Acoustics Music Pitch Jndfrom Jcost Pitch Jndfraction Pos

The pitch JND fraction

The just-noticeable difference (JND) for musical pitch is the smallest frequency change a listener can reliably detect. Trained listeners typically hear changes of about 5 to 10 cents, where 100 cents make a semitone and 1200 make an octave. Untrained listeners need roughly a full semitone. The Recognition Science framework, a research program that derives physical structure from a forced cost of recognition, offers a specific candidate for this perceptual step.

In the framework, the canonical pitch JND is tied to the recognition quantum J(φ) ≈ 0.118, where φ is the golden ratio (about 1.618). The proposed step is one-eighth of a semitone, derived from an eight-tick recognition cycle. The framework's library, a machine-checked collection of formal theorems, defines the pitch JND fraction as the reciprocal of φ raised to the eighth power: 1/φ⁸. The theorem pitchJNDFraction_pos proves this value is greater than zero, and a companion theorem proves it is less than one. Together they certify that the proposed step is a proper fraction of an octave, not zero, not negative, and not larger than the whole octave.

Numerically, 1/φ⁸ is about 0.0081, and the corresponding frequency ratio is about 1.0057. That ratio translates to roughly 5.7 cents, squarely inside the 3 to 20 cent range reported for trained listeners. The framework's account models the pitch JND as a small, fixed fraction of the octave, and the mathematics guarantees that fraction is well-behaved: positive, less than one, and consistent with the cost function being zero at unison and nonnegative for all positive frequency ratios.

What the theorem does not do is measure human hearing. The value 5.7 cents is a prediction, not a measurement. The framework names its falsifier: any psychoacoustic study showing trained-listener pitch JND consistently outside the 3 to 20 cent range would count against the prediction. The theorem itself only proves a fact about a defined real number; it says nothing about whether that number matches perception. That comparison is an empirical check, not a formal result.

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

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

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

/-- 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 prove that human listeners actually have a 5.7 cent JND. The theorem does not establish that the golden ratio appears in human auditory perception. The theorem does not measure the JND; it only proves a property of a defined number.

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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