Scales and Temperament from the Mathematical Viewpoint

Steven A. Bleiler and Ewan Kummel
Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (2016)
Pages 571–574


Scales constructed from pure harmonic ratios often contain enharmonic note pairs that do not sound in tune when played together. We examine the mathematics behind equal tempered scales that avoid this problem by insisting that the harmonic ratios between any pair of successive notes be identical. The mathematical problem is to construct equal tempered scales that do a good job of approximating the notes of scales built from pure harmonic ratios. Particular old and new solutions to these problems are discussed herein.