Nonlinearities and Synchronization in Musical Acoustics and Music Psychology

474 Pages·2013·32.06 MB·English

by  Rolf Bader (auth.)

Current Research in Systematic Musicology 2
Editor-in-Chief
Prof.Dr.RolfBader
MusikwissenschaftlichesInstitut
UniversitätHamburg
NeueRabenstr.13
20354Hamburg
Germany
E-mail:[email protected]
Prof.Dr.MarcLeman
IPEM–Dep.ofMusicology
UniversityofGhent
Blandijnberg2
9000Ghent
Belgium
E-Mail:[email protected]
Prof.Dr.RolfIngeGodøy
DepartmentofMusicology
P.O.Box1017
Blindern0315
Oslo
Norway
E-Mail:[email protected]
Forfurthervolumes:
http://www.springer.com/series/11684
Rolf Bader
Nonlinearities
and Synchronization
in Musical Acoustics
and Music Psychology
ABC
Author
Prof.Dr.RolfBader
MusikwissenschaftlichesInstitut
UniversitätHamburg
Hamburg
Germany
Additionalmaterialtothisbookcanbedownloadedfromhttp://extras.springer.com
ISSN1860-949X e-ISSN1860-9503
ISBN978-3-642-36097-8 e-ISBN978-3-642-36098-5
DOI10.1007/978-3-642-36098-5
SpringerHeidelbergNewYorkDordrechtLondon
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(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Contents
1 Introduction 1
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
I Signal Processing 9
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Frequency Representations 13
2.1 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Correlogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 CorrelogramSimilarity Function . . . . . . . . . . . . 21
2.3 Spectral Centroid . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 Temporal Integration Path . . . . . . . . . . . . . . . 29
2.3.3 Gestalt Pattern . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Spectral Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 Temporal Integration Path . . . . . . . . . . . . . . . 41
2.4.3 Entropy Gestalt Pattern . . . . . . . . . . . . . . . . . 45
2.5 Karhunen-Loeve Decomposition . . . . . . . . . . . . . . . . . 45
2.5.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6 Microphone Array Techniques . . . . . . . . . . . . . . . . . . 51
3 Embedding Representations 57
3.1 Dimensionality and Log-Log Plots . . . . . . . . . . . . . . . 58
3.2 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.3 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.1 Periodicity and Autocorrelation . . . . . . . . . . . . . 65
3.4 Pseudo Phase Plots. . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 75
3.5.1 Capacity or Box-Counting Dimension . . . . . . . . . 76
3.5.2 Information Dimension. . . . . . . . . . . . . . . . . . 81
3.5.3 Multi-fractal Spectrum. . . . . . . . . . . . . . . . . . 85
VI Contents
3.5.4 Information Structure . . . . . . . . . . . . . . . . . . 88
3.5.5 Correlation Dimension . . . . . . . . . . . . . . . . . . 94
3.5.6 Fractal CorrelationDimension and Transients . . . . . 98
3.5.7 Global Fractal Dimension . . . . . . . . . . . . . . . . 100
3.5.8 Entropy from Autocorrelation Matrix . . . . . . . . . 102
3.6 Lyapunov Exponent and Poincaré Maps . . . . . . . . . . . . 103
II Physical Modelling 109
4 Applications to Musical Instruments 113
5 Finite-Difference Simulation 117
5.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2 Static Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3 Eigenvalue Solution. . . . . . . . . . . . . . . . . . . . . . . . 120
5.4 Time Dependent Solution . . . . . . . . . . . . . . . . . . . . 122
5.5 External and Internal Damping . . . . . . . . . . . . . . . . . 124
5.6 Example of a Horn Radiation . . . . . . . . . . . . . . . . . . 125
6 Finite-Element Simulation 133
6.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2 Static Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3 Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . 145
6.4 Time-Dependent Solutions . . . . . . . . . . . . . . . . . . . . 146
6.5 Moving Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.6 Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . 147
6.7 Nonlinear Solver . . . . . . . . . . . . . . . . . . . . . . . . . 148
III Musical Acoustics 155
7 Musical Instruments 157
7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
7.1.1 Pitch Winner of Coupled Oscillator. . . . . . . . . . . 158
7.1.2 Damping as Cause of Self-organization . . . . . . . . . 158
