V. Fiber Bundle Structure
Fiber
The four inversional positions (voicings) of a single chord, obtained via Tymoczko’s interscalar transposition. Motion within the fiber changes registral distribution without changing pitch-class content — “color, not function.”
Coloration
A specific fiber position (inversion) of a chord in B. Following Persichetti: colorations vary the registral distribution of a chord without altering its harmonic identity.
Fiber Class
Class A: 1 of 4 inversions lands back in [6,8]. (11 orbits) Class B: 2 of 4 inversions land back in [6,8]. (3 orbits: Saddle Q686, Narrows Q676, Precipice Q688)
Class B membership correlates with period-2 step-sequence symmetry (Step Symmetry Conjecture — computationally verified, not yet analytically proved).
Universal L1 Law
Every inversion step costs exactly 12 semitones (L1 displacement) across all 14 orbits. The full four-step inversion cycle costs 72 semitones in the pattern [12, 12, 12, 36]. This universality justifies treating the fiber as metrically orthogonal to the base space.
Quartal/Quintal Duality Theorem
The quartal traversal (t₋₁) visits the same four voicings as the quintal traversal (t₁) in reverse order. Calling a chord “quartal” or “quintal” names the direction of fiber traversal — a ℤ₂ orientation symmetry on the ℤ₄ fiber.
Double Stability Theorem
The Saddle orbit [6,8,6] is the unique orbit satisfying both: (a) maximum degree 8 and maximum betweenness centrality in B, and (b) two inversions landing in [6,8] (maximum among all degree-8 orbits). No other orbit achieves both criteria simultaneously.
Octave-Transfer Incompatibility
The naïve cyclic inversion operator ρ (moving the bottom note up an octave) is compatible with [6,8] only for the all-tritone chord [6,6,6] — which is not in B. For all chords in B, ρ exits the [6,8] space. This is why Tymoczko’s interscalar transposition is the correct inversion mechanism for OTH.