Foundation Universal Forcing Strict Music Strict Music Realization

A strict musical realization

In music, the octave is the interval where one note's frequency is exactly twice another's. The perfect fifth sits at a frequency ratio of 3/2, and the perfect fourth at 4/3. These three intervals are the classical building blocks of Western tuning, and they are the starting point for a formal construction called strictMusicRealization, which the Recognition Science framework defines as a concrete instance of its StrictLogicRealization structure.

The construction works by taking positive frequency ratios, meaning any positive real number that represents how many times faster one note vibrates than another. It assigns a cost, a measure of how different two ratios are, with the simplest possible rule: the cost is 0 if the ratios are identical and 1 otherwise. This cost function is symmetric, so the cost of going from ratio a to ratio b equals the cost from b to a, and it is zero exactly when the two ratios match. The framework proves these two properties as theorems in its machine-checked library of formal theorems.

The musical content enters through octave stacking. The framework defines the octave as the ratio 2/1, the perfect fifth as 3/2, and the perfect fourth as 4/3. It then constructs strictMusicRealization by using octave stacking as the canonical generator, meaning the entire structure is built from repeated multiplication by 2. This choice makes the octave the fundamental unit of the system, with all other intervals derived from it.

The key result is that this musical structure is not just a collection of intervals. The framework proves that the arithmetic inside strictMusicRealization is equivalent to the standard natural numbers, via a definition called music_arith_equiv_logicNat. This means the musical system, built purely from frequency ratios and octave stacking, contains the full arithmetic of counting numbers. The framework also notes that this strict pass uses the simple equality-cost on ratios, and that richer psychoacoustic dissonance costs could refine the model later.

What strictMusicRealization does not claim is substantial. It does not model human perception of consonance or dissonance; the cost function is purely formal, not psychological. It does not derive the specific intervals 3/2 and 4/3 from any deeper principle; they are chosen as definitions. And it does not claim that octave stacking is the only way to build such a structure, only that it is the canonical generator used here. The construction is a proof of concept: a minimal, machine-checked demonstration that musical intervals can carry full arithmetic structure.

/-- Strict musical realization using octave stacking as the canonical generator. -/
noncomputable def strictMusicRealization : StrictLogicRealization where
  Carrier := FrequencyRatio
  Cost := Nat
  zeroCost := inferInstance
  compare := ratioCost
  compose := fun a b => ⟨a.1 * b.1, mul_pos a.2 b.2⟩
  one := ⟨1, one_pos⟩
  generator := octave
  identity_law := ratioCost_self
  non_contradiction_law := ratioCost_symm
  excluded_middle_law := True
  composition_law := True
  invariance_law := True
  nontrivial_law := by
    have hne : octave ≠ (⟨1, one_pos⟩ : FrequencyRatio) := by
      intro h
      have hv := congrArg Subtype.val h
      norm_num [octave] at hv
    simp [ratioCost, hne]

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

noncomputable def music_arith_equiv_logicNat :
    (StrictLogicRealization.arith strictMusicRealization).peano.carrier
      ≃ ArithmeticFromLogic.LogicNat :=
  (StrictLogicRealization.toLightweight strictMusicRealization).orbitEquivLogicNat

def octave : FrequencyRatio := ⟨2, by norm_num⟩

noncomputable def perfectFifth : FrequencyRatio := ⟨(3 : ℝ) / 2, by norm_num⟩

noncomputable def perfectFourth : FrequencyRatio := ⟨(4 : ℝ) / 3, by norm_num⟩

What this page does not claim

The model does not represent human perception of consonance or dissonance. The intervals 3/2 and 4/3 are chosen definitions, not derived from any deeper principle. Octave stacking is the canonical generator used here, not the only possible one.

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/Strict/Music.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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