Markov chains model context-dependent musical choices by making each event depend probabilistically on prior states

A Markov chain defines a set of states (e.g. notes, chords, rhythmic values) and a transition probability table: for each current state, what is the probability of moving to each next state? A first-order Markov chain looks one step back; a kth-order chain looks k steps back. The transition table grows exponentially with order: 3 states at 2nd-order requires 9 entries. In algorithmic music, Markov chains are trained from a corpus of existing music to capture stylistic tendencies, or designed by hand to reflect compositional logic. The Markov model addresses the limitation of memoryless uniform distributions: in tonal music, the leading note typically resolves upward rather than randomly.

Examples

A Markov chain trained on Bach chorales generates harmonically plausible sequences. David Cope’s EMI system uses Markov-like statistical models over large corpora. JamFactory (1987) was an early real-time Markov interactive music system.

Assessment

What limitation of a simple uniform probability distribution does a Markov chain address? How many transition probabilities are required for a 2nd-order Markov chain with 4 states?