Music and Acoustics

Philippe Guillaume is a Professor at INSA Toulouse, France where he teaches applied mathematics and musical acoustics.He is a researcher within the Laboratory of Mathematics for Industry and Physics in the same Institute. He worked for many years in his youth as a piano tuner for famous pianists.

• Foreword 13Chapter 1. Sounds 171.1. Sound propagation 171.1.1. A look at the physical models 171.1.1.1. Mass conservation 181.1.1.2. The Euler equation 201.1.1.3. The state equation 211.1.2. The wave equation 211.1.3. The Helmholtz equation 231.1.4. Sound intensity 251.2. Music theory interlude 271.2.1. Intervals, octave 281.2.2. Scientific pitch notation 281.2.3. Dividing the octave into twelve semitones 281.2.4. Diatonic scales 291.2.4.1. Major scale 291.2.4.2. Minor scales 301.3. Different types of sounds 311.3.1. Periodic sounds 321.3.1.1. Fourier series 341.3.2. Sounds with partials 351.3.3. Continuous spectrum sounds 361.3.4. Noise 371.4. Representation of sound 411.4.1. Time or frequency analysis, discrete Fourier transform 421.4.2. Time-frequency analysis, the spectrogram 441.5. Filtering 491.5.1. Discrete spectrum 491.5.1.1. Transfer function 501.5.1.2. Impulse response 511.5.2. Continuous spectrum 531.5.3. Ideal low-pass, band-pass and all-pass filters 531.6. Study problems 561.6.1. Normal reflection on a wall (*) 561.6.2. Comb filtering produced by a microphone located near a wall (**) 561.6.3. Summing intensities (***) 571.6.4. Intensity of a standing wave (**) 581.6.5. Sound of a siren (*) 581.7. Practical computer applications 581.7.1. First sound, vectors 581.7.2. Modifying the parameters: the command file 591.7.3. Creating more complex sounds: using functions 601.7.3.1. Noise and siren interlude 611.7.4. Analysis 611.7.4.1. Time analysis 621.7.4.2. Frequency analysis 621.7.4.3. Time-frequency analysis 631.7.5. Filtering 63Chapter 2. Music Instruments 652.1. Strings 662.1.1. Free vibrations of a string 662.1.2. Beats, chords and consonance 692.2. Bars 722.2.1. Bar fixed at both ends 732.2.2. Bar clamped at one end 752.3. Membranes 772.4. Tubes 792.4.1. Pressure control 812.4.1.1. Response to a harmonic excitation 812.4.1.2. The resonance effect 822.4.1.3. Natural modes 842.4.1.4. The resulting sound 842.4.2. Speed control 852.4.2.1. Response to a harmonic excitation 852.4.2.2. Resonance and natural modes 872.4.2.3. Comments on phases 872.4.3. Tuning 882.5. Timbre of instruments 892.5.1. Nature of the spectrum 892.5.1.1. Harmonics or partials, the piano’s inharmonicity 902.5.1.2. Richness in higher harmonics 912.5.1.3. Different harmonics distributions 932.5.1.4. The purpose of the resonator 942.5.2. Envelope of the sound 952.5.2.1. Calculation of the envelope 962.5.2.2. Using several envelopes 972.6. Study problems 982.6.1. Vibrations of a string (general case) (**) 982.6.2. Plucked string (*) 1002.6.3. Bow drawn across a string (*) 1002.6.4. String reduced to one degree of freedom (**) 1012.6.5. Coupled string-bridge system and the remanence effect (***) 1022.6.6. Calculation of the inharmonicity of a real string (***) 1042.6.7. Coincidence frequency of a wave in a board (***) 1062.6.8. Resonance of the bourdon (**) 1072.6.9. Resonance of a cylindrical dual controlled tube (**) 1082.6.10. Resonance of a conical tube (1) (**) 1092.6.11. Resonance of a conical