Foundation Universal Forcing Music Realization Music Cost

The interval cost

In music, an interval is the distance between two pitches, commonly described by the number of scale steps it spans, such as a second or a fifth. The Recognition Science framework builds a mathematical model of recognition events, where a ledger is a discrete record of such events. In its musical realization, the framework models an interval as a natural number, the count of steps in the interval. The declaration musicCost then defines the cost of moving from one interval to another: the cost is zero if the two intervals are the same, and one if they differ. This is the simplest possible comparison, a binary distinction between sameness and difference.

The framework proves two basic properties of this cost. First, the cost of comparing an interval with itself is always zero, which is immediate from the definition. Second, the cost is symmetric: the cost of moving from interval A to interval B equals the cost of moving from B to A. These are not empirical observations about music; they are formal consequences of the definition, checked in the framework's machine-checked library of formal theorems. The definition also connects to the broader framework by interpreting logical natural numbers as interval steps, so that the musical realization fits into the same structure as other realizations.

What the declaration does not claim is more limited than it might appear. It does not claim that this cost function is the unique or forced cost for music, nor that it derives from the deeper cost function J(x) = (x + 1/x)/2 - 1 that the framework proves elsewhere. The music cost is a simple model, a definitional choice for how to compare intervals, not a theorem about what music must be. It does not encode harmony, consonance, or any aesthetic property; it only records whether two intervals are the same step count. The musical realization is a lightweight illustration of how the framework's abstract structures can be instantiated, not a theory of music itself.

The practical upshot is that the framework can represent musical intervals in a way that is consistent with its general treatment of recognition costs, but the representation is deliberately minimal. A reader should not expect this declaration to explain why certain intervals sound pleasant or why scales are built as they are. Those questions remain outside the scope of this formal definition, which is concerned only with the arithmetic of step counts and the cost of changing from one count to another.

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

@[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']

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

What this page does not claim

The music cost is not derived from the framework's forced cost function J(x). The declaration does not model harmony, consonance, or any aesthetic property of music. The musical realization does not explain why certain intervals sound pleasant.

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)

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