xensieve

An implementation of the Xenakis Sieve, providing a Sieve from a string expression that filters integer sequences into iterators of integers, Boolean states, or interval widths. Sieves are built from Residuals, defined as a modulus (M) and a shift (S), notated M@S. Sieve string expressions, and Sieve structs, support complementation, intersection, symmetric difference, and union operations on Residuals with operators !, &, ^ and |, respectively.

The Xenakis Sieve is a tool for generating discrete interval patterns. Such patterns have boundless applications in creative domains: the Xenakis Sieve can be used to generate scales or multi-octave pitch sequences, rhythms and polyrhythms, and used to control countless other aspects of pictorial or architectural design.

This new Rust implementation (and Python wrapper) follows the Python implementation in Ariza (2005), with significant performance and interface enhancements: https://direct.mit.edu/comj/article/29/2/40/93957

• Code (Rust): https://github.com/ampullator/xensieve-rs
• Docs (Rust): https://docs.rs/xensieve
• Crate: https://crates.io/crates/xensieve
• Code (Python): https://github.com/ampullator/xensieve-py
• Packages: https://pypi.org/project/xensieve

Strategies for Creating Sieves

First, we can examine the output of Sieves built from a single Residual. As shown above, a Residual is defined as a modulus (M) and a shift (S), notated M@S. In the diagram below, three Residuals are shown: 5@0, 4@2, and 30@10. As can be seen, for every M units, a value is articulated at the shift S. The final example shows an application of the unary inversion operator !30@10.

Complex Sieves combine Residuals with logical operators such as complementation, intersection, symmetric difference, and union. In the example below, Residuals 5@0 and 4@2 are combined by union with the expression 5@0|4@2. Combining many Residuals by union is a practical approach to building sequences. The final example, (5@0|4@2)&!30@10, shows "removing" selected values from these unioned components by intersecting them with an inverted Residual (!30@10)

While all Sieves are, by definition, periodic, combinations of Residuals can result in sequences with great local complexity and inner patterning.

The xensieve.Sieve (Python)

The Sieves shown above can be created with xensieve.Sieve and used to produce iterators of integers, Boolean states, or interval widths. The Sieve constructor accepts arbitrarily complex Sieve expressions.

>>> from xensieve import Sieve

>>> s1 = Sieve("5@0")
>>> s2 = Sieve("30@10")
>>> s3 = Sieve("(5@0|4@2)&!30@10")

The iter_value() method takes a range (defined by start and stop integers) that can be used to "drive" the Sieve. The iterator yields the subset of integers contained within the Sieve.

>>> s1.iter_value(0, 50)
<builtins.IterValue object at 0x7f538abdb9c0>
>>> list(s1.iter_value(0, 50))
[0, 5, 10, 15, 20, 25, 30, 35, 40, 45]
>>> list(s2.iter_value(0, 50))
[10, 40]
>>> list(s3.iter_value(0, 50))
[0, 2, 5, 6, 14, 15, 18, 20, 22, 25, 26, 30, 34, 35, 38, 42, 45, 46]

The xensieve.Sieve features two alternative iterators to permit using Sieves in different contexts. The iter_state() iterator returns, for each provided integer, the resulting Boolean state.

>>> list(s1.iter_state(0, 10))
[True, False, False, False, False, True, False, False, False, False]
>>> list(s3.iter_state(0, 10))
[True, False, True, False, False, True, True, False, False, False]

The iter_interval() iterator returns, for sequential pairs of provided integers that are within the Sieve, the resulting interval.

>>> list(s2.iter_interval(0, 50))
[30]
>>> list(s3.iter_interval(0, 50))
[2, 3, 1, 8, 1, 3, 2, 2, 3, 1, 4, 4, 1, 3, 4, 3, 1]

The xensieve.Sieve instance implements __contains__() such that in can be used to test if arbitrary integers are contained within the Sieve:

>>> 5 in s1
True
>>> 6 in s1
False
>>> 10 in s3
False
>>> 30 in s3
True

The xensieve.Sieve instance supports the same operators permitted in Sieve expressions, such that instances can be combined to build complex Sieves.

