Dicot family: Difference between revisions

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The [[5-limit]] parent [[comma]] for the dicot family is 25/24, the [[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]].
The [[5-limit]] parent [[comma]] for the dicot family is 25/24, the [[chromatic semitone]]. Its [[monzo]] is {{monzo| -3 -1 2 }}, and flipping that yields {{wedgie| 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 {{val|24 38 55}} (24c) and [[31edo]] using the val {{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]].


==Seven limit children==
==Seven limit children==
The second comma of the [[Normal_lists|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.
The second comma of the [[Normal_lists|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot, with wedgie {{wedgie|2 1 3 -3 -1 4}} adds 36/35, sharp with wedgie {{wedgie|2 1 6 -3 4 11}} adds 28/27, and dichotic with wedgie {{wedgie|2 1 -4 -3 -12 -12}} ads 64/63, all retaining the same period and generator. Decimal with wedgie {{wedgie|4 2 2 -6 -8 -1}} adds 49/48, sidi with wedgie {{wedgie|4 2 9 -3 6 15}} adds 245/243, and jamesbond with wedgie {{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.


=Dicot=
=Dicot=
Line 9: Line 9:
[[POTE tuning|POTE generator]]: ~5/4 = 348.594
[[POTE tuning|POTE generator]]: ~5/4 = 348.594


[[Map]]: [<1 1 2|, <0 2 1|]
[[Map]]: [{{val|1 1 2}}, {{val|0 2 1}}]


{{Vals|legend=1| 3, 4, 7, 17, 24c, 31c }}
{{Vals|legend=1| 3, 4, 7, 17, 24c, 31c }}
Line 20: Line 20:
[[POTE tuning|POTE generator]]: ~5/4 = 336.381
[[POTE tuning|POTE generator]]: ~5/4 = 336.381


[[Map]]: [<1 1 2 2|, <0 2 1 3|]
[[Map]]: [{{val|1 1 2 2}}, {{val|0 2 1 3}}]


Wedgie: <<2 1 3 -3 -1 4||
Wedgie: {{wedgie|2 1 3 -3 -1 4}}


{{Vals|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Vals|legend=1| 3d, 4, 7, 18bc, 25bccd }}
Line 33: Line 33:
POTE generator: ~5/4 = 342.125
POTE generator: ~5/4 = 342.125


Map: [<1 1 2 2 2|, <0 2 1 3 5|]
Map: [{{val|1 1 2 2 2}}, {{val|0 2 1 3 5}}]


Vals: {{Vals| 3de, 4e, 7 }}
Vals: {{Vals| 3de, 4e, 7 }}
Line 44: Line 44:
POTE generator: ~5/4 = 336.051
POTE generator: ~5/4 = 336.051


Map: [<1 1 2 2 4|, <0 2 1 3 -2|]
Map: [{{val|1 1 2 2 4}}, {{val|0 2 1 3 -2}}]


Vals: {{Vals| 3d, 4, 7, 18bc, 25bccd }}
Vals: {{Vals| 3d, 4, 7, 18bc, 25bccd }}
Line 55: Line 55:
POTE generator: ~5/4 = 338.846
POTE generator: ~5/4 = 338.846


Map: [<1 1 2 2 4 4|, <0 2 1 3 -2 -1|]
Map: [{{val|1 1 2 2 4 4}}, {{val|0 2 1 3 -2 -1}}]


Vals: {{Vals| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
Vals: {{Vals| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
Line 66: Line 66:
[[POTE tuning|POTE generator]]: ~5/4 = 331.916
[[POTE tuning|POTE generator]]: ~5/4 = 331.916


[[Map]]: [<1 1 2 3|, <0 2 1 -1|]
[[Map]]: [{{val|1 1 2 3}}, {{val|0 2 1 -1}}]


Wedgie: <<2 1 -1 -3 -7 -5||
Wedgie: {{wedgie|2 1 -1 -3 -7 -5}}


{{Vals|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Vals|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
Line 79: Line 79:
POTE generator: ~5/4 = 337.532
POTE generator: ~5/4 = 337.532


Map: [<1 1 2 3 4|, <0 2 1 -1 -2|]
Map: [{{val|1 1 2 3 4}}, {{val|0 2 1 -1 -2}}]


