Difference tones in " non-Pythagorean" scales based on logarithms. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Morrill, Thomas
• Is Part Of: Journal of mathematics and music. Volume 14:Issue 1(2020)
• Page Start: 48
• Page End: 57
• Publication Date: 2020-01-02
• Subjects: creative process -- pitch -- intervals -- scales -- tonality; 00A65; Music theory -- Periodicals; Music -- Mathematical models -- Periodicals; Music -- Mathematics -- Periodicals; 781.2
• DOI: 10.1080/17459737.2019.1704084
• Languages: English
• ISSNs: 1745-9737

Abstract

Abstract : In order to explore tonality outside of the "Pythagorean" paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different degrees of the scale are themselves octave equivalents of notes in the scale. In doing so, we prove that a scale which contains all its difference tones in this way must consist solely of integer ratios. With this in mind, we present some methods for producing logarithmic scales which contain many, but not all, of the difference tones they generate.