Geometric Construction of Pythagorean and Just Musical Scales and Commas

Abstract

To your average pianist, G-sharp and A-flat are the same. They are located at the same black key between the white G and A keys, but with different names. For other musicians, however, for example violinists, they are different notes. They would claim that the piano represents a compromise, since it is often tuned to the equal-tempered scale, while other musical scales, based on simple harmonic ratios, often appear more harmonious to the ear. This article explains and illustrates this dilemma geometrically. In these scales, G-sharp and A-flat will no longer be the same. The difference between them is called the Pythagorean comma, and the name hints at the long history of music theory.