Dicot family: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 15: Line 15:
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]


{{Val list|legend=1| 3, 4, 7, 17, 24c, 31c }}
{{Optimal ET sequence|legend=1| 3, 4, 7, 17, 24c, 31c }}


[[Badness]]: 0.013028
[[Badness]]: 0.013028
Line 33: Line 33:
[[POTE generator]]: ~5/4 = 336.381
[[POTE generator]]: ~5/4 = 336.381


{{Val list|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}


[[Badness]]: 0.019935
[[Badness]]: 0.019935
Line 46: Line 46:
POTE generator: ~5/4 = 342.125
POTE generator: ~5/4 = 342.125


Optimal GPV sequence: {{Val list| 3de, 4e, 7 }}
{{Optimal ET sequence|legend=1| 3de, 4e, 7 }}


Badness: 0.019854
Badness: 0.019854
Line 59: Line 59:
POTE generator: ~5/4 = 336.051
POTE generator: ~5/4 = 336.051


Optimal GPV sequence: {{Val list| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}


Badness: 0.027114
Badness: 0.027114
Line 72: Line 72:
POTE generator: ~5/4 = 338.846
POTE generator: ~5/4 = 338.846


Optimal GPV sequence: {{Val list| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
{{Optimal ET sequence|legend=1| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}


Badness: 0.023828
Badness: 0.023828
Line 87: Line 87:
[[POTE generator]]: ~5/4 = 331.916
[[POTE generator]]: ~5/4 = 331.916


{{Val list|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}


[[Badness]]: 0.025381
[[Badness]]: 0.025381
Line 100: Line 100:
POTE generator: ~5/4 = 337.532
POTE generator: ~5/4 = 337.532


Optimal GPV sequence: {{Val list| 3, 4, 7d }}
{{Optimal ET sequence|legend=1| 3, 4, 7d }}


Badness: 0.024988
Badness: 0.024988
Line 113: Line 113:
POTE generator: ~5/4 = 341.023
POTE generator: ~5/4 = 341.023


Optimal GPV sequence: {{Val list| 3, 4, 7d }}
{{Optimal ET sequence|legend=1| 3, 4, 7d }}


Badness: 0.023420
Badness: 0.023420
Line 128: Line 128:
{{Multival|legend=1|2 1 6 -3 4 11}}
{{Multival|legend=1|2 1 6 -3 4 11}}


{{Val list|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
{{Optimal ET sequence|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}


[[Badness]]: 0.028942
[[Badness]]: 0.028942
Line 141: Line 141:
POTE generator: ~5/4 = 356.106
POTE generator: ~5/4 = 356.106


Optimal GPV sequence: {{Val list| 3de, 7d, 10, 17d, 27cde }}
{{Optimal ET sequence|legend=1| 3de, 7d, 10, 17d, 27cde }}


Badness: 0.022366
Badness: 0.022366
Line 156: Line 156:
[[POTE generator]]: ~7/6 = 251.557
[[POTE generator]]: ~7/6 = 251.557


{{Val list|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}


[[Badness]]: 0.028334
[[Badness]]: 0.028334
Line 169: Line 169:
POTE generator: ~7/6 = 253.493
POTE generator: ~7/6 = 253.493


Optimal GPV sequence: {{Val list| 10, 14c, 24c, 38ccd, 52cccde }}
{{Optimal ET sequence|legend=1| 10, 14c, 24c, 38ccd, 52cccde }}


Badness: 0.026712
Badness: 0.026712
Line 182: Line 182:
POTE generator: ~7/6 = 255.066
POTE generator: ~7/6 = 255.066


Optimal GPV sequence: {{Val list| 4, 10e, 14c }}
{{Optimal ET sequence|legend=1| 4, 10e, 14c }}


Badness: 0.031456
Badness: 0.031456
Line 195: Line 195:
POTE generator: ~8/7 = 243.493
POTE generator: ~8/7 = 243.493


Optimal GPV sequence: {{Val list| 4, 6, 10 }}
{{Optimal ET sequence|legend=1| 4, 6, 10 }}


Badness: 0.032385
Badness: 0.032385
Line 210: Line 210:
[[POTE generator]]: ~5/4 = 356.264
[[POTE generator]]: ~5/4 = 356.264


