Acoustics Music Pitch Jndfrom Jcost Pitch Jndfraction

The auditory quantum

The just-noticeable difference (JND) for musical pitch is the smallest change in frequency a listener can reliably hear. Classical psychoacoustics places this threshold for trained listeners between about 5 and 10 cents, where 100 cents make a semitone and 1200 cents make an octave. Untrained listeners need roughly a full semitone. A recognition framework, one that treats perception as a discrete record of events with a forced cost, derives a specific candidate for this threshold from its own structure rather than from a fit to hearing data.

In Recognition Science, the framework's library defines the pitch JND fraction as the reciprocal of the golden ratio raised to the eighth power, written 1/φ⁸. The golden ratio φ ≈ 1.618 comes from the framework's central cost function, and the eighth power follows from an eight-tick recognition cycle. This fraction of the octave equals 2^(1/φ⁸) ≈ 2^(0.0081) ≈ 1.0057, which is about 5.7 cents. The definition is a model choice, a way the framework chooses to map its internal quantum of recognition onto the physical scale of musical intervals. The library proves the fraction is positive and less than one, and it proves the associated cost function is zero when two frequencies match and never negative otherwise. These are formal theorems about the definition's internal consistency, not measurements of human hearing.

The framework's claim is that this 5.7 cent value lands squarely inside the trained-listener range of 5 to 10 cents. That is a prediction with a named falsifier: any psychoacoustic study showing trained listeners consistently outside a 3 to 20 cent window would break the correspondence. The definition itself does not prove that humans hear this way. It establishes a number and a set of formal properties; the empirical check against real listeners remains a separate, open question. The declaration does not claim that the golden ratio directly sets pitch perception, nor that the eight-tick cycle is a discovered feature of the auditory system. It claims only that a particular fraction, derived from the framework's internal logic, matches a known experimental range well enough to warrant a hearing test.

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

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

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

No claim that 1/φ⁸ is a measured property of human hearing. No claim that the golden ratio directly determines pitch perception. No claim that the eight-tick cycle is a discovered anatomical feature.

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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Derived articles

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