Theory and Technique of Electronic Music

333 Pages·2006·1.85 MB·english

by  M. Puckette

The Theory and Technique of Electronic
Music
DRAFT: September 24, 2006
Miller Puckette
Copyright c2006 Miller Puckette
(cid:176)
All rights reserved.
Contents
Introduction 1
1 Sinusoids, amplitude and frequency 3
1.1 Measures of Amplitude. . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Units of Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Controlling Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Synthesizing a sinusoid . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Superposing Signals . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Periodic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 About the Software Examples . . . . . . . . . . . . . . . . . . . . 17
Quick Introduction to Pd . . . . . . . . . . . . . . . . . . . . . . 17
How to flnd and run the examples . . . . . . . . . . . . . . . . . 19
1.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Constant amplitude scaler . . . . . . . . . . . . . . . . . . . . . . 19
Amplitude control in decibels . . . . . . . . . . . . . . . . . . . . 20
Smoothed amplitude control with an envelope generator . . . . . 23
Major triad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Conversion between frequency and pitch . . . . . . . . . . . . . . 24
More additive synthesis . . . . . . . . . . . . . . . . . . . . . . . 25
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Wavetables and samplers 29
2.1 The Wavetable Oscillator . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Enveloping samplers . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Timbre stretching. . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Wavetable oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 49
Wavetable lookup in general . . . . . . . . . . . . . . . . . . . . . 50
Using a wavetable as a sampler . . . . . . . . . . . . . . . . . . . 52
Looping samplers . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Overlapping sample looper. . . . . . . . . . . . . . . . . . . . . . 56
iii
iv CONTENTS
Automatic read point precession . . . . . . . . . . . . . . . . . . 58
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
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 Audio signals as control . . . . . . . . . . . . . . . . . . . . . . . 73
3.8 Operations on control streams . . . . . . . . . . . . . . . . . . . . 76
3.9 Control operations in Pd . . . . . . . . . . . . . . . . . . . . . . . 79
3.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Sampling and foldover . . . . . . . . . . . . . . . . . . . . . . . . 80
Converting controls to signals . . . . . . . . . . . . . . . . . . . . 82
Non-looping wavetable player . . . . . . . . . . . . . . . . . . . . 83
Signals to controls . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Analog-style sequencer . . . . . . . . . . . . . . . . . . . . . . . . 85
MIDI-style synthesizer . . . . . . . . . . . . . . . . . . . . . . . . 85
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Automation and voice management 91
4.1 Envelope Generators . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.2 Linear and Curved Amplitude Shapes . . . . . . . . . . . . . . . 94
4.3 Continuous and discontinuous control changes . . . . . . . . . . . 96
Muting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Switch-and-ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4 Polyphony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5 Voice allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.6 Voice tags . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.7 Encapsulation in Pd . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
ADSR envelope generator . . . . . . . . . . . . . . . . . . . . . . 105
Transfer functions for amplitude control . . . . . . . . . . . . . . 108
Additive synthesis: Risset’s bell . . . . . . . . . . . . . . . . . . . 109
Additive synthesis: spectral envelope control . . . . . . . . . . . 112
Polyphonic synthesis: sampler . . . . . . . . . . . . . . . . . . . . 113
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5 Modulation 121
5.1 Taxonomy of spectra . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Multiplying audio signals . . . . . . . . . . . . . . . . . . . . . . 124
5.3 Waveshaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 Frequency and phase modulation . . . . . . . . . . . . . . . . . . 134
5.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
CONTENTS v
Ring modulation and spectra . . . . . . . . . . . . . . . . . . . . 136
Octave divider and formant adder . . . . . . . . . . . . . . . . . 137
Waveshaping and difierence tones . . . . . . . . . . . . . . . . . . 140
Waveshaping using Chebychev polynomials . . . . . . . . . . . . 141
Waveshaping using an exponential function . . . . . . . . . . . . 142
Sinusoidal waveshaping: evenness and oddness . . . . . . . . . . 143
Phase modulation and FM. . . . . . . . . . . . . . . . . . . . . . 143
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6 Designer spectra 149
6.1 Carrier/modulator model . . . . . . . . . . . . . . . . . . . . . . 150
6.2 Pulse trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.2.1 Pulse trains via waveshaping . . . . . . . . . . . . . . . . 153
6.2.2 Pulse trains via wavetable stretching . . . . . . . . . . . . 154
