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

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

let es = [(2,3),(2,5)
         ,(3,4),(3,5),(3,8)
         ,(4,7),(4,9),(4,12),(4,15)
         ,(5,6),(5,7),(5,8),(5,9),(5,11),(5,12),(5,13),(5,16)
         ,(6,7),(6,13)
         ,(7,8),(7,9),(7,10),(7,12),(7,15),(7,16),(7,17),(7,18)
         ,(8,17),(8,19)
         ,(9,14),(9,16),(9,22),(9,23)
         ,(11,12),(11,24)
         ,(13,24)
         ,(15,34)]
in map (\e -> let e' = bjorklund e in (e,xdot e',iseq_str e')) es

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

xdot :: [Bool] -> String

xdot notation for pattern.

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

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)"