Every note is secretly a chord. Pluck one string and it sounds a fundamental f and a stack of overtones at 2f, 3f, 4f, 5f… — exact integer multiples. Their spacing carves the just-intonation ratios (octave 2, fifth 3/2, fourth 4/3, third 5/4); their relative loudness IS the timbre. Rendered, not quoted.
A string clamped at both ends can only vibrate in modes whose half-wavelengths divide its length: 1, 2, 3, … loops. Mode n resonates at fₙ = n·f₁ — the harmonic series.
overtoneFreq(n,f) = n·f
interval(n) = fₙ₊₁/fₙ = (n+1)/n
timbre(n) = amplitude of harmonic n
The intervals fall out of the integers: 2/1, 3/2, 4/3, 5/4 … the closer harmonics, the smaller the step. Pure ratios, not tuning conventions — the physics comes first.
The hidden chord inside a single note. The overtone series — integer multiples of the fundamental — is the physical origin of the-just-intonation's whole-number ratios and of every instrument's distinct timbre.
Neighbour sphere: the-just-intonation. There the ratios are the object; here they are derived from the spacing of the partials a real string already produces.
Live re-check: runs the engine's own selfcheck() against the panel again, confirms every harmonic is still n·f and every interval still exact — flips red the instant the Tamper (6) perturbs it.
A fundamental frequency and a chosen timbre (which harmonics, how loud).
The proven result: the partials and the intervals they define.
"Real strings are inharmonic — stiffness sharpens the upper partials, so n·f is a lie." True, and bounded: for an ideal flexible string the partials are exact integers; on a piano the inharmonicity coefficient B is ~1e-4, shifting the 16th partial under a semitone. The idealization is the limit, and the ear hears the fused pitch from the near-integer spacing regardless.
"Overtones are added on top of the note by the instrument's body."
No. The overtones ARE the note — a single plucked string produces the whole series at once; the body only
re-weights their amplitudes (the timbre).
"The fifth is 3/2 because we chose to tune it that way."
Backwards. 3/2 is harmonic 3 over harmonic 2 — a ratio the string hands you before anyone tunes anything.
"A missing fundamental means you can't hear the pitch."
False. Drop harmonic 1: the spacing between 2f,3f,4f is still f, and the pitch survives — a telephone trick.
The disclosed planted void: replace the integer ladder n·f with an inharmonic one, nₐ1.05·f. The intervals stop being 2, 3/2, 4/3 — and the Witness (7) catches it live.