https://orcid.org/0000-0001-8786-5573
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Abstract
The logarithm mapping of natural numbers is a sum of products of coefficients. If these coefficients are arbitrary parameters, a new mapping of natural numbers to some subset of real numbers appears. This mapping preserves some crucial logarithm properties and constructs a new musical sound with a spectrum of inharmonic overtones. The simplest one-parameter mapping to a subset of polynomials with integer coefficients is constructed. The parameter defines a new “perfect fifth” similarly to the meantone temperament. It is an interesting case when the mapping parameter is defined from the linear relation between the new “major third” and the new “perfect fifth.” The most interesting case of a linear relation is the condition of zeroing the syntonic comma. Here, the new meantone temperament, based on the new “perfect fifth,” simultaneously coincides with Pythagorean- and five-limit-like tunings.
2010 Mathematics Subject Classification::
Acknowledgments
The author thanks Vasily V. Kabachenko for his great patience and constructive comments on previous versions of this paper. The author would also like to offer many thanks to Co-Editors-in-Chief Jason Yust and Emmanuel Amiot. Section 7 was not present in the th version and was added later as the answer to questions by reviewers. The author thanks the reviewers for helpful information, numerous helpful comments and good questions. The author would also like to thank Editage (www.editage.com) for English language editing.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Supplemental data
Supplemental data for this article can be accessed online at https://doi.org/10.1080/17459737.2023.2233972.