Music.Theory.Xenakis.Sieve

"Sieves" by Iannis Xenakis and John Rahn Perspectives of New Music Vol. 28, No. 1 (Winter, 1990), pp. 58-78

Documentation

type I = Integer

Synonym for Integer

data Sieve

A Sieve.

Constructors

Empty

Empty Sieve

L (I, I)

Primitive Sieve of modulo and index

Union Sieve Sieve

Union of two Sieves

Intersection Sieve Sieve

Intersection of two Sieves

Complement Sieve

Complement of a Sieve

Instances

Eq Sieve

Methods

(==) :: Sieve -> Sieve -> Bool

(/=) :: Sieve -> Sieve -> Bool

Show Sieve

Methods

showsPrec :: Int -> Sieve -> ShowS

show :: Sieve -> String

showList :: [Sieve] -> ShowS

union :: [Sieve] -> Sieve

The Union of a list of Sieves, ie. foldl1 Union.

intersection :: [Sieve] -> Sieve

The Intersection of a list of Sieves, ie. foldl1 Intersection.

(∪) :: Sieve -> Sieve -> Sieve infixl 3

Unicode synonym for Union.

(∩) :: Sieve -> Sieve -> Sieve infixl 4

Unicode synonym for Intersection.

c :: Sieve -> Sieve

Synonym for Complement.

sieve_pp :: Sieve -> String

Pretty-print sieve. Fully parenthesised.

l :: I -> I -> Sieve

Variant of L, ie. curry L.

l 15 19 == L (15,19)

(⋄) :: I -> I -> Sieve infixl 5

unicode synonym for l.

normalise :: Sieve -> Sieve

In a normal Sieve m is > i.

normalise (L (15,19)) == L (15,4)
normalise (L (11,13)) == L (11,2)

is_normal :: Sieve -> Bool

Predicate to test if a Sieve is normal.

is_normal (L (15,4)) == True
is_normal (L (11,13)) == False

element :: Sieve -> I -> Bool

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]

i_complement :: [I] -> [I]

build :: Sieve -> [I]

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]
    ;r = d ++ map (+ 12) d}
in take 14 (build (union (map (l 12) d))) == r

buildn :: Int -> Sieve -> [I]

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 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
    ;s' = 24⋄23 ∪ 30⋄3 ∪ 104⋄70}
in 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]
in 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]
in 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]
in 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)
    ;b = 11⋄2
    ;c' = c (11⋄4 ∪ 11⋄8)
    ;d = 13⋄9
    ;e = 13⋄0 ∪ 13⋄1 ∪ 13⋄6
    ;f = (a ∩ b) ∪ (c' ∩ d) ∪ e}
in 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

let {n = [0,1,2,6,9,13,14,19,22,24,26,27,32]
    ;r = "C C𝄲 C♯ D♯ E𝄲 F𝄰 G A𝄲 B C C♯ C𝄰 E"}
in 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]
in differentiate (buildn 25 (union s))

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]
in differentiate (buildn 24 (union s))

Major scale:

let s = (c(3⋄2) ∩ 4⋄0) ∪ (c(3⋄1) ∩ 4⋄1) ∪ (3⋄2 ∩ 4⋄2) ∪ (c(3⋄0) ∩ 4⋄3)
in 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) ;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]} in buildn 32 s == r

Psappha (Flint):

let {s = union [(8⋄0∪8⋄1∪8⋄7)∩(5⋄1∪5⋄3)
               ,(8⋄0∪8⋄1∪8⋄2)∩5⋄0
               ,8⋄3∩(5⋄0∪5⋄1∪5⋄2∪5⋄3∪5⋄4)
               ,8⋄4∩(5⋄0∪5⋄1∪5⋄2∪5⋄3∪5⋄4)
               ,(8⋄5∪8⋄6)∩(5⋄2∪5⋄3∪5⋄4)
               ,8⋄1∩5⋄2
               ,8⋄6∩5⋄1]
    ;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]}
in buildn 27 s == r

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

let {s = union [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)]
    ;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]}
in buildn 42 s == r

differentiate :: Num a => [a] -> [a]

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]

euclid :: Integral a => a -> a -> a

Euclid's algorithm for computing the greatest common divisor.

euclid 1989 867 == 51

de_meziriac :: Integral a => a -> a -> a

Bachet De Méziriac's algorithm.

de_meziriac 15 4 == 3 && euclid 15 4 == 1

reduce_intersection :: Integral t => (t, t) -> (t, t) -> Maybe (t, t)

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 :: Sieve -> Sieve

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
in 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
in reduce s == (24⋄23 ∪ 30⋄3 ∪ 104⋄70)