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Table of contents
1 Introduction
1.1 Motivations
1.2 Scope of the Thesis
1.3 Relevant Music Theoretic Concepts and Terminology
1.3.1 Notes
1.3.2 Key and Scales
1.3.3 Chords
1.3.4 Metrical Structure
1.4 Applications
1.5 Objectives
1.6 Overview of the Thesis
1.7 Main Thesis Contributions
2 Databases and Evaluation Measures Used in This Dissertation
2.1 Introduction
2.2 About Evaluation
2.3 Music Collections for Evaluation
2.3.1 Signal Experiment Test-set
2.3.2 Popular Music: The Beatles Test-set
2.3.2.1 Chord Annotations
2.3.2.2 Key Annotations
2.3.2.3 Metric Structure Annotation
2.3.3 Classical Music: The Piano Mozart test-set
2.3.4 Detailed Analysis of the Databases
2.4 Evaluation measures
2.4.1 Beat and Downbeat Tracking Evaluation Measure
2.4.2 Chord Evaluation Measures
2.4.2.1 Label Accuracy
2.4.2.2 Segmentation Accuracy
2.4.2.3 Neighboring Chords Confusions
2.4.3 Keys
2.4.3.1 Main Key
2.4.3.2 Local Keys
2.4.4 About Statistical Significance Testing
2.4.5 About Evaluation of Algorithms Based on Training
3 Towards a Signal Representation for Harmonic Content Analysis
3.1 Introduction
3.2 A Representation of Audio for Harmonic Content Analysis
3.2.1 Music Transcription-Based Approaches
3.2.2 Chroma Representation, an Alternative to Transcription
3.2.2.1 Definition
3.2.3 Representation of Music Signals, Notations
3.2.4 About Acoustic Signal Representation
3.2.4.1 Fourier Transform
3.2.4.2 Frequency Resolution Versus Time Resolution
3.2.4.3 Constant-Q Transform
3.3 Chroma Representation, Background
3.3.1 Chromagram Based on the Fourier Transform
3.3.2 Considering the Harmonics in the Pitch Class Profiles
3.3.3 Constant-Q Profiles
3.3.4 Chromagram Based on multi-f0s
3.3.5 Filter bank
3.3.6 Extension: the Tonal Centroid
3.3.7 Why Using Chroma Features for Harmonic Content Analysis? .
3.4 Derivation of Chroma Features
3.4.1 Chroma Based on a Spectral Representation
3.4.1.1 Tuning
3.4.1.2 Frequency Region Selection for Chroma Computation
3.4.1.3 Computation of a Semitone Pitch Spectrum
3.4.1.4 Smoothing
3.4.1.5 Chroma Spectrum
3.4.1.6 Post-processing: Normalization
3.4.2 Chroma Based on multiple f0s
3.5 Two Problems Related to the Chroma Features
3.5.1 Chroma Features and Harmonics
3.5.2 Beat-Synchronous Analysis
3.5.2.1 Towards a Beat-Synchronous Analysis
3.5.2.2 Problem of Mixing Harmonies
3.5.2.3 Influence of the Position of an Adaptive Window
3.6 Selecting a Feature Vector for Harmonic Analysis
3.6.1 Defining a Measure to Compare Various Features
3.6.1.1 Previously proposed measures
3.6.1.2 Proposed Measure for Chroma Feature Comparison .
3.6.2 Database for Feature Selection
3.6.3 On the use of a Beat-Synchronous Analysis
3.6.3.1 Beat-Synchronous Versus Frame-by-Frame Analysis .
3.6.3.2 Influence of the Position of an Adaptive Window
3.6.3.3 Conclusion on Beat-Synchronous Analysis
3.6.4 Fixed versus Multi-resolution Analysis
3.6.5 Multi-f0s Versus Spectral Representation
3.7 Summary and Conclusion
4 Chord Progression Estimation From an Audio File
4.1 Introduction
4.2 Previous Work on Chord Estimation
4.2.1 Features That Describe the Harmonic Content
4.2.2 Statistical Machine Learning Techniques for Chord Estimation
4.2.2.1 HMM-based Baseline Approaches
4.2.2.2 Chords and Musical Context
4.2.2.3 Introducing Language Modeling, N-grams
4.2.2.4 Other Statistical Modeling Approaches
4.2.3 Pattern Matching Approaches
4.2.4 Real-Time Implementation for Chord Estimation
4.2.5 Summary of Chords Estimation Techniques
4.2.5.1 Summary of the Above-Presented Methods
4.2.5.2 Summary of the MIREX Chord Recognition Systems .
4.3 Proposed Approach for Chord Estimation
4.3.1 Hidden Markov Models
4.3.2 On the Use of HMM for Chord Estimation
