The Mathematical Structure of Pythagorean-like Tuning Systems

The Pythagorean tuning system is an outstanding cultural achievement of the ancient world that remains significant today. Its history has been treated by quite a few authors; see, for example, [2]. Here we reveal its underlying mathematical structure. Our treatment covers Pythagorean tuning along with all its developments, from meantone and equal temperaments to irregular ones. On the surface, it is not apparent that together, they satisfy formative mathematical relations. In the section after next we focus on the significant instance in which the scale is generated by a single fifth. The general case is treated in a later section, where we also come to the conclusion that any scale may be considered an outcome of the Pythagorean setup. The main results offer formulas of welcome simplicity for the frequency ratios of the scales’ intervals and for the order of the participating notes and semitones. As an application, we address approximations by means of multidivision scales. We begin with an introduction to the subject, and we shall end with proofs in the final section.