tube (2) (**) 1102.7. Practical computer applications 1102.7.1. Create your synthesizer 1102.7.1.1. Write your instrument function 1112.7.1.2. Add an envelope 1112.7.1.3. And play your instrument 1122.7.2. Modify the timbre of your instrument 1122.7.3. Remanent sound 112Chapter 3. Scales and Temperaments 1153.1. The Pythagorean scale 1163.2. The Zarlino scale 1173.3. The tempered scales 1183.3.1. Equal temperament 1193.3.2. A historical temperament 1193.3.3. Equal temperament with perfect fifth 1203.3.4. The practice of tuners 1203.3.5. The practice of musicians 1213.4. A brief history of A4 1213.5. Giving names to notes 1223.6. Other examples of scales 1233.7. Study problems 1233.7.1. Frequencies of a few scales (***) 1233.7.2. Beats of the fifths and the major thirds (*) 1233.8. Practical computer applications 1253.8.1. Building a few scales 1253.8.2. Listening to beats 125Chapter 4. Psychoacoustics 1274.1. Sound intensity and loudness 1274.1.1. The phon 1284.1.2. The sone 1294.2. The ear 1304.3. Frequency and pitch 1314.3.1. The mel scale 1324.3.2. Composed sounds 1334.3.2.1. Pitch of sounds composed of harmonics 1334.3.2.2. Pitch of sounds composed of partials 1334.3.3. An acoustic illusion 1344.4. Frequency masking 1364.5. Study problems 1384.5.1. Equal-loudness levels (**) 1384.5.2. Frequency masking (**) 1384.5.3. Perpetually ascending sound (**) 1384.6. Practical computer applications 1384.6.1. Frequency masking 1384.6.2. Perpetually ascending scale 138Chapter 5. Digital Sound 1415.1. Sampling 1435.1.1. The Nyquist criterion and the Shannon theorem 1455.1.1.1. Case of a sinusoidal signal 1455.1.1.2. General case 1475.1.1.3. Consequences 1485.1.1.4. Theoretical impossibility 1485.1.1.5. What happens if the Nyquist criterion is not met? 1485.1.2. Quantization 1525.1.2.1. Error due to quantization 1535.1.3. Reconstruction of the sound signal 1545.2. Audio compression 1555.2.1. Psychoacoustic compression 1555.2.2. Entropy compression 1595.3. Digital filtering and the Z-transform 1605.3.1. Digital filtering 1605.3.2. The Z-transform 1615.3.2.1. Definition 1615.3.2.2. Effect of a delay 1625.3.2.3. Filtering and Z-transform 1635.4. Study problems 1645.4.1. Nyquist criterion (*) 1645.4.2. Aliasing of an ascending sound (*) 1655.4.3. Another example of reconstruction (***) 1655.4.4. Elementary filter bank (**) 1665.5. Practical computer applications 1675.5.1. Spectrum aliasing 1675.5.2. Quantization noise 168Chapter 6. Synthesis and Sound Effects 1696.1. Synthesis of musical sounds 1706.1.1. Subtractive synthesis 1706.1.2. Additive synthesis 1716.1.3. FM synthesis 1716.1.4. Synthesis based on the use of sampled sounds 1746.2. Time effects: echo and reverberation 1756.2.1. Simple echo 1756.2.2. Multiple echo 1766.2.3. Reverberation 1776.2.3.1. Using the impulse response 1776.2.3.2. Using echoes and all-pass filters 1786.3. Effects based on spectrum modification 1796.3.1. The ‘Wah-wah’ effect 1806.3.1.1. An example of a band-pass filter 1806.3.2. AM or FM type sound effects 1826.3.2.1. Vibrato 1836.3.2.2. Leslie effect 1846.4. Study problems 1866.4.1. The Doppler effect (**) 1866.4.2. FM and Chowning (***) 1876.5. Practical computer applications 1886.5.1. Sound synthesis 1886.5.2. Chowning synthesis 1886.5.3. Reverberation 1896.5.4. Vibrato 1896.5.5. The Leslie effect 189Bibliography 191Index 193