{
  "abstract": "Transposition, inversion, and retrograde are fundamental operations in music theory, from Schoenberg's twelve-tone technique to Forte's set classes and Lewin's transformational theory, yet their use in melodies has rarely been quantified. We read a melodic line through the orbits and stabilizers of its windows of $m = 3-6$ notes under the group generated by these operations and cyclic shift. The window entropy splits into an orbit entropy (motivic vocabulary) and a within-orbit entropy normalized by orbit size (deployment), and the group elements relating windows of one orbit give the motivic transformations. In 852 keyboard pieces spanning four centuries, at least 97% of melodies have lower orbit entropy than surrogates preserving their pitch content, spectrum, or one-step transitions ($m \\geq 4$). Periods differ in vocabulary and symmetry types; musical forms differ in deployment. Recurring motifs are transposed 1.6 times more often than in random permutations of their forms, and transposed by a fifth or fourth 2.8 times more often, a preference that weakens from the Baroque to the 20th century and from Bach's fugues to Shostakovich's. Folk phrases and instrumental themes are rarely symmetric (2-10%), but are 1.6-5 times more often symmetric than note rearrangements. The framework extends to any symbolic sequence with a symmetry group.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40303",
  "author_total": 1,
  "authors": [
    {
      "name": "Alfredo González-Espinoza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/77v0aayrxdjya.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40303",
  "fulltext_url": null,
  "id": "77v0aayrxdjya",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40303",
  "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",
      "cs.SD"
    ],
    [
      "arxiv",
      "physics.soc-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40303",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/77v0aayrxdjya.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:28Z",
      "native_id": "2610.40303",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40303"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.soc-ph",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Symmetric Structure and Motivic Transformation in Melodies, Phrases, and Themes",
  "title_text": "Symmetric Structure and Motivic Transformation in Melodies, Phrases, and Themes",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:47Z",
  "url": "https://research.draconitic.com/paper/77v0aayrxdjya",
  "venue": "arXiv",
  "versions": []
}