The Euler Space

Guerino Mazzola, