sequence: {{Val list| 19, 27, 46, 157df, 203cdff, 249cddff }}
Badness: 0.022908


=== Sensis ===
=== Sensis ===
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POTE generator: ~9/7 = 443.962
POTE generator: ~9/7 = 443.962


Vals: {{Val list| 8d, 19, 27e, 73ee }}
Optimal GPV sequence: {{Val list| 8d, 19, 27e, 73ee }}


Badness: 0.028680
Badness: 0.028680
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POTE generator: ~9/7 = 443.945
POTE generator: ~9/7 = 443.945


Vals: {{Val list| 19, 27e, 46e, 73ee }}
Optimal GPV sequence: {{Val list| 19, 27e, 46e, 73ee }}


Badness: 0.020017
Badness: 0.020017
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POTE generator: ~9/7 = 443.626
POTE generator: ~9/7 = 443.626


Vals: {{Val list| 19e, 27e, 46, 119c, 165c }}
Optimal GPV sequence: {{Val list| 19e, 27e, 46, 119c, 165c }}


Badness: 0.029486
Badness: 0.029486
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POTE generator: ~9/7 = 443.559
POTE generator: ~9/7 = 443.559


Vals: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}
Optimal GPV sequence: {{Val list| 19e, 27e, 46, 165cf, 211bccf, 257bccff, 303bccdff }}


Badness: 0.020789
Badness: 0.020789
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POTE generator: ~9/7 = 443.518
POTE generator: ~9/7 = 443.518


Vals: {{Val list| 19e, 27, 46ee }}
Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}


Badness: 0.036835
Badness: 0.036835
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POTE generator: ~9/7 = 443.506
POTE generator: ~9/7 = 443.506


Vals: {{Val list| 19e, 27, 46ee }}
Optimal GPV sequence: {{Val list| 19e, 27, 46ee }}


Badness: 0.023258
Badness: 0.023258
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POTE generator: ~25/22 = 221.605
POTE generator: ~25/22 = 221.605


Vals: {{Val list| 27e, 65, 157de, 222cde }}
Optimal GPV sequence: {{Val list| 27e, 65, 157de, 222cde }}


Badness: 0.048714
Badness: 0.048714
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POTE generator: ~35/32 = 156.883
POTE generator: ~35/32 = 156.883


Vals: {{Val list| 46, 84, 130, 306, 436ce }}
Optimal GPV sequence: {{Val list| 46, 84, 130, 306, 436ce }}


Badness: 0.037132
Badness: 0.037132
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POTE generator: ~35/32 = 156.904
POTE generator: ~35/32 = 156.904


Vals: {{Val list| 46, 84, 130, 566ce, 596cef }}
Optimal GPV sequence: {{Val list| 46, 84, 130, 566ce, 596cef }}


Badness: 0.023504
Badness: 0.023504
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POTE generator: ~11/8 = 547.631
POTE generator: ~11/8 = 547.631


Vals: {{Val list| 46, 103, 149, 252e, 401bdee }}
Optimal GPV sequence: {{Val list| 46, 103, 149, 252e, 401bdee }}


Badness: 0.042412
Badness: 0.042412
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POTE generator: ~11/8 = 547.629
POTE generator: ~11/8 = 547.629


Vals: {{Val list| 46, 103, 149, 252ef, 401bdeef }}
Optimal GPV sequence: {{Val list| 46, 103, 149, 252ef, 401bdeef }}


Badness: 0.025779
Badness: 0.025779
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POTE generator: ~11/8 = 547.635
POTE generator: ~11/8 = 547.635


Vals: {{Val list| 46, 103, 149, 252ef, 401bdeef }}
Optimal GPV sequence: {{Val list| 46, 103, 149, 252ef, 401bdeef }}


Badness: 0.018479
Badness: 0.018479
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POTE generator: ~11/8 = 547.614
POTE generator: ~11/8 = 547.614


Vals: {{Val list| 46, 103h, 149h, 252efhh }}
Optimal GPV sequence: {{Val list| 46, 103h, 149h, 252efhh }}


Badness: 0.019005
Badness: 0.019005

Revision as of 23:50, 28 December 2021


Temperaments of the sensipent family temper out the sensipent comma, 78732/78125, also known as medium semicomma. This temperament family includes sensi, sensei, bison, and heinz.

