Theory and Techniques of Electronic Music

This book is about using electronic techniques to record, synthesize, process, and analyze musical sounds, a practice which came into its modern form in the years 1948-1952, but whose technological means and artistic uses have undergone several revolutions since then. Nowadays most electronic music is made using computers, and this book will focus exclusively on what used to be called computer music", but which should really now be called \electronic music using a computer".



1 Sinusoids, amplitude and frequency 3
1.1 Measures of Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Units of Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Controlling Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Synthesizing a Sinusoid . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Superposing Signals . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Periodic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8 About the Software Examples . . . . . . . . . . . . . . . . . . . . 14
Quick Introduction to Pd . . . . . . . . . . . . . . . . . . . . . . 17
How to Find and run the examples . . . . . . . . . . . . . . . . . 19

1.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Constant amplitude scaler . . . . . . . . . . . . . . . . . . . . . . 19
Amplitude control in decibels . . . . . . . . . . . . . . . . . . . . 21
Smoothed amplitude control with an envelope generator . . . . . 23
Major triad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Conversion between frequency and pitch . . . . . . . . . . . . . . 24
More additive synthesis . . . . . . . . . . . . . . . . . . . . . . . 25
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Wavetables and samplers 29
2.1 The Wavetable Oscillator . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Enveloping samplers . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 Timbre stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.6.1 wavetable oscillator . . . . . . . . . . . . . . . . . . . . . 49
2.6.2 wavetable lookup in general . . . . . . . . . . . . . . . . . 50
2.6.3 using a wavetable as a sampler . . . . . . . . . . . . . . . 52
2.6.4 looping samplers . . . . . . . . . . . . . . . . . . . . . . . 54
2.6.5 Overlapping sample looper . . . . . . . . . . . . . . . . . 56
2.6.6 Automatic read point precession . . . . . . . . . . . . . . 58

iv CONTENTS
3 Audio and control computations 61
3.1 The sampling theorem . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.3 Control streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Converting from audio signals to numeric control streams . . . . 69
3.5 Control streams in block diagrams . . . . . . . . . . . . . . . . . 70
3.6 Event detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.7 Control computation using audio signals directly . . . . . . . . . 73
3.8 Operations on control streams . . . . . . . . . . . . . . . . . . . . 75
3.9 Control operations in Pd . . . . . . . . . . . . . . . . . . . . . . . 77
3.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.10.1 Sampling and foldover . . . . . . . . . . . . . . . . . . . . 79
3.10.2 Converting controls to signals . . . . . . . . . . . . . . . . 81
3.10.3 Non-looping sample player . . . . . . . . . . . . . . . . . . 82
3.10.4 Signals to controls . . . . . . . . . . . . . . . . . . . . . . 84
3.10.5 Analog-style sequencer . . . . . . . . . . . . . . . . . . . . 84
3.10.6 MIDI-style synthesizer . . . . . . . . . . . . . . . . . . . . 86
4 Automation and voice management 89
4.1 Envelope Generators . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2 Linear and Curved Amplitude Shapes . . . . . . . . . . . . . . . 92
4.3 Continuous and discontinuous control changes . . . . . . . . . . . 94
4.3.1 Muting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3.2 Switch-and-ramp . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 Polyphony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Voice allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.6 Voice tags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.7 Encapsulation in Pd . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.8.1 ADSR envelope generator . . . . . . . . . . . . . . . . . . 103
4.8.2 Transfer functions for amplitude control . . . . . . . . . . 106
4.8.3 Additive synthesis: Risset's bell . . . . . . . . . . . . . . . 107
4.8.4 Additive synthesis: spectral envelope control . . . . . . . 110
4.8.5 Polyphonic synthesis: sampler . . . . . . . . . . . . . . . . 113
5 Modulation 119
5.1 Taxonomy of spectra . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Multiplying audio signals . . . . . . . . . . . . . . . . . . . . . . 122
5.3 Waveshaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 Frequency and phase modulation . . . . . . . . . . . . . . . . . . 132
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5.1 Ring modulation and spectra . . . . . . . . . . . . . . . . 135
5.5.2 Octave divider and formant adder . . . . . . . . . . . . . 137
5.5.3 Waveshaping and dierence tones . . . . . . . . . . . . . . 138
5.5.4 Waveshaping using Chebychev polynomials . . . . . . . . 139
5.5.5 Waveshaping using an exponential function . . . . . . . . 140