{{Val list|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}


[[Badness]]: 0.037565
[[Badness]]: 0.037565
Line 223: Line 223:
POTE generator: ~5/4 = 354.262
POTE generator: ~5/4 = 354.262


Optimal GPV sequence: {{Val list| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}


Badness: 0.030680
Badness: 0.030680
Line 236: Line 236:
POTE generator: ~5/4 = 354.365
POTE generator: ~5/4 = 354.365


Optimal GPV sequence: {{Val list| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}


Badness: 0.021674
Badness: 0.021674
Line 249: Line 249:
POTE generator: ~5/4 = 354.073
POTE generator: ~5/4 = 354.073


Optimal GPV sequence: {{Val list| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}


Badness: 0.031719
Badness: 0.031719
Line 262: Line 262:
POTE generator: ~5/4 = 354.313
POTE generator: ~5/4 = 354.313


Optimal GPV sequence: {{Val list| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}


Badness: 0.022741
Badness: 0.022741
Line 275: Line 275:
POTE generator: ~5/4 = 360.659
POTE generator: ~5/4 = 360.659


Optimal GPV sequence: {{Val list| 3, 7e, 10 }}
{{Optimal ET sequence|legend=1| 3, 7e, 10 }}


Badness: 0.041361
Badness: 0.041361
Line 288: Line 288:
POTE generator: ~5/4 = 360.646
POTE generator: ~5/4 = 360.646


Optimal GPV sequence: {{Val list| 3, 7e, 10 }}
{{Optimal ET sequence|legend=1| 3, 7e, 10 }}


Badness: 0.027938
Badness: 0.027938
Line 303: Line 303:
[[POTE generator]]: ~8/7 = 258.139
[[POTE generator]]: ~8/7 = 258.139


{{Val list|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 7, 14c }}


[[Badness]]: 0.041714
[[Badness]]: 0.041714
Line 316: Line 316:
POTE generator: ~8/7 = 258.910
POTE generator: ~8/7 = 258.910


Optimal GPV sequence: {{Val list| 7, 14c }}
{{Optimal ET sequence|legend=1| 7, 14c }}


Badness: 0.023524
Badness: 0.023524
Line 329: Line 329:
POTE generator: ~8/7 = 250.764
POTE generator: ~8/7 = 250.764


Optimal GPV sequence: {{Val list| 7, 14c }}
{{Optimal ET sequence|legend=1| 7, 14c }}


Badness: 0.023003
Badness: 0.023003
Line 342: Line 342:
POTE generator: ~8/7 = 247.445
POTE generator: ~8/7 = 247.445


Optimal GPV sequence: {{Val list| 7, 14cf }}
{{Optimal ET sequence|legend=1| 7, 14cf }}


Badness: 0.022569
Badness: 0.022569
Line 357: Line 357:
[[POTE generator]]: ~9/7 = 427.208
[[POTE generator]]: ~9/7 = 427.208


{{Val list|legend=1| 3d, 14c, 45cc, 59bcccd }}
{{Optimal ET sequence|legend=1| 3d, 14c, 45cc, 59bcccd }}


[[Badness]]: 0.056586
[[Badness]]: 0.056586
Line 370: Line 370:
POTE generator: ~9/7 = 427.273
POTE generator: ~9/7 = 427.273


Optimal GPV sequence: {{Val list| 3de, 14c, 45cce, 59bcccdee }}
{{Optimal ET sequence|legend=1| 3de, 14c, 45cce, 59bcccdee }}


Badness: 0.032957
Badness: 0.032957
Line 385: Line 385:
[[POTE generator]]: ~8/7 = 324.482
[[POTE generator]]: ~8/7 = 324.482


{{Val list|legend=1| 4 }}
{{Optimal ET sequence|legend=1| 4 }}


[[Badness]]: 0.045911
[[Badness]]: 0.045911

Revision as of 16:18, 7 May 2023

The 5-limit parent comma for the dicot family is 25/24, the classic chromatic semitone. Its monzo is [-3 -1 2⟩, and flipping that yields ⟨⟨2 1 -3]] for the wedgie. This tells us the generator is a third (major and minor mean the same thing), and that two thirds gives a fifth. In fact, (5/4)^2 = 3/2 * 25/24. Possible tunings for dicot are 7edo, 24edo using the val ⟨24 38 55] (24c) and 31edo using the val ⟨31 49 71] (31c). In a sense, what dicot is all about is using neutral thirds and pretending that's 5-limit, and like any temperament which seems to involve pretending, dicot is at the edge of what can sensibly be called a temperament at all. In other words, it is an exotemperament.

