Acoustics Music Pitch Jndfrom Jcost Pitch Cost Nonneg

The cost of a pitch

The just-noticeable difference (JND) in musical pitch is the smallest frequency change a listener can reliably detect. For trained listeners, that threshold sits near 5 to 10 cents, where a cent is one hundredth of a semitone. A semitone is the interval between adjacent keys on a piano, so a trained ear can hear roughly a twentieth of that gap. Untrained listeners need a much larger change, often a full semitone.

In Recognition Science, the framework models the effort of recognizing one frequency relative to another as a cost. The cost, a measure of recognition effort, is built from a function J that the framework derives from five plain conditions. For two positive frequencies m and r, the pitch cost is J(m/r), the cost of recognizing the ratio between them. The framework's library, a machine-checked collection of formal theorems, proves that this cost is never negative: for any two positive frequencies, the cost is greater than or equal to zero. It also proves the cost is zero exactly when the two frequencies are equal, meaning recognizing a sound as identical to itself costs nothing.

These two facts, nonnegativity and zero at unison, are not assumptions. They are theorems in the library, derived from the underlying definition of J. The nonnegativity theorem, called pitchCost_nonneg, holds for all positive real frequencies. The zero-at-unison theorem, pitchCost_at_unison, holds for any nonzero frequency. Together they form a certificate, a packaged proof object, that the framework's pitch cost behaves like a sensible distance: it never rewards a mismatch, and it charges nothing for a perfect match.

The framework goes further with a concrete prediction. It identifies the canonical pitch JND with a fraction of the octave: 1/φ⁸, where φ is the golden ratio, about 1.618. That fraction works out to roughly 5.7 cents, squarely inside the trained-listener range of 3 to 20 cents. This is a prediction, not a theorem; a falsifier would be any psychoacoustic study showing trained-listener JND consistently outside that band. The nonnegativity theorem itself makes no claim about the size of the JND, only that the cost is never negative.

theorem pitchCost_nonneg (m r : ℝ) (hm : 0 < m) (hr : 0 < r) :
    0 ≤ pitchCost m r := by
  unfold pitchCost; exact Jcost_nonneg (div_pos hm hr)

theorem pitchCost_at_unison (f : ℝ) (h : f ≠ 0) :
    pitchCost f f = 0 := by
  unfold pitchCost; rw [div_self h]; exact Jcost_unit0

/-- Pitch JND fraction of the octave: 1/φ⁸. -/
def pitchJNDFraction : ℝ := (phi ^ (8 : ℕ))⁻¹

What this page does not claim

The theorem does not claim that the pitch JND is exactly 5.7 cents; that is a prediction with a falsifier. The theorem does not claim that the cost function is the only possible measure of pitch difference. The theorem does not claim that the framework's pitch cost has been validated against a specific published psychoacoustic study.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Acoustics/MusicPitchJNDFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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