Sieve.hs

-- | \"Sieves\" by Iannis Xenakis and John Rahn
-- /Perspectives of New Music/
-- Vol. 28, No. 1 (Winter, 1990), pp. 58-78
module Music.Theory.Xenakis.Sieve where

import qualified Data.List as L
import Music.Theory.List

-- | A Sieve.
data Sieve = Empty -- ^ 'Empty' 'Sieve'
           | L (Integer, Integer) -- ^ Primitive 'Sieve' of /modulo/ and /index/
           | Union Sieve Sieve -- ^ 'Union' of two 'Sieve's
           | Intersection Sieve Sieve -- ^ 'Intersection' of two 'Sieve's
           | Complement Sieve -- ^ 'Complement' of a 'Sieve'
             deriving (Eq,Show)

-- | The 'Union' of a list of 'Sieve's, ie. 'foldl1' 'Union'.
union :: [Sieve] -> Sieve
union = foldl1 Union

-- | The 'Intersection' of a list of 'Sieve's, ie. 'foldl1' 'Intersection'.
intersection :: [Sieve] -> Sieve
intersection = foldl1 Intersection

-- | Unicode synonym for 'Union'.
(∪) :: Sieve -> Sieve -> Sieve
(∪) = Union

-- | Unicode synonym for 'Intersection'.
(∩) :: Sieve -> Sieve -> Sieve
(∩) = Intersection

-- | Synonym for 'Complement'.
c :: Sieve -> Sieve
c = Complement

-- | Pretty-print sieve.  Fully parenthesised.
sieve_pp :: Sieve -> String
sieve_pp s =
    case s of
      Empty -> "∅"
      L (p,q) -> concat [show p,".",show q]
      Union p q -> concat ["(",sieve_pp p," ∪ ",sieve_pp q,")"]
      Intersection p q -> concat ["(",sieve_pp p," ∩ ",sieve_pp q,")"]
      Complement p -> concat ["(∁ ",sieve_pp p,")"]

-- | Variant of 'L', ie. 'curry' 'L'.
--
-- > l 15 19 == L (15,19)
l :: Integer -> Integer -> Sieve
l = curry L

-- | unicode synonym for 'l'.
(⋄) :: Integer -> Integer -> Sieve
(⋄) = l

infixl 3 ∪
infixl 4 ∩
infixl 5 ⋄

{- | In a /normal/ 'Sieve' /m/ is '>' /i/.

> normalise (L (15,19)) == L (15,4)
> normalise (L (11,13)) == L (11,2)
-}
normalise :: Sieve -> Sieve
normalise s =
    case s of
      Empty -> Empty
      L (m,i) -> L (m,i `mod` m)
      Union s0 s1 -> Union (normalise s0) (normalise s1)
      Intersection s0 s1 -> Intersection (normalise s0) (normalise s1)
      Complement s' -> Complement (normalise s')

{- | Predicate to test if a 'Sieve' is /normal/.

> is_normal (L (15,4)) == True
> is_normal (L (11,13)) == False
-}
is_normal :: Sieve -> Bool
is_normal s = s == normalise s

{- | Predicate to determine if an 'I' is an element of the 'Sieve'.

> map (element (L (3,1))) [1..4] == [True,False,False,True]
> map (element (L (15,4))) [4,19 .. 49] == [True,True,True,True]
-}
element :: Sieve -> Integer -> Bool
element s n =
    case s of
      Empty -> False
      L (m,i) -> n `mod` m == i `mod` m && n >= i
      Union s0 s1 -> element s0 n || element s1 n
      Intersection s0 s1 -> element s0 n && element s1 n
      Complement s' -> not (element s' n)

{- | 'I' not in set.

> take 9 (i_complement [1,3..]) == [0,2..16]
-}
i_complement :: [Integer] -> [Integer]
i_complement =
    let f x s = case s of
                [] -> [x ..]
                e:s' -> case compare x e of
                          LT -> x : f (x + 1) s
                          EQ -> f (x + 1) s'
                          GT -> error "i_complement"
    in f 0

{- | Construct the sequence defined by a 'Sieve'.  Note that building
     a sieve that contains an intersection clause that has no elements
     gives @_|_@.

> let d = [0,2,4,5,7,9,11]
> let r = d ++ map (+ 12) d
> take 14 (build (union (map (l 12) d))) == r
-}
build :: Sieve -> [Integer]
build s =
    let u_f = map head . L.group
        i_f = let g [x,_] = [x]
                  g _ = []
              in concatMap g . L.group
    in case s of
         Empty -> []
         L (m,i) -> [i, i+m ..]
         Union s0 s1 -> u_f (merge (build s0) (build s1))
         Intersection s0 s1 -> i_f (merge (build s0) (build s1))
         Complement s' -> i_complement (build s')

{- | Variant of 'build' that gives the first /n/ places of the 'reduce' of 'Sieve'.

