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Title Page Copy PHYSICAL ACOUSTICS OF MUSIC PERCEPTION R.B. Moore McGill University © 1995 CONTENTS 1. INTRODUCTION 1 2. THE PERCEPTION OF SOUND IN A ROOM 5 2.1 The distinguishing of Direct Sound from Reflected Sound 5 2.2 The Perception of Sound Direction 7 2.3 The Perception of Reverberant Sound 8 2.3.1 The Importance of Reverberant Sound 8 2.3.2 Room Radius 9 3. CONCERT HALL ACOUSTICS 11 3.1 Desired Properties of Concert Halls 12 3.1.1 Reverberation Times 12 3.1.2 Variation of Reverberation Time with Frequency 13 3.1.3 The Importance of Early Sounds 14 3.1.4 Stage Geometry Considerations 15 3.2 Achieving Good Hall Acoustics 15 3.2.1 Diffuse Reflector Ceilings 15 3.2.2 Electroacoustical Apparatus 16 3.3 The Use of Computers in Concert Hall Acoustics 17 3.3.1 Diagnosing Concert Hall Acoustic Problems 17 3.3.2 Determining Listener Preference in Concert Hall Acoustics 17 3.3.3 Digital Processing of Sound 18 3.3.4 General Uses of Computers in Music 19 4. THE SOUND OF A POINT SOURCE IN A ROOM 23 4.1 The Direct Sound Wave 23 4.1.1 Plane Sound Waves 23 4.1.2 Analogy With Water Waves 26 4.1.3 Speed of Propagation of Sound in Air 27 4.1.4 The Connection Between Pressure and Air Velocity in a Sound Wave 28 4.1.5 Power in a Sound Wave 28 4.1.6 The dB Scale of Sound Intensity 31 4.1.7 The RMS Pressure in Sound 32 4.1.8 The Energy in a Sound Wave 32 4.1.9 Spherical Sound Waves 33 4.2 Sound Reflection 34 4.2.1 Plane Wave Reflection 34 4.2.2 Spherical Wave Reflection 36 4.3 Multiple Sound Reflections 36 4.4 Reverberant Sound 37 Appendix 41 ii Contents 5. THE SOUND OF A VIBRATING SURFACE 45 5.1 Polar Diagrams 46 5.1.1 Decibel Plots 46 5.1.2 Directivity Plots 47 5.2 Radiation Patterns from Some Standard Sources 48 5.2.1 The Radiation Pattern from a Vibrating Circular Surface 48 5.2.2 The Radiation Pattern from a Vibrating Cylindrical Segment 49 5.2.3 The Radiation Pattern from Musical Instruments 50 6. SOUND DIRECTION AND RELATIVE PHASE 53 6.1 The Superposition of Four Isotropic Sources 53 6.2 Directing Sound in a Desired Direction 55 6.3 Directional Microphones 56 6.4 Phase Relationships and Wave Direction 56 7. SOUND WAVE DIFFRACTION 59 7.1 The Treatment of Oscillations as Phasors 59 7.2 Phasor Treatment of a Four Speaker System 61 7.3 Phasor Treatment of a Dipole Radiator 62 7.4 Slit Diffraction 63 7.5 Diffraction Through a Circular Opening 64 7.6 Some Consequences of Wave Diffraction 65 8. THE RADIATION PATTERN OF MUSICAL INSTRUMENTS 67 9. THE FREQUENCY SPECTRUM OF SOUNDS 69 9.1 Frequency Components of the Human Voice 69 9.2 The Frequency Spectra of Some Simple Tones 70 9.2.1 A Pure Tone 70 9.2.2 A Square Wave-form Oscillation 70 9.2.3 A Triangular Wave-form Oscillation 70 9.3 TheSynthesis of Some Simple Tones 71 9.3.1 A Square Tone 71 9.3.2 A Triangular Tone plus Others 72 9.4 The Frequency Spectrum of Sharp Pulses 73 9.5 The Connection Between Time and Frequency Spectra 79 9.6 The Addition of a Bundle of Close Frequencies 80 9.7 The Fourier Transform 80 10. THE ORIGINS OF MUSICAL SOUNDS 83 10.1 The Helmholtz Oscillator 83 10.2 Standing Waves on a String 85 10.3 Standing Waves in Air in a Tube 87 10.4 The Modes of Vibration of Surfaces 89 10.5 The Modes of Vibration of Air in an Enclosure 91 10.6 Some General Aspects of Standing Waves; The Concept of Normal Modes 94 Appendix 97 Contents iii 11. THE GENERATION OF MUSICAL SOUNDS 99 11.1 The Excitation of Normal Modes by Resonance 99 11.2 The Growth of Normal Modes when Excited by Resonance 100 11.3 The Oscillation Amplitude of Normal Modes when Excited by Resonance 101 11.4 The Q of Oscillators 102 11.4.1 Relationship of Q to Amplitude at Resonance 102 11.4.2 Relationship of Q to Width of the Resonance Curve 102 11.4.3 Relationship of Q to Rate of Energy Loss of an Oscillator 103 11.4.4 Relationship of Q to the Growth of an Oscillation 104 11.4.5 The Growth of an Oscillation Driven Off-Resonance 104 11.5 Exciting a Multi-mode System by Resonance 105 11.6 The Excitation of Normal Modes by Impulse 106 11.6.1 The Excitation of Normal Modes by a Single Impulse 106 11.6.2 The Excitation of Normal Modes by Successive Impulses 108 11.7 The Excitation of Normal Modes by Feedback 109 11.7.1 The Concept of Feedback 109 11.7.2 Oscillations in Fed-back Systems 109 11.7.3 The Factors Leading to the Selection of a Particular Mode for Feedback Oscillation 