Equal Temperament and Pythagorean Tuning: a geometrical interpretation in the plane

摘要

We show the following: Pythagorean Tuning (17 valued) and Equal Temperament (12 valued) can be canonically represented as discrete sets of the plane and there is a natural parallel projection of Pythagorean Tuning to the points of Equal Temperament. This fact implies particularly that when performing a composition on two instruments simultaneously in both Pythagorean Tuning and Equal Temperature, then the Garbuzov zones (supports of tone fuzzy sets) can be considered as segments on parallel lines in the plane.