Electronic Instruments
These technical notes trace a lineage not of devices (products) but of how instrument sound is represented mathematically, period by period. Approximating an instrument tone by a sum of sinusoids gives additive synthesis; approximating it by the output of a linear system gives subtractive synthesis; approximating it by the frequency modulation of a sinusoid gives FM synthesis, and so on.
The whole picture
This single chart shows, end to end, which piece of mathematics became which instrument.
How to read these notes
When a human chose the coefficients
In this period the coefficients entering the synthesis formula were chosen by ear, by the designer or the performer. Each change of representation changed how many coefficients there were and what they meant.
Additive Synthesis — Turning the Fourier Series into a Machine
Reading $x(t)=\sum_k a_k\cos(k\omega_0 t+\phi_k)$ as a blueprint. The sawtooth coefficients $a_k=1/k$ are obtained by integration, and the Gibbs overshoot $\frac1\pi\mathrm{Si}(\pi)-\frac12$ that follows from truncating the series is derived. Telharmonium, Michelson's harmonic analyser, the Hammond drawbars and the Soviet ANS.
1822 / 1897 / 1935 / 1958 PrincipleHeterodyning — Only Nonlinearity Creates New Frequencies
An eigenfunction argument shows that a linear time-invariant system cannot produce frequencies absent from its input, and the product-to-sum identity gives frequency translation. The theremin's sensitivity, whereby a capacitance change of order $10^{-4}$ spans several octaves, is computed by differentiating the beat frequency.
1920 SynthesisSubtractive Synthesis — The Source–Filter Model and the Placement of Poles
Instead of specifying partials one by one, a wideband excitation $E$ is carved by a filter $H$ so that $Y(z)=H(z)E(z)$, approximating only the shape of the spectral envelope. The poles of the four-pole ladder filter are solved in closed form as $s=\omega_c(k^{1/4}e^{j(2m+1)\pi/4}-1)$, showing that at $k=4$ they reach the imaginary axis and the filter self-oscillates. Why the 1 V/oct exponential law matches the exponential characteristic of a transistor.
1939 / 1964 PrincipleThe Sampling Theorem — Sound Becomes a Sequence
Convolution with a Dirac comb shows that sampling periodises the spectrum and that aliasing is the overlap of those copies. A worked computation shows how a naive sawtooth produces inharmonic components, followed by band-limited additive synthesis and BLEP correction. The RCA Mark II and its punched paper tape.
1948 / 1955 / 1957 SynthesisFM Synthesis — When Bessel Functions Became an Instrument
The Jacobi–Anger expansion $e^{jI\sin\theta}=\sum_n J_n(I)e^{jn\theta}$ is derived from the generating function, showing that the sideband amplitudes are $J_n(I)$. Carson's bandwidth rule, power conservation via $\sum_n J_n^2=1$, the reflection of negative orders, and the weakness that the inverse problem cannot be solved.
1967 / 1983When analysis chose the coefficients
In this period the coefficients are estimated by measuring real sounds. The synthesis formula itself barely changes; what moves is where the coefficients come from — from human hands to analysis, and then to learning.
Analysis/Synthesis — Measure First, Then Build
The STFT, windows and the uncertainty relation $\Delta t\,\Delta f\ge 1/(4\pi)$, the phase vocoder that recovers instantaneous frequency from phase differences, the McAulay–Quatieri sinusoidal tracks, the deterministic-plus-residual decomposition of SMS, and the normal equations of LPC solved by Levinson–Durbin.
1939 / 1966 / 1986 / 1990 RepresentationGranular Synthesis — Tiling the Time–Frequency Plane with Gabor Atoms
The Fourier basis is localised in frequency but not in time, while the Dirac impulse is localised in time but not in frequency. Gabor atoms, localised in both, are derived in between, and the Gaussian window is shown to be the unique shape attaining the uncertainty bound up to the trivial transformations (translation, modulation and scaling). Approximating timbre by the parameters of a probability distribution shifts the criterion from "the waveforms are close" to "the statistics agree".
1946 / 1970s / 1990s SynthesisPhysical Modelling — Playing the Wave Equation
The wave equation is derived from tension and linear density, and the two travelling waves of the d'Alembert solution are shown to become delay lines. The poles of Karplus–Strong are solved as the roots of $2z^{L+1}-z-1=0$, confirming numerically that higher partials decay faster without being told to.
1983 / 1987 / 1994 Analysis/SynthesisDifferentiable Synthesis — Letting Learning Choose the Coefficients
The cost of generating waveforms with autoregressive models, and DDSP, which writes the synthesiser itself as differentiable layers. Following the gradient of the multi-scale STFT loss back to the additive coefficients $a_k$ returns us to the $\sum a_k\cos(\cdot)$ of 120 years earlier.
2016 / 2020Frequently asked questions
Q1. In what order should these notes be read?
Start with the overview and then read the chapters in order, because each chapter takes as its starting point whatever was expensive in the previous one. Additive synthesis needs one control curve per partial, subtractive synthesis makes it hard to specify individual partials independently, and FM synthesis admits no general closed-form inverse mapping from a target spectrum to its parameters: the limitation of one chapter motivates the next. Each chapter is also written to stand on its own if only one method is of interest.
Q2. Do these notes explain how to play or program the instruments?
No. They trace a lineage of mathematical representations of instrument sound rather than a lineage of devices or products. Individual instruments appear as implementations of the sound model adopted in their period. Instead of operating instructions or sound design tips, the notes derive what mathematical representation each synthesis method rests on and what it approximates.
Q3. What mathematical background is assumed?
Fourier series and complex exponentials, linear ordinary differential equations, and an introduction to the $z$-transform or the Laplace transform are enough for most chapters. The FM chapter uses the generating function of the Bessel functions, the analysis/synthesis chapter uses the Cauchy–Schwarz inequality, and the physical modelling chapter uses a change of variables in a partial differential equation, but each of these is derived in place.
Related notes
Fourier Analysis
The foundation of additive synthesis and the sampling theorem: series expansions, transforms and the convolution theorem.
Signal ProcessingFilter Design
The mathematics of designing the $H(z)$ used in subtractive synthesis, from analogue prototypes to optimal digital designs.
PhysicsPhysics — Vibrating Strings and Membranes
The vibrations that physical modelling takes as its object: longitudinal and transverse waves on a piano string, the inharmonic modes of a circular membrane, and the three-dimensional wave equation.