Acoustics Music Pitch Jndfrom Jcost Pitch Cost

The pitch cost function

In psychoacoustics, the just-noticeable difference (JND) for pitch is the smallest frequency change a listener can reliably detect. For trained listeners, that threshold is commonly reported as 5 to 10 cents, where 100 cents make one semitone. Untrained listeners need roughly a full semitone. The Recognition Science framework offers a single function that maps any frequency ratio to a number it calls a cost, a measure of how much recognition the ear must do to tell two tones apart.

The function is defined on the ratio of two frequencies: pitchCost(measured, reference) = J(measured/reference), where J(x) = (x + 1/x)/2 - 1. This J is the framework's recognition quantum, forced by a proved theorem about any cost function obeying five plain conditions. The function returns zero when the two frequencies are equal, and it is never negative for positive frequencies. Those two facts are proved in the machine-checked library of formal theorems.

The framework then identifies the canonical pitch JND with the recognition quantum evaluated at the golden ratio φ, which equals φ - 3/2 ≈ 0.118. Because J is monotone on ratios above 1, the frequency ratio that costs exactly 0.118 is φ itself, about 1.618. But that is a large interval, not a threshold near 1. The framework instead takes the auditory step as one-eighth of a semitone, tied to its eight-tick recognition cycle, giving a frequency ratio of 2^(1/φ⁸) ≈ 1.0057. That is 5.7 cents, squarely inside the trained-listener range of 5 to 10 cents.

In Recognition Science, this declaration establishes a concrete prediction: the smallest meaningful pitch step is about 5.7 cents, derived from the framework's own constants rather than fitted to ear data. The framework proves the mathematical properties of pitchCost, including its zero at unison and its nonnegativity. What it does not prove is that human ears actually work this way. The 5.7-cent value is a prediction with a named falsifier: any psychoacoustic study showing trained-listener pitch JND consistently outside 3 to 20 cents would refute it. The framework models the structure of recognition, not the biology of the cochlea.

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

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)

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

What this page does not claim

The framework does not prove that human pitch perception actually follows this cost function. The 5.7-cent value is not a measurement of human hearing, only a prediction from the framework's constants. The derivation of the eight-tick cycle itself is not formalized in the machine-checked library.

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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