Dicot family: Difference between revisions
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The [[5-limit]] parent [[comma]] for the dicot family is 25/24, the [[ | 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== | ==Seven limit children== | ||
Revision as of 07:25, 22 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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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|]
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||
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|]
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|]
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|]
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||
Badness: 0.045911