Resolution Geometry and the Origin of Musical Consonance: Deriving Harmony as the Minimization of Topological Cost

Here’s something that’s been bugging people since antiquity; when you hear two notes played together, some combinations sound stable and pleasant… consonant… and others sound tense and unstable… dissonant. Ever wonder why?

The question has been open for roughly 2,500 years:

what is the structure beneath consonance and dissonance? What is your ear actually computing?

This companion paper demonstrates that musical consonance is not merely an aesthetic preference, but a quantifiable measure of computational efficiency on a finite-capacity scaffold.  By modeling the auditory system as a ledger processing distinct identities (frequencies), we derive the hierarchy of harmonic intervals from two axioms: Exchange-Consistency and Finite Distinguishability.

Consonance occurs when two periodic signals (f₁, f₂) form a simple closed loop in phase space, allowing the system to write a Single Joint Receipt for the pair. Dissonance is Receipt Overflow; the failure of the system to resolve the combined waveform into a unified identity within the available temporal integration window.

We derive the "Cost of Closure" metric C(p,q) = p + q − 1 for frequency ratios p:q directly from the topology of torus knots on the phase scaffold. 

This metric accurately predicts the standard consonance hierarchy (Octave < Fifth < Fourth < Major Third < Minor Third).

Furthermore, we identify the 5-Limit Threshold as the "Geometric Horizon" of Western harmony, analogous to topological frustration boundaries in physical systems.