Music.Theory.Bjorklund

Godfried T. Toussaint et. al. "The distance geometry of music" Journal of Computational Geometry: Theory and Applications Volume 42, Issue 5, July, 2009 (http://dx.doi.org/10.1016/j.comgeo.2008.04.005)

Documentation

type STEP a = ((Int, Int), ([[a]], [[a]]))

left :: STEP a -> STEP a

right :: STEP a -> STEP a

bjorklund' :: STEP a -> STEP a

bjorklund :: (Int, Int) -> [Bool]

Bjorklund's algorithm to construct a binary sequence of n bits with k ones such that the k ones are distributed as evenly as possible among the (n - k) zeroes.

bjorklund (5,9) == [True,False,True,False,True,False,True,False,True]
map xdot (bjorklund (5,9)) == "x.x.x.x.x"

let {es = [(2,[3,5]),(3,[4,5,8]),(4,[7,9,12,15]),(5,[6,7,8,9,11,12,13,16])
          ,(6,[7,13]),(7,[8,9,10,12,15,16,17,18]),(8,[17,19])
          ,(9,[14,16,22,23]),(11,[12,24]),(13,[24]),(15,[34])]
    ;es' = concatMap (\(i,j) -> map ((,) i) j) es}
in mapM_ (putStrLn . euler_pp') es'

> E(2,3) [××·] (12)
> E(2,5) [×·×··] (23)
> E(3,4) [×××·] (112)
> E(3,5) [×·×·×] (221)
> E(3,8) [×··×··×·] (332)
> E(4,7) [×·×·×·×] (2221)
> E(4,9) [×·×·×·×··] (2223)
> E(4,12) [×··×··×··×··] (3333)
> E(4,15) [×···×···×···×··] (4443)
> E(5,6) [×××××·] (11112)
> E(5,7) [×·××·××] (21211)
> E(5,8) [×·××·××·] (21212)
> E(5,9) [×·×·×·×·×] (22221)
> E(5,11) [×·×·×·×·×··] (22223)
> E(5,12) [×··×·×··×·×·] (32322)
> E(5,13) [×··×·×··×·×··] (32323)
> E(5,16) [×··×··×··×··×···] (33334)
> E(6,7) [××××××·] (111112)
> E(6,13) [×·×·×·×·×·×··] (222223)
> E(7,8) [×××××××·] (1111112)
> E(7,9) [×·×××·×××] (2112111)
> E(7,10) [×·××·××·××] (2121211)
> E(7,12) [×·××·×·××·×·] (2122122)
> E(7,15) [×·×·×·×·×·×·×··] (2222223)
> E(7,16) [×··×·×·×··×·×·×·] (3223222)
> E(7,17) [×··×·×··×·×··×·×·] (3232322)
> E(7,18) [×··×·×··×·×··×·×··] (3232323)
> E(8,17) [×·×·×·×·×·×·×·×··] (22222223)
> E(8,19) [×··×·×·×··×·×·×··×·] (32232232)
> E(9,14) [×·××·××·××·××·] (212121212)
> E(9,16) [×·××·×·×·××·×·×·] (212221222)
> E(9,22) [×··×·×··×·×··×·×··×·×·] (323232322)
> E(9,23) [×··×·×··×·×··×·×··×·×··] (323232323)
> E(11,12) [×××××××××××·] (11111111112)
> E(11,24) [×··×·×·×·×·×··×·×·×·×·×·] (32222322222)
> E(13,24) [×·××·×·×·×·×·××·×·×·×·×·] (2122222122222)
> E(15,34) [×··×·×·×·×··×·×·×·×··×·×·×·×··×·×·] (322232223222322)

bjorklund_r :: Int -> (Int, Int) -> [Bool]

rotate_right of bjorklund.

map xdot' (bjorklund_r 2 (5,16)) == "··×··×··×··×··×·"

euler_pp_f :: (Bool -> Char) -> (Int, Int) -> String

Pretty printer, generalise.

euler_pp' :: (Int, Int) -> String

Unicode form, ie. ×·.

euler_pp' (7,12) == "E(7,12) [×·××·×·××·×·] (2122122)"

euler_pp :: (Int, Int) -> String

ASCII form, ie. x..

euler_pp (7,12) == "E(7,12) [x.xx.x.xx.x.] (2122122)"

xdot :: Bool -> Char

xdot notation for pattern.

map xdot (bjorklund (5,9)) == "x.x.x.x.x"

xdot' :: Bool -> Char

Unicode variant.

map xdot' (bjorklund (5,12)) == "×··×·×··×·×·"
map xdot' (bjorklund (5,16)) == "×··×··×··×··×···"

iseq :: [Bool] -> [Int]

The iseq of a pattern is the distance between True values.

iseq (bjorklund (5,9)) == [2,2,2,2,1]

iseq_str :: [Bool] -> String

iseq of pattern as compact string.

iseq_str (bjorklund (5,9)) == "(22221)"