Editing Markov chain (section)

===Music===
Markov chains are employed in [[algorithmic composition|algorithmic music composition]], particularly in [[software]] such as [[Csound]], [[Max (software)|Max]], and [[SuperCollider]]. In a first-order chain, the states of the system become note or pitch values, and a [[probability vector]] for each note is constructed, completing a transition probability matrix (see below). An algorithm is constructed to produce output note values based on the transition matrix weightings, which could be [[MIDI]] note values, frequency ([[Hertz|Hz]]), or any other desirable metric.<ref>{{cite journal |title=Making Music with Algorithms: A Case-Study System |author1=K McAlpine |author2=E Miranda |author3=S Hoggar |journal=Computer Music Journal |issue=2 |year=1999 |volume=23 |doi=10.1162/014892699559733 |pages=19–30 }}</ref>

{| class="wikitable" style="float: left"
|+ 1st-order matrix
! Note !! A !! C{{music|sharp}} !! E{{music|flat}}
|-
! A
| 0.1 || 0.6 || 0.3
|-
! C{{music|sharp}}
| 0.25 || 0.05 || 0.7
|-
! E{{music|flat}}
| 0.7 || 0.3 || 0
|}

{| class="wikitable" style="float: left; margin-left: 1em"
|+ 2nd-order matrix
! Notes !! A !! D !! G
|-
! AA
| 0.18 || 0.6 || 0.22
|-
! AD
| 0.5 || 0.5 || 0
|-
! AG
| 0.15 || 0.75 || 0.1
|-
! DD
| 0 || 0 || 1
|-
! DA
| 0.25 || 0 || 0.75
|-
! DG
| 0.9 || 0.1 || 0
|-
! GG
| 0.4 || 0.4 || 0.2
|-
! GA
| 0.5 || 0.25 || 0.25
|-
! GD
| 1 || 0 || 0
|}
{{Clear}}

A second-order Markov chain can be introduced by considering the current state ''and'' also the previous state, as indicated in the second table. Higher, ''n''th-order chains tend to "group" particular notes together, while 'breaking off' into other patterns and sequences occasionally. These higher-order chains tend to generate results with a sense of [[phrase (music)|phrasal]] structure, rather than the 'aimless wandering' produced by a first-order system.<ref name="Roads">{{cite book|editor=Curtis Roads |title=The Computer Music Tutorial |year=1996|publisher=MIT Press|isbn= 978-0-262-18158-7}}</ref>

Markov chains can be used structurally, as in Xenakis's Analogique A and B.<ref>Xenakis, Iannis; Kanach, Sharon (1992) ''Formalized Music: Mathematics and Thought in Composition'', Pendragon Press. {{ISBN|1576470792}}</ref> Markov chains are also used in systems which use a Markov model to react interactively to music input.<ref>{{Cite web|url=http://www.csl.sony.fr/~pachet/|archive-url=https://web.archive.org/web/20120713235933/http://www.csl.sony.fr/~pachet/|url-status=dead |title=Continuator|archive-date=July 13, 2012}}</ref>

Usually musical systems need to enforce specific control constraints on the finite-length sequences they generate, but control constraints are not compatible with Markov models, since they induce long-range dependencies that violate the Markov hypothesis of limited memory. In order to overcome this limitation, a new approach has been proposed.<ref>Pachet, F.; Roy, P.; Barbieri, G. (2011) [http://www.csl.sony.fr/downloads/papers/2011/pachet-11b.pdf "Finite-Length Markov Processes with Constraints"] {{webarchive|url=https://web.archive.org/web/20120414183247/http://www.csl.sony.fr/downloads/papers/2011/pachet-11b.pdf |date=2012-04-14}}, ''Proceedings of the 22nd International Joint Conference on Artificial Intelligence'', IJCAI, pages 635–642, Barcelona, Spain, July 2011</ref>