[<<][math][>>][..]
Tue Sep 4 17:51:46 CEST 2012

Diagrams of x/11

   0 1 2 3 4 5 6 7 8 9 T 0 1 2 3 4 5 6 7 8 9 T
 0 . . . . . . . . . . . . . . . . . . . . . .
 1 . . . . . . . . . . X . . . . . . . . . . X
 2 . . . . . X . . . . X . . . . . X . . . . X
 3 . . . X . . . X . . X . . . X . . . X . . X
 4 . . X . . X . . X . X . . X . . X . . X . X
 5 . . X . X . X . X . X . . X . X . X . X . X
 6 . X . X . X . X . X X . X . X . X . X . X X
 7 . X . X X . X X . X X . X . X X . X X . X X
 8 . X X . X X X . X X X . X X . X X X . X X X
 9 . X X X X . X X X X X . X X X X . X X X X X 
10 . X X X X X X X X X X . X X X X X X X X X X
11 X X X X X X X X X X X X X X X X X X X X X X



 4/11 0  4  8  1  5  9  2  6 10  3  7  0 
-7       0  0  1  0  0  1  0  0  1  0  1

 5/11 0  5 10  4  9  3  8  2  7  1  6  0
-6       0  0  1  0  1  0  1  0  1  0  1


Two symmetries, one obvious: the 0-1 duality (top/bottom) and one
"almost" symmetry around 5: mirrors except 0/T, which I can't explain.
It has to do with rounding, i.e. the last column T has all the
"fractional bits".

This might be an artifact of 11 = 12 - 1 with 12 being highly
composite: 2*2*3

A Haskell program for visualization:

-- Output sequences of Sigma/Delta modulator (1st order non-leaky
-- accumulator) approximating constant rational.

bit False = 0    
bit True  = 1

-- Bit sequence
sd q p = f 0 where
  f s = b : f s'' where
    s' = s + q
    b  = bit $ s' >= p
    s'' = s' - b * p
    

-- N periods
sdn n q p = take (fromInteger $ n * p) $ sd q p    

-- Print
ps = concat . (map f) where
  f 0 = "."
  f _ = "X"
  
-- Build a spectrum: wave forms for values from 0/p to p/p  
spec n p = concat $ map (++ "\n") lines where
  lines = map (\q -> ps $ sdn n q p) [0..p]
s n p = putStr $ spec n p



Example: 2 periods of all x/37 waveforms
*Main> s 2 37
..........................................................................
....................................X....................................X
..................X.................X..................X.................X
............X...........X...........X............X...........X...........X
.........X........X........X........X.........X........X........X........X
.......X......X.......X......X......X.......X......X.......X......X......X
......X.....X.....X.....X.....X.....X......X.....X.....X.....X.....X.....X
.....X....X....X.....X....X....X....X.....X....X....X.....X....X....X....X
....X....X...X....X....X...X....X...X....X....X...X....X....X...X....X...X
....X...X...X...X...X...X...X...X...X....X...X...X...X...X...X...X...X...X
...X...X...X..X...X...X..X...X...X..X...X...X...X..X...X...X..X...X...X..X
...X..X...X..X..X...X..X..X...X..X..X...X..X...X..X..X...X..X..X...X..X..X
...X..X..X..X..X..X..X..X..X..X..X..X...X..X..X..X..X..X..X..X..X..X..X..X
..X..X..X..X..X..X.X..X..X..X..X..X.X..X..X..X..X..X..X.X..X..X..X..X..X.X
..X..X.X..X..X.X..X..X.X..X..X.X..X.X..X..X.X..X..X.X..X..X.X..X..X.X..X.X
..X.X..X.X..X.X..X.X..X.X..X.X..X.X.X..X.X..X.X..X.X..X.X..X.X..X.X..X.X.X
..X.X.X..X.X.X..X.X.X..X.X.X..X.X.X.X..X.X.X..X.X.X..X.X.X..X.X.X..X.X.X.X
..X.X.X.X.X..X.X.X.X.X.X..X.X.X.X.X.X..X.X.X.X.X..X.X.X.X.X.X..X.X.X.X.X.X
..X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X..X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X
.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.XX.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.X.XX
.X.X.X.X.X.XX.X.X.X.X.X.XX.X.X.X.X.XX.X.X.X.X.X.XX.X.X.X.X.X.XX.X.X.X.X.XX
.X.X.X.XX.X.X.XX.X.X.XX.X.X.XX.X.X.XX.X.X.X.XX.X.X.XX.X.X.XX.X.X.XX.X.X.XX
.X.X.XX.X.XX.X.XX.X.XX.X.XX.X.XX.X.XX.X.X.XX.X.XX.X.XX.X.XX.X.XX.X.XX.X.XX
.X.XX.X.XX.XX.X.XX.XX.X.XX.XX.X.XX.XX.X.XX.X.XX.XX.X.XX.XX.X.XX.XX.X.XX.XX
.X.XX.XX.XX.XX.XX.X.XX.XX.XX.XX.XX.XX.X.XX.XX.XX.XX.XX.X.XX.XX.XX.XX.XX.XX
.XX.XX.XX.XX.XX.XX.XX.XX.XX.XX.XX.XXX.XX.XX.XX.XX.XX.XX.XX.XX.XX.XX.XX.XXX
.XX.XX.XXX.XX.XX.XXX.XX.XX.XXX.XX.XXX.XX.XX.XXX.XX.XX.XXX.XX.XX.XXX.XX.XXX
.XX.XXX.XXX.XX.XXX.XXX.XX.XXX.XXX.XXX.XX.XXX.XXX.XX.XXX.XXX.XX.XXX.XXX.XXX
.XXX.XXX.XXX.XXX.XXX.XXX.XXX.XXX.XXXX.XXX.XXX.XXX.XXX.XXX.XXX.XXX.XXX.XXXX
.XXX.XXXX.XXX.XXXX.XXXX.XXX.XXXX.XXXX.XXX.XXXX.XXX.XXXX.XXXX.XXX.XXXX.XXXX
.XXXX.XXXX.XXXX.XXXXX.XXXX.XXXX.XXXXX.XXXX.XXXX.XXXX.XXXXX.XXXX.XXXX.XXXXX
.XXXXX.XXXXX.XXXXX.XXXXX.XXXXX.XXXXXX.XXXXX.XXXXX.XXXXX.XXXXX.XXXXX.XXXXXX
.XXXXXX.XXXXXX.XXXXXXX.XXXXXX.XXXXXXX.XXXXXX.XXXXXX.XXXXXXX.XXXXXX.XXXXXXX
.XXXXXXXX.XXXXXXXX.XXXXXXXX.XXXXXXXXX.XXXXXXXX.XXXXXXXX.XXXXXXXX.XXXXXXXXX
.XXXXXXXXXXX.XXXXXXXXXXX.XXXXXXXXXXXX.XXXXXXXXXXX.XXXXXXXXXXX.XXXXXXXXXXXX
.XXXXXXXXXXXXXXXXX.XXXXXXXXXXXXXXXXXX.XXXXXXXXXXXXXXXXX.XXXXXXXXXXXXXXXXXX
.XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX.XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX




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