> buildn 6 (union (map (l 8) [0,3,6])) == [0,3,6,8,11,14]
> buildn 12 (L (3,2)) == [2,5,8,11,14,17,20,23,26,29,32,35]
> buildn 9 (L (8,0)) == [0,8,16,24,32,40,48,56,64]
> buildn 3 (L (3,2) ∩ L (8,0)) == [8,32,56]
> buildn 12 (L (3,1) ∪ L (4,0)) == [0,1,4,7,8,10,12,13,16,19,20,22]
> buildn 14 (5⋄4 ∪ 3⋄2 ∪ 7⋄3) == [2,3,4,5,8,9,10,11,14,17,19,20,23,24]
> buildn 6 (3⋄0 ∪ 4⋄0) == [0,3,4,6,8,9]
> buildn 8 (5⋄2 ∩ 2⋄0 ∪ 7⋄3) == [2,3,10,12,17,22,24,31]
> buildn 12 (5⋄1 ∪ 7⋄2) == [1,2,6,9,11,16,21,23,26,30,31,36]
> buildn 19 (L (3,2) ∪ L (7, 1)) == [1, 2, 5, 8, 11, 14, 15, 17, 20, 22, 23, 26, 29, 32, 35, 36, 38, 41, 43]
> buildn 19 (3⋄0 ∪ 7⋄0) == [0, 3, 6, 7, 9, 12, 14, 15, 18, 21, 24, 27, 28, 30, 33, 35, 36, 39, 42]

> buildn 10 (3⋄2 ∩ 4⋄7 ∪ 6⋄9 ∩ 15⋄18) == [3,11,23,33,35,47,59,63,71,83]

> let s = 3⋄2∩4⋄7∩6⋄11∩8⋄7 ∪ 6⋄9∩15⋄18 ∪ 13⋄5∩8⋄6∩4⋄2 ∪ 6⋄9∩15⋄19
> let s' = 24⋄23 ∪ 30⋄3 ∪ 104⋄70
> buildn 16 s == buildn 16 s'

> buildn 10 (24⋄23 ∪ 30⋄3 ∪ 104⋄70) == [3,23,33,47,63,70,71,93,95,119]

> let r = [2,3,4,5,8,9,10,11,14,17,19,20,23,24,26,29,31]
> buildn 17 (5⋄4 ∪ 3⋄2 ∪ 7⋄3) == r

> let r = [0,1,3,6,9,10,11,12,15,16,17,18,21,24,26,27,30]
> buildn 17 (5⋄1 ∪ 3⋄0 ∪ 7⋄3) == r

> let r = [0,2,3,4,6,7,9,11,12,15,17,18,21,22,24,25,27,30,32]
> buildn 19 (5⋄2 ∪ 3⋄0 ∪ 7⋄4) == r

Agon et. al. p.155

> let a = c (13⋄3 ∪ 13⋄5 ∪ 13⋄7 ∪ 13⋄9)
> let b = 11⋄2
> let c' = c (11⋄4 ∪ 11⋄8)
> let d = 13⋄9
> let e = 13⋄0 ∪ 13⋄1 ∪ 13⋄6
> let f = (a ∩ b) ∪ (c' ∩ d) ∪ e
> buildn 13 f == [0,1,2,6,9,13,14,19,22,24,26,27,32]

> differentiate [0,1,2,6,9,13,14,19,22,24,26,27,32] == [1,1,4,3,4,1,5,3,2,2,1,5]

> import Music.Theory.Pitch {- hmt -}

> let n = [0,1,2,6,9,13,14,19,22,24,26,27,32]
> let r = "C C𝄲 C♯ D♯ E𝄲 F𝄰 G A𝄲 B C C♯ C𝄰 E"
> unwords (map (pitch_class_pp . pc24et_to_pitch . (`mod` 24)) n) == r

Jonchaies

> let s = map (17⋄) [0,1,4,5,7,11,12,16]
> let r = [1,3,1,2,4,1,4,1,1,3,1,2,4,1,4,1,1,3,1,2,4,1,4,1]
> differentiate (buildn 25 (union s)) == r
> let a2 = octpc_to_midi (2,9)
> let m = scanl (+) a2 r
> import Music.Theory.Pitch.Spelling.Table {- hmt -}
> let p = "A2 A#2 C#3 D3 E3 G#3 A3 C#4 D4 D#4 F#4 G4 A4 C#5 D5 F#5 G5 G#5 B5 C6 D6 F#6 G6 B6 C7"
> unwords (map (pitch_pp_iso . midi_to_pitch pc_spell_sharp) m) == p