7.1.3 General Musical Instrument Models . . . . . . . . . . 159
7.2 Guitar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7.2.1 Initial Transient . . . . . . . . . . . . . . . . . . . . . 164
7.2.2 String/Body Coupling Higher Order Terms . . . . . . 170
7.2.3 Coupling of Body Bending to Longitudinal Waves . . 171
7.2.4 Nonlinear Coupling between String Modes . . . . . . . 174
7.2.5 Large String Displacement. . . . . . . . . . . . . . . . 179
7.2.6 Radiation Patterns of Forced Oscillations . . . . . . . 181
7.2.7 Fractal Dimensions of Guitar Tones . . . . . . . . . . 189
Contents VII
7.3 Violin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7.3.1 Bow – String Interaction. . . . . . . . . . . . . . . . . 198
7.3.2 Initial Transient of Bowing . . . . . . . . . . . . . . . 202
7.3.3 Finite-Difference Violin String/Bow
and Body Model . . . . . . . . . . . . . . . . . . . . . 203
7.3.4 Bow-String Finite-Difference Model . . . . . . . . . . 207
7.3.5 Fractal Dimensions of Violin Tones . . . . . . . . . . . 214
7.4 Wind Instruments . . . . . . . . . . . . . . . . . . . . . . . . 220
7.4.1 Reed Instruments. . . . . . . . . . . . . . . . . . . . . 220
7.4.2 Reed Characteristics . . . . . . . . . . . . . . . . . . . 223
7.4.3 Multiphonics in Reed Instruments . . . . . . . . . . . 226
7.4.4 Fractal Dimensions of Multiphonics . . . . . . . . . . 227
7.4.5 Mirlitons . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.4.6 Fractal Dimensions of Saxophone Tones . . . . . . . . 229
7.4.7 Tone Transition with a Clarinet. . . . . . . . . . . . . 230
7.4.8 Labium and Self-sustained Oscillation . . . . . . . . . 237
7.4.9 The Labium as a Raynolds-AveragedNavier-Stokes
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.4.10 The k−(cid:2) Model . . . . . . . . . . . . . . . . . . . . . 244
7.4.11 RANS Finite-Element Model of the Flute . . . . . . . 247
7.4.12 Self-organizationOrder Parameter . . . . . . . . . . . 252
7.4.13 Synchronization of Organ Pipes . . . . . . . . . . . . . 253
7.4.14 Shock Waves in Brass Instruments . . . . . . . . . . . 254
7.5 Piano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7.6 Singing Voice . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.6.1 Onset Bifurcation . . . . . . . . . . . . . . . . . . . . 261
7.6.2 Vocal Folds Oscillation as Hopf Bifurcation . . . . . . 263
7.6.3 Two-Mass Model of Vocal Folds . . . . . . . . . . . . . 265
7.6.4 Asymmetric Vocal Folds: Rough Voice and
Subharmonics . . . . . . . . . . . . . . . . . . . . . . . 266
7.6.5 Hysteresis Loop of Singing Onset and Offset. . . . . . 267
7.6.6 Global Fractal Dimensions of Vocal Fold
Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 270
7.7 Percussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
7.7.1 Mode Coupling in Gongs, Cymbals, or Genders . . . . 274
7.7.2 Subharmonics and Chaotic Motion in Gongs
and Cymbals . . . . . . . . . . . . . . . . . . . . . . . 276
7.7.3 Pitch Glides with Gongs . . . . . . . . . . . . . . . . . 278
7.7.4 Pitch Glides in Drums . . . . . . . . . . . . . . . . . . 282
7.7.5 Friction Instruments . . . . . . . . . . . . . . . . . . . 282
7.7.6 Energy Transfer from Drum Membranes to Their
Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
VIII Contents
8 Impulse Pattern Formulation 285
8.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
8.1.1 Instantaneous Interaction – Fix Point . . . . . . . . . 287
8.1.2 Delayed Interaction . . . . . . . . . . . . . . . . . . . 288
8.1.3 Multi Delayed Interaction . . . . . . . . . . . . . . . . 290
8.2 Transient Behaviour . . . . . . . . . . . . . . . . . . . . . . . 294
8.2.1 Constant Control Parameter . . . . . . . . . . . . . . 294
8.2.2 Changing Control Parameter . . . . . . . . . . . . . . 294
9 Examples of Impulse Pattern Formulation 297
9.1 Guitar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.1.1 Impulse Patterns . . . . . . . . . . . . . . . . . . . . . 297
9.2 Violin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
9.2.1 Impulse Pattern . . . . . . . . . . . . . . . . . . . . . 303
9.2.2 Violin Ribs . . . . . . . . . . . . . . . . . . . . . . . . 303
9.2.3 Violin String . . . . . . . . . . . . . . . . . . . . . . . 304
