Algorithmic composition is the technique of using

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

s to create

music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...

. Algorithms (or, at the very least, formal sets of rules) have been used to compose music for centuries; the procedures used to plot voice-leading in Western

counterpoint In music, counterpoint is the relationship between two or more musical lines (or voices) which are harmonically interdependent yet independent in rhythm and melodic contour. It has been most commonly identified in the European classical tra ...

, for example, can often be reduced to algorithmic determinacy. The term can be used to describe music-generating techniques that run without ongoing human intervention, for example through the introduction of chance procedures. However through live coding and other interactive interfaces, a fully human-centric approach to algorithmic composition is possible. Some algorithms or data that have no immediate musical relevance are used by composers as creative inspiration for their music. Algorithms such as

fractals In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...

L-system An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into som ...

statistical models A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, ...

, and even arbitrary

data In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpret ...

(e.g.

census A census is the procedure of systematically acquiring, recording and calculating information about the members of a given population. This term is used mostly in connection with national population and housing censuses; other common censuses inc ...

figures, GIS coordinates, or

magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...

measurements) have been used as source materials.

Models for algorithmic composition

Compositional algorithms are usually classified by the specific programming techniques they use. The results of the process can then be divided into 1) music composed by computer and 2) music composed with the aid of computer. Music may be considered composed by computer when the algorithm is able to make choices of its own during the creation process. Another way to sort compositional algorithms is to examine the results of their compositional processes. Algorithms can either 1) provide notational information (

sheet music Sheet music is a handwritten or printed form of musical notation that uses musical symbols to indicate the pitches, rhythms, or chords of a song or instrumental musical piece. Like its analogs – printed books or pamphlets in English, ...

MIDI MIDI (; Musical Instrument Digital Interface) is a technical standard that describes a communications protocol, digital interface, and electrical connectors that connect a wide variety of electronic musical instruments, computers, and ...

) for other instruments or 2) provide an independent way of sound synthesis (playing the composition by itself). There are also algorithms creating both notational data and sound synthesis. One way to categorize compositional algorithms is by their structure and the way of processing data, as seen in this model of six partly overlapping types: * translational models * mathematical models * knowledge-based systems * grammars * optimization approaches * evolutionary methods * systems which learn * hybrid systems

Translational models

This is an approach to music synthesis that involves "translating" information from an existing non-musical medium into a new sound. The translation can be either rule-based or

stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...

. For example, when translating a picture into sound, a

jpeg JPEG ( ) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and imag ...

image of a horizontal line may be interpreted in sound as a constant pitch, while an upwards-slanted line may be an ascending scale. Oftentimes, the software seeks to extract concepts or metaphors from the medium, (such as height or sentiment) and apply the extracted information to generate songs using the ways music theory typically represents those concepts. Another example is the translation of text into music, which can approach composition by extracting sentiment (positive or negative) from the text using

machine learning Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine ...

methods like

sentiment analysis Sentiment analysis (also known as opinion mining or emotion AI) is the use of natural language processing, text analysis, computational linguistics, and biometrics to systematically identify, extract, quantify, and study affective states and subjec ...

and represents that sentiment in terms of chord quality such as minor (sad) or major (happy) chords in the musical output generated.

Mathematical models

Mathematical models are based on mathematical

equations In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...

and random events. The most common way to create compositions through mathematics is

stochastic processes In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...

. In stochastic models a piece of music is composed as a result of non-

deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...

methods. The compositional process is only partially controlled by the composer by weighting the possibilities of random events. Prominent examples of stochastic algorithms are Markov chains and various uses of Gaussian distributions. Stochastic algorithms are often used together with other algorithms in various decision-making processes. Music has also been composed through natural phenomena. These chaotic models create compositions from the

harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...

and inharmonic phenomena of nature. For example, since the 1970s

have been studied also as models for algorithmic composition. As an example of deterministic compositions through mathematical models, the

On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to t ...

provides an option to play an

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...

12-tone equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That result ...

music. (It is initially set to convert each integer to a note on an 88-key

musical keyboard A musical keyboard is the set of adjacent depressible levers or keys on a musical instrument. Keyboards typically contain keys for playing the twelve notes of the Western musical scale, with a combination of larger, longer keys and smaller, s ...

by computing the integer

modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...

