Foundation Universal Forcing Music Realization Musical Interval Step

The interval step

In music, an interval is the distance between two pitches. The Recognition Science declaration MusicalIntervalStep defines that distance as a natural number: 0 for the same pitch, 1 for a step, 2 for two steps, and so on. This is the classical notion of a scale degree count, the kind of number a musician uses when saying "up a third" or "down a fifth." The declaration is an abbreviation for the natural numbers, so every interval is just a nonnegative integer.

The framework's musical realization then adds a cost structure. The cost of moving from one interval step to another is 0 if they are the same step, and 1 if they differ. This is a discrete ledger: a record of whether a change happened, not how large the change was. Two theorems in the machine-checked library of formal theorems confirm that this cost is symmetric and that moving from a step to itself costs nothing. The arithmetic of the realization is provably equivalent to the natural numbers, so the musical reading does not introduce new structure; it renames counting.

In Recognition Science, this is one realization among many. The framework proves that any cost function meeting five conditions must equal J(x) = (x + 1/x)/2 - 1, and the musical realization is a lightweight instance where the carrier records interval steps. The semantic reading is pitch-ratio stacking: intervals compose by adding steps, which is the same arithmetic as multiplying frequency ratios. The forced arithmetic is the iteration count of interval composition, nothing more.

What the declaration does not claim is important. It does not define a specific tuning system, such as equal temperament or just intonation, because it only counts steps and never assigns them frequencies. It does not claim that music theory derives from the framework's cost theorem; the musical realization is a definitional choice, not a proved consequence. And it does not assert that the number of steps in an octave is fixed, since the natural numbers allow any count. The declaration establishes a vocabulary for talking about intervals as counts, and the cost theorems establish the minimal properties of comparison, but the rich structure of actual music remains untouched.

abbrev MusicalIntervalStep := Nat

def musicCost (a b : MusicalIntervalStep) : Nat :=
  if a = b then 0 else 1

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']

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

noncomputable def music_arith_equiv_nat :
    (arithmeticOf musicRealization).peano.carrier ≃ LogicNat :=
  musicRealization.orbitEquivLogicNat

What this page does not claim

The declaration does not define a specific tuning system such as equal temperament or just intonation. The musical realization is a definitional choice, not a proved consequence of the framework's cost theorem. The number of steps in an octave is not fixed by this declaration.

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.

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