# Sieves

## Introduction

“…sieve theory is the study of the internal symmetries of a series of points either constructed intuitively, given by observation, or invented completely from moduli of repetition.”

- Xenakis, Formalized Music, pp. 276

score.sieves provides sieve functions based on the work of Iannis Xenakis. Sieves are represented as a 2-vector of [modulo index]. U and I are used to create Union and Intersection Sieves. Sieves may also be nil, though are factored out when a Sieve is simplified, and generates no sequence when gen-sieve is called.

## Usage

Sieves are used to generate a series of numbers that have been sieved according to values given by the user. The process starts with the series of positive numbers starting from zero. The numbers are then filtered against the given sieves.

In score.sieves, the gen-sieves function is used to generate a sequence using sieves. The function takes in the number of steps to generate and a sieve to use.

The basic sieve is a 2-vector of [modulo index]. The modulo gives the period of the sieve, and the index gives the offset. For example, a sieve [4 1] would start at index 1, and repeat every 4:

```
user=> (gen-sieve 12 [4 1])
(1 5 9 13 17 21 25 29 33 37 41 45)
```

A Union filters according to a set of sieves. Any value that satisfies any of the sieves in the union will be passed through. This is equivalent to a logical or:

```
user=> (gen-sieve 12 (U [4 1] [3 2]) )
(1 5 9 13 17 21 25 29 33 37 41 45)
```

An Intersection filters according to a set of a other sieves. However, unlike a Union, a value is passed through only if it satisfies all sieves within the intersection. This is equivalent to a logical and:

```
user=> (gen-sieve 12 (I [4 1] [3 2]))
(5 17 29 41 53 65 77 89 101 113 125 137)
```

More complex sieves can be made up of nested Unions and Intersections:

```
user=> (gen-sieve 12 (U [3 2] (I [3 2] [2 0])))
(2 5 8 11 14 17 20 23 26 29 32 35)
```

When gen-sieves is called, sieves are first normalized, then simplified, before generating values. You can view the simplified version of a sieve by using the simplified function:

```
user=> (simplified (U [3 2] (I [3 2] [2 0])))
#score.sieves.Union{:l [3 2], :r [6 2]}
```

## Analysis

analyze-sieve is a function that given a sequence of numbers, returns a sieve analysis map. The map has three keys:

- :analysis - vector of 3-vector sieve analyses, where the valus are modulus, index, and number of items covered by that sieve
- :sieve - a Union sieve derived from the :analysis, ready for use to generate sieve sequences
- :period - period of repetition of the analyzed sieve

For example:

```
user=> (analyze-sieve [0 2 3 5 8 11])
{ :analysis [[8 0 2] [3 2 4] [5 3 2]],
:sieve #score.sieves.Union{:l #score.sieves.Union{:l [8 0], :r [3 2]}, :r [5 3]},
:period 120}
```

The sequence of [0 2 3 5 8 11] is analyzed to yield a Union sieve of [8 0], [3 2], and [5 3], which repeats every 120 steps.

## Usage

Sieves may be used for any purpose, such as a list of pitch intervals, rhythms, scales, and so on. They can be used together with Score’s gen-notes as well as scale and pitch generation functions. For example:

```
(gen-notes
1 0.0 3.0
(map #(pch->freq (pch-add base-pch %))
(gen-sieve 7 [2 0]))
0.25)
```

would yield:

```
([1 0.0 3.0 261.6255653005986 0.25]
[1 0.0 3.0 293.6647679174076 0.25]
[1 0.0 3.0 329.6275569128699 0.25]
[1 0.0 3.0 369.9944227116344 0.25]
[1 0.0 3.0 415.3046975799451 0.25]
[1 0.0 3.0 466.1637615180899 0.25]
[1 0.0 3.0 523.2511306011972 0.25])
```

The [2 0] sieve used here with gen-sieve and an assumed 12-tone equal temperament tuning, would yield the first 7 scale degrees of a whole-tone scale. The above score may then be used to generate a whole-tone chord, starting at time 0.0, with duration 3.0, and amplitude of 0.25 for each note of the chord.

## References

### Implementations

- C Code from “Sieves” article below.
- Haskell Music Theory
- athenaCL

### Literature

- Xenakis and Rahn. “Sieves”. Perspectives of New Music, Vol. 28, No. 1 (Winter, 1990), pp. 58-78.
- Xenakis. “Formalized Music”. pp. 268-288.