The Euler Space
In this chapter, we will discuss music theories from two mathematical perspectives. In the first part, we will explain what is a tuning and how the Euler Space relates to Western tuning systems. We will use Euler’s theory of tuning to explain three typical tunings: equal temperament tuning, Pythagorean tuning, and just tuning. In the second part, we will use a mathematical torus model ℤ3 × ℤ4 and its symmetries to explain counterpoint and its application in the RUBATO® music software.
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