9.3 Reed Instruments . . . . . . . . . . . . . . . . . . . . . . . . . 306
9.3.1 Impulse Pattern . . . . . . . . . . . . . . . . . . . . . 306
9.3.2 Multiphonics . . . . . . . . . . . . . . . . . . . . . . . 306
9.4 Flute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
9.4.1 Stability Analysis. . . . . . . . . . . . . . . . . . . . . 310
9.5 Sound Production . . . . . . . . . . . . . . . . . . . . . . . . 311
9.5.1 Trumpet . . . . . . . . . . . . . . . . . . . . . . . . . . 313
9.5.2 Trombone . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.5.3 Violin . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
IV Music Psychology 317
10 Psychoacoustics 319
10.1 Logic of Psychoacoustics . . . . . . . . . . . . . . . . . . . . . 319
10.2 Auditory Time Scale . . . . . . . . . . . . . . . . . . . . . . . 323
11 Timbre 329
11.1 Timbre as Multidimensional Space . . . . . . . . . . . . . . 330
11.2 Instrument Identification . . . . . . . . . . . . . . . . . . . . 351
11.3 Self-organized Neural Maps . . . . . . . . . . . . . . . . . . 351
11.4 Initial Transients and Formants . . . . . . . . . . . . . . . . 353
11.5 Articulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
11.6 Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
11.7 Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
11.8 Profile Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 357
11.9 NeurologicalCorrelates of Timbre . . . . . . . . . . . . . . . 359
11.10 Efficient Auditory Coding . . . . . . . . . . . . . . . . . . . 362
Contents IX
11.11 Free-Energy Model of Perception . . . . . . . . . . . . . . . 366
11.12 Material Properties . . . . . . . . . . . . . . . . . . . . . . . 372
11.13 Conclusions - Structure of Timbre. . . . . . . . . . . . . . . 373
12 Rhythm 381
12.1 Time-Keeper Model . . . . . . . . . . . . . . . . . . . . . . . 382
12.2 Synergetic Model of Finger Movements . . . . . . . . . . . . 383
12.3 Polyrhythms . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
12.4 Fractal Bownian Motion Model of Long-Term Rhythmic
Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
12.5 Continuous and Discrete Movements. . . . . . . . . . . . . . 393
12.6 Rhythm and Timbre. . . . . . . . . . . . . . . . . . . . . . . 396
12.7 Participatory Discrepancies . . . . . . . . . . . . . . . . . . . 396
12.8 Resonance Model of Rhythm Perception . . . . . . . . . . . 397
12.9 Neuronal Representation of Rhythm Perception . . . . . . . 397
13 Pitch, Melody, Tonality 403
13.1 Pitch Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
13.2 Pitch as Tonal Fusion . . . . . . . . . . . . . . . . . . . . . . 405
13.3 The Logic of Tonal Systems . . . . . . . . . . . . . . . . . . 408
13.4 Tonality as Pitch and Interval Counting . . . . . . . . . . . . 413
13.5 Tonality as Gestalt Principles . . . . . . . . . . . . . . . . . 414
13.6 Neural Net Models of Tonality . . . . . . . . . . . . . . . . . 414
13.6.1 Self-Organizing Maps . . . . . . . . . . . . . . . . . . 415
13.6.2 Hidden-Layer Neural Nets . . . . . . . . . . . . . . . 419
13.7 Bayes Models of Tonality . . . . . . . . . . . . . . . . . . . . 425
13.8 Fractal CorrelationDimension as Music
Event Density . . . . . . . . . . . . . . . . . . . . . . . . . . 426
13.8.1 Example 1: Ligeti – Musica Ricercata VIII, Piano
Sonata Galamb Boring . . . . . . . . . . . . . . . . . 427
13.8.2 Example 2: Even Parker – Broken Wing . . . . . . . 432
14 CD Tracks 439
14.1 CD Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
14.1.1 Audio Tracks. . . . . . . . . . . . . . . . . . . . . . . 439
14.1.2 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . 439
14.1.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . 441
Index 443
List of Figures
2.1 WaveletTransformofaguitartoneforthe first100msfrom
20 Hz to 4 KHz. The shape of the attack is rounded by the
changing scaling length of the integration. . . . . . . . . . . 15
2.2 Wavelettransformofanartificiallybuiltsignalconsistingof
three sinusodials of 250 Hz, 1000 Hz, and 3000 Hz, lasting
100ms. The amplitude values of the frequencies are stable
anddonotfluctuate overtime, asappearswith the Wavelet
plot of the guitar, shown above.. . . . . . . . . . . . . . . . 16
2.3 Correlogramofapuresinewaveof300Hzfromatimeseries
of 100ms. A large repetition window is chosen to show the
