Music.Theory.Tiling.Canon

Documentation

type S = [Int]

Sequence.

type R = (Int, S, [Int], [Int])

Canon of (period,sequence,multipliers,displacements).

type V = [Int]

Voice.

type T = [[Int]]

Tiling (sequence)

p_cycle :: Int -> [Int] -> [Int]

Cycle at period.

 take 9 (p_cycle 18 [0,2,5]) == [0,2,5,18,20,23,36,38,41]

type E = (S, Int, Int)

Element of (sequence,multiplier,displacement).

e_to_seq :: E -> [Int]

Resolve sequence from E.

 e_to_seq ([0,2,5],2,1) == [1,5,11]
 e_to_seq ([0,1],3,4) == [4,7]
 e_to_seq ([0],1,2) == [2]

e_from_seq :: [Int] -> E

Infer E from sequence.

 e_from_seq [1,5,11] == ([0,2,5],2,1)
 e_from_seq [4,7] == ([0,1],3,4)
 e_from_seq [2] == ([0],1,2)

r_voices :: R -> [V]

Set of V from R.

rr_voices :: [R] -> [V]

concatMap of r_voices.

t_retrograde :: T -> T

Retrograde of T, the result T is sorted.

 let r = [[0,7,14],[1,5,9],[2,4,6],[3,8,13],[10,11,12]]
 in t_retrograde [[0,7,14],[1,6,11],[2,3,4],[5,9,13],[8,10,12]] == r

t_normal :: T -> T

The normal form of T is the min of t and it's t_retrograde.

 let r = [[0,7,14],[1,5,9],[2,4,6],[3,8,13],[10,11,12]]
 in t_normal [[0,7,14],[1,6,11],[2,3,4],[5,9,13],[8,10,12]] == r

r_from_t :: T -> [R]

Derive set of R from T.

 let {r = [(21,[0,1,2],[10,8,2,4,7,5,1],[0,1,2,3,5,8,14])]
     ;t = [[0,10,20],[1,9,17],[2,4,6],[3,7,11],[5,12,19],[8,13,18],[14,15,16]]}
 in r_from_t t == r

Construction

fromList :: MonadPlus m => [a] -> m a

msum . map return.

 observeAll (fromList [1..7]) == [1..7]

perfect_tilings_m :: MonadPlus m => [S] -> [Int] -> Int -> Int -> m T

Search for perfect tilings of the sequence S using multipliers from m to degree n with k parts.

perfect_tilings :: [S] -> [Int] -> Int -> Int -> [T]

t_normal of observeAll of perfect_tilings_m.

 perfect_tilings [[0,1]] [1..3] 6 3 == []

 let r = [[[0,7,14],[1,5,9],[2,4,6],[3,8,13],[10,11,12]]]
 in perfect_tilings [[0,1,2]] [1,2,4,5,7] 15 5 == r

 length (perfect_tilings [[0,1,2]] [1..12] 15 5) == 1

 let r = [[[0,1],[2,5],[3,7],[4,6]]
         ,[[0,1],[2,6],[3,5],[4,7]]
         ,[[0,2],[1,4],[3,7],[5,6]]]
 in perfect_tilings [[0,1]] [1..4] 8 4 == r

 let r = [[[0,1],[2,5],[3,7],[4,9],[6,8]]
         ,[[0,1],[2,7],[3,5],[4,8],[6,9]]
         ,[[0,2],[1,4],[3,8],[5,9],[6,7]]
         ,[[0,2],[1,5],[3,6],[4,9],[7,8]]
         ,[[0,3],[1,6],[2,4],[5,9],[7,8]]]
 in perfect_tilings [[0,1]] [1..5] 10 5 == r

Johnson 2004, p.2

 let r = [[0,6,12],[1,8,15],[2,11,20],[3,5,7],[4,9,14],[10,13,16],[17,18,19]]
 in perfect_tilings [[0,1,2]] [1,2,3,5,6,7,9] 21 7 == [r]

 let r = [[0,10,20],[1,9,17],[2,4,6],[3,7,11],[5,12,19],[8,13,18],[14,15,16]]
 in perfect_tilings [[0,1,2]] [1,2,4,5,7,8,10] 21 7 == [t_retrograde r]

Display

elemOrd :: Ord a => a -> [a] -> Bool

Variant of elem for ordered sequences, which can therefore return False when searching infinite sequences.

 5 `elemOrd` [0,2..] == False && 10 `elemOrd` [0,2..] == True

v_dot_star :: Int -> V -> String

A .* diagram of n places of V.

 v_dot_star 18 [0,2..] == "*.*.*.*.*.*.*.*.*."

v_space_ix :: Int -> V -> String

A white space and index diagram of n places of V.

>>> mapM_ (putStrLn . v_space_ix 9) [[0,2..],[1,3..]]
>
>  0   2   4   6   8
>    1   3   5   7

with_bars :: Int -> String -> String

Insert | every n places.

 with_bars 6 (v_dot_star 18 [0,2..]) == "*.*.*.|*.*.*.|*.*.*."

v_dot_star_m :: Int -> Int -> V -> String

Variant with measure length m and number of measures n.

 v_dot_star_m 6 3 [0,2..] == "*.*.*.|*.*.*.|*.*.*."

v_print :: Int -> [V] -> IO ()

Print .* diagram.

v_print_m :: Int -> Int -> [V] -> IO ()

Variant to print | at measures.

v_print_m_from :: Int -> Int -> Int -> [V] -> IO ()

Variant that discards first k measures.