Dicot

Subgroup: 2.3.5

Comma list: 25/24

Mapping: [⟨1 1 2], ⟨0 2 1]]

POTE generator: ~5/4 = 348.594

Tuning ranges:

  • 5-odd-limit diamond monotone: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
  • 5-odd-limit diamond tradeoff: ~5/4 = [315.641, 386.314]
  • 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]

Optimal ET sequence: 3, 4, 7, 17, 24c, 31c

Badness: 0.013028

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal dicot, with wedgie ⟨⟨2 1 3 -3 -1 4]] adds 36/35, sharp with wedgie ⟨⟨2 1 6 -3 4 11]] adds 28/27, and dichotic with wedgie ⟨⟨2 1 -4 -3 -12 -12]] ads 64/63, all retaining the same period and generator. Decimal with wedgie ⟨⟨4 2 2 -6 -8 -1]] adds 49/48, sidi with wedgie ⟨⟨4 2 9 -3 6 15]] adds 245/243, and jamesbond with wedgie ⟨⟨0 0 7 0 11 16]] adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.

Septimal dicot

Subgroup: 2.3.5.7

Comma list: 15/14, 25/24

Mapping: [⟨1 1 2 2], ⟨0 2 1 3]]

Wedgie: ⟨⟨2 1 3 -3 -1 4]]

POTE generator: ~5/4 = 336.381

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness: 0.019935

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 25/24

Mapping: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

POTE generator: ~5/4 = 342.125

Optimal ET sequence: 3de, 4e, 7

Badness: 0.019854

Eudicot

Subgroup: 2.3.5.7.11

Comma list: 15/14, 25/24, 33/32

Mapping: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

POTE generator: ~5/4 = 336.051

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness: 0.027114

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 15/14, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 2 4 4], ⟨0 2 1 3 -2 -1]]

POTE generator: ~5/4 = 338.846

Optimal ET sequence: 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef

Badness: 0.023828

Flat

Subgroup: 2.3.5.7

Comma list: 21/20, 25/24

Mapping: [⟨1 1 2 3], ⟨0 2 1 -1]]

Wedgie: ⟨⟨ 2 1 -1 -3 -7 -5 ]]

POTE generator: ~5/4 = 331.916

Optimal ET sequence: 3, 4, 7d, 11cd, 18bcddd

Badness: 0.025381

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

POTE generator: ~5/4 = 337.532

Optimal ET sequence: 3, 4, 7d

Badness: 0.024988

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 14/13, 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4 4], ⟨0 2 1 -1 -2 -1]]

POTE generator: ~5/4 = 341.023

Optimal ET sequence: 3, 4, 7d

Badness: 0.023420

Sharp

Subgroup: 2.3.5.7

Comma list: 25/24, 28/27

Mapping: [⟨1 1 2 1], ⟨0 2 1 6]]

POTE generator: ~5/4 = 357.938

Wedgie: ⟨⟨ 2 1 6 -3 4 11 ]]

Optimal ET sequence: 3d, 7d, 10, 37cd, 47bccd, 57bccdd

Badness: 0.028942

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 28/27, 35/33

Mapping: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

POTE generator: ~5/4 = 356.106

Optimal ET sequence: 3de, 7d, 10, 17d, 27cde

Badness: 0.022366

Decimal

Subgroup: 2.3.5.7

Comma list: 25/24, 49/48

Mapping: [⟨2 0 3 4], ⟨0 2 1 1]]

Wedgie: ⟨⟨ 4 2 2 -6 -8 -1 ]]

