Waveforms and harmonics
Every repeating wave is a stack of sines. Which ones, and how loud, is the whole story of tone color.
In the 1820s Joseph Fourier proved a fact synthesis still runs on: any repeating wave, however jagged, can be built by adding plain sine waves — and only particular ones. There is a sine at the repeat rate itself, called the fundamental, plus sines at exact whole-number multiples of it (2×, 3×, 4× and so on), called harmonics. A sawtooth at 110 Hz and a cello at 110 Hz contain sines at the same frequencies; they differ only in how loud each member of the stack is. That loudness profile is the sound’s recipe, and it is what your ear reads as tone color.
Adding sines: 1×, plus 3× at a third of the level, plus 5× at a fifth. The sum on the right is already recognizably square-ish. Feed in the rest of the odd series and the corners keep sharpening.
Hearing inside the stack
The harmonics do not sound like separate notes. Locked to whole-number multiples, they fuse with the fundamental into one pitch that carries a character. With practice you can still hear into the stack, and it helps to know where the members sit musically: the 2nd harmonic is one octave above the fundamental, the 3rd an octave plus a fifth, the 4th two octaves. Words like “thin”, “hollow”, “nasal”, and “brassy” are everyday descriptions of which members are strong and which are missing.
The four classic recipes
- Sine — the fundamental alone. The reference tone of hearing tests, and the sub-bass layer under club tracks. On its own it sounds small and disappears easily in a mix, because there is nothing above the fundamental for the ear to hold on to.
- Triangle — odd harmonics only, falling off steeply at 1/n²: the 3rd harmonic at 1/9 the level, the 5th at 1/25. It reads as a sine with a faint reedy edge; classic material for soft flutes and chiptune leads.
- Square — odd harmonics at 1/n: a strong 3rd and 5th. That gives the hollow, woody, clarinet-like character; the absent even harmonics are exactly what makes it hollow rather than full.
- Sawtooth — every harmonic at 1/n, the fullest of the four. Strings, brass, and most classic synth basses start from a saw for a practical reason: the filter you will meet in lesson 4 can only remove harmonics, so the raw material needs to arrive rich.
The four shapes with their recipes. Bars show harmonics 1 through 7 at their relative levels: one bar for sine, odd-only for triangle and square, the full series for saw.
Watch the spectrum, not the scope. Count bars per shape: sine one; square at 1, 3, 5, 7; saw a full fence. Then sweep the frequency slowly — the whole comb slides as one piece, because every harmonic is a multiple of the fundamental you are moving.
Hardware rarely hands you a bare four-waveform menu; it gives machines that morph between recipes continuously. Two of them below, plus the sampler view of the same idea: retuning a recording slides its entire recipe up or down, exactly like the frequency slider above.