Cloudy clan: Difference between revisions

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Line 34: Line 34:
POTE generator: ~21/20 = 73.264
POTE generator: ~21/20 = 73.264


Vals: {{Val list| 15, 35, 50, 65, 115d }}
Optimal GPV sequence: {{Val list| 15, 35, 50, 65, 115d }}


Badness: 0.045070
Badness: 0.045070


== Decic ==
== Decic ==
{{main| Decic }}
The ''decic'' temperament (10&50) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit, as well as the [[15/14ths equal temperament|linus comma]], {{monzo|11 -10 -10 10}}. This temperament is supported by [[10edo|10]], [[50edo|50]], and [[60edo|60]] EDOs, and can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.
The ''decic'' temperament (10&50) tempers out the [[cloudy comma]], 16807/16384 and the [[225/224|marvel comma]], 225/224 in the 7-limit, as well as the [[15/14ths equal temperament|linus comma]], {{monzo|11 -10 -10 10}}. This temperament is supported by [[10edo|10]], [[50edo|50]], and [[60edo|60]] EDOs, and can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.


Line 53: Line 55:
[[POTE generator]]: ~3/2 = 698.696
[[POTE generator]]: ~3/2 = 698.696


[[Tuning ranges]]:
{{Val list|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
* 7-odd-limit [[diamond monotone]]: ~3/2 = [680.000, 720.000] (34\60 to 6\10)
* 9-odd-limit diamond monotone: ~3/2 = [696.000, 720.000] (29\50 to 6\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [693.129, 702.512]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [693.129, 702.512]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [696.000, 702.512]
 
{{Val list|legend=1| 10, 50, 60, 110d, 170cdd }}


[[Badness]]: 0.089135
[[Badness]]: 0.089135
Line 75: Line 70:
POTE generator: ~3/2 = 696.791
POTE generator: ~3/2 = 696.791


Tuning ranges:
Optimal GPV sequence: {{Val list| 10, 30be, 40, 50 }}
* 11-odd-limit diamond monotone: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* 11-odd-limit diamond tradeoff: ~3/2 = [689.363, 702.512]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.000, 700.000]
 
Vals: {{Val list| 10, 50 }}


Badness: 0.063900
Badness: 0.063900
Line 95: Line 85:
POTE generator: ~3/2 = 696.993
POTE generator: ~3/2 = 696.993


Tuning ranges:
Optimal GPV sequence: {{Val list| 10, 30be, 40, 50 }}
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [689.363, 702.512]
* 13- and 15-odd-limit diamond monotone and tradeoff: ~3/2 = [696.000, 700.000]
 
Vals: {{Val list| 10, 50 }}


Badness: 0.036880
Badness: 0.036880
Line 115: Line 100:
POTE generator: ~3/2 = 697.085
POTE generator: ~3/2 = 697.085


Tuning ranges:
Optimal GPV sequence: {{Val list| 10, 30beg, 40, 50 }}
* 17-odd-limit diamond monotone: ~3/2 = [696.000, 700.000] (29\50 to 35\60)
* 17-odd-limit diamond tradeoff: ~3/2 = [689.363, 704.955]
* 17-odd-limit diamond monotone and tradeoff: ~3/2 = [696.000, 700.000]
 
Vals: {{Val list| 10, 50 }}


Badness: 0.025064
Badness: 0.025064
Line 165: Line 145:
POTE generator: ~3/2 = 702.733
POTE generator: ~3/2 = 702.733


Vals: {{Val list| 15, 45, 60, 75de, 135de }}
Optimal GPV sequence: {{Val list| 15, 45, 60, 75de, 135de }}


Badness: 0.077420
Badness: 0.077420
Line 180: Line 160:
POTE generator: ~3/2 = 701.715
POTE generator: ~3/2 = 701.715


Vals: {{Val list| 15, 45f, 60, 135de, 195cddee }}
Optimal GPV sequence: {{Val list| 15, 45f, 60, 135de, 195cddee }}


Badness: 0.051740
Badness: 0.051740
Line 195: Line 175:
POTE generator: ~3/2 = 700.318
POTE generator: ~3/2 = 700.318


Vals: {{Val list| 15, 45e, 60e }}
Optimal GPV sequence: {{Val list| 15, 45e, 60e }}


Badness: 0.044405
Badness: 0.044405
Line 210: Line 190:
POTE generator: ~3/2 = 701.647
POTE generator: ~3/2 = 701.647


Vals: {{Val list| 15f, 45e, 60e }}
Optimal GPV sequence: {{Val list| 15f, 45e, 60e }}


Badness: 0.055361
Badness: 0.055361
Line 238: Line 218:
POTE generator: ~5/4 = 393.438
POTE generator: ~5/4 = 393.438


Vals: {{Val list| 15, 40, 55, 125cd, 180ccdde }}
Optimal GPV sequence: {{Val list| 15, 40, 55, 125cd, 180ccdde }}


Badness: 0.065192
Badness: 0.065192
Line 251: Line 231:
POTE generator: ~5/4 = 392.798
POTE generator: ~5/4 = 392.798


Vals: {{Val list| 15, 40, 55 }}
Optimal GPV sequence: {{Val list| 15, 40, 55 }}


Badness: 0.048440
Badness: 0.048440

Revision as of 08:40, 5 February 2022

The cloudy clan tempers out the cloudy comma, 16807/16384, thus splitting the octave into 5 equal parts and mapping the harmonic 7 to 4\5.

