Statistics in Musicology

Table of Contents......Page 0
Statistics in Musicology......Page 1
Table of Contents......Page 3
Preface......Page 5
1.1 General background......Page 7
1.2.2 Definitions and results......Page 13
1.3.1 The Mathieu group......Page 18
1.3.3 Representation of music......Page 19
1.3.4 Classification of circular chords and other musical objects......Page 20
1.3.6 Transformations......Page 21
2.1 Musical motivation......Page 29
2.2.1 Definitions......Page 31
2.3.1 Tempo curves......Page 35
2.3.2 Notes modulo 12......Page 38
Correlation......Page 43
Regression......Page 45
Regression smoothing......Page 47
Interpolation......Page 49
2.5.1 Empirical tempo-acceleration......Page 51
2.5.2 Interpolated and smoothed tempo curves–velocity and acceleration......Page 52
2.5.3 Tempo–hierarchical decomposition by smoothing......Page 54
2.5.4 Tempo curves and melodic indicator......Page 59
2.5.5 Tempo and loudness......Page 60
2.5.7 Melodic tempo-sharpening......Page 65
2.6.1Definitions......Page 69
2.7.1 Distribution of notes–Chernoff faces......Page 70
2.7.2 Distribution of notes–star plots......Page 71
2.7.3 Joint distribution of interval steps of envelopes......Page 75
2.7.4 Pitch distribution–symbol plots with circles......Page 76
2.7.5 Pitch distribution–symbol plots with rectangles......Page 79
2.7.6 Pitch distribution–sym ol plots with stars......Page 80
2.7.7 Pitch distribution–profile plots......Page 82
3.2.1 Measuring information and randomness......Page 86
Omnibus metric, melodic, and harmonic indicators......Page 91
Specific indicators......Page 92
3.2.3 Measuring dimension......Page 94
3.3.1 Entropy of melodic shapes......Page 98
3.3.2 Spectral entropy of local interval variability......Page 101
3.3.3 Omnibus metric, melodic, and harmonic indicators for compositions by Bach, Schumann, and Webern......Page 103
3.3.4 Specific melodic indicators for Schumann’s Traumerei......Page 104
4.2.1 Deterministic and random components, basic definitions......Page 111
4.2.2 Sampling of continuous-time time series......Page 115
4.2.4 Special models......Page 117
4.2.5 Fitting parametric models......Page 121
4.2.6 Fitting non- and semiparametric models......Page 123
4.2.7 Spectral estimation......Page 124
4.2.8 The harmonic regression model......Page 126
4.2.9 Dominating frequencies in random series......Page 127
4.3.1 Analysis and modeling of musical instruments......Page 128
4.3.2 Licklider’s theory of pitch perception......Page 135
4.3.3 Identifiation of pitch, tone separation and purity of intonation......Page 136
4.3.4 Music as 1/f noise?......Page 137
5.2.1 Hierarchical aggregation and decomposition......Page 142
5.2.2 Hierarchical regression......Page 143
5.2.3 Hierarchical smoothing......Page 144
5.2.4 Hierarchical wavelet models......Page 147
5.3.1 Hierarchical decomposition of metric, melodic, and harmonic weights......Page 160
5.3.2 HIREG models of the relationship between tempo and melodic curves......Page 161
5.3.3 HISMOOTH models for the relationship between tempo and structural curves......Page 164
5.3.5 Wavelet analysis of tempo curves......Page 167
5.3.6 HIWAVE models of the relationship between tempo and melodic curves......Page 169
6.2.1 Definition of Markov chains......Page 176
6.2.2 Transience, persistence, irreducibility, periodicity, and stationarity......Page 177
6.2.4 Parameter estimation for Markov and hidden Markov models......Page 181
6.3.1 Stationary distribution of intervals modulo 12......Page 182
6.3.2 Stationary distribution of interval torus values......Page 185
6.3.3 Classification by hidden Markov models......Page 187
6.3.4 Reconstructing scores from acoustic signals......Page 192
7.2.1 Some descriptive statistics......Page 194
7.2.2 Correlation and autocorrelation......Page 197
7.2.3 Probability distributions......Page 198
7.2.4 Statistical inference......Page 200
7.3.1 Variability and autocorrelation of notes modulo 12......Page 201
7.3.2 Variability and autocorrelation of note intervals modulo 12......Page 203
7.3.3 Notes and intervals on the circle of fourths......Page 211
8.2.1 Definition of PCA for multivariate probability dist ibutions......Page 212
8.2.2 Definition of PCA for observed data......Page 214
8.2.3 Scale invariance?......Page 215
8.2.4 Choosing important principal components......Page 216
8.3.1 PCA of tempo skewness......Page 217
8.3.2 PCA of entropies......Page 221
9.2.1 Allocation rules......Page 227
Discriminant analysis with prior group probabilities –the Bayesian rule......Page 228
Which rule is better?......Page 229
9.2.3 Case II: Population dist ibution form known, parameters unknown......Page 230
9.2.4 Case III: Population distributions completely unknown......Page 231
9.2.5 How good is an empirical discriminant rule?......Page 232
9.3.1 Identification of pitch, tone separation, and purity of intonation......Page 233
9.3.2 Identification of historic periods......Page 234
10.2.1 Maximum likelihood classification......Page 239
10.2.2 Hierarchical clustering......Page 241
10.3.1 Distribution of notes......Page 244
10.3.3 Tempo curves......Page 248
10.3.4 Tempo curves and melodic structure......Page 249
11.2.1 Basic definitions......Page 252
11.2.2 Metric MDS......Page 253
11.2.4 Chronological ordering......Page 254
11.3.2 Perception and music psychology......Page 255
List of figures......Page 259
References......Page 269