{
  "abstract": "A cycle of nine pitch classes in twelve-tone equal temperament can be arranged so that its nine consecutive three-note windows realize, exactly once each, every trichord type containing a semitone -- types taken up to transposition but not inversion. Such cycles exist in abundance, and generalize to n-tone equal temperament, where the windows realize the n-3 semitone-containing types. In every one of the 1764 such cycles in twelve-tone equal temperament, the chromatic trichord 012 appears scalarly, as a direct chromatic run, and never in a broken \"zigzag\" contour such as 1-0-2. Why is the zigzag missing? We show that the zigzag presentation behaves as a conserved topological charge of an underlying interval-flow network: it occurs if and only if 3 divides n and n is neither 6 nor 12. The forward obstruction is a Z/3-valued conservation law, proved for all n by a weight-function argument; the two exceptional temperaments -- one of them the common Western tuning -- are excluded by finite certificates; and existence for every other multiple of 3 follows from an explicit construction. We also quantify rarity, and give exact censuses well beyond the reach of direct search. The classification is machine verified in Lean 4 in both directions. The paper is written for two audiences. Musical sections require no more than pitch-class set vocabulary; the mathematical development is self-contained and elementary, yet fully rigorous.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26114",
  "author_total": 1,
  "authors": [
    {
      "name": "David Victor Feldman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0a4ee82vfr25s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26114",
  "fulltext_url": null,
  "id": "0a4ee82vfr25s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26114",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "1210|Musicology and Musical Analysis"
    ]
  ],
  "openalex": "W7214080127",
  "pdf_url": "https://arxiv.org/pdf/2609.26114",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0a4ee82vfr25s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26114",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26114"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "The Missing Zigzag: Cycles of Semitone Trichords and a Conservation Law in Equal Temperament",
  "title_text": "The Missing Zigzag: Cycles of Semitone Trichords and a Conservation Law in Equal Temperament",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/0a4ee82vfr25s",
  "venue": "arXiv",
  "versions": []
}