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

data Sieve

A Sieve.

Constructors

Empty

Empty Sieve

L (Integer, Integer)

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

Show Sieve

Defined in Music.Theory.Xenakis.Sieve

Methods

showsPrec :: Int -> Sieve -> ShowS

show :: Sieve -> String

showList :: [Sieve] -> ShowS

Eq Sieve

Defined in Music.Theory.Xenakis.Sieve

Methods

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

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

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 :: Integer -> Integer -> Sieve

Variant of L, ie. curry L.

l 15 19 == L (15,19)

(⋄) :: Integer -> Integer -> 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 -> Integer -> 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 :: [Integer] -> [Integer]

I not in set.

take 9 (i_complement [1,3..]) == [0,2..16]

build :: Sieve -> [Integer]

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

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

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 

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 
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 
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

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
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)

Literature

psappha_flint_c :: [Sieve]

psappha_flint :: Sieve

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

a_r_squibbs_c :: [Sieve]

a_r_squibbs :: Sieve

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