Classification of melody by composer using hidden Markov models Greg Eustace MUMT 614: Music Information Acquisition, Preservation, and Retrieval

Overview Project description Software and dataset Representations of melody HMM parameters The learning process Training and testing HMMs Summary

Project description The goal of this project is to use Hidden Markov Models (HMM) for the automatic classification of symbolic melodic data by composer. Research questions: Are there significant statistical differences between melodies written by different composers? How do different representations of melody affect the performance of the classifier? How do different types of HMMs affect the performance?

Representations of melody C major scale: 1. Absolute pitch (normalised to the octave). (e.g. 1, 3, 5, 6, 8, 10, 11, 12) 2. Absolute pitch with rhythm (e.g. 1, 1, 3, 5, 6, 8, 10, 10, 11, 11, 12, 12) The note number is given once if equal to a quarter note, twice for a half note, etc. (Chai, and Vercoe 2001). 3. Interval (e.g. 2, 2, 1, 2, 2, 2, 1)

Representations of melody 4. Melodic Contour (e.g. 2, 2, 2, 2, 2, 2, 2) The convention is that intervals of a unison = 1, intervals of second = 2 and all other intervals are Alternative contour representations?

Software and dataset Extraction of melodic data uses ad hoc algorithms developed in MAX/MSP and MATLAB. Classification uses the HMM Toolbox by Kevin Murphy. A large collection of links to MIDI files has been compiled by Cory McKay: The melodic data is the soprano line extracted from a type 1 MIDI file. It is hoped that composers of contrasting style will show the greatest statistical differences.

Hidden Markov models “A Hidden Markov model (HMM) is a doubly imbedded stochastic process with an underlying stochastic process that is not observable (i.e. is hidden) but can only be observed through another set of stochastic processes that produce the sequence of observations” (Rabiner 1989). An HMM is defined by the parameters: M = number of distinct observation symbols (e.g. for absolute pitch these are the numbers 1-13 corresponding to the 12 notes of the octave). N = number of states in the model (these may not have physical significance). A = {a ij } = state transition probability distribution B = {b j (k)} = observation symbol probability distribution π = {π i } = the initial state distribution The set of hidden model parameters are given by λ = (A, B, π).

The learning process The Baum-Welch learning algorithm is used to find the hidden parameters (λ) of the HMM. This process uses maximum likelihood parameter estimation. In general, the likelihood is maximized when a given test sequence corresponds to a specific model. It is also common to attempt to maximize the logarithm of the likelihood.

Training and testing HMMs The training procedure corresponds to one of the three fundamental problems associated with HMMs as defined by Rabiner (1989). That is, for a given observation sequence O = {O1, O2,… On} and a model with parameters λ = (A, B, π) what is the value of λ that maximizes the probability of the observation sequence P(O| λ)? The first fundamental problem is essential to the classification of an unknown sequence. Given an observation sequence O = {O1, O2,… On} and a model λ = (A, B, π) how do we efficiently compute P(O|λ) (Rabiner 1989)?

Training and testing HMMs The process can be summarised as: Train a different model on data from each composer. Once the models have been trained compute the log-likelihood for a test sequence for each model. Classify the training data according to the model which gives the highest value for the log-likelihood. Repeat the process for all representations of melody

Training and testing HMMs Using different types of HMMs may affect classification Fully connected (ergodic) models: every state is connected to every other state. Left-right (Bakis) models: states are connected to themselves and to the adjacent state (proceeding from left to right). These are typically used for modelling time varying signals.

Summary A summary of variables which may affect classification: Composer Size of the dataset Nature of the pieces used MIDI transcription Melodic extraction Melodic representation Type of HMM Number of model states

References Chai, W., and B. Vercoe. Folk music classification using hidden Markov models Proceedings of the international conference on artificial intelligence. Rabiner, L A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE 77: 257–86.