Temperaments discussed elsewhere include:

Sensipent

Subgroup: 2.3.5

Comma list: 78732/78125

Mapping: [⟨1 -1 -1], ⟨0 7 9]]

POTE generator: 162/125 = 443.058

Template:Val list

Badness: 0.035220

Sensi

Main article: Sensi
See also: Sensamagic clan § Sensi

Sensi tempers out 686/675, 245/243 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 of size 11, 19 and 27 are available. The name "sensi" is a play on the words "semi-" and "sixth."

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

Wedgie: ⟨⟨ 7 9 13 -2 1 5 ]]

POTE generator: ~9/7 = 443.383

Minimax tuning:

Eigenmonzos (unchanged intervals): 2, 7
Eigenmonzos (unchanged intervals): 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]
  • 7-odd-limit diamond monotone and tradeoff: ~9/7 = [442.179, 445.628]
  • 9-odd-limit diamond monotone and tradeoff: ~9/7 = [442.105, 444.444]

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

Template:Val list

Badness: 0.025622

Sensation

Subgroup: 2.3.5.7.13

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

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

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

Gencom: [2 9/7; 91/90 126/125 169/168]

POTE generator: ~9/7 = 443.322

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~9/7 = 443.294

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~9/7 = 443.321

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~9/7 = 443.365

Optimal GPV sequence: Template:Val list

Badness: 0.022908

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]]

POTE generator: ~9/7 = 443.962

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~9/7 = 443.945

Optimal GPV sequence: Template:Val list

Badness: 0.020017

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]]

POTE generator: ~9/7 = 443.626

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~9/7 = 443.559

Optimal GPV sequence: Template:Val list

Badness: 0.020789

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]]

POTE generator: ~9/7 = 443.518

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~9/7 = 443.506

Optimal GPV sequence: Template:Val list

Badness: 0.023258

Hemisensi

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]]

POTE generator: ~25/22 = 221.605

Optimal GPV sequence: Template:Val list

Badness: 0.048714

Sensei

Subgroup: 2.3.5.7

Comma list: 225/224, 78732/78125

Mapping: [⟨1 -1 -1 -9], ⟨0 7 9 32]]

Wedgie: ⟨⟨ 7 9 32 -2 31 49 ]]

POTE generator: ~125/81 = 757.245

Template:Val list

Badness: 0.059218

Bison

Subgroup: 2.3.5.7

Comma list: 6144/6125, 78732/78125

Mapping: [⟨2 5 7 3], ⟨0 -7 -9 10]]

Wedgie: ⟨⟨ 14 18 -20 -4 -71 -97 ]]

POTE generator: ~35/32 = 156.925

Template:Val list

Badness: 0.070375

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 5 7 3 3], ⟨0 -7 -9 10 15]]

POTE generator: ~35/32 = 156.883

Optimal GPV sequence: Template:Val list

Badness: 0.037132

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 5 7 3 3 4], ⟨0 -7 -9 10 15 13]]

POTE generator: ~35/32 = 156.904

Optimal GPV sequence: Template:Val list

Badness: 0.023504

Heinz

Subgroup: 2.3.5.7

Comma list: 1029/1024, 78732/78125

Mapping: [⟨1 -8 -10 6], ⟨0 21 27 -7]]

POTE generator: ~48/35 = 546.815

Template:Val list

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]]

POTE generator: ~11/8 = 547.631

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~11/8 = 547.629

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~11/8 = 547.635

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~11/8 = 547.614

Optimal GPV sequence: Template:Val list

Badness: 0.019005

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