Foundation Universal Forcing Strict Music Music Is Positive Ratio Subrealization

A music of pure ratios

Music is built from ratios. When one note has twice the frequency of another, we hear an octave; a ratio of 3/2 gives a perfect fifth, and 4/3 a perfect fourth. These are not arbitrary labels. They are positive real numbers, and the space of all such ratios has a natural structure: you can compare two ratios, and you can say whether they are the same or different.

The Recognition Science framework asks whether this musical structure can serve as a concrete instance of its abstract logic. The declaration music_is_positive_ratio_subrealization answers yes. It establishes that the set of positive frequency ratios, with a cost function that charges 0 for equality and 1 for any difference, forms a valid subrealization of the framework's strict logic. This is a formal claim, checked by machine: the definition is a proof of the statement True, meaning the construction is accepted by the system.

The cost function here is deliberately simple. It is not a model of human perception or psychoacoustic dissonance. It merely records whether two ratios are equal or not. The framework's own documentation notes this is a strict pass, and that richer cost functions based on dissonance could refine the model later. The octave, perfect fifth, and perfect fourth are defined as specific ratios, and the whole structure is shown to be equivalent to the framework's arithmetic logic.

What this declaration does not claim is important. It does not say that this cost function is the correct or only way to model music. It does not claim that the ratio 3/2 is inherently more consonant than 4/3, or that any particular tuning system is privileged. It does not assert anything about how humans perceive intervals. The claim is narrower: that this mathematical structure, with this simple equality cost, is a legitimate instance of the framework's logic, and that this instance is equivalent to the framework's arithmetic.

The consequence is that music, in its most abstract form as a space of frequency ratios, is not separate from the framework's arithmetic logic. The two are shown to be equivalent structures. This opens the door to asking whether richer musical cost functions, ones that might capture something about consonance or dissonance, could also be realized within the framework, and what that would imply about the structure of music itself.

def music_is_positive_ratio_subrealization : True := trivial

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

What this page does not claim

This does not claim that the equality cost function is a model of human musical perception. This does not claim that any particular tuning system is privileged over another. This does not claim that the framework derives the consonance of specific intervals.

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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