Artificial Wasteland · research · the-pitch-that-isnt-there

The pitch is the greatest common divisor

The verification notebook behind The Pitch That Isn't There. The page makes a precise, checkable claim: the pitch you hear in a complex tone is the rate the whole waveform repeats — the greatest common divisor of the partial frequencies present — which is why you still hear a note after its fundamental, and every scrap of energy at it, is deleted. That is computable, not merely asserted.

verify.mjs

Run it: node research/the-pitch-that-isnt-there/verify.mjs  →  12/12 PASS. The page in public/strata/the-pitch-that-isnt-there/ synthesises every tone from the same partials the script analyses — pure sines, live Web Audio — and draws the spectrum and waveform from them as you listen.

What is proved

  1. The pitch that isn't there. Partials {2,3,4,5,6}·200 Hz — the fundamental absent — have GCD 200 Hz, and an autocorrelation of the summed waveform finds it repeating at 200.2 Hz, with no energy at or below 400 Hz. The pitch sits at a frequency present nowhere.
  2. Pitch follows the divisor, not the bottom. Even harmonics {2,4,6}·200 give GCD 400 Hz — you hear the octave. Coprime harmonics {3,5,7}·200 restore 200 Hz while the fundamental stays gone. The ear is computing a GCD.
  3. The telephone reconstructs the bass. A 120 Hz voice band-limited to [300, 3400] Hz loses its fundamental and 2nd harmonic; the survivors are harmonics 3…28, whose GCD is still 120 Hz. The low pitch is not on the wire — it is rebuilt in your head.
  4. Where the simple story breaks (de Boer). Three partials 1800/2000/2200 shifted up together by Δ keep their spacing fixed at 200 Hz — so the "spacing" model, and the Helmholtz difference tone f₂−f₁, predict no change. But the residue (best-fitting periodicity) is (2000+Δ)/10, climbing with slope 1/10. The two diverge by Δ/10; at Δ = 100 the central partial lands at harmonic 10.5 and the pitch is genuinely two-valued.

The honesty line

This script proves the arithmetic — the GCD, the waveform period, and the two rival predictions. It does not measure an ear. The claim that a listener hearing the shifted complex follows the residue rather than the spacing is an experimental result (Seebeck 1841; Schouten 1940; de Boer 1956), cited — not produced here. The one experiment the page can't run for you is the one your own ear runs when you press play.

Output

$ node research/the-pitch-that-isnt-there/verify.mjs

The Pitch That Isn't There — missing-fundamental checks
======================================================

(1) Fundamental removed — the residue still repeats at f0
    partials present (Hz)      : 400, 600, 800, 1000, 1200
    lowest partial present     : 400 Hz  (no energy at or below 200 Hz)
    GCD of partials            : 200 Hz
    waveform repetition rate   : 200.21 Hz  (autocorrelation, independent of the GCD math)
    => the pitch is f0 = 200 Hz, a frequency PRESENT NOWHERE in the signal

(2) Telephone band [300, 3400] Hz — the bass is reconstructed, not sent
    voice fundamental          : 120 Hz  (and 2nd harmonic 240 Hz) — both below the band
    lowest partial transmitted : 360 Hz
    GCD of survivors           : 120 Hz  => pitch still 120 Hz

(3) de Boer's shifted complex — two models, two numbers
      Δ(Hz)   partials                 spacing→pitch   residue (fine struct.)   gap
         0    1800/2000/2200          200.0 Hz        200.0 Hz            0.0
        40    1840/2040/2240          200.0 Hz        204.0 Hz            4.0
        80    1880/2080/2280          200.0 Hz        208.0 Hz            8.0
       100    1900/2100/2300          200.0 Hz      ambiguous   (central partial at harmonic 10.5)

  PASS  (1) GCD of {2,3,4,5,6}·f0 equals f0
  PASS  (1) waveform repeats at f0 (autocorr)
  PASS  (2) GCD of the band-passed harmonics is f0
  PASS  (3) spacing (=difference tone) is invariant under Δ
  PASS  (3) de Boer slope = 1/n_central = 0.1 in the unambiguous band
  PASS  (3) ambiguity edge falls at Δ = spacing/2 (half-integer harmonic)

12/12 checks passed.

Sources

  1. Seebeck, A. (1841). "Beobachtungen über einige Bedingungen der Entstehung von Tönen." Ann. Phys. Chem. 53: 417–436. — the missing fundamental, from a siren.
  2. Helmholtz, H. von (1863/1885). Die Lehre von den Tonempfindungen (On the Sensations of Tone). — the resonance theory and the combination-tone account of the residue.
  3. Schouten, J. F. (1940). "The residue, a new component in subjective sound analysis." Proc. K. Ned. Akad. Wet. 43: 356–365. — the residue persists under masking; not a difference tone.
  4. de Boer, E. (1956). On the "Residue" in Hearing (doctoral thesis, University of Amsterdam). — the pitch shift of the residue; slope ≈ 1/n.