Cost of Closure on the Phase Torus II - From Triads to Progressions - Deriving Tonal Harmony, Cadence Asymmetry, and the Pachelbel Disjunction from a Single Closure-Cost Formula

A companion paper to Resolution Geometry and the Origin of Musical Consonance (Connerty, 2026), in which the closure-cost formula C(p,q) = p + q − 1 was derived from the topology of phase loops on a 2D torus, predicting the consonance hierarchy of two-note intervals. The present work extends C(p,q) to chords, pivot chords, and chord progressions. Five results follow.

First, the cost-min center of a triad — the chord tone whose pairwise closure cost is lowest — falls on the root for major triads (4:5:6) and on the fifth for minor triads (10:12:15). This asymmetry is a derivation of Hugo Riemann's harmonic dualism, a music-theoretic claim that has been disputed since the late nineteenth century.

Second, the V7 → i cadence in septimal-tuned harmonic minor produces a unison closure: the dominant seventh's cost-min center and the tonic minor's cost-min center occupy the same pitch class. This is the structurally tightest possible cadence under the framework, and it corresponds to the perceptual fact that V7 → i in minor often feels more decisive than V7 → I in major.

Third, the cost-tilt of a pivot chord — the differential between its closure cost to source-key tonic versus destination-key tonic — encodes harmonic function quantitatively. Predominant, mediant, dominant, and tonic-substitute pivots show distinct tilt signatures, and a structural rising-fifth/falling-fifth asymmetry emerges that is invisible to conventional chord-naming.

Fourth, applying the framework to multi-chord progressions yields three measurable progression-level invariants: implied tonic (where the closure-cost minimum sits across all chords), tonic strength (the gap between best and second-best implied tonic), and mean disjunction (the average gap between voice-leading distance and anchor-cost distance per transition). These invariants distinguish natural minor from harmonic minor, identify the tonal anchor of modal progressions, treat pop-axis rotations as the same key on different trajectories, and quantify the structural property that makes Pachelbel's canon sound "lush."

Fifth, total chord cost — the sum of all pairwise closure costs in a chord — orders altered dominants monotonically by harmonic tension and identifies V7♭5 as a chord with split cost-min anchor, providing a structural derivation of jazz harmony's tritone-substitution equivalence.

Methodology note: chord-internal closure costs use a directional convention (faithful to the chord's canonical voicing); inter-chord anchor jumps use a symmetric convention (minimum closure cost between pitch classes). The directional convention is empirically validated against major-key tonic identification on a five-progression test set (4 of 4 correct; symmetric fails 4 of 4). Both conventions are needed; using either alone produces systematic errors.

Together, these results extend the closure-cost framework from interval-level perception to the architecture of tonal harmony, deriving structural features of music theory that have historically been described rather than explained.