4.3.3 The Problem of the Harmonics
4.4 Chord Estimation From the Chroma Vectors Using a HMM
4.4.1 Model
4.4.1.1 Chord Lexicon
4.4.1.2 Overview of the Proposed Model
4.4.2 Initial State Distribution
4.4.3 Observation Symbol Probability Distribution
4.4.3.1 Method 1
4.4.3.2 Method 2
4.4.3.3 Method 3
4.4.4 State Transition Probability Distribution
4.4.4.1 Method A
4.4.4.2 Method B
4.4.4.3 Method C
4.4.4.4 Method D
4.4.5 Chord Progression Detection Over Time
4.5 Evaluation and Results
4.5.1 Test Set and Protocol
4.5.2 Results
4.5.3 Analysis of Results
4.5.3.1 Chord Estimation Method
4.5.3.2 Transition Matrix
4.5.3.3 Number of Harmonics
4.5.4 Discussion
4.5.4.1 Chord Confusions Due to Ambiguous Mapping
4.5.4.2 Neighboring Triad Confusions
4.5.4.3 Passing or Missing Tones
4.5.4.4 Limitation for Inharmonic Sounds
4.6 Conclusion
5 Joint Estimation of Chords and Downbeats
5.1 Introduction
5.2 Related Work
5.3 Proposed Approach
5.4 Model
5.4.1 Extraction of Beat-Synchronous Chroma Features
5.4.2 Overview of the Model
5.4.3 Initial State Distribution π
5.4.4 Observation Probabilities
5.4.4.1 Observation pim Probability Distribution
5.4.4.2 Observation Chord Symbol Probability Distribution
5.4.5 State Transition Probability Distribution
5.4.5.1 Distribution of Chord Changes
5.4.5.2 Transition Matrix for a Constant 4/4 Meter
5.4.5.3 Transition Matrix for a Variable Meter
5.4.6 Simultaneous Estimation of Chords and Downbeats
5.5 Evaluation Method
5.6 Analysis of the Results
5.6.1 Chords and Downbeats Interaction
5.6.2 Downbeat Position Estimation
5.6.2.1 Semi-automatic Downbeat Position Estimation
5.6.2.2 Estimated Beats Versus Theoretical Beats
5.6.2.3 Comparison With the State-of-the-art
5.6.2.4 Handling Variable Meter
5.6.2.5 Handling Insertion or Deletion of Beats
5.6.3 Chord Estimation
5.6.3.1 MIREX 2008 “Audio Chord Detection”
5.6.3.2 MIREX 2009 “Audio Chord Detection”
5.6.3.3 Chord Segmentation
5.6.3.4 Analysis of Chord Detection Errors
5.6.3.5 Tactus-synchronous Versus Tactum-synchronous Analysis .
5.6.4 Case Study Examples
5.6.4.1 Boundary Errors
5.6.4.2 Chord Changes
5.7 Conclusion
6 Interaction Between Chords, Downbeats and Keys
6.1 Introduction
6.1.1 Organization of the chapter:
6.2 Related work
6.2.1 Global Key
6.2.1.1 Template-Based Key-finding Models
6.2.1.2 Key-finding Models Based on HMMs
6.2.1.3 The Spiral Array Model
6.2.2 Local Key
6.2.3 Key Estimation Methods Based on Chord Progression
6.2.4 Summary of the Works on Key Estimation
6.3 Interaction between Chords, Meter and Global Key
6.3.1 Overview of the Model
6.3.2 Musical Key Information in the Transition Matrix
6.3.3 Simultaneous estimation of key, chords and downbeats
6.3.3.1 Key Selection
6.3.3.2 Post-processing Key Estimation Step
6.3.4 Test-Set and Evaluation Measure
6.3.5 Overall Results
6.3.6 Analysis of the Results
6.4 Interaction between Chords, Meter and Local Key
6.4.1 The Problem of the Analysis Window length
6.4.2 Model
6.4.3 Extraction of Key Observation Vectors
6.4.4 Key Estimation From Chords Using Hidden Markov Models .
6.4.4.1 Initial State Distribution
6.4.4.2 Observation Probabilities of Keys
6.4.4.3 State Transition Probability Distribution
6.4.4.4 Local Key Estimation
6.4.5 Evaluation
6.4.5.1 Test-set and evaluation measures
6.4.5.2 Results and discussion
6.4.5.3 Relationship Between Chords and Local Key
6.4.5.4 Importance of the Metrical Structure
6.4.5.5 Effect of the Length of the Analysis Window
6.4.5.6 Effect of the Choice of the Key Templates
6.4.5.7 Smooth Modulations:
6.5 Conclusion of the Chapter
7 Conclusion
7.1 Thesis Contributions
7.1.1 Features
7.1.2 Chords
7.1.3 Downbeat
7.1.4 Key
7.2 Future Works
Annexe A – List of the Beatles songs
Annexe B – List of publications
Bibliography
