Abstract:While motivated by structural problems in mathematical music theory, this article introduces a novel combinatorial framework that advances the classification of cyclic cubic bipartite graphs. We extend the classical study of Levi graphs by endowing their vertices with an internal algebraic anatomy -- specifically, treating them not as empty geometric nodes, but as defined subsets of a cyclic base space Z_n. This internal structure allows us to formalize and classify a highly restricted class of graph isomorphisms: those strictly induced by global affine bijections f(x) = ax+b (mod n) operating directly on the underlying base set. By applying this framework to generalized tone networks (Tonnetze) unrolled via the Chinese Remainder Theorem in composite dimensions -- specifically the classic 12-TET (3x4) and the decaphonic 10-TET (2x5) -- we reveal absolute geometric anchors for these spaces, namely the (9,4) and (6,5) systems respectively. We completely classify the topological orbits of these structured graphs, proving a fundamental architectural dichotomy: while the isomorphic landscape of 12-TET splits into an orientation-preserving family and an orientation-reversing chiral mirror (providing a rigorous foundation for musical Negative Harmony), the 10-TET space is unconditionally orientation-preserving. Finally, we demonstrate that these abstract combinatorial properties manifest as rigidly coherent, parallel auditory universes through explicit structural voice-leading maps and acoustic physical modeling synthesis.
| Comments: | This is a new, standalone article. While it addresses the harmonic context explored in arXiv:2601.02271, it introduces a fundamentally distinct combinatorial framework using the Chinese Remainder Theorem and structured Levi graphs |
| Subjects: | Combinatorics (math.CO); History and Overview (math.HO) |
| Cite as: | arXiv:2605.25649 [math.CO] |
| (or arXiv:2605.25649v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25649 arXiv-issued DOI via DataCite (pending registration) |
From: Pawel Nurowski [view email]
[v1]
Mon, 25 May 2026 09:53:26 UTC (29 KB)
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