The Physics of Musical Instruments

The history of musical instruments is nearly as old as the history of civilization itself, and the aesthetic principles upon which judgments of musical quality are based are intimately connected with the whole culture within which the instruments have evolved. An educated modem Western player or listener can make critical judgments about particular instruments or particular per­ formances but, to be valid, those judgments must be made within the appro­ priate cultural context. The compass of our book is much less sweeping than the first paragraph might imply, and indeed our discussion is primarily confined to Western musical instruments in current use, but even here we must take account of centuries of tradition. A musical instrument is designed and built for the playing of music of a particular type and, conversely, music is written to be performed on particular instruments. There is no such thing as an "ideal" instrument, even in concept, and indeed the unbounded possibilities of modem digital sound-synthesis really require the composer or performer to define a whole set of instruments if the result is to have any musical coherence. Thus, for example, the sound and response of a violin are judged against a mental image of a perfect violin built up from experience of violins playing music written for them over the centuries. A new instrument may be richer in sound quality and superior in responsiveness, but if it does not fit that image then it is not a better violin.

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I Vibrating Systems.- 1 Free and Forced Vibrations of Simple Systems.- 1.1. Simple Harmonic Motion in One Dimension.- 1.2. Complex Amplitudes.- 1.3. Superposition of Two Harmonic Motions in One Dimension.- 1.4. Energy.- 1.5. Damped Oscillations.- 1.6. Other Simple Vibrating Systems.- 1.7. Forced Oscillations.- 1.8. Transient Response of an Oscillator.- 1.9. Two-Dimensional Harmonic Oscillator.- 1.10. Graphical Representations of Vibrations: Lissajous Figures.- 1.11. Normal Modes of Two-Mass Systems.- 1.12. Nonlinear Vibrations of a Simple System.- A.1. Alternative Ways of Expressing Harmonic Motion.- A.2. Equivalent Electrical Circuit for a Simple Oscillator.- References.- 2 Continuous Systems in One Dimension: Strings and Bars.- 2.1. Linear Array of Oscillators.- 2.2. Transverse Wave Equation for a String.- 2.3. General Solution of the Wave Equation: Traveling Waves.- 2.4. Reflection at Fixed and Free Ends.- 2.5. Simple Harmonic Solutions to the Wave Equation.- 2.6. Standing Waves.- 2.7. Energy of a Vibrating String.- 2.8. Plucked String: Time and Frequency Analyses.- 2.9. Struck String.- 2.10. Bowed String.- 2.11. Driven String: Impedance.- 2.12. Motion of the End Supports.- 2.13. Damping.- 2.14. Longitudinal Vibrations of a String or Thin Bar.- 2.15. Bending Waves in a Bar.- 2.16. Bars with Fixed and Free Ends.- 2.17. Vibrations of Thick Bars: Rotary Inertia and Shear Deformation.- 2.18. Vibrations of a Stiff String.- 2.19. Dispersion in Stiff and Loaded Strings: Cutoff Frequency.- 2.20. Torsional Vibrations of a Bar.- References.- 3 Two-Dimensional Systems: Membranes and Plates.- 3.1. Wave Equation for a Rectangular Membrane.- 3.2. Square Membranes: Degeneracy.- 3.3. Circular Membranes.- 3.4. Real Membranes: Stiffness and Air Loading.- 3.5. Waves in a Thin Plate.- 3.6. Circular Plates.- 3.7. Elliptical Plates.- 3.8. Rectangular Plates.- 3.9. Square Plates.- 3.10. Square and Rectangular Plates with Clamped Edges.- 3.11. Rectangular Wood Plates.- 3.12. Bending Stiffness in a Membrane.- 3.13. Shallow Spherical Shells.- 3.14. Nonlinear Vibrations in Plates and Shallow Shells.- 3.15. Driving Point Impedance.- References.- 4 Coupled Vibrating Systems.- 4.1. Coupling Between Two Identical Vibrators.- 4.2. Normal Modes.- 4.3. Weak and Strong Coupling.- 4.4. Forced Vibrations.- 4.5. Coupled Electrical Circuits.- 4.6. Forced Vibration of a Two-Mass System.- 4.7. Systems with Many Masses.