CONTENTS v
5.5.6 Sinusoidal waveshaping: evenness and oddness . . . . . . 141
5.5.7 Phase modulation and FM . . . . . . . . . . . . . . . . . 143
6 Designer spectra 147
6.1 Carrier/modulator model . . . . . . . . . . . . . . . . . . . . . . 148
6.2 Pulse trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.2.1 Pulse trains via waveshaping . . . . . . . . . . . . . . . . 151
6.2.2 Pulse trains via wavetable stretching . . . . . . . . . . . . 152
6.2.3 Resulting spectra . . . . . . . . . . . . . . . . . . . . . . . 152
6.3 Movable ring modulation . . . . . . . . . . . . . . . . . . . . . . 156
6.4 Phase-aligned formant (PAF) generator . . . . . . . . . . . . . . 158
6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.5.1 Wavetable pulse train . . . . . . . . . . . . . . . . . . . . 163
6.5.2 Simple formant generator . . . . . . . . . . . . . . . . . . 166
6.5.3 Two-cosine carrier signal . . . . . . . . . . . . . . . . . . . 167
6.5.4 The PAF generator . . . . . . . . . . . . . . . . . . . . . . 168
6.5.5 Stretched wavetables . . . . . . . . . . . . . . . . . . . . . 172
7 Time shifts 173
7.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.1.1 Sinusoids as geometric series . . . . . . . . . . . . . . . . 176
7.2 Time shifts and phase changes . . . . . . . . . . . . . . . . . . . 178
7.3 Delay networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4 Recirculating delay networks . . . . . . . . . . . . . . . . . . . . 183
7.5 Power conservation and complex delay networks . . . . . . . . . 187
7.6 ArtiFicial reverberation . . . . . . . . . . . . . . . . . . . . . . . . 192
7.6.1 Controlling reverberators . . . . . . . . . . . . . . . . . . 194
7.7 Variable and fractional shifts . . . . . . . . . . . . . . . . . . . . 196
7.8 Accuracy and frequency response of interpolating delay lines . . 199
7.9 Pitch shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.10.1 Fixed, noninterpolating delay line . . . . . . . . . . . . . 206
7.10.2 Recirculating comb Filter . . . . . . . . . . . . . . . . . . . 207
7.10.3 Variable delay line . . . . . . . . . . . . . . . . . . . . . . 208
7.10.4 Order of execution and lower limits on delay times . . . . 209
7.10.5 Order of execution in non-recirculating delay lines . . . . 211
7.10.6 Non-recirculating comb Filter as octave doubler . . . . . . 213
7.10.7 Time-varying complex comb Filter: shakers . . . . . . . . 214
7.10.8 Reverberator . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.10.9 Pitch shifter . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.10.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

