Foundation Universal Forcing Music Realization Music Interpret

The musical reading

Music, in the Recognition Science framework, is treated as a discrete record of interval steps. A ledger, a discrete record of events, can be read musically: a step is a single move from one pitch to the next, and an interval is the number of such steps between two pitches. The declaration musicInterpret is the bridge that makes this reading precise. It takes a logical number, the framework's internal counting object, and maps it to a natural number, the ordinary counting number that measures interval steps.

This mapping is not a claim about how music sounds. It is a definitional choice, a way of modeling one kind of structure inside another. The framework models a musical interval as a count of steps, and musicInterpret is the function that carries out that modeling. The definition is simple: the logical number n becomes the natural number n. Nothing is lost in translation, because the framework's logical numbers are already built to correspond to ordinary counting numbers.

The surrounding declarations show what this musical reading supports. A cost function, musicCost, assigns 0 when two interval steps are equal and 1 when they differ. The framework proves that this cost is symmetric, meaning the cost of comparing two intervals does not depend on their order. These are small, exact results: the cost of comparing an interval with itself is 0, and comparing two different intervals costs 1. The musical realization is complete as a formal object, with the logical arithmetic matching ordinary natural number arithmetic.

What musicInterpret does not claim is any empirical fact about music theory, acoustics, or perception. It does not say that human listeners hear intervals as step counts, nor does it derive any property of actual musical scales or tuning systems. The declaration establishes a formal correspondence, nothing more. It is a definitional bridge inside a machine-checked library of formal theorems, not a statement about the physics of sound or the psychology of hearing.

def musicInterpret (n : LogicNat) : MusicalIntervalStep :=
  LogicNat.toNat n

@[simp] theorem musicCost_self (a : MusicalIntervalStep) : musicCost a a = 0 := by
  simp [musicCost]

theorem musicCost_symm (a b : MusicalIntervalStep) : musicCost a b = musicCost b a := by
  by_cases h : a = b
  · subst h; simp [musicCost]
  · have h' : b ≠ a := by intro hb; exact h hb.symm
    simp [musicCost, h, h']

What this page does not claim

No claim that human listeners perceive musical intervals as step counts. No derivation of any property of actual musical scales, tuning systems, or acoustics. No claim that the musical reading is unique or preferred over other readings of the ledger.

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/Foundation/UniversalForcing/MusicRealization.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

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