Temperaments discussed elsewhere are:

Coblack

See also: Trisedodge family § Coblack

Subgroup: 2.3.5.7

Comma list: 126/125, 16807/16384

Mapping: [⟨5 7 11 14], ⟨0 3 2 0]]

Wedgie: ⟨⟨ 15 10 0 -19 -42 -28 ]]

POTE generator: ~21/20 = 73.044

Template:Val list

Badness: 0.107282

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 385/384

Mapping: [⟨5 7 11 14 17], ⟨0 3 2 0 1]]

POTE generator: ~21/20 = 73.264

Optimal GPV sequence: Template:Val list

Badness: 0.045070

Decic

Main article: Decic

The decic temperament (10&50) tempers out the cloudy comma, 16807/16384 and the marvel comma, 225/224 in the 7-limit, as well as the linus comma, [11 -10 -10 10⟩. This temperament is supported by 10, 50, and 60 EDOs, and can be extended naturally to the 11-, 13-, and 17-limit by adding 385/384, 105/104, and 170/169 to the comma list in this order.

Subgroup: 2.3.5.7

Comma list: 225/224, 16807/16384

Mapping: [⟨10 0 39 28], ⟨0 1 -1 0]]

Mapping generators: ~15/14, ~3

Wedgie: ⟨⟨ 10 -10 0 -39 -28 28 ]]

POTE generator: ~3/2 = 698.696

Template:Val list

Badness: 0.089135

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 3087/3025

Mapping: [⟨10 0 39 28 3], ⟨0 1 -1 0 2]]

Mapping generators: ~15/14, ~3

POTE generator: ~3/2 = 696.791

Optimal GPV sequence: Template:Val list

Badness: 0.063900

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 2200/2197

Mapping: [⟨10 0 39 28 3 37], ⟨0 1 -1 0 2 0]]

Mapping generators: ~14/13, ~3

POTE generator: ~3/2 = 696.993

Optimal GPV sequence: Template:Val list

Badness: 0.036880

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 144/143, 170/169, 196/195, 221/220

Mapping: [⟨10 0 39 28 3 37 25], ⟨0 1 -1 0 2 0 1]]

Mapping generators: ~14/13, ~3

POTE generator: ~3/2 = 697.085

Optimal GPV sequence: Template:Val list

Badness: 0.025064

Pentadecal

The pentadecal temperament tempers out the 15-5/3-comma (Quintriyo, pentadecatonic minor thirds comma), [-11 -15 15⟩. This temperament can be described as 15&60 temperament, tempering out the cloudy comma, 16807/16384 and the keema, 875/864 in the 7-limit.

Subgroup: 2.3.5

Comma list: 30517578125/29386561536

Mapping: [⟨15 0 11], ⟨0 1 1]]

POTE generator: ~3/2 = 702.652

Template:Val list

Badness: 0.692042

7-limit

Subgroup: 2.3.5.7

Comma list: 875/864, 16807/16384

Mapping: [⟨15 0 11 42], ⟨0 1 1 0]]

Mapping generators: ~21/20, ~3

POTE generator: ~3/2 = 700.223

Template:Val list

Badness: 0.114833

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 1372/1331

Mapping: [⟨15 0 11 42 52], ⟨0 1 1 0 0]]

Mapping generators: ~21/20, ~3

POTE generator: ~3/2 = 702.733

Optimal GPV sequence: Template:Val list

Badness: 0.077420

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 1372/1331

Mapping: [⟨15 0 11 42 52 8], ⟨0 1 1 0 0 2]]

Mapping generators: ~21/20, ~3

POTE generator: ~3/2 = 701.715

Optimal GPV sequence: Template:Val list

Badness: 0.051740

Quindecal

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 875/864

Mapping: [⟨15 0 11 42 28], ⟨0 1 1 0 1]]

Mapping generators: ~21/20, ~3

POTE generator: ~3/2 = 700.318

Optimal GPV sequence: Template:Val list

Badness: 0.044405

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 352/351, 875/864

Mapping: [⟨15 0 11 42 28 103], ⟨0 1 1 0 1 -2]]

Mapping generators: ~21/20, ~3

POTE generator: ~3/2 = 701.647

Optimal GPV sequence: Template:Val list

Badness: 0.055361

Quinkee

The quinkee temperament (15&40) has a period of 1/5 octave and tempers out the keega (1029/1000), and from this it derives its name.

Subgroup: 2.3.5.7

Comma list: 1029/1000, 6144/6125

Mapping: [⟨5 9 12 14], ⟨0 -3 -1 0]]

POTE generator: ~5/4 = 393.439

Template:Val list

Badness: 0.184222

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 1029/1000

Mapping: [⟨5 9 12 14 18], ⟨0 -3 -1 0 -2]]

POTE generator: ~5/4 = 393.438

Optimal GPV sequence: Template:Val list

Badness: 0.065192

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 637/625

Mapping: [⟨5 9 12 14 18 20], ⟨0 -3 -1 0 -2 -4]]

POTE generator: ~5/4 = 392.798

Optimal GPV sequence: Template:Val list

Badness: 0.048440

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