POTE generator: ~7/6 = 251.557

Optimal ET sequence: 4, 10, 14c, 24c, 38ccd, 62cccdd

Badness: 0.028334

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 49/48

Mapping: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

POTE generator: ~7/6 = 253.493

Optimal ET sequence: 10, 14c, 24c, 38ccd, 52cccde

Badness: 0.026712

Decimated

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 49/48

Mapping: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

POTE generator: ~7/6 = 255.066

Optimal ET sequence: 4, 10e, 14c

Badness: 0.031456

Decibel

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 49/48

Mapping: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

POTE generator: ~8/7 = 243.493

Optimal ET sequence: 4, 6, 10

Badness: 0.032385

Dichotic

Subgroup: 2.3.5.7

Comma list: 25/24, 64/63

Mapping: [⟨1 1 2 4], ⟨0 2 1 -4]]

Wedgie: ⟨⟨ 2 1 -4 -3 -12 -12 ]]

POTE generator: ~5/4 = 356.264

Optimal ET sequence: 3, 7, 10, 17, 27c, 37c, 64bccc

Badness: 0.037565

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 64/63

Mapping: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

POTE generator: ~5/4 = 354.262

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness: 0.030680

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 40/39, 45/44, 64/63

Mapping: [⟨1 1 2 4 2 4], ⟨0 2 1 -4 5 -1]]

POTE generator: ~5/4 = 354.365

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness: 0.021674

Dichotomic

Subgroup: 2.3.5.7.11

Comma list: 22/21, 25/24, 33/32

Mapping: [⟨1 1 2 4 4], ⟨0 2 1 -4 -2]]

POTE generator: ~5/4 = 354.073

Optimal ET sequence: 3, 7, 10e, 17e

Badness: 0.031719

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 4 4 4], ⟨0 2 1 -4 -2 -1]]

POTE generator: ~5/4 = 354.313

Optimal ET sequence: 3, 7, 10e, 17e

Badness: 0.022741

Dichosis

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 64/63

Mapping: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

POTE generator: ~5/4 = 360.659

Optimal ET sequence: 3, 7e, 10

Badness: 0.041361

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 35/33, 40/39, 64/63

Mapping: [⟨1 1 2 4 5 4], ⟨0 2 1 -4 -5 -1]]

POTE generator: ~5/4 = 360.646

Optimal ET sequence: 3, 7e, 10

Badness: 0.027938

Jamesbond

Subgroup: 2.3.5.7

Comma list: 25/24, 81/80

Mapping: [⟨7 11 16 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨ 0 0 7 0 11 16 ]]

POTE generator: ~8/7 = 258.139

Optimal ET sequence: 7, 14c

Badness: 0.041714

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 45/44

Mapping: [⟨7 11 16 0 24], ⟨0 0 0 1 0]]

POTE generator: ~8/7 = 258.910

Optimal ET sequence: 7, 14c

Badness: 0.023524

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 27/26, 33/32, 40/39

Mapping: [⟨7 11 16 0 24 26], ⟨0 0 0 1 0 0]]

POTE generator: ~8/7 = 250.764

Optimal ET sequence: 7, 14c

Badness: 0.023003

Septimal

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 33/32, 45/44, 65/63

Mapping: [⟨7 11 16 0 24 6], ⟨0 0 0 1 0 1]]

POTE generator: ~8/7 = 247.445

Optimal ET sequence: 7, 14cf

Badness: 0.022569

Sidi

Subgroup: 2.3.5.7

Comma list: 25/24, 245/243

Mapping: [⟨1 3 3 6], ⟨0 -4 -2 -9]]

Wedgie: ⟨⟨ 4 2 9 -12 3 15 ]]

POTE generator: ~9/7 = 427.208

Optimal ET sequence: 3d, 14c, 45cc, 59bcccd

Badness: 0.056586

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 99/98

Mapping: [⟨1 3 3 6 7], ⟨0 -4 -2 -9 -10]]

POTE generator: ~9/7 = 427.273

Optimal ET sequence: 3de, 14c, 45cce, 59bcccdee

Badness: 0.032957

Quad

Subgroup: 2.3.5.7

Comma list: 9/8, 25/24

Mapping: [⟨4 6 9 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨ 0 0 4 0 6 9 ]]

POTE generator: ~8/7 = 324.482

Optimal ET sequence: 4

Badness: 0.045911

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