Vals: {{Vals| 3, 4, 7d }}
Vals: {{Vals| 3, 4, 7d }}
Line 90: Line 90:
POTE generator: ~5/4 = 341.023
POTE generator: ~5/4 = 341.023


Map: [<1 1 2 3 4 4|, <0 2 1 -1 -2 -1|]
Map: [{{val|1 1 2 3 4 4}}, {{val|0 2 1 -1 -2 -1}}]


Vals: {{Vals| 3, 4, 7d }}
Vals: {{Vals| 3, 4, 7d }}
Line 101: Line 101:
[[POTE tuning|POTE generator]]: ~5/4 = 357.938
[[POTE tuning|POTE generator]]: ~5/4 = 357.938


[[Map]]: [<1 1 2 1|, <0 2 1 6|]
[[Map]]: [{{val|1 1 2 1}}, {{val|0 2 1 6}}]


Wedgie: <<2 1 6 -3 4 11||
Wedgie: {{wedgie|2 1 6 -3 4 11}}


{{Vals|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
{{Vals|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
Line 114: Line 114:
POTE generator: ~5/4 = 356.106
POTE generator: ~5/4 = 356.106


Map: [<1 1 2 1 2|, <0 2 1 6 5|]
Map: [{{val|1 1 2 1 2}}, {{val|0 2 1 6 5}}]


Vals: {{Vals| 3de, 7d, 10, 17d, 27cde }}
Vals: {{Vals| 3de, 7d, 10, 17d, 27cde }}
Line 125: Line 125:
[[POTE tuning|POTE generator]]: ~7/6 = 251.557
[[POTE tuning|POTE generator]]: ~7/6 = 251.557


[[Map]]: [<2 0 3 4|, <0 2 1 1|]
[[Map]]: [{{val|2 0 3 4}}, {{val|0 2 1 1}}]


Wedgie: <<4 2 2 -6 -8 -1||
Wedgie: {{wedgie|4 2 2 -6 -8 -1}}


{{Vals|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
{{Vals|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
Line 138: Line 138:
POTE generator: ~7/6 = 253.493
POTE generator: ~7/6 = 253.493


Map: [<2 0 3 4 -1|, <0 2 1 1 5|]
Map: [{{val|2 0 3 4 -1}}, {{val|0 2 1 1 5}}]


Vals: {{Vals| 10, 14c, 24c, 38ccd, 52cccde }}
Vals: {{Vals| 10, 14c, 24c, 38ccd, 52cccde }}
Line 149: Line 149:
POTE generator: ~7/6 = 255.066
POTE generator: ~7/6 = 255.066


Map: [<2 0 3 4 10|, <0 2 1 1 -2|]
Map: [{{val|2 0 3 4 10}}, {{val|0 2 1 1 -2}}]


Vals: {{Vals| 4, 10e, 14c }}
Vals: {{Vals| 4, 10e, 14c }}
Line 160: Line 160:
POTE generator: ~8/7 = 243.493
POTE generator: ~8/7 = 243.493


Map: [<2 0 3 4 7|, <0 2 1 1 0|]
Map: [{{val|2 0 3 4 7}}, {{val|0 2 1 1 0}}]


Vals: {{Vals| 4, 6, 10 }}
Vals: {{Vals| 4, 6, 10 }}
Line 171: Line 171:
[[POTE tuning|POTE generator]]: ~5/4 = 356.264
[[POTE tuning|POTE generator]]: ~5/4 = 356.264


[[Map]]: [<1 1 2 4|, <0 2 1 -4|]
[[Map]]: [{{val|1 1 2 4}}, {{val|0 2 1 -4}}]


Wedgie: <<2 1 -4 -3 -12 -12||
Wedgie: {{wedgie|2 1 -4 -3 -12 -12}}


{{Vals|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
{{Vals|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
Line 184: Line 184:
POTE generator: ~5/4 = 354.262
POTE generator: ~5/4 = 354.262


Map: [<1 1 2 4 2|, <0 2 1 -4 5|]
Map: [{{val|1 1 2 4 2}}, {{val|0 2 1 -4 5}}]


Vals: {{Vals| 7, 10, 17, 27ce, 44cce }}
Vals: {{Vals| 7, 10, 17, 27ce, 44cce }}
Line 195: Line 195:
POTE generator: ~5/4 = 360.659
POTE generator: ~5/4 = 360.659


