Foundation Universal Forcing Strict Music Perfect Fourth

The perfect fourth as a ratio

The perfect fourth is one of the oldest intervals in music theory. It is the distance between two notes where the higher note vibrates at four thirds the frequency of the lower one. If a string sounds a low C, the F a perfect fourth above it vibrates 4/3 times as fast. Ancient Greek theorists such as Pythagoras and Aristoxenus measured it by ear and by arithmetic, and the ratio 4:3 appears in their accounts of consonance. The interval sits between the unison (ratio 1:1) and the perfect fifth (ratio 3:2), and it inverts to the fifth: a note a fourth above a lower note is a fifth below the upper octave.

The ratio has a simple mathematical shape. Doubling a frequency raises a note by an octave, so the octave is the ratio 2:1. Stacking a perfect fourth (4:3) on top of a perfect fifth (3:2) gives exactly an octave, because (4/3) × (3/2) = 2. This multiplicative structure is what makes the interval useful for building scales. In just intonation, the system of tuning that uses whole-number ratios, the fourth and fifth are the two basic building blocks alongside the octave. Equal temperament, the modern standard, approximates the 4:3 ratio with a value close to 1.3348, slightly flat compared to the pure 1.3333.

In Recognition Science, the declaration perfectFourth (a named value in the framework's machine-checked library of formal theorems) defines the interval as the positive frequency ratio 4/3, the same number musicians have used for millennia. The framework models recognition events with a discrete ledger, a record of distinct outcomes, and assigns a cost to comparing two frequencies: zero if they are equal, one if they differ. This equality-cost rule is symmetric, meaning the cost of comparing A to B equals the cost of comparing B to A, a property the library proves for its ratio cost. The perfect fourth then appears as one specific point in this cost structure, a ratio that the framework can use to build a musical realization of its logic.

What the declaration does not do is more important than what it names. The perfectFourth definition does not prove that 4:3 is the only consonant interval, nor does it derive the ratio from any principle of recognition. It simply chooses the classical value and plugs it into the framework's machinery. The framework's larger forcing chain, which derives constants like the golden ratio and the number of spatial dimensions from the cost function, does not reach the perfect fourth. The interval is an input to the framework's musical example, not an output of its theorems. A reader who wants a proof that recognition forces 4:3 will not find one here, because none exists.

The practical consequence is modest but concrete. The framework can now speak about music in its own terms: it has a domain of positive frequency ratios, a symmetric cost on them, and a named octave, fifth, and fourth. That lets a formal library state and check claims about intervals without inventing a new vocabulary. The perfect fourth remains the same interval Pythagoras heard, now carried into a machine-checked setting where its ratio is explicit and its role is clear.

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

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 ratioCost (a b : FrequencyRatio) : Nat :=
  if a = b then 0 else 1

What this page does not claim

The perfectFourth declaration does not prove that 4:3 is the only consonant interval. The framework's forcing chain does not derive the perfect fourth from the cost function. The ratio 4:3 is an input to the framework's musical example, not an output of its theorems.

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)

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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