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.

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