112 11.7.4 The Feedback Process in Air Flow Over Surfaces 113 11.6 Growth in Feedback Oscillations 114 11.7 Normal Modes, Feedback, Resonance and Harmonics 114 Appendix 115 12. THE CHARACTERISTICS OF MUSICAL INSTRUMENTS 119 12.1 Percussion Instruments 119 12.1.1 General Features of the Class 119 12.1.2 Definite-pitch Percusssion Instruments 119 12.1.3 Indefinite-pitch Percusssion Instruments 121 12.2 String Instruments 122 12.2.1 General Features of the Class 122 12.2.2 Plucked and Struck String Instruments 123 12.2.2 Bowed String Instruments 125 12.3 Wind Instruments 127 12.3.1 Double Reed Wind Instruments 128 12.3.2 Human Voice (vocal-cord reed) 128 12.3.3 Lip Reed Instruments 130 12.3.4 Single Mechanical Reed Instruments 130 12.3.5 Air Reed Instruments 130 12.3.6 Pipe Organ 130 13. ACOUSTIC IMPEDANCE 131 13.1 TheAbsorption of a Wave 131 13.1.1 The Absorption of a Transverse Wave 131 13.1.2 The Absorption of a Sound Wave 134 13.2 Acoustic Impedance of Standing Waves 135 13.2.1 The Ratio of Pressure to Velosity in a Standing Wave 135 13.2.2 Acoustic Reactance 137 13.2.3 The General Concept of Acoustic Impedance 137 13.3 Acoustic Impedance and Acoustic Power 138 13.3.1 Acoustic Impedance vs Characteristic Acoustic Impedance 138 13.3.2 The Acoustic Power of a Vibrating Disk; An Example of the Use of Acoustic Impedance 140 13.3.3 Acoustic Impednace of a Trumpet 142 13.3.4 The Q of a Helmholtx Resonator 143 13.4 Analysis of Systems Using Acoustic Impedance 144 Appendix 146 iv Contents CHAPTER 1 INTRODUCTION These notes are intended to help music students understand physical acoustics. The material assembled here was done so at the request of the Faculty of Music HEADSETS of McGill University for students intending to enter a graduate program in music recording. While this means that the subject matter is approached from a definite point of view, it is hoped that it does not mean with a limited perspective. An attempt is made to LISTENER emphasize the broad fundamentals of physical PLAYBACK UNIT acoustics, particularly in their importance to understanding the acoustic environment of a human listener. Fig. 1.3 System analysis of listening to recorded music. Why is an understanding of the fundamentals of physical acoustics so important to music recording? A simple analysis would indicate that music recording is By careful engineering of the dummy head, the straight-forward. The perception of live sound involves microphones, the recorder and the playback system it a source, a medium to propagate the sound from the would seen that we should be able to exactly recreate source to the listener, and the receiver (the human ear- the sound pressure sequences in the original music, at brain system). least to within the human ability to detect any difference. From the systems point of view the process can be represented by a simple diagram. ROOM SOURCE LISTENER WAVE TRANSMISSION SOURCE LISTENER SYSTEM (ROOM) (RECEIVER) Fig. 1.1 System analysis of a live sound Fig. 1.4 "Black Box" representation of live experience. musical experience. Recording the sound for future playback could be done by replacing the human listener with a dummy head of the same physical consistency as a real head and in which the human ears are replaced by sensitive ROOM microphones, the electric output of which are led to an SOURCE LISTENER electronic recorder. MICROPHONES IN DUMMY HEAD RECORDER PLAYBACK WAVE TRANSMISSION SOURCE SYSTEM (ROOM) DUMMY HEAD RECORDER Fig. 1.2 System analysis of sound recording. Fig. 1.4 "Black Box" representation of live The original acoustic experience can then be duplicated musical experience. by a playback system which recreates the original sound pressure patterns using headphones attached to A lot of engineering effort has indeed been directed at a real persons head. constructing the necessary devices and very faithful reproduction of sound signals is now possible. Therefore it would seem that there is no need for the typical recording engineer activity of placing a myriad 2 Physical Acoustics of Music Perception of microphones and a tangle of wires in a symphony Of course a good record requires a good original hall and of tampering with the various recorded signals performance; a bad performance can never yield a good with complex electronics until a satisfactory result is record.
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