Classification of melody by composer using hidden Markov models

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Classification of melody by composer using hidden
Markov models

• Greg Eustace
• MUMT 614 Music Information Acquisition,
  Preservation, and Retrieval

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Overview

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

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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?

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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)

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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 3.
• 5. Alternative contour representations?

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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.
• http//www.cs.ubc.ca/murphyk/Software/HMM/hmm.htm
  l
• A large collection of links to MIDI files has
  been compiled by Cory McKay
• http//www.music.mcgill.ca/cmckay/midi.html
• 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.

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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 aij state transition probability
  distribution
• B bj(k) observation symbol probability
  distribution
• p pi the initial state distribution
• The set of hidden model parameters are given by ?
  (A, B, p).

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The learning process

• The Baum-Welch learning algorithm is used to find
  the hidden parameters (?) 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.

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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 ? (A, B, p) what is the
  value of ? that maximizes the probability of the
  observation sequence P(O ?)?
• The first fundamental problem is essential to the
  classification of an unknown sequence. Given an
  observation sequence O O1, O2, On and a
  model ? (A, B, p) how do we efficiently compute
  P(O?) (Rabiner 1989)?

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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

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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.

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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

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References

• Chai, W., and B. Vercoe. Folk music
  classification using hidden Markov models. 2001.
  Proceedings of the international conference on
  artificial intelligence.
• Rabiner, L. 1989. A tutorial on hidden Markov
  models and selected applications in speech
  recognition. Proceedings of the IEEE 77 25786.