Morris_1974.hs

-- | Robert Morris and D. Starr. \"The Structure of All-Interval Series\".
-- /Journal of Music Theory/, 18:364-389, 1974.
module Music.Theory.Z.Morris_1974 where

import qualified Control.Monad.Logic as L {- logict -}

-- | 'L.msum' '.' 'map' 'return'.
--
-- > L.observeAll (fromList [1..7]) == [1..7]
fromList :: L.MonadPlus m => [a] -> m a
fromList = L.msum . map return

-- | Interval from /i/ to /j/ in modulo-/n/.
--
-- > let f = int_n 12 in (f 0 11,f 11 0) == (11,1)
int_n :: Integral a => a -> a -> a -> a
int_n n i j = abs ((j - i) `mod` n)

-- | 'L.MonadLogic' all-interval series.
--
-- > map (length . L.observeAll . all_interval_m) [4,6,8,10] == [2,4,24,288]
-- > [0,1,3,2,9,5,10,4,7,11,8,6] `elem` L.observeAll (all_interval_m 12)
-- > length (L.observeAll (all_interval_m 12)) == 3856
all_interval_m :: L.MonadLogic m => Int -> m [Int]
all_interval_m n =
    let recur k p q = -- k = length p, p = pitch-class sequence, q = interval set
            if k == n
            then return (reverse p)
            else do i <- fromList [1 .. n - 1]
                    L.guard (i `notElem` p)
                    let j:_ = p
                        m = int_n n i j
                    L.guard (m `notElem` q)
                    recur (k + 1) (i : p) (m : q)
    in recur 1 [0] []

{- | 'L.observeAll' of 'all_interval_m'.

> let r = [[0,1,5,2,4,3],[0,2,1,4,5,3],[0,4,5,2,1,3],[0,5,1,4,2,3]]
> all_interval 6 == r

> d_dx_n n l = zipWith (int_n n) l (tail l)
> map (d_dx_n 6) r == [[1,4,3,2,5],[2,5,3,1,4],[4,1,3,5,2],[5,2,3,4,1]]

-}
all_interval :: Int -> [[Int]]
all_interval = L.observeAll . all_interval_m