Map: [<1 1 2 4 5|, <0 2 1 -4 -5|]
Map: [{{val|1 1 2 4 5}}, {{val|0 2 1 -4 -5}}]


Vals: {{Vals| 3, 7e, 10 }}
Vals: {{Vals| 3, 7e, 10 }}
Line 206: Line 206:
[[POTE tuning|POTE generator]]: ~8/7 = 258.139
[[POTE tuning|POTE generator]]: ~8/7 = 258.139


[[Map]]: [<7 11 16 0|, <0 0 0 1|]
[[Map]]: [{{val|7 11 16 0}}, {{val|0 0 0 1}}]


Wedgie: <<0 0 7 0 11 16||
Wedgie: {{wedgie|0 0 7 0 11 16}}


{{Vals|legend=1| 7, 14c }}
{{Vals|legend=1| 7, 14c }}
Line 219: Line 219:
POTE generator: ~8/7 = 258.910
POTE generator: ~8/7 = 258.910


Map: [<7 11 16 0 24|, <0 0 0 1 0|]
Map: [{{val|7 11 16 0 24}}, {{val|0 0 0 1 0}}]


Vals: {{Vals| 7, 14c }}
Vals: {{Vals| 7, 14c }}
Line 230: Line 230:
POTE generator: ~8/7 = 250.764
POTE generator: ~8/7 = 250.764


Map: [<7 11 16 0 24 26|, <0 0 0 1 0 0|]
Map: [{{val|7 11 16 0 24 26}}, {{val|0 0 0 1 0 0}}]


Vals: {{Vals| 7, 14c }}
Vals: {{Vals| 7, 14c }}
Line 241: Line 241:
POTE generator: ~8/7 = 247.445
POTE generator: ~8/7 = 247.445


Map: [<7 11 16 0 24 6|, <0 0 0 1 0 1|]
Map: [{{val|7 11 16 0 24 6}}, {{val|0 0 0 1 0 1}}]


Vals: {{Vals| 7, 14cf }}
Vals: {{Vals| 7, 14cf }}
Line 252: Line 252:
[[POTE tuning|POTE generator]]: ~9/7 = 427.208
[[POTE tuning|POTE generator]]: ~9/7 = 427.208


[[Map]]: [<1 3 3 6|, <0 -4 -2 -9|]
[[Map]]: [{{val|1 3 3 6}}, {{val|0 -4 -2 -9}}]


Wedgie: <<4 2 9 -12 3 15||
Wedgie: {{wedgie|4 2 9 -12 3 15}}


{{Vals|legend=1| 3d, 14c, 45cc, 59bcccd }}
{{Vals|legend=1| 3d, 14c, 45cc, 59bcccd }}
Line 265: Line 265:
POTE generator: ~9/7 = 427.273
POTE generator: ~9/7 = 427.273


Map: [<1 3 3 6 7|, <0 -4 -2 -9 -10|]
Map: [{{val|1 3 3 6 7}}, {{val|0 -4 -2 -9 -10}}]


Vals: {{Vals| 3de, 14c, 17, 45cce, 59bcccdee }}
Vals: {{Vals| 3de, 14c, 17, 45cce, 59bcccdee }}
Line 276: Line 276:
[[POTE tuning|POTE generator]]: ~8/7 = 324.482
[[POTE tuning|POTE generator]]: ~8/7 = 324.482


[[Map]]: [<4 6 9 0|, <0 0 0 1|]
[[Map]]: [{{val|4 6 9 0}}, {{val|0 0 0 1}}]


Wedgie: <<0 0 4 0 6 9||
Wedgie: {{wedgie|0 0 4 0 6 9}}


{{Vals|legend=1| 4 }}
{{Vals|legend=1| 4 }}

Revision as of 16:24, 26 April 2021

The 5-limit parent comma for the dicot family is 25/24, the 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.

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.

