plr-harmony
based on neo-Riemannian PLR transformations and triadic harmony
Source
C maj
C · E · G
Target
E maj
E · G# · B
Unweighted Shortest Path
Distance:2
Path:
LP
Weighted Path (P=1, L=1, R=2)
Cost:2
Path:
LP
Path Realization
C majC · E · G
L
E minE · G · B
P
E majE · G# · B
All Minimal Paths (showing 1 of max 16)
1.L P
PLR Operations
Parallel: flips mode, preserves root & fifth (±1 semitone on 3rd)
Leittonwechsel: flips mode, preserves 3rd & 5th (±1 semitone on root)
Relative: flips mode, preserves root & 3rd (±2 semitones on 5th)
Algebraic Structure
The PLR group is dihedral of order 24, acting simply transitively on major/minor triads. Each operation is an involution (self-inverse), maximizing pitch-class intersection with parsimonious voice leading. The group is dual to the T/I (transposition/inversion) group.
Harmony as Distance
The PLR metric measures harmonic distance as the shortest word length between triads. This correlates with voice-leading economy—P and L move by semitone, R by whole tone. Hexatonic cycles emerge from <P,L>, octatonic from <R,P>.