repetition behaviour of the calculation. In later analysis we
only need to look at one periodicity of the sound. . . . . . . 18
2.4 Correlogram of a white noise signal from a time series of
100ms. Again, a large repetition window is chosen to show
large scale behaviour of the calculation (see text for details). 19
2.5 Correlogramof a classicalguitar tone of1 s length. The ini-
tialtransientisclearlynoisy,wheretheperiodicityincreases
exponentially to a steady value. . . . . . . . . . . . . . . . . 20
2.6 Correlogram of the initial transient of the classical guitar
tone of Fig. 2.5, first 100 ms . . . . . . . . . . . . . . . . . 21
2.7 Correlogramsimilarityfunction SC for the sinusodialsound
used above in Fig. 2.3. As all correlogram vectors SC are
ti
equal,allpossibletimecorrelationsshowaperfectsimilarity
of SC =1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
ti
2.8 Correlogramsimilarityfunction SC forthe noisesignalused
beforeinFig.2.4.Thevectorsaremuchlesscorrelatedasin
Fig.2.7.Stillofcoursethe scalarproductofeachtime point
vector with itself is SC =1. . . . . . . . . . . . . . . . . 23
ti,ti
2.9 Correlogram similarity function SC for the guitar tone of
Fig. 2.5 in two plot style versions of the same data of SC. . 24
2.10 Correlogram similarity function SC for the initial transient
of the guitar tone of the correlogramof Fig. 2.6 in two plot
style versions of the same data of SC . . . . . . . . . . . . . 25
XII List of Figures
2.11 Time dependent spectral centroid of a pure sine wave of
300 Hz, calculated by a Wavelet Transform of a 100 ms
constructed time series(for Wavelet Transforms see section
above) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.12 Time dependent spectral centroid of white noise, calculated
by a Wavelet Transform of a 100 ms time series . . . . . . 28
2.13 Time developmentof the spectral centroid of a plucked gui-
tar tone of 1 s, where Fig. 2.14 is the initial transient . . . 29
2.14 Time development of the spectral centroid from a Wavelet
Transform of an initial transient of a plucked guitar tone of
100 ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.15 Spectral Centroid Integration Path of the spectral centroid
curveofFig.2.11fora300Hzsinusodial.Asnofluctuations
appear, the integration up to the highest level is 300 Hz
everywhere. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.16 Spectral Centroid Integration Path of the spectral centroid
curve of Fig.2.12 for white noise. Here up to the highest
levelthefluctuationpathdoesnotsmoothoutandfluctuates
constantly. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.17 Spectral Centroid Integration path of the spectral centroid
curve of Fig. 2.13 of a plucked guitar tone . . . . . . . . . 32
2.18 Spectral Centroid Integration path of the spectral centroid
curve of Fig. 2.14 of the initial transient for the guitar tone
in Fig. 2.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.19 Coefficientsofd(u)forthedifferenceequationofeightsorder
in E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
mic
2.20 Array plot of D for a difference equation of eights order in
E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
mic
2.21 Gestaltpatternofa)a sinewaveof300Hz,b)a white noise
signal, c) a guitar tone of 1 second and d) the same guitar
tone within its initial transient of 100 ms as a force plots of
the spectral centroid curves in comparison to the centroid
curves itself and an averaged centroid curves (see text for
details) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.22 Time dependent spectralentropy of a pure sine wave of 300
Hz, calculated from a Wavelet Transform of a 100 ms time
series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.23 Time dependent spectral entropy of white noise, calculated
from a Wavelet Transform of a 100 ms time series . . . . . 39
2.24 Time dependent spectral entropy of the guitar tone used in
the spectral centroid section. . . . . . . . . . . . . . . . . . 41
2.25 Timedependentspectralentropyoftheinitialtransient(100
ms) of the guitar tone used of Fig. 2.24 . . . . . . . . . . . 41
2.26 Temporalintegrationpathofspectralentropyofthe 300Hz
sinusodial . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42