Nekuïa

> let s = [24⋄0,14⋄2,22⋄3,31⋄4,28⋄7,29⋄9,19⋄10,25⋄13,24⋄14,26⋄17,23⋄21,24⋄10,30⋄9,35⋄17,29⋄24,32⋄25,30⋄29,26⋄21,30⋄17,31⋄16]
> let r = [2,1,1,3,2,1,3,1,2,1,4,3,1,4,1,4,1,3,1,4,1,3,1,4,1,4,1,1,3,1,3,1,2,3,1,4,1,4,4,1]
> differentiate (buildn 41 (union s)) == r
> let a0 = octpc_to_midi (0,9)
> let m = scanl (+) a0 r
> import Music.Theory.Pitch.Spelling.Table {- hmt -}
> let p = "A0 B0 C1 C#1 E1 F#1 G1 A#1 B1 C#2 D2 F#2 A2 A#2 D3 D#3 G3 G#3 B3 C4 E4 F4 G#4 A4 C#5 D5 F#5 G5 G#5 B5 C6 D#6 E6 F#6 A6 A#6 D7 D#7 G7 B7 C8"
> unwords (map (pitch_pp_iso . midi_to_pitch pc_spell_sharp) m) == p

> let s = [8⋄0∩3⋄0,2⋄0∩7⋄2,2⋄1∩11⋄3,31⋄4,4⋄3∩7⋄0,29⋄9,19⋄10,25⋄13,8⋄6∩3⋄2,2⋄1∩13⋄4,23⋄21,8⋄2∩3⋄1,2⋄1∩3⋄0∩5⋄4,5⋄2∩7⋄3,29⋄24,32⋄25,2⋄1∩3⋄2∩5⋄4,2⋄1∩13⋄8,2⋄1∩3⋄2∩5⋄2,31⋄16]
> differentiate (buildn 41 (union s)) == r

Major scale:

> let s = (c(3⋄2) ∩ 4⋄0) ∪ (c(3⋄1) ∩ 4⋄1) ∪ (3⋄2 ∩ 4⋄2) ∪ (c(3⋄0) ∩ 4⋄3)
> buildn 7 s == [0,2,4,5,7,9,11]

Nomos Alpha:

let s = (c (13⋄3 ∪ 13⋄5 ∪ 13⋄7 ∪ 13⋄9) ∩ 11⋄2) ∪ (c (11⋄4 ∪ 11⋄8) ∩ 13⋄9) ∪ (13⋄0 ∪ 13⋄1 ∪ 13⋄6)
let r = [0,1,2,6,9,13,14,19,22,24,26,27,32,35,39,40,45,52,53,58,61,65,66,71,78,79,84,87,90,91,92,97]
buildn 32 s == r

-}
buildn :: Int -> Sieve -> [Integer]
buildn n = take n . build . reduce

{- | Standard differentiation function.

> differentiate [1,3,6,10] == [2,3,4]
> differentiate [0,2,4,5,7,9,11,12] == [2,2,1,2,2,2,1]
-}
differentiate :: (Num a) => [a] -> [a]
differentiate x = zipWith (-) (tail x) x

{- | Euclid's algorithm for computing the greatest common divisor.

> euclid 1989 867 == 51
-}
euclid :: (Integral a) => a -> a -> a
euclid i j =
    let k = i `mod` j
    in if k == 0 then j else euclid j k

{- | Bachet De Méziriac's algorithm.

> de_meziriac 15 4 == 3 && euclid 15 4 == 1
-}
de_meziriac :: (Integral a) => a -> a -> a
de_meziriac i j =
    let f t = if (t * i) `mod` j /= 1
              then f (t + 1)
              else t
    in if j == 1 then 1 else f 1

{- | Attempt to reduce the 'Intersection' of two 'L' nodes to a singular 'L' node.

> reduce_intersection (3,2) (4,7) == Just (12,11)
> reduce_intersection (12,11) (6,11) == Just (12,11)
> reduce_intersection (12,11) (8,7) == Just (24,23)
-}
reduce_intersection :: (Integral t) => (t,t) -> (t,t) -> Maybe (t,t)
reduce_intersection (m1,i1) (m2,i2) =
    let d = euclid m1 m2
        i1' = i1 `mod` m1
        i2' = i2 `mod` m2
        c1 = m1 `div` d
        c2 = m2 `div` d
        m3 = d * c1 * c2
        t = de_meziriac c1 c2
        i3 = (i1' + t * (i2' - i1') * c1) `mod` m3
    in if d /= 1 && (i1' - i2') `mod` d /= 0
       then Nothing
       else Just (m3,i3)

{- | Reduce the number of nodes at a 'Sieve'.