- 4.8. Graphical Representation of Frequency Response Functions.- 4.9. Vibrating String Coupled to a Soundboard.- 4.10. Two Strings Coupled by a Bridge.- A.1. Structural Dynamics and Frequency Response Functions.- A.2. Modal Analysis.- A.3. Finite Element Analysis.- References.- 5 Nonlinear Systems.- 5.1. A General Method of Solution.- 5.2. Illustrative Examples.- 5.3. The Self-Excited Oscillator.- 5.4. Multimode Systems.- 5.5. Mode Locking in Self-Excited Systems.- References.- II Sound Waves.- 6 Sound Waves in Air.- 6.1. Plane Waves.- 6.2. Spherical Waves.- 6.3. Sound Pressure Level and Intensity.- 6.4. Reflection, Diffraction, and Absorption.- 6.5. Acoustic Components at Low Frequencies.- References.- 7 Sound Radiation.- 7.1. Simple Multipole Sources.- 7.2. Pairs of Point Sources.- 7.3. Arrays of Point Sources.- 7.4. Radiation from a Spherical Source.- 7.5. Line Sources.- 7.6. Radiation from a Plane Source in a Baffle.- 7.7. Unbaffled Radiators.- References.- 8 Pipes and Horns.- 8.1. Infinite Cylindrical Pipes.- 8.2. Wall Losses.- 8.3. Finite Cylindrical Pipes.- 8.4. Radiation from a Pipe.- 8.5. Impedance Curves.- 8.6. Horns.- 8.7. Finite Conical and Exponential Horns.- 8.8. Bessel Horns.- 8.9. Compound Horns.- 8.10. Perturbations.- 8.11. Numerical Calculations.- 8.12. The Time Domain.- References.- III String Instruments.- 9 Guitars and Lutes.- 9.1. Design and Construction of Guitars.- 9.2. The Guitar as a System of Coupled Vibrators.- 9.3. Force Exerted by the String.- 9.4. Modes of Vibration of Component Parts.- 9.5. Coupling of the Top Plate to the Air Cavity: Two-Oscillator Model.- 9.6. Coupling to the Back Plate: Three-Oscillator Model.- 9.7. Resonances of a Guitar Body.- 9.8. Response to String Forces.- 9.9. Sound Radiation.- 9.10. Resonances, Radiated Sound, and Quality.- 9.11. Electric Guitars.- 9.12. Frets and Compensation.- 9.13. Lutes.- 9.14. Other Plucked String Instruments.- References.- 10 Bowed String Instruments.- 10.1. A Brief History.- 10.2. Research on Violin Acoustics.- 10.3. Construction of the Violin.- 10.4. Motion of Bowed Strings.- 10.5. Violin Body Vibrations.- 10.6. The Bridge.- 10.7. Sound Radiation.- 10.8. The Bow.- 10.9. The Wolf Tone.- 10.10. Tonal Quality of Violins.- 10.11. Viola, Cello, and Double Bass.- 10.12. Viols.- 10.13. A New Violin Family.- References.- 11 Harps, Harpsichords, and Clavichords.- 11.1. The Koto.- 11.2. The Harp.- 11.3. The Harpsichord.- 11.4. Harpsichord Design Considerations.- 11.5. Harpsichord Characteristics.- 11.6. The Clavichord.- References.- 12 The Piano.- 12.1. General Design of Pianos.- 12.2. Piano Action.- 12.3. Piano Strings.- 12.4. String Excitation by the Hammer.- 12.5. The Soundboard.- 12.6. Sound Decay: Interaction of Strings, Bridge, and Soundboard.- 12.7. Tuning and Inharmonicity.- 12.8. Timbre.- 12.9. Electric Pianos.- References.- IV Wind Instruments.- 13 Sound Generation by Reed and Lip Vibrations.- 13.1. Basic Reed Generators.- 13.2. Detailed Discussion of a Reed Generator.- 13.3. Effect of Reservoir Impedance.- 13.4. Reed Generators Coupled to Horns.- 13.5. Nonlinearity.- 13.6. Time-Domain Approach.- References.- 14 Lip-Driven Brass Instruments.- 14.1. Historical Development of Brass Instruments.- 14.2. Horn Profiles.- 14.3. Mouthpieces.- 14.4. Radiation.- 14.5. Slides and Valves.- 14.6. Small-Amplitude Nonlinearity.- 14.7. Large-Amplitude Nonlinearity.- 14.8. Input Impedance Curves.- 14.9. Transients.- 14.10. Acoustic Spectra.- 14.11. Mutes.- 14.12. Performance Technique.- References.- 15 Woodwind Reed Instruments.- 15.1. Woodwind Bore Shapes.- 15.2. Finger Holes.- 15.3. Impedance Curves.- 15.4. Reed and Air Column Interaction.- 15.5. Directionality.- 15.6. Performance Technique.- 15.7. The Clarinet.- 15.8. The Oboe.- 15.9. The Bassoon.- 15.10. The Saxophone.- 15.11. Construction Materials.- References.- 16 Flutes and Flue Organ Pipes.- 16.1. Dynamics of an Air Jet.- 16.2. Disturbance of an Air Jet.- 16.3. Jet-Resonator Interaction.- 16.4. The Regenerative Excitation Mechanism.- 16.5. Jet Drive Nonlineari…