Algorithmic composition is the technique of using algorithms to create music.
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, for example, can often be reduced to algorithmic determinacy. The term is usually reserved, however, for the use of formal procedures to make music without human intervention, either through the introduction of chance procedures or the use of computers.
There is no universal method to sort different compositional algorithms into categories. One way to do this is to look at the way an algorithm takes part in the compositional process. 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) 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 categorise compositional algorithms is by their structure and the way of processing data, as seen in this model of six partly overlapping types[according to whom?]:
Mathematical models are based on mathematical equations and random events. The most common way to create compositions through mathematics is stochastic processes. In stochastic models a piece of music is composed as a result of non-deterministic 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 and inharmonic phenomena of nature. For example, since the 1970s fractals have been studied also as models for algorithmic composition.
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.
Music can also be examined as a language with a distinctive grammar 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 and rhythm, rather than single notes.
Evolutionary methods of composing music are based on genetic algorithms. The composition is being built by the means of evolutionary process. Through mutation and natural selection, 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.
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.
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.
Algorithmic Composition: Computational Thinking in Music by Michael Edwards. Communications of the ACM, Vol. 54 No. 7, Pages 58-67 10.1145/1965724.1965742. From the abstract: "This article outlines the history of algorithmic composition from the pre- and post-digital computer age, concentrating, but not exclusively, on how it developed out of the avant-garde Western classical tradition in the second half of the 20th century. This survey is more illustrative than all-inclusive, presenting examples of particular techniques and some of the music that has been produced with them."
Gerhard Nierhaus: Algorithmic Composition - Paradigms of Automated Music Generation. Springer 2008. - ISBN 978-3-211-75539-6
Wooller, Rene, Brown, Andrew R, Miranda, Eduardo, Diederich, Joachim, & Berry, Rodney (2005) A framework for comparison of process in algorithmic music systems. In: Generative Arts Practice, 5–7 December 2005, Sydney, Australia. 
Samples of algorithmic music
Complex algorithmic compositions made largely with the software Fractmus posted on sound cloud: Coq Freisenberger.
Musical Algorithms An interactive exploration of the relationship between music and mathematical formulas funded by the Northwest Academic Computing Consortium, project directed by Jonathan N. Middleton.