Harmonics & timbre

Physics classrooms love the tuning fork because it is the one sound-maker that behaves: strike it and it rings at almost exactly one frequency, a single clean spike on the spectrum. Almost nothing else in the world is that polite. Pluck a guitar string and it does not pick one motion. It vibrates along its whole length, and in two halves, and in three thirds, and in ever-smaller subdivisions, all at the same time. Each of those shapes rings at its own rate: the whole string at some frequency f₀, the halves at exactly 2·f₀, the thirds at 3·f₀, and so on up. One plucked note is not one frequency. It is a stack.

The members of the stack are called harmonics: whole-number multiples of the fundamental, f₀. (Everything above the fundamental also goes by overtones, the same idea counted from one rung up.) Point the FFT from chapter 03 at any real instrument and this is what you get: not a lone spike but a picket fence, spikes marching up the spectrum at f₀, 2·f₀, 3·f₀, each one shorter or taller than its neighbors depending on what made the sound.

And that dependence is the whole story of why things sound like themselves. Your ear assigns the note from the fundamental — a stack built on 220 Hz is heard as the A below middle C no matter what rides above it. Everything else about the sound’s character lives in the recipe: the relative heights of the harmonics. That recipe is timbre. A violin and a flute playing the same A have the same fundamental, yet you will never confuse them, because the violin pours energy high into the stack and the flute keeps its tone nearly pure. Your friend’s voice is your friend’s voice for the same reason: their vocal folds buzz out a harmonic stack, and the particular shape of their throat and mouth boosts some harmonics and swallows others. The pitch is the message; the recipe is the signature.

A few recipes are famous enough to have names, and the math is simple enough to hear. Take only the odd harmonics at heights 1/n and the sum squares off into a square wave, the buzzy tone of early video games. Take every harmonic at 1/n and you get the brassy rip of a sawtooth. Odd harmonics again, but falling off as 1/n², so there’s much less energy up high, and you get the soft, hollow triangle. The figure below builds each recipe the honest way: eight ToneSpecs summed by tones() from @sonoglyph/dsp, then handed to the same FFT the pipeline runs.

Switch between recipes and watch both views. The waveform reshapes completely: smooth curve, squared shoulders, ramps, ripples. The spectrum tells you exactly why, one spike per harmonic, heights tracking the recipe. Now press play after each switch. The pitch never moves. Every one of these waveforms repeats 220 times per second, so every one is the same A; only the voice saying it changes. The brightness slider is a tone knob: it scales harmonic k by brightness^(k−1), dimming the top of the stack first, and at zero every recipe collapses back into the tuning fork’s pure sine.

One ingredient is conspicuously missing from the recipe: phase. tones() starts every harmonic at zero, but you could slide each one’s starting angle anywhere — the waveform would contort into something unrecognizable, and it would sound almost exactly the same. To a good first approximation the ear reads the magnitudes of the stack and shrugs at their alignment, which is why spectrum displays, and most recognizers, throw phase away without regret.

Why a recognizer must expect the stack

This matters to Sonoglyph because real signals arrive wearing their overtones. The playground’s keypad synthesizes mathematically pure sine pairs, but a human whistling or humming a tone near 697 Hz (the top row of a telephone keypad) delivers a stack: real energy at 1,394 Hz, some at 2,091 Hz, trailing on up. A naive recognizer that treats any strong spike as an independent tone would read that 1,394 Hz harmonic as a second signal, sitting squarely inside DTMF’s high-frequency band. It lands on none of the four high-group frequencies, though. That is no accident. The engineers who chose DTMF’s eight frequencies in the 1950s picked them so that no tone’s harmonic falls on any other tone, a piece of numerical craftsmanship chapter 08 unpacks in full.

So a recognizer worthy of the name holds two ideas at once: a spike at 2·f₀ is evidence about the tone at f₀, not a second tone; and the frequencies worth trusting are the ones that survive scrutiny — persistent, well-placed, sensibly proportioned against their neighbors. The DTMF recognizer in this codebase checks that the two frequencies it accepts come one from each designated group, and even compares their heights (the “twist”), because a legitimate keypress has a predictable balance and a stray harmonic does not.

Every claim in this chapter is one glance away in the spectrum view. It is the instrument that makes the invisible stack visible. Open the playground, hum at the microphone, and your own overtone stack marches across the display in real time. The next question is how a machine reads that picture: turning a forest of spikes into a short, honest list of frequencies is chapter 05, and it is harder than it looks — partly because, as you now know, the tallest spikes are not all independent tones.