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The Music of Proportions

In: Proportions and Their Music

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  • Karlheinz Schüffler

    (Heinrich-Heine-Universität Düsseldorf, Mathematik)

Abstract

In this chapter (The Music of Proportions), the most important musical structures that can be developed from the idea of proportions are presented in a building manner. This includes the fundamental relation between proportion and interval as understood from the monochord. We present the most important methods of using the possibilities of proportion-mathematics to obtain a system for generating musical systems of intervals and tones. We then go into some detail about three systems-pythagorean, just diatonic, and ecmelic-and their most important interval structures-including their semitonia and commas. Symmetries of proportion chains and their corresponding chords generalize the major-minor idea and a wealth of proportion chains with their sounding chord equivalents emerges. On questions such as: How do the semitonia of just diatonics relate to each other? we find our way to the tetrachords of Greek antiquity, touch upon Gregorian chant, and in the final section reach the applications of the theory of proportions in the tonal world of the organ. Here we develop a foot-number arithmetic of the organ stops, guided by the laws of proportion.

Suggested Citation

  • Karlheinz Schüffler, 2024. "The Music of Proportions," Springer Books, in: Proportions and Their Music, chapter 5, pages 195-263, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-65336-4_5
    DOI: 10.1007/978-3-662-65336-4_5
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