Acoustics Music Pitch Jndfrom Jcost Pitch Jndfraction Lt One

A bounded step in pitch

The just-noticeable difference (JND) for musical pitch is the smallest change in frequency a listener can reliably detect. For trained listeners, psychoacoustic studies place this threshold at roughly 5 to 10 cents, where a cent is one hundredth of a semitone and a semitone is one twelfth of an octave. Untrained listeners typically need a larger change, often around a full semitone.

In Recognition Science (RS), the framework models a ledger, a discrete record of recognition events, and derives a cost function J(x) = (x + 1/x)/2 - 1 that measures the forced cost of recognizing a ratio x. The framework identifies a canonical quantum of recognition, J(φ), where φ is the golden ratio, and applies this to pitch perception. The relevant quantity is the pitch JND fraction, defined as the reciprocal of the eighth power of φ, written 1/φ⁸.

The declaration pitchJNDFraction_lt_one is a proved theorem in the framework's machine-checked library of formal theorems. It establishes that this fraction is strictly less than 1: 1/φ⁸ < 1. This is a structural result, not a numerical fit. The theorem also comes with a companion proof that the fraction is positive, and together these place the pitch JND fraction in the open interval (0, 1), a bounded step within the octave.

This fraction corresponds to a frequency ratio of 2^(1/φ⁸), approximately 1.0057, which is about 5.7 cents. That value sits squarely within the 5 to 10 cent range measured for trained listeners. The framework's account models this as the auditory phi-step, one eighth of a semitone, tied to an eight-tick recognition cycle. The theorem itself, however, only proves the inequality; the mapping to cents and the comparison with human data is a separate, empirical check.

What the theorem does not claim is just as important. It does not prove that any human listener has a JND of exactly 5.7 cents. It does not establish the measured range of 5 to 10 cents; that is a fact from psychoacoustics, not from the framework. The theorem only certifies that the defined fraction is a positive number less than 1, a necessary condition for it to represent a small perceptual step. The framework's specific prediction, that trained-listener JNDs fall within 3 to 20 cents, is a falsifiable hypothesis, not a consequence of this inequality.

theorem pitchJNDFraction_lt_one : pitchJNDFraction < 1 := by
  unfold pitchJNDFraction
  rw [inv_lt_one_iff₀]
  right
  apply one_lt_pow₀ one_lt_phi
  norm_num

theorem pitchJNDFraction_lt_one : pitchJNDFraction < 1 := by
  unfold pitchJNDFraction
  rw [inv_lt_one_iff₀]
  right
  apply one_lt_pow₀ one_lt_phi
  norm_num

theorem pitchJNDFraction_pos : 0 < pitchJNDFraction := by
  unfold pitchJNDFraction
  apply inv_pos.mpr
  apply pow_pos Constants.phi_pos

What this page does not claim

The theorem does not prove any claim about measured human pitch discrimination. The theorem does not establish the numerical value of the pitch JND in cents. The theorem does not assert that the pitch JND fraction is the only possible model for pitch perception.

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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