The Structural Harmonic Spectrum (SHS): mapping atomic-scale architecture onto pitch-class sets

Abstract

Full preprint of the Structural Harmonic Spectrum (SHS) study: the complete account, with the historical and methodological discussion, the proofs and the full result tables. The source code, the computed corpus of 288 structures over 581 crystallographically independent sites, the derived match listings and the rendered audio are deposited separately: https://doi.org/10.5281/zenodo.22873432. The Structural Harmonic Spectrum (SHS) maps the coordination geometry of an experimentally determined crystal structure onto musical pitch-class sets. For every crystallographically independent site the neighbourhood is partitioned into coordination shells by a stated geometric criterion; the shell radii are mapped to relative frequency by the convention f ∝ 1/r and folded into a pitch-class set in cents. No tuning system enters the construction, and every shell of this acoustic core is retained whether or not it approximates a musical interval, so the resulting sets are in general inharmonic and not confined to the 12-TET grid. The method is applied to the known allotropes of the elementary substances and to the structure types of inorganic compounds with the largest numbers of isotypic representatives, and yields several hundred distinct pitch-class sets; for orientation these are compared with the theoretical scales of a cross-cultural database and with chord templates, with deviations reported in cents. Each site, and the overlay of all sites of a structure, is rendered as a MIDI file. Version 1.1 (26 September 2026) supersedes the anonymous version deposited for review. It names the author and carries an ORCID iD on the title page; it adds a declaration of generative AI use; it defines the scattering vector where it is first used (q = 4π sin θ/λ), which the earlier version left implicit; the section on data and code availability has been rewritten for open access and renamed accordingly; and the typesetting no longer carries the identifiers of any journal. No result, figure, table or numerical value changed. Version 1.2 (27 September 2026) adds Section 3.10 with Table 8 and Appendix C: in ideal close-packed polytypes the coordination numbers of the six core shells depend only on the number of adjacent h-type layers, checked on the 119 close-packed sites of the corpus, and the exact closures at 1200 cents are examined against the input geometry, using two scripts published with version 1.1.0 of the corpus deposit. It also cites a reciprocal-lattice interval mapping (Leisegang et al., 2018), pymatgen and Jagodzinski (1949), corrects the author order and bibliographic details of Sethares and Staley (1992), and updates the data and code availability statement.