Science of String Instruments (eBook)

Tom Rossing has taught musical acoustics for nearly 50 years, and has been active in research in this area for at least 30 years. In 1992 he was awarded the Silver Medal in Musical Acoustics by the Acoustical Society of America, and his biography is included in The New Grove Dictionary of Music and Musicians. He is also the editor of the 2006 Springer Handbook of Acoustics. In 2009 he was awarded the ASA Gold Medal by the Acoustical Society of America for contributions to musical acoustics, leadership in science education, and service to the Society.

The Science of String Instruments 3
Contents 5
Contributors 7
Chapter 1: Introduction 9
1.1 A Brief History of the Science of String Instruments 9
1.1.1 Bowed String Instruments 10
1.1.2 Lutes and Guitars 11
1.1.3 Harpsichords, Clavichords, and Dulcimers 13
1.1.4 Piano 14
1.1.5 Electric and Virtual String Instruments 15
1.2 Modal Analysis of String Instruments 15
1.2.1 Experimental Modal Testing 15
1.2.2 Mathematical Modal Analysis 16
1.2.3 Sound Field Analysis 17
1.2.4 Holographic Modal Analysis 17
References 17
Chapter 2: Plucked Strings 19
2.1 Transverse Waves on a String 19
2.1.1 Impulsive Waves, Reflection, and Interference 20
2.1.2 Standing Waves 20
2.2 Plucked String: Time and Frequency Analyses 22
2.3 Force Exerted by the String 23
2.4 Plucking 24
References 26
Chapter 3: Guitars and Lutes 27
3.1 Acoustic Guitars 27
3.1.1 The Guitar as a System of Coupled Vibrators 28
3.1.2 Force Exerted by the Vibrating String 28
3.1.3 Frequency Response of Guitars 29
3.2 Vibrations of the Guitar Body 30
3.2.1 Normal Modes of Vibration 30
3.2.2 Modes of Component Parts 31
3.2.3 Coupling of the Top Plate to the Air Cavity: Two-Oscillator Model 34
3.2.4 Coupling to the Back Plate: Three-Oscillator Model 35
3.2.5 Low-Frequency Resonances of a Guitar Body 35
3.2.6 Modal Shapes 36
3.3 String Forces 37
3.4 Sound Radiation 38
3.5 Quality 40
3.5.1 Influence of Design and Construction 41
3.5.2 The Bridge 41
3.5.3 Thickness of the Top Plate and Braces 41
3.5.4 Asymmetrical and Radial Bracing 42
3.6 A Family of Scaled Guitars 43
3.7 Synthetic Materials 45
3.8 Other Families of Guitars 45
3.9 Electric Guitars 45
3.9.1 Body Vibrations and Dead Spots 47
3.9.2 Electric Bass 47
3.10 Lutes 48
3.10.1 Acoustics of the European Short Lute 49
3.10.2 Acoustics of the Turkish Long-Necked Lute 49
3.11 Concluding Remarks 52
References 52
Chapter 4: Portuguese Guitar 54
4.1 Origins 54
4.2 Types and Characteristics 55
4.3 Vibroacoustic Behavior 56
4.4 Subjective Acoustical Quality Evaluation 59
4.4.1 Objective Parameters 60
4.4.2 Listening Tests 60
4.4.3 Test Conditions 61
4.4.3.1 Subjective Parameters Used 61
4.4.3.2 Conditions of the Guitars 62
4.4.4 Test Procedure 62
4.5 Results 63
4.5.1 Subjective Tests 63
4.5.2 Objective Tests 63
References 64
Chapter 5: Banjo 65
5.1 Introduction 65
5.2 Banjo Anatomy 66
5.3 Banjo Sound 67
5.4 Head Modes 71
5.5 Harmonics Analysis 73
5.6 Resonators 74
5.7 Bridges 77
5.8 Tone Rings, Rims, and Neck 79
5.9 Summary 80
References 81
Chapter 6: Mandolin Family Instruments 82
6.1 Introduction 82
6.2 Types of Mandolins 83
6.2.1 Neapolitan Mandolins 84
6.2.2 Flatback Mandolins 85
6.2.3 Cylinderback Mandolins and Other Unique Designs 86
6.2.4 Archtop Mandolins, Oval Sound Hole 86
6.2.5 Archtop Mandolins, f-Holes 86
6.2.6 Mandolas, Octave Mandolins, and Mandocellos 87
6.3 Normal Modes of Vibration and Holographic Interferometry 88
6.4 Normal Mode Shapes in Mandolins 89
6.5 Normal Mode Frequencies in Different Types of Mandolins 92
6.6 Sustain in Mandolins 97