Dicot

Comma: 25/24

POTE generator: ~5/4 = 348.594

Map: [⟨1 1 2], ⟨0 2 1]]

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

Badness: 0.013028

7-limit

Comma list: 15/14, 25/24

POTE generator: ~5/4 = 336.381

Map: [⟨1 1 2 2], ⟨0 2 1 3]]

Wedgie: ⟨⟨2 1 3 -3 -1 4]]

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

Badness: 0.019935

11-limit

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

POTE generator: ~5/4 = 342.125

Map: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

Vals: 3de, 4e, 7

Badness: 0.019854

Eudicot

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

POTE generator: ~5/4 = 336.051

Map: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

Vals: 3d, 4, 7, 18bc, 25bccd

Badness: 0.027114

13-limit

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

POTE generator: ~5/4 = 338.846

Map: [⟨1 1 2 2 4 4], ⟨0 2 1 3 -2 -1]]

Vals: 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef

Badness: 0.023828

Flat

Comma list: 21/20, 25/24

POTE generator: ~5/4 = 331.916

Map: [⟨1 1 2 3], ⟨0 2 1 -1]]

Wedgie: ⟨⟨2 1 -1 -3 -7 -5]]

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

Badness: 0.025381

11-limit

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

POTE generator: ~5/4 = 337.532

Map: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

Vals: 3, 4, 7d

Badness: 0.024988

13-limit

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

POTE generator: ~5/4 = 341.023

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

Vals: 3, 4, 7d

Badness: 0.023420

Sharp

Comma list: 25/24, 28/27

POTE generator: ~5/4 = 357.938

Map: [⟨1 1 2 1], ⟨0 2 1 6]]

Wedgie: ⟨⟨2 1 6 -3 4 11]]

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

Badness: 0.028942

11-limit

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

POTE generator: ~5/4 = 356.106

Map: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

Vals: 3de, 7d, 10, 17d, 27cde

Badness: 0.022366

Decimal

Comma list: 25/24, 49/48

POTE generator: ~7/6 = 251.557

Map: [⟨2 0 3 4], ⟨0 2 1 1]]

Wedgie: ⟨⟨4 2 2 -6 -8 -1]]

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

Badness: 0.028334

11-limit

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

POTE generator: ~7/6 = 253.493

Map: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

Vals: 10, 14c, 24c, 38ccd, 52cccde

Badness: 0.026712

Decimated

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

POTE generator: ~7/6 = 255.066

Map: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

Vals: 4, 10e, 14c

Badness: 0.031456

Decibel

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

POTE generator: ~8/7 = 243.493

Map: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

Vals: 4, 6, 10

Badness: 0.032385

Dichotic

Comma list: 25/24, 64/63

POTE generator: ~5/4 = 356.264

Map: [⟨1 1 2 4], ⟨0 2 1 -4]]

Wedgie: ⟨⟨2 1 -4 -3 -12 -12]]

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

Badness: 0.037565

11-limit

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

POTE generator: ~5/4 = 354.262

Map: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

Vals: 7, 10, 17, 27ce, 44cce

Badness: 0.030680

Dichosis

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

POTE generator: ~5/4 = 360.659

Map: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

Vals: 3, 7e, 10

Badness: 0.041361

Jamesbond

Comma list: 25/24, 81/80

POTE generator: ~8/7 = 258.139

Map: [⟨7 11 16 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨0 0 7 0 11 16]]

Optimal ET sequence: 7, 14c

Badness: 0.041714

11-limit

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

POTE generator: ~8/7 = 258.910

Map: [⟨7 11 16 0 24], ⟨0 0 0 1 0]]

Vals: 7, 14c

Badness: 0.023524

13-limit

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

POTE generator: ~8/7 = 250.764

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

Vals: 7, 14c

Badness: 0.023003

Septimal

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

POTE generator: ~8/7 = 247.445

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

Vals: 7, 14cf

Badness: 0.022569

Sidi

Comma list: 25/24, 245/243

POTE generator: ~9/7 = 427.208

Map: [⟨1 3 3 6], ⟨0 -4 -2 -9]]

Wedgie: ⟨⟨4 2 9 -12 3 15]]

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

Badness: 0.056586

11-limit

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

POTE generator: ~9/7 = 427.273

Map: [⟨1 3 3 6 7], ⟨0 -4 -2 -9 -10]]

Vals: 3de, 14c, 17, 45cce, 59bcccdee

Badness: 0.032957

Quad

Comma list: 9/8, 25/24

POTE generator: ~8/7 = 324.482

Map: [⟨4 6 9 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨0 0 4 0 6 9]]

Optimal ET sequence: 4

Badness: 0.045911

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