Spectral Music: Building a Sound One Overtone at a Time

One pitch, infinitely many sounds

Play middle C on a piano and on a clarinet and you hear the same pitch, instantly recognisable as "the same note" — and yet nobody would ever confuse the two instruments. The pitch comes from the fundamental frequency, roughly 262 Hz in both cases. Everything that lets you tell them apart — what musicians and acousticians call timbre — comes from a second, richer layer sitting on top of that single number: the relative strength of every overtone above it.

The harmonic series

A vibrating string or air column does not oscillate at just one frequency. It vibrates simultaneously at the fundamental f and at a whole ladder of harmonics sitting at exact integer multiples: 2f, 3f, 4f, and so on, forming the natural harmonic series that has been the backbone of Western tuning theory since Pythagoras noticed that a string divided in simple integer ratios produces consonant intervals.

f = 220 Hz  (A3, the fundamental)

partial 1:  1×f = 220 Hz   — the fundamental, sets the pitch
partial 2:  2×f = 440 Hz   — an octave above
partial 3:  3×f = 660 Hz   — an octave + a perfect fifth
partial 4:  4×f = 880 Hz   — two octaves above
partial 5:  5×f = 1100 Hz  — two octaves + a major third

Each partial is a pure sine wave with its own amplitude and, in a real instrument, its own decay time. What our ear does automatically — and what makes music perceptually possible at all — is fuse this whole stack of sine waves back into a single perceived pitch and a single perceived timbre, because the partials share one common fundamental. Break that harmonic relationship and the fusion collapses: play the same five sine waves at slightly mismatched, non-integer frequencies and the ear stops hearing "one note" and starts hearing a cluster of separate tones, or a bell-like clang.

Additive synthesis: timbre from a recipe

Additive synthesis builds a complex tone the direct way: sum a bank of sine oscillators, one per harmonic, each with an independently controllable amplitude and its own envelope. It is the oldest synthesis method — pipe organs approximated it centuries ago by physically mixing ranks of pipes tuned to octaves and fifths — and it is also the most transparent, because every knob maps to a physical quantity you can reason about directly.

signal(t) = Σ  A_n · sin(2π · n·f · t + φ_n)
            n=1..N

A_n = amplitude of the n-th harmonic (the "recipe" — this IS the timbre)
φ_n = phase offset of the n-th harmonic (usually inaudible for a steady tone)

A pure sine wave is the recipe with only A_1 nonzero — it sounds hollow and flute-like, because a real flute's tone is in fact close to a sine wave with weak upper partials. A square wave uses only the odd harmonics with amplitude falling as 1/n; a sawtooth uses every harmonic falling as 1/n, which is why it sounds bright and buzzy — analogue synthesizers built entire genres on exactly that waveform. A clarinet, physically closed at one end, is close to a real-world square wave: it is missing even harmonics almost entirely, which gives its distinctive hollow, woody character compared to an oboe or violin that produces the full harmonic ladder.

Formants, envelopes and why instruments aren't static

A single fixed recipe of amplitudes only gets you a static, slightly synthetic-sounding tone. Real instruments do two more things that matter enormously to the ear. First, each harmonic has its own envelope — a plucked string's upper harmonics decay faster than the fundamental, which is why a guitar note sounds bright at the attack and mellows within a second. Second, the instrument's body has fixed resonances called formants that boost whichever harmonics happen to land near them, regardless of the note played — this is why a singer's vowel sound stays recognisable across a wide vocal range, and why brass and woodwind instruments have a characteristic "colour" that persists as you change pitch.

Inharmonicity: when the ladder gets crooked

Not every partial sits at a clean integer multiple. A piano string has physical stiffness, which pulls its upper partials slightly sharp of the ideal harmonic series — an effect called inharmonicity that piano tuners deliberately compensate for by stretching the octaves. Bells, drums and other stiff or two-dimensional resonators diverge much further: their partials follow the physics of plate and shell vibration rather than a simple string, landing at ratios with no simple relationship to the fundamental at all, which is exactly why bells sound like bells and not like plucked strings — pitch becomes ambiguous and the ear hears colour and clang instead of a single clear note.

Frequently asked questions

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