Pourquoi la gamme a sept notes? Une math{\'e}matique des notes et des gammes

TL;DR

This paper introduces an algebraic framework for constructing musical notes and scales, revealing why certain note ranges like seven notes are mathematically natural, and classifies scales using multiplicative structures.

Contribution

It provides a novel algebraic approach to music theory, linking scale construction to multiplicative structures and clarifying the mathematical basis for common musical note ranges.

Findings

Classification of scales with fixed number of notes

Introduction of multiplicative structures to define tonality

Detailed analysis of five- and seven-note scales

Abstract

We present an algebraic construction of music notes and show how to associate them inseveral ways to construct music ranges. Then a family of ranges emerge with a fixed number of notes: two, three, five, seven, twelve, seventeen, etc. A classification of these scales is simple with the concept of multiplicative structure. The action of such a multiplicative structure on a music note introduces the definition of tonality. Multiplicative structure classification is then straightforward with the notions of type and mode. We detail the case of music scales with five and seven notes and propose musical examples for a wide range of modes.