88, at a steady rhythm. Thus 123456, the natural numbers, equals half of a chromatic scale.) As another example, the all-interval series has been used for computer-aided composition

Knowledge-based systems

One way to create compositions is to isolate the aesthetic code of a certain musical genre and use this code to create new similar compositions. Knowledge-based systems are based on a pre-made set of arguments that can be used to compose new works of the same style or genre. Usually this is accomplished by a set of tests or rules requiring fulfillment for the composition to be complete.

Grammars

Music can also be examined as a language with a distinctive

grammar In linguistics, the grammar of a natural language is its set of structural constraints on speakers' or writers' composition of clauses, phrases, and words. The term can also refer to the study of such constraints, a field that includes doma ...

set. Compositions are created by first constructing a musical grammar, which is then used to create comprehensible musical pieces. Grammars often include rules for macro-level composing, for instance

harmonies In music, harmony is the process by which individual sounds are joined together or composed into whole units or compositions. Often, the term harmony refers to simultaneously occurring frequencies, pitches ( tones, notes), or chords. Howev ...

and

rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular re ...

, rather than single notes.

Optimization approaches

When generating well defined styles, music can be seen as a combinatorial optimization problem, whereby the aim is to find the right combination of notes such that the objective function is minimized. This objective function typically contains rules of a particular style, but could be learned using machine learning methods such as Markov models. Researchers have generated music using a myriad of different optimization methods, including integer programming, variable neighbourhood search, and evolutionary methods as mentioned in the next subsection.

Evolutionary methods

Evolutionary methods of composing music are based on genetic algorithms. The composition is being built by the means of

evolutionary Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variati ...

process. Through

mutation In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, m ...

and

natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Cha ...

, different solutions evolve towards a suitable musical piece. Iterative action of the algorithm cuts out bad solutions and creates new ones from those surviving the process. The results of the process are supervised by the critic, a vital part of the algorithm controlling the quality of created compositions.

Evo-Devo approach

Evolutionary Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variati ...

methods, combined with developmental processes, constitute the ''evo-devo'' approach for generation and optimization of complex structures. These methods have also been applied to music composition, where the musical structure is obtained by an iterative process that transform a very simple composition (made of a few notes) into a complex fully-fledged piece (be it a score, or a MIDI file).

Systems that learn

Learning systems are programs that have no given knowledge of the genre of music they are working with. Instead, they collect the learning material by themselves from the example material supplied by the user or programmer. The material is then processed into a piece of music similar to the example material. This method of algorithmic composition is strongly linked to algorithmic modeling of style, machine improvisation, and such studies as cognitive science and the study of

neural networks A neural network is a network or circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up of biological ...

. Assayag and Dubnov proposed a variable length Markov model to learn motif and phrase continuations of different length. Marchini and Purwins presented a system that learns the structure of an audio recording of a rhythmical percussion fragment using unsupervised clustering and variable length Markov chains and that synthesizes musical variations from it.

Hybrid systems

Programs based on a single algorithmic model rarely succeed in creating aesthetically satisfying results. For that reason algorithms of different type are often used together to combine the strengths and diminish the weaknesses of these algorithms. Creating hybrid systems for music composition has opened up the field of algorithmic composition and created also many brand new ways to construct compositions algorithmically. The only major problem with hybrid systems is their growing complexity and the need of resources to combine and test these algorithms. Another approach, which can be called ''computer-assisted composition'', is to algorithmically create certain structures for finally "hand-made" compositions. As early as in the 1960s, Gottfried Michael Koenig developed computer programs ''Project 1'' and ''Project 2'' for aleatoric music, the output of which was sensibly structured "manually" by means of performance instructions. In the 2000s, Andranik Tangian developed a computer algorithm to determine the time event structures for rhythmic canons and rhythmic fugues, which were then worked out into harmonic compositions ''Eine kleine Mathmusik I'' and ''Eine kleine Mathmusik II''; for scores and recordings see.

References

External links

*
Algorithmic Composer
series of algorithmic composition tutorials {{Computer music Computer music Music theory Markov models Procedural generation