The Butterfly's Voice — a note's pitch IS its FFT bin

keyboard + waveform · sound

|X[k]| spectrum · number

A note's pitch IS the exact FFT bin its spectrum peaks in. Pick a note: the pitch that names it — semiToFreq(s), the same function that voices the Carillon's bells — predicts the integer bin c where the imported radix-2 FFT places its peak. Under coherent sampling the line lands in one bin: k_measured = k_predicted = c, exactly.

The note — pick a pitch

Click a key on the strip (or use ◀ ▶ / the arrow keys). The picked key gilds; the waveform and the spectrum both update from the SAME synthesised buffer. Default: middle C (s=0).

The sampling — break it yourself

Coherent

Device-rate (48kHz)

In coherent mode we DERIVE the sample rate so exactly c whole cycles fit the window — the tone IS the DFT basis vector at bin c, so the match is geometry, not luck. Flip to device-rate and the cycles don't close: the line LEAKS across bins, the peak is only the NEAREST bin, and k·Δf recovers f to ±Δf/2, not exactly.

The window — N (power of two)

1024

4096

16384

Exactness is (f, N, fs) aligning, not "more samples." Every choice here is a power of two — the butterfly's radix-2 requirement. A non-power-of-two N makes the transform throw (self-test LEG D).

The audio — muted by default

Sound off

Sound on

This is a PURE tone at the note's fundamental — not the Carillon's bell timbre (a real bell is inharmonic; its FFT peak isn't its fundamental). What you HEAR is the SAME buffer the FFT analyses. Honours the estate-wide ws:pref:muted key; first sound waits for a click.

One number, three faces

note—

f = semiToFreq(s)—

c = predicted bin—

bin c

measured peak k—

recovered k·Δf—

k = c — EXACT

recovered f

Coherent: k·Δf IS the note's pitch f, to machine precision — the bin the transform places it in is the frequency that names it.

Legend

the synthesised waveform (slate)

the period-comb · bin c · pitch f (gilt)

k_predicted = c (gilt diamond, drawn first)

k_measured (slate dot — lands ON c when coherent)

device-rate / leakage (violet)

↔ the cross

← The Butterfly · the transform the radix-2 FFT this bench runs on the note — imported, no re-typed math; here pointed at a single pure pitch. ← The Carillon · the voice where this note's pitch is BORN: semiToFreq(semi), the same generating fn the bells ring. The Sampling Theorem · why the teeth bite → here the device-rate line leaks (a smear) — below Nyquist, no fold. Drop f_s below 2f and the smear becomes a fold: same samples, a different tone. Leakage and aliasing, two faces of one ceiling.