6.7 Other Mandolin Family Instruments: Normal Modes in Two Mandolas 100
6.8 Mandocellos 101
6.9 Summary and Conclusions 102
References 102
Chapter 7: Psalteries and Zithers 104
7.1 Introduction 104
7.2 Influence of Stresses in Strings on the Instrument´s Shape 105
7.3 Plucking Stiffness, and Strength of a Plucked String 105
7.4 String Materials 106
7.5 Acoustical Study of Carved Baltic Psalteries 107
7.5.1 History of the Carved Baltic Psaltery 108
7.5.2 Playing Techniques 109
7.5.3 Body Resonances of Some Carved Baltic Psalteries 110
7.5.4 Coupling of Strings to Body Resonances 114
7.5.5 Experiments with Distribution of Sound Holes 116
7.5.6 Some Conclusions and Applications 117
7.5.7 Features of Proposed New Traditional-Style Designs 118
7.5.8 A More Radical Design from Finland 119
7.6 Zithers 120
7.6.1 Zithers Without Fretboard 120
7.6.2 Fretted (Alpine) Zithers 121
7.7 Hammered Dulcimers 122
7.8 Modernized Baltic Psalteries 122
7.8.1 Diatonically Tuned Versions 122
7.8.2 Chromatic Baltic Psalteries 123
References 126
Chapter 8: Harpsichord and Clavichord 128
8.1 Introduction 128
8.2 The Harpsichord 129
8.2.1 General Design 129
8.2.2 Plucked Strings 132
8.2.3 Soundboard and Radiation 135
8.2.4 Acoustic Balance 137
8.2.5 Design Extensions 139
8.3 The Clavichord 141
8.3.1 General Design 141
8.3.2 String Excitation in the Clavichord 144
8.4 Keyboard Tuning 145
8.5 Conclusion 147
References 148
Chapter 9: Harp 149
9.1 Introduction 149
9.2 Overview 149
9.2.1 Origins and Development 149
9.2.2 Structure 150
9.3 Strings 154
9.3.1 History 154
9.3.1.1 Diatonic Versus Chromatic Stringing 155
9.3.1.2 Sharping Mechanisms 155
9.3.2 Basic String Considerations 156
9.3.3 String Motion and Its Influence on the Sound Spectrum 157
9.3.4 String Motion and Temporal Development of the Sound 160
9.4 Soundboard and Soundbox 160
9.4.1 Evolution of the Soundboard 160
9.4.2 Vibrational Behavior of the Soundboard 162
9.4.3 Helmholtz and Pipe Resonances of the Soundbox 162
9.4.4 Vibroacoustic Behavior of the Soundbox 164
9.5 The Harp as a Whole 167
9.5.1 Strings and Soundbox 167
9.5.2 Sound Radiation 168
9.5.3 The Sound of the Harp 169
9.6 Conclusion 169
References 170
Chapter 10: Burmese Arched Harp 171
10.1 History 171
10.2 Construction and Playing Techniques 172
10.3 Scales and Tunings 173
10.4 Measurements of Plucked Tones 174
References 175
Chapter 11: Plucked String Instruments in Asia 176
11.1 Classification of Asian Musical Instruments Based on Construction Material 176
11.2 Japanese Satsuma Biwa 179
11.2.1 Structural Response 181
11.2.2 Sawari Mechanisms and Their Effects on High-Frequency Emphasis 183
11.2.3 Examples of Characteristic Sounds 188
11.2.4 Brief Comparison with the Chinese Pipa 189
11.3 Japanese Shamisen 189
11.3.1 Shamisen as an Overall String-Bridge-Membrane System 190
11.3.2 Sawari and Its Effect on the Tuning 192
11.4 Japanese Koto and Korean Gayageum 193
11.5 Concluding Remarks 196
References 196
Chapter 12: Bowed Strings 199
12.1 Kinematics of the Bowed String 199
12.2 Dynamics of the Bowed String 204
12.3 Bowing to Achieve Anomalous Low Frequencies 205
References 209
Chapter 13: Violin 211
13.1 History 211
13.2 Research 212
13.3 Evaluating Violins 213
13.4 Sound Analysis 214
13.5 Frequency Response 215
13.6 Tone Quality 215
13.6.1 Sizzle 217
13.6.2 Directional Tone Color 217
13.6.3 Projection 218
13.7 Playability 219
13.7.1 Helmholtz Motion 220
13.7.2 Bow Force Limits 221
13.7.3 Damping and Playability 222
13.8 Violin Body Vibrations 222
13.8.1 Normal Modes of Vibration 223
13.8.2 Vibrational Models 224
13.8.3 A Three-Dimensional Model of Vibration 226
13.8.4 Modal Analysis 226