6.2.3 Resulting spectra . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 Movable ring modulation . . . . . . . . . . . . . . . . . . . . . . 158
6.4 Phase-aligned formant (PAF) generator . . . . . . . . . . . . . . 160
6.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Wavetable pulse train . . . . . . . . . . . . . . . . . . . . . . . . 165
Simple formant generator . . . . . . . . . . . . . . . . . . . . . . 168
Two-cosine carrier signal . . . . . . . . . . . . . . . . . . . . . . . 169
The PAF generator . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Stretched wavetables . . . . . . . . . . . . . . . . . . . . . . . . . 174
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7 Time shifts and delays 175
7.1 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.1.1 Complex sinusoids . . . . . . . . . . . . . . . . . . . . . . 178
7.2 Time shifts and phase changes . . . . . . . . . . . . . . . . . . . 179
7.3 Delay networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.4 Recirculating delay networks . . . . . . . . . . . . . . . . . . . . 184
7.5 Power conservation and complex delay networks . . . . . . . . . 189
7.6 Artiflcial reverberation . . . . . . . . . . . . . . . . . . . . . . . . 193
7.6.1 Controlling reverberators . . . . . . . . . . . . . . . . . . 196
7.7 Variable and fractional shifts . . . . . . . . . . . . . . . . . . . . 196
7.8 Fidelity of interpolating delay lines . . . . . . . . . . . . . . . . . 201
7.9 Pitch shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Fixed, noninterpolating delay line . . . . . . . . . . . . . . . . . . 208
Recirculating comb fllter . . . . . . . . . . . . . . . . . . . . . . . 209
Variable delay line . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Order of execution and lower limits on delay times . . . . . . . . 211
Order of execution in non-recirculating delay lines . . . . . . . . 213
Non-recirculating comb fllter as octave doubler . . . . . . . . . . 215
Time-varying complex comb fllter: shakers . . . . . . . . . . . . . 216
Reverberator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
vi CONTENTS
Pitch shifter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
8 Filters 223
8.1 Taxonomy of fllters . . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.1.1 Low-pass and high-pass fllters . . . . . . . . . . . . . . . . 224
8.1.2 Band-pass and stop-band fllters . . . . . . . . . . . . . . . 226
8.1.3 Equalizing fllters . . . . . . . . . . . . . . . . . . . . . . . 227
8.2 Elementary fllters . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
8.2.1 Elementary non-recirculating fllter . . . . . . . . . . . . . 229
8.2.2 Non-recirculating fllter, second form . . . . . . . . . . . . 231
8.2.3 Elementary recirculating fllter. . . . . . . . . . . . . . . . 232
8.2.4 Compound fllters . . . . . . . . . . . . . . . . . . . . . . . 232
8.2.5 Real outputs from complex fllters . . . . . . . . . . . . . . 233
8.2.6 Two recirculating fllters for the price of one . . . . . . . . 234
8.3 Designing fllters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.3.1 One-pole low-pass fllter . . . . . . . . . . . . . . . . . . . 236
8.3.2 One-pole, one-zero high-pass fllter . . . . . . . . . . . . . 237
8.3.3 Shelving fllter . . . . . . . . . . . . . . . . . . . . . . . . . 238
8.3.4 Band-pass fllter . . . . . . . . . . . . . . . . . . . . . . . . 239
8.3.5 Peaking and stop-band fllter . . . . . . . . . . . . . . . . 240
8.3.6 Butterworth fllters . . . . . . . . . . . . . . . . . . . . . . 240
8.3.7 Stretching the unit circle with rational functions . . . . . 243
8.3.8 Butterworth band-pass fllter . . . . . . . . . . . . . . . . 244
8.3.9 Time-varying coe–cients . . . . . . . . . . . . . . . . . . 245
8.3.10 Impulse responses of recirculating fllters . . . . . . . . . . 246
8.3.11 All-pass fllters . . . . . . . . . . . . . . . . . . . . . . . . 249
8.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.4.1 Subtractive synthesis . . . . . . . . . . . . . . . . . . . . . 250
8.4.2 Envelope following . . . . . . . . . . . . . . . . . . . . . . 252
8.4.3 Single Sideband Modulation . . . . . . . . . . . . . . . . . 254
8.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Prefabricated low-, high-, and band-pass fllters . . . . . . . . . . 256
Prefabricated time-varying band-pass fllter . . . . . . . . . . . . 256
Envelope followers . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Single sideband modulation . . . . . . . . . . . . . . . . . . . . . 259
Using elementary fllters directly: shelving and peaking . . . . . . 259
Making and using all-pass fllters . . . . . . . . . . . . . . . . . . 261
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
9 Fourier analysis and resynthesis 263
9.1 Fourier analysis of periodic signals . . . . . . . . . . . . . . . . . 263
9.1.1 Periodicity of the Fourier transform . . . . . . . . . . . . 264
9.1.2 Fourier transform as additive synthesis . . . . . . . . . . . 265
9.2 Properties of Fourier transforms . . . . . . . . . . . . . . . . . . 265
9.2.1 Fourier transform of DC . . . . . . . . . . . . . . . . . . . 266
CONTENTS vii
9.2.2 Shifts and phase changes . . . . . . . . . . . . . . . . . . 267
9.2.3 Fourier transform of a sinusoid . . . . . . . . . . . . . . . 269
9.3 Fourier analysis of non-periodic signals . . . . . . . . . . . . . . . 270
9.4 Fourier analysis and reconstruction of audio signals . . . . . . . . 273