> reduce (L (3,2) ∪ Empty) == L (3,2)
> reduce (L (3,2) ∩ Empty) == L (3,2)
> reduce (L (3,2) ∩ L (4,7)) == L (12,11)
> reduce (L (6,9) ∩ L (15,18)) == L (30,3)

> let s = 3⋄2∩4⋄7∩6⋄11∩8⋄7 ∪ 6⋄9∩15⋄18 ∪ 13⋄5∩8⋄6∩4⋄2 ∪ 6⋄9∩15⋄19
> reduce s == (24⋄23 ∪ 30⋄3 ∪ 104⋄70)

> putStrLn $ sieve_pp (reduce s)

> let s = 3⋄2∩4⋄7∩6⋄11∩8⋄7 ∪ 6⋄9∩15⋄18 ∪ 13⋄5∩8⋄6∩4⋄2 ∪ 6⋄9∩15⋄19
> reduce s == (24⋄23 ∪ 30⋄3 ∪ 104⋄70)
-}
reduce :: Sieve -> Sieve
reduce s =
    let f g s1 s2 =
            let s1' = reduce s1
                s2' = reduce s2
                s' = g s1' s2'
            in if s1 == s1' && s2 == s2'
               then s'
               else reduce s'
    in case s of
         Empty -> Empty
         L _ -> s
         Union s1 Empty -> s1
         Union s1 s2 -> f Union s1 s2
         Intersection s1 Empty -> s1
         Intersection (L p) (L q) -> maybe Empty L (reduce_intersection p q)
         Intersection s1 s2 -> f Intersection s1 s2
         Complement s' -> Complement (reduce s')

-- * Literature

psappha_flint_c :: [Sieve]
psappha_flint_c =
  let s0 = (8⋄0∪8⋄1∪8⋄7)∩(5⋄1∪5⋄3)
      s1 = (8⋄0∪8⋄1∪8⋄2)∩5⋄0
      s2 = 8⋄3∩(5⋄0∪5⋄1∪5⋄2∪5⋄3∪5⋄4)
      s3 = 8⋄4∩(5⋄0∪5⋄1∪5⋄2∪5⋄3∪5⋄4)
      s4 = (8⋄5∪8⋄6)∩(5⋄2∪5⋄3∪5⋄4)
      s5 = 8⋄1∩5⋄2
      s6 = 8⋄6∩5⋄1
  in [s0, s1, s2, s3, s4, s5, s6]

{- | /Psappha/ (Flint)

> let r = [0,1,3,4,6,8,10,11,12,13,14,16,17,19,20,22,23,25,27,28,29,31,33,35,36,37,38]
> buildn 27 psappha_flint == r
-}
psappha_flint :: Sieve
psappha_flint = union psappha_flint_c

a_r_squibbs_c :: [Sieve]
a_r_squibbs_c =
  [8⋄0∩(11⋄0∪11⋄4∪11⋄5∪11⋄6∪11⋄10)
  ,8⋄1∩(11⋄2∪11⋄3∪11⋄6∪11⋄7∪11⋄9)
  ,8⋄2∩(11⋄0∪11⋄1∪11⋄2∪11⋄3∪11⋄5∪11⋄10)
  ,8⋄3∩(11⋄1∪11⋄2∪11⋄3∪11⋄4∪11⋄10)
  ,8⋄4∩(11⋄0∪11⋄4∪11⋄8)
  ,8⋄5∩(11⋄0∪11⋄2∪11⋄3∪11⋄7∪11⋄9∪11⋄10)
  ,8⋄6∩(11⋄1∪11⋄3∪11⋄5∪11⋄7∪11⋄8∪11⋄9)
  ,8⋄7∩(11⋄1∪11⋄3∪11⋄6∪11⋄7∪11⋄8∪11⋄10)]

{- | À R. (Hommage à Maurice Ravel) (Squibbs, 1996)

let r = [0,2,3,4,7,9,10,13,14,16,17,21,23,25,29,30,32,34,35,38,39,43,44,47,48,52,53,57,58,59,62,63,66,67,69,72,73,77,78,82,86,87]
buildn 42 a_r_squibbs == r
-}
a_r_squibbs :: Sieve
a_r_squibbs = union a_r_squibbs_c