Music.Theory.Tiling.Canon

Contents

Synopsis

# Documentation

type S = [Int]Source

Sequence.

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

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

type V = [Int]Source

Voice.

type T = [[Int]]Source

Tiling (sequence)

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

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)Source

Element of (sequence,multiplier,displacement).

e_to_seq :: E -> [Int]Source

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] -> ESource

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]Source

Set of `V` from `R`.

rr_voices :: [R] -> [V]Source

`concatMap` of `r_voices`.

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]]
```

t_normal :: T -> TSource

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]Source

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 aSource

`msum` `.` `map` `return`.

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

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

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]Source

`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] -> BoolSource

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
```

A `.*` diagram of n places of `V`.

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

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
```

Insert `|` every n places.

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

v_dot_star_m :: Int -> Int -> V -> StringSource

Variant with measure length m and number of measures n.

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

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

Print `.*` diagram.

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

Variant to print `|` at measures.

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

Variant that discards first k measures.