Foundation Universal Forcing Strict Music Perfect Fifth

The perfect fifth in music theory

The perfect fifth is the musical interval between two notes whose frequencies stand in a 3 to 2 ratio. If one note vibrates at 300 hertz and another at 450 hertz, the higher note is a perfect fifth above the lower. This ratio is the oldest and most stable consonance in Western music: it is the interval between the first and fifth notes of a major scale, and it appears in the overtone series of every naturally vibrating string or air column. The ancient Greeks, including Pythagoras and his school, recognized this ratio as a fundamental building block of tuning, and it remains the anchor of just intonation and the basis for the circle of fifths.

The ratio 3/2 is not arbitrary. It is the simplest ratio, after the octave's 2/1, that produces a stable, consonant interval, and it is the only ratio that can generate the entire diatonic scale through repeated stacking and reduction by octaves. In equal temperament, the standard modern tuning, the perfect fifth is slightly narrowed to about 1.4983, but the pure 3/2 ratio remains the theoretical ideal. The interval's mathematical simplicity and its central role in harmony have made it a recurring subject in physics, psychology, and number theory, where it connects to the study of rational approximations and continued fractions.

In Recognition Science, the framework models the perfect fifth as a frequency ratio, a positive real number representing the quotient of two frequencies. The framework's library, a machine-checked collection of formal theorems, defines the perfect fifth as the ratio 3/2 and records a cost, a measure of recognition effort, that assigns zero cost to identical ratios and unit cost to different ratios. This cost function is symmetric, meaning the effort to recognize one ratio from another does not depend on direction. The framework proves this symmetry and the self-equality of the cost, and it places the perfect fifth inside a broader musical realization that uses octave stacking as its canonical generator. This realization is an example, not a derivation: it chooses the 3/2 ratio as a definitional starting point and shows how it fits into the framework's logic.

What the declaration does not claim is more important than what it does. It does not claim that the perfect fifth is uniquely forced by the framework's principles, nor that the 3/2 ratio emerges from any deeper cost minimization. It does not claim that the framework explains why humans prefer this interval, or that equal temperament is wrong. The declaration simply names a familiar musical object in the framework's language and verifies that the basic properties of its cost function hold for it. The framework's later results about forced structure, such as the golden ratio and the eight-tick cycle, do not depend on this musical example; it is a demonstration of how the framework can express a classical concept, not a proof about music itself.

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

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

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

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']

@[simp] theorem ratioCost_self (a : FrequencyRatio) : ratioCost a a = 0 := by
  simp [ratioCost]

What this page does not claim

The perfect fifth is not derived from the framework's forcing principles. The 3/2 ratio is not claimed to be unique among consonant intervals. No claim is made about human preference or the psychology of consonance.

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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