vi CONTENTS
8 Filters 221
8.1 Taxonomy of Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 222
8.1.1 Low-pass and high-pass Filters . . . . . . . . . . . . . . . . 222
8.1.2 Band-pass and stop-band Filters . . . . . . . . . . . . . . . 224
8.1.3 Equalizing Filters . . . . . . . . . . . . . . . . . . . . . . . 224
8.2 Designing Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.2.1 Elementary non-recirculating Filter . . . . . . . . . . . . . 227
8.2.2 Non-recirculating Filter, second form . . . . . . . . . . . . 228
8.2.3 Elementary recirculating Filter . . . . . . . . . . . . . . . . 231
8.2.4 Compound Filters . . . . . . . . . . . . . . . . . . . . . . . 231
8.2.5 Real outputs from complex Filters . . . . . . . . . . . . . . 232
8.3 Designing Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
8.3.1 One-pole low-pass Filter . . . . . . . . . . . . . . . . . . . 234
8.3.2 One-pole, one-zero high-pass Filter . . . . . . . . . . . . . 235
8.3.3 Shelving Filter . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.3.4 Band-pass Filter . . . . . . . . . . . . . . . . . . . . . . . . 238
8.3.5 Peaking and band-stop Filter . . . . . . . . . . . . . . . . 238
8.3.6 Butterworth Filters . . . . . . . . . . . . . . . . . . . . . . 239
8.3.7 Stretching the unit circle with rational functions . . . . . 241
8.3.8 Butterworth band-pass Filter . . . . . . . . . . . . . . . . 244
8.3.9 Time-varying coe±cients . . . . . . . . . . . . . . . . . . 245
8.3.10 Impulse responses of recirculating Filters . . . . . . . . . . 247
8.3.11 All-pass Filters . . . . . . . . . . . . . . . . . . . . . . . . 247
8.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.4.1 Subtractive synthesis . . . . . . . . . . . . . . . . . . . . . 250
8.4.2 Envelope following . . . . . . . . . . . . . . . . . . . . . . 250
8.4.3 Single Sideband Modulation . . . . . . . . . . . . . . . . . 253
8.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8.5.1 Prefabricated low-, high-, and band-pass Filters . . . . . . 255
8.5.2 Prefabricated time-variable band-pass Filter . . . . . . . . 256
8.5.3 Envelope followers . . . . . . . . . . . . . . . . . . . . . . 258
8.5.4 Single sideband modulation . . . . . . . . . . . . . . . . . 258
8.5.5 Using elementary Filters directly: shelving and peaking . . 261
8.5.6 Making and using all-pass Filters . . . . . . . . . . . . . . 261
9 Fourier analysis and resynthesis 265
9.1 Fourier analysis of periodic signals . . . . . . . . . . . . . . . . . 265
9.1.1 Fourier transform as additive synthesis . . . . . . . . . . . 267
9.1.2 Periodicity of the Fourier transform . . . . . . . . . . . . 267
9.2 Properties of Fourier transforms . . . . . . . . . . . . . . . . . . 267
9.2.1 Fourier transform of DC . . . . . . . . . . . . . . . . . . . 268
9.2.2 Shifts and phase changes . . . . . . . . . . . . . . . . . . 269
9.2.3 Fourier transform of a sinusoid . . . . . . . . . . . . . . . 271
9.3 Fourier analysis of non-periodic signals . . . . . . . . . . . . . . . 272
9.4 Fourier analysis and reconstruction of audio signals . . . . . . . . 275
9.4.1 Narrow-band companding . . . . . . . . . . . . . . . . . . 277

CONTENTS vii
9.4.2 Timbre stamping (classical vocoder) . . . . . . . . . . . . 279
9.5 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
9.5.1 Phase relationships between channels . . . . . . . . . . . . 285
9.6 Phase bashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
9.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
10 Classical waveforms 297
10.1 Symmetries and Fourier series . . . . . . . . . . . . . . . . . . . . 299
10.1.1 Sawtooth waves and symmetry . . . . . . . . . . . . . . . 300
10.2 Decomposing the classical waveforms into sawtooth and parabolic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
10.3 Fourier series of the elementary waveforms . . . . . . . . . . . . . 304
10.4 Predicting and controlling foldover . . . . . . . . . . . . . . . . . 309
10.4.1 Oversampling . . . . . . . . . . . . . . . . . . . . . . . . . 309
10.4.2 Sneaky triangle waves . . . . . . . . . . . . . . . . . . . . 310
10.4.3 Transition splicing . . . . . . . . . . . . . . . . . . . . . . 311
10.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314