>>> s4 = (Sieve("5@0") | Sieve("4@2")) & ~Sieve("30@10")
>>> s4
Sieve{5@0|4@2&!(30@10)}
>>> list(s4.iter_value(0, 100)) == list(s3.iter_value(0, 100))
True

The xensieve::Sieve Interface (Rust)

The Sieves shown above can be created with xensieve.Sieve and used to produce iterators of integers, Boolean states, or interval widths. The Sieve::new constructor accepts arbitrarily complex Sieve expressions. Sieve is generic over unsigned integer types and defaults to u64; annotate the binding to select a different type.

use xensieve::Sieve;

let s1: Sieve<u64> = Sieve::new("5@0");
let s2: Sieve<u64> = Sieve::new("30@10");
let s3: Sieve<u64> = Sieve::new("(5@0|4@2)&!30@10");

To construct a Sieve over a different unsigned integer type, annotate the binding accordingly:

use xensieve::Sieve;

let s: Sieve<u32> = Sieve::new("3@0|5@1|5@4");
assert_eq!(s.to_string(), "Sieve{3@0|5@1|5@4}");
assert!(s.contains(6));

The iter_value() method takes an iterator if integers that can be used to "drive" the Sieve, either with ordered contiguous integers or arbitrary sequences. The iterator yields the subset of integers contained within the Sieve.

use xensieve::Sieve;

assert_eq!(s1.iter_value(0..50).collect::<Vec<_>>(), vec![0, 5, 10, 15, 20, 25, 30, 35, 40, 45]);
assert_eq!(s2.iter_value(0..50).collect::<Vec<_>>(), vec![10, 40]);
assert_eq!(s3.iter_value(0..50).collect::<Vec<_>>(), vec![0, 2, 5, 6, 14, 15, 18, 20, 22, 25, 26, 30, 34, 35, 38, 42, 45, 46]);

The xensieve.Sieve features two alternative iterators to permit using Sieves in different contexts. The iter_state() iterator returns, for each provided integer, the resulting Boolean state.

assert_eq!(s1.iter_state(0..10).collect::<Vec<_>>(), vec![true, false, false, false, false, true, false, false, false, false]);
assert_eq!(s3.iter_state(0..10).collect::<Vec<_>>(), vec![true, false, true, false, false, true, true, false, false, false]);

The iter_interval() iterator returns, for sequential pairs of provided integers that are within the Sieve, the resulting interval.

assert_eq!(s2.iter_interval(0..50).collect::<Vec<_>>(), vec![30]);
assert_eq!(s3.iter_interval(0..50).collect::<Vec<_>>(), vec![2, 3, 1, 8, 1, 3, 2, 2, 3, 1, 4, 4, 1, 3, 4, 3, 1]);

The contains() method can be used to test if arbitrary integers are contained within the Sieve:

assert_eq!(s1.contains(5), true);
assert_eq!(s1.contains(6), false);
assert_eq!(s3.contains(10), false);
assert_eq!(s3.contains(30), true);

The xensieve.Sieve instance supports the same operators permitted in Sieve expressions, such that instances can be combined to build complex Sieves.

let s4: Sieve<u64> = (Sieve::new("5@0") | Sieve::new("4@2")) & !Sieve::new("30@10");
assert_eq!(s4.to_string(), "Sieve{5@0|4@2&!(30@10)}");
assert_eq!(s3.iter_value(0..100).collect::<Vec<_>>(), s4.iter_value(0..100).collect::<Vec<_>>());

What is New in xensieve

1.0.0

Improved memory management of nested Sieve using Rc.

Made residual class integer representations generic.

0.8.0

Documentation and CI improvements.

0.7.0

Documentation and CI improvements.

0.6.0

CI improvements.

0.5.0

Documentation improvements.

0.4.0

Implemented operator support for &Sieve.

Improved documentation.

0.3.0

Code cleanup and improved doc strings.

0.2.0

Implemented symmetric difference with the ^ operator on Sieve.

0.1.0

First release.