Markov chains

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Figure 19.13

Figure 19.14

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源自《计算机音乐教程》、罗玆、人民音乐出版社、2011。

An important subclass of stochastic algorithms are the family of Markov chain techniques. The Markov chain is one of the earliest and most popular strategies for algorithmic composition of music. First formulated in 1906 by the Russian mathematician A. A. Markov (1856 1922), a Markov chain is a probability system in which the likelihood of a future event is determined by the state of one or more events in the immediate past.

The probabilities in a Markov chain can be laid out in a state-transition matrix. The state-transition matrix in figure 19.13 is a Markov chain for a simple melody composition, using the notes of a pentatonic scale: G, A, C, D, and E. The cells show the probability of a next pitch, given a current pitch shown at the left of the row.

Figure 19.14 shows a possible sequence of pitches generated by this matrix. One can increase the sophistication of this method by increasing the order of the chain. The order of a Markov chain indicates the number of prior states that are taken into consideration.

- Events in a zeroth-order chain (equivalent to a regular probability table with no looking back at previous states) are independent of one another.

- Events in a first-order chain have one predecessor (such as the example in figure 19.13).

- Events in a second-order chain look back two states, and so on.

Advanced Markov Techniques for Music: an interesting extension involves turning the Markov chain into an interactive composition and performance game. Playing with the control parameters in real time, musicians can change the state of the Markovian process on a moment-to-moment basis, as music is generated.

Hierarchical Markov Chains: one chain that generates high-level structure could select one of a number of types of sections, for example, a fast movement, a slow movement, a movement in a particular key, or a movement that relied on a particular pitch row. Within a section intermediate-level chains might select the sequence of phrases contained in the section. The details of each phrase type might be filled in by a low-level chain.

(Source: Curtis Roads, The Computer Music Tutorial. MIT Press. 1995)

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