13.8.5 What Modes Can a Maker Control? 227
13.9 Component Parts 228
13.9.1 Top and Back Plates 228
13.9.2 Tap Tones 229
13.9.3 The Mass of a Violin 231
13.9.4 Enclosed Air 232
13.9.5 Bridge 232
13.9.6 Ribs 235
13.9.7 Fingerboard 235
13.9.8 Bass bar and Soundpost 236
13.10 Measuring Sound Radiation 237
13.11 Low-Frequency Radiation 240
13.12 High-Frequency Radiation 241
13.13 Radiation Damping 242
13.14 Electric and Virtual Violins 243
References 244
Chapter 14: Cello 247
14.1 The Cello 247
14.2 Modal Analysis of Cellos 249
14.2.1 Frequency Response 249
14.2.2 Modes of Vibration 250
14.2.3 Observing the Modes 251
14.2.4 Labeling the Resonances 251
14.3 Modes of Component Parts 252
14.3.1 Cello Plate Modes 252
14.3.2 Cello Air Cavity Modes 252
14.4 Cello Body Modes 253
14.4.1 Comparison with Violin Resonances 256
14.5 Sound Spectra of the Cello 256
14.6 Mobility (Admittance) at the Bridge 257
14.7 The ``New Violin Family´´ 258
14.8 Conclusion 258
References 258
Chapter 15: Double Bass 260
15.1 Modes of Vibration 260
15.1.1 The Modes in Playing 261
15.1.2 Mobility Curves and Instrument Identity 262
15.2 The Double Bass Compared to the Violin and Cello 264
15.3 Double Basses of Different Quality 265
15.4 The Violin Octet 269
15.5 The Player´s Support 270
15.6 Scaling 271
15.7 Body Size and Radiated Sound 272
15.8 Stage Risers 274
15.9 Directional Radiation 276
15.10 Further Reading 278
References 278
Chapter 16: Bows, Strings, and Bowing 279
16.1 The Bow 279
16.1.1 Effect of Camber on Transverse Hair Stiffness 279
16.1.2 Wood 281
16.1.3 Tonal Quality 281
16.1.4 Effect of Hair Elasticity and Surface Roughness 282
16.1.5 Rosin/Friction 282
16.2 Strings 283
16.2.1 The Concept of Wave Resistance or Wave Impedance 283
16.2.2 Tension 284
16.2.3 Damping 285
16.2.4 Torsion 287
16.3 Bowing Techniques 287
16.3.1 The Main Three Bowing Parameters 287
16.3.2 Flautando 290
16.3.3 Harmonics 290
16.3.4 Harmonics and Intonation 291
16.3.5 Double Stops 292
16.3.6 Tone Onsets, Attacks 292
16.3.7 Détaché 294
16.3.8 Martelé 295
16.3.9 Light Bowing 295
16.3.10 Spiccato/Sautillé/Ricochet 295
16.3.11 Bouncing Rate 296
16.3.12 Parameters That Affect the String´s Spectrum 297
References 299
Chapter 17: Viols and Other Historic Bowed String Instruments 300
17.1 Medieval Bowed String Instruments 301
17.1.1 Medieval Fiddles 301
17.1.2 Rebecs 302
17.1.3 Acoustics of Medieval Bowed String Instruments 303
17.1.3.1 Acoustical Properties of the Medieval Fiddle 304
17.1.3.2 Acoustical Properties of the Rebec 305
17.2 Renaissance Viols 306
17.2.1 The Development of the Renaissance Viol 306
17.2.2 Acoustics of Renaissance Viols 308
17.3 Baroque Viols 309
17.3.1 Development of the Baroque Viol 310
17.3.2 Acoustics of the Baroque Viol 311
17.3.2.1 The Baroque Treble Viol 311
17.3.2.2 The Baroque Tenor Viol 312
17.3.2.3 The Baroque Bass Viol 312
References 313
Chapter 18: The Hutchins-Schelleng Violin Octet After 50 Years 315
18.1 Introduction 315
18.2 Brief Octet History 317
18.2.1 Identification of Important Resonances 318
18.3 What Do We Know Now? 319
18.3.1 Summary of Octet-Related Developments, 1964-2007 319
18.3.2 How Bowed-String Instruments Radiate 322
18.3.3 Where Do Materials Come in? 323
18.3.4 A1 Radiation in the B1 Region 324
18.4 Scaling Basics 325
18.4.1 Scaling Assumptions 325
18.4.2 The Practicalities 326
18.4.3 Flat Plate Scaling Equations 327
18.4.4 Important A0 Scaling Equation Modification 328
18.4.5 Similarity of Shape 329
18.5 Modal and Acoustical Analyses 330
18.5.1 Modal and Acoustical Analyses of the Octet 331
18.5.2 A0 and A1: Coupling 334
