xiv
Contents
5.3.1
Analysis of Brass Performance Using Simulations . . . . . . . . . . 233
5.3.2
Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.4 Going Further: From Linear Stability Analysis to Oscillation
Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
5.4.1
Introduction: A Van der Pol Self-Sustained Oscillator . . . . . . 241
5.4.2
State-Space Representations of the Elementary Brass
Playing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.4.3
Linear Stability Analysis Applied to Brass Instruments . . . . . 251
5.4.4
The Trombone Pedal Note Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
5.4.5
Bifurcation Diagrams of Reed and Brass Instruments . . . . . . . 257
5.4.6
Multiphonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
6 Shocks and Surprises: Refining the Elementary Model . . . . . . . . . . . . . . . . . 271
6.1 Why Brass Instruments Sound Brassy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.1.1
Brassy Sounds in Music. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
6.1.2
Experimental Evidence for Shock Waves in Brass
Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
6.1.3
To Infinity and Beyond: Nonlinear Propagation in Tubes . . . 276
6.1.4
The Brassiness Potential Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 280
6.1.5
Elephants, Exhausts and Angels: Some Surprising
Sources of Brassy Sounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
6.2 Going Further: Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
6.2.1
From the Fundamental Fluid Dynamic Equations to
the Nonlinear Wave Propagation Equation . . . . . . . . . . . . . . . . . . . 288
6.2.2
The Burgers Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
6.2.3
Brassiness of Flaring Bells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
6.3 The Player’s Windway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
6.3.1
Coupling of Upstream and Downstream Resonances . . . . . . . 296
6.3.2
Tuning of Windway Impedance Peaks . . . . . . . . . . . . . . . . . . . . . . . 298
6.3.3
Other Effects of the Player’s Windway . . . . . . . . . . . . . . . . . . . . . . . 299
6.3.4
Respiratory Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.4 Improving the Lip Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.4.1
Evidence from Mechanical Response Measurements . . . . . . . 302
6.4.2
Evidence from Measurements of Threshold Playing
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
6.4.3
Models with More Than One Degree of Freedom . . . . . . . . . . . 308
6.5 Playing Frequencies of Brass Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
6.6 The Influence of Wall Material on Brass Instrument Performance . . . 311
6.6.1
Factors Affecting the Choice of Wall Material. . . . . . . . . . . . . . . 312
6.6.2
Experimental Studies of Brass Instrument Wall Vibrations . 312
6.6.3
Pathological Wall Vibration Effects in Wind Instruments . . . 316
6.6.4
Frequency-Localised and Broadband Effects of
Structural Resonances in Brass Instruments . . . . . . . . . . . . . . . . . 317
6.6.5
Mechanical Vibration at the Lip-Mouthpiece Interface . . . . . 325
Contents
5.3.1
Analysis of Brass Performance Using Simulations . . . . . . . . . . 233
5.3.2
Bifurcation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.4 Going Further: From Linear Stability Analysis to Oscillation
Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
5.4.1
Introduction: A Van der Pol Self-Sustained Oscillator . . . . . . 241
5.4.2
State-Space Representations of the Elementary Brass
Playing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.4.3
Linear Stability Analysis Applied to Brass Instruments . . . . . 251
5.4.4
The Trombone Pedal Note Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
5.4.5
Bifurcation Diagrams of Reed and Brass Instruments . . . . . . . 257
5.4.6
Multiphonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
6 Shocks and Surprises: Refining the Elementary Model . . . . . . . . . . . . . . . . . 271
6.1 Why Brass Instruments Sound Brassy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
6.1.1
Brassy Sounds in Music. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
6.1.2
Experimental Evidence for Shock Waves in Brass
Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
6.1.3
To Infinity and Beyond: Nonlinear Propagation in Tubes . . . 276
6.1.4
The Brassiness Potential Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 280
6.1.5
Elephants, Exhausts and Angels: Some Surprising
Sources of Brassy Sounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
6.2 Going Further: Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
6.2.1
From the Fundamental Fluid Dynamic Equations to
the Nonlinear Wave Propagation Equation . . . . . . . . . . . . . . . . . . . 288
6.2.2
The Burgers Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
6.2.3
Brassiness of Flaring Bells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
6.3 The Player’s Windway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
6.3.1
Coupling of Upstream and Downstream Resonances . . . . . . . 296
6.3.2
Tuning of Windway Impedance Peaks . . . . . . . . . . . . . . . . . . . . . . . 298
6.3.3
Other Effects of the Player’s Windway . . . . . . . . . . . . . . . . . . . . . . . 299
6.3.4
Respiratory Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
6.4 Improving the Lip Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
6.4.1
Evidence from Mechanical Response Measurements . . . . . . . 302
6.4.2
Evidence from Measurements of Threshold Playing
Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
6.4.3
Models with More Than One Degree of Freedom . . . . . . . . . . . 308
6.5 Playing Frequencies of Brass Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
6.6 The Influence of Wall Material on Brass Instrument Performance . . . 311
6.6.1
Factors Affecting the Choice of Wall Material. . . . . . . . . . . . . . . 312
6.6.2
Experimental Studies of Brass Instrument Wall Vibrations . 312
6.6.3
Pathological Wall Vibration Effects in Wind Instruments . . . 316
6.6.4
Frequency-Localised and Broadband Effects of
Structural Resonances in Brass Instruments . . . . . . . . . . . . . . . . . 317
6.6.5
Mechanical Vibration at the Lip-Mouthpiece Interface . . . . . 325