Music.Theory.Xenakis.Sieve

Description

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

Synopsis

# Documentation

type I = IntegerSource

Synonym for `Integer`

data Sieve Source

A Sieve.

Constructors

 Empty `Empty` `Sieve` L (I, I) Primitive `Sieve` of modulo and index Union Sieve Sieve `Union` of two `Sieve`s Intersection Sieve Sieve `Intersection` of two `Sieve`s

Instances

 Eq Sieve Show Sieve

union :: [Sieve] -> SieveSource

The `Union` of a list of `Sieve`s, ie. `foldl1` `Union`.

intersection :: [Sieve] -> SieveSource

The `Intersection` of a list of `Sieve`s, ie. `foldl1` `Intersection`.

(∪) :: Sieve -> Sieve -> SieveSource

Unicode synonym for `Union`.

(∩) :: Sieve -> Sieve -> SieveSource

Unicode synonym for `Intersection`.

l :: I -> I -> SieveSource

Variant of `L`, ie. `curry` `L`.

``` l 15 19 == L (15,19)
```

(⋄) :: I -> I -> SieveSource

unicode synonym for `l`.

In a normal `Sieve` m is `>` i.

``` normalise (L (15,19)) == L (15,4)
```

Predicate to test if a `Sieve` is normal.

``` is_normal (L (15,4)) == True
```

element :: Sieve -> I -> BoolSource

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]
```

build :: Sieve -> [I]Source

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]
in take 7 (build (union (map (l 12) d))) == d
```

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

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
in buildn 16 s == buildn 16 (24⋄23 ∪ 30⋄3 ∪ 104⋄70)
```
``` buildn 10 (24⋄23 ∪ 30⋄3 ∪ 104⋄70) == [3,23,33,47,63,70,71,93,95,119]
```

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

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 -> aSource

Euclid's algorithm for computing the greatest common divisor.

``` euclid 1989 867 == 51
```

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

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)Source

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 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)
```
``` 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)
```