18.5.3 Wall Compliance and Cavity Mode Frequencies 335
18.5.4 Rib Heights and Pressure Ratios 336
18.5.5 Clarifying A1 Status 336
18.5.6 Fat Bottoms, Wall Compliance, and Pressure Ratios 338
18.6 Future of the Violin Octet 340
18.7 Conclusions 341
References 342
Chapter 19: Hammered Strings 344
19.1 Dynamics of the Hammer-String Interaction 344
19.2 Piano Hammers 346
19.3 String Excitation by a Piano Hammer 347
19.4 Hammer Position on the String 348
19.5 String Excitation in a Hammered Dulcimer 349
References 349
Chapter 20: Some Remarks on the Acoustics of the Piano 350
20.1 Introduction 350
20.2 History of the Instrument 350
20.3 Overall Design 352
20.4 Vibrating Strings 353
20.5 The Hammers 357
20.6 The Soundboard as a Speaker 360
20.7 How We Perceive Piano Tones 364
20.8 Modeling of the Piano 365
20.9 Lessons 366
References 367
Chapter 21: Hammered Dulcimer 368
21.1 History 368
21.2 The Basic Instrument 369
21.3 Inharmonicity and Scaling 371
21.4 Lateral Stability 372
21.5 Instrument Warp 373
21.6 Tuning Stability 375
21.6.1 Tuning Stability: Temperature 375
21.6.2 Tuning Stability: Humidity 377
21.6.3 String-Bridge Friction 379
21.7 The Percussive Sound: Hammer and String Interaction 379
21.8 Hammers and Course Spacing 381
21.9 String Coupling and Resonance Time 382
21.10 Sound Board and Body Modes 384
21.11 Sound Board Materials, Back Plates, and Design 386
21.12 Bridges, Bridge Caps, and Bridge Vibrations 386
21.13 Pin Blocks, Pins, and Hitch Pins 387
21.14 Sound Radiation Patterns 388
21.15 Unimportant Characteristics: Sound Holes, Special Finishes, Peglegs, and Perfect Fifths 388
References 389
Chapter 22: Electric Guitar and Violin 390
22.1 Historical Background 391
22.2 The Electric Guitar 392
22.3 The Electric Violin 396
22.4 Acoustic, Magnetic, and Piezoelectric Pickups 399
22.4.1 Acoustic Pickups 399
22.4.2 Magnetic Pickups 400
22.4.3 Special Sound Effects 403
22.4.4 Piezoelectric Pickups 404
22.4.5 Other Types of Pickups 407
22.5 The Electric Violin as a Research Tool 408
22.5.1 Multiresonant Filter Characteristics 409
22.5.2 Sound Perception and Acoustical Properties 410
22.5.3 Real-Time Synthesis of Cremonese Instruments 411
References 412
23: Virtual String Synthesis 413
23.1 Introduction 413
23.2 Nomenclature 414
23.2.1 Digital Signals 415
23.2.1.1 Sampling 415
23.2.1.2 Sum of Sinusoids 416
23.2.2 Digital Filtering 418
23.3 Elements of Stringed Instruments 418
23.3.1 Vibrating String 419
23.3.1.1 D´Alembert´s Wave Equation 419
23.3.1.2 The Delay Line 421
23.3.1.3 Digital Waveguide Models 422
23.3.1.4 Natural Decay of the String 423
23.3.1.5 Modeling Two Planes of Vibration 424
23.3.1.6 Varying the Digital Waveguide 425
23.3.2 Plucking the String 426
23.3.2.1 Theoretical Plucks 426
23.3.2.2 Complexities of Real Plucks 427
23.3.3 Body Resonance 428
23.3.3.1 Driving-Point Admittance 428
23.3.3.2 Filtering with the Driving-Point Admittance 428
23.3.3.3 Bidirectional Interaction 429
23.3.3.4 String-Body Scattering Junction 430
23.3.4 Pressure Radiation 431
23.4 Measurements 432
23.4.1 String Vibration 432
23.4.2 Bridge Impedance 433
23.4.3 Body Vibration 434
23.4.4 Pressure Radiation 435
23.5 Parameter Estimation 435
23.5.1 Short-Time Fourier Transform 435
23.5.2 Excitation 436
23.5.2.1 The Statistical Spectral Interpolation Method 438
23.5.3 String 440
23.5.3.1 Loop Filter Estimation 440
23.5.4 Body Resonator 443
23.5.4.1 Low-Order Filter Implementations 445
23.5.5 Radiated Sound Pressure 446
23.5.5.1 Low-Order Filter Implementations 447
23.5.5.2 Combining and Interpolating Between Measurements 448
23.6 Summary and Conclusion 449
References 449
Index 452