9.4.1 Narrow-band companding . . . . . . . . . . . . . . . . . . 277
9.4.2 Timbre stamping (classical vocoder) . . . . . . . . . . . . 279
9.5 Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.5.1 Phase relationships between channels. . . . . . . . . . . . 283
9.6 Phase bashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
9.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Fourier analysis and resynthesis in Pd . . . . . . . . . . . . . . . 286
Narrow-band companding: noise suppression . . . . . . . . . . . 288
Timbre stamp (\vocoder") . . . . . . . . . . . . . . . . . . . . . . 290
Phase vocoder time bender . . . . . . . . . . . . . . . . . . . . . 292
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
10 Classical waveforms 295
10.1 Symmetries and Fourier series . . . . . . . . . . . . . . . . . . . . 297
10.1.1 Sawtooth waves and symmetry . . . . . . . . . . . . . . . 298
10.2 Dissecting classical waveforms . . . . . . . . . . . . . . . . . . . . 300
10.3 Fourier series of the elementary waveforms . . . . . . . . . . . . . 302
10.3.1 Sawtooth wave . . . . . . . . . . . . . . . . . . . . . . . . 303
10.3.2 Parabolic wave . . . . . . . . . . . . . . . . . . . . . . . . 304
10.3.3 Square and symmetric triangle waves. . . . . . . . . . . . 304
10.3.4 General (non-symmetric) triangle wave. . . . . . . . . . . 305
10.4 Predicting and controlling foldover . . . . . . . . . . . . . . . . . 307
10.4.1 Over-sampling . . . . . . . . . . . . . . . . . . . . . . . . 307
10.4.2 Sneaky triangle waves . . . . . . . . . . . . . . . . . . . . 309
10.4.3 Transition splicing . . . . . . . . . . . . . . . . . . . . . . 309
10.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Combining sawtooth waves . . . . . . . . . . . . . . . . . . . . . 313
Strategies for bandlimiting sawtooth waves . . . . . . . . . . . . 314
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
Index 319
Bibliography 323
viii CONTENTS
Introduction
This is a book 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 under-
gone several revolutions since then. Nowadays most electronic music is made
using computers, and this book will focus exclusively on what used to be called
\computermusic",butwhichshouldreallynowbecalled\electronicmusicusing
a computer".
Mostofthecomputermusictoolsavailabletodayhaveantecedentsinearlier
generations of equipment. The computer, however, is relatively cheap and the
results of using one are easy to document and re-create. In these respects at
least, the computer makes the ideal electronic music instrument|it is hard to
see what future technology could displace it.
The techniques and practices of electronic music can be studied (at least
in theory) without making explicit reference to the current state of technology.
Still, it’s important to provide working examples. So each chapter starts with
theory (avoiding any reference to implementation) and ends with a series of
examples realized in a currently available software package.
Theidealreaderofthisbookisanyonewhoknowsandlikeselectronicmusic
of any genre, has plenty of facility with computers in general, and who wants
to learn how to make electronic music from the ground up, starting with the
humbleoscillatorandcontinuingthroughsampling, FM,flltering, waveshaping,
delays, and so on. This will take plenty of time.
Thisbookdoesn’ttaketheeasyrouteofrecommendingpre-cookedsoftware
to try out these techniques; instead, the emphasis is on learning how to use a
general-purpose computer music environment to realize them yourself. Of the
several such packages available, we’ll use Pd, but that shouldn’t stop you from
usingthesesametechniquesinotherenvironmentssuchasCsoundorMax/MSP.
To read this book you must understand mathematics through intermediate
algebra and trigonometry; starting in Chapter 7, complex numbers also make
an appearance, although not complex analyis. (For instance, complex numbers
are added, multiplied, and conjugated, but there are no complex exponentials.)
A review of mathematics for computer music by F. Richard Moore appears in
[Str85, pp. 1-68].
Although the \level" of mathematics is not high, the mathematics itself
is sometimes quite challenging. All sorts of cool mathematics is in the reach
1
2 CONTENTS
of any student of algebra or geometry. In the service of computer music, for
instance, we’ll run into Bessel functions, Chebychev polynomials, the Central
Limit Theorem, and, of course, Fourier analysis.
You don’t need much background in music as it is taught in the West; in
particular, Western written music notation is not needed. Some elementary
bits of Western music theory are used, such as the tempered scale, the A-B-
C system of naming pitches, and terms like \note" and \chord". Also you
shouldbefamiliarwithtermsofmusicalacousticssuchassinusoids, amplitude,
frequency, and the overtone series.
Each chapter starts with a theoretical discussion of some family of tech-
niques or theoretical issues, followed by a a series of examples realized in Pd
to illustrate them. The examples are included in the Pd distribution, so you
can run them and/or edit them into your own spinofis. In addition, all the flg-
ures were created using Pd patches, which appear in an electronic supplement.
Thesearen’tcarefullydocumentedbutinprinciplecouldbeusedasanexample
of Pd’s drawing capabilities for anyone interested in that.