Acoustics Music Pitch Jndfrom Jcost Pitch Cost At Unison

The cost of a perfect match

In the Recognition Science framework, a ledger, a discrete record of recognition events, assigns a cost to comparing two frequencies. The pitch cost is defined as the framework's J-cost applied to the ratio of the measured frequency to a reference frequency. The declaration pitchCost_at_unison establishes the simplest possible case: when the measured frequency equals the reference frequency, so the ratio is exactly 1, the cost is exactly 0. This is a proved theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The theorem is a formal statement about the framework's own cost function, not a claim about human hearing. It says that a perfect match, a unison, carries zero recognition cost. This follows directly from the definition of J-cost, which is zero when its input is 1. The framework also proves that the cost is never negative for positive frequencies, so the zero at unison is a genuine minimum, not just an arbitrary value.

The framework's larger claim about musical pitch is separate and weaker. It predicts that the just-noticeable difference (JND) for trained listeners, the smallest pitch change they can detect, is about 5.7 cents, based on a proposed auditory step of one-eighth of a semitone. This prediction is tagged as a hypothesis with a named falsifier: any psychoacoustic study showing trained-listener JND consistently outside the range of 3 to 20 cents would refute it. The theorem about unison cost does not establish this prediction; it only provides a foundation for the cost model.

What the theorem does not claim is equally clear. It does not say that human listeners perceive a perfect match as 'free' or effortless. It does not say that the cost of a near match is proportional to the frequency difference in cents. The cost function is a mathematical object with specific properties, and the theorem describes those properties, not the psychology of hearing. The framework's contribution is a precise, machine-checked statement about its own model, which can then be compared against empirical data.

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 theorem does not claim that human listeners perceive a perfect match as effortless. The theorem does not claim that the cost of a near match is proportional to the frequency difference in cents. The prediction of 5.7 cents is not a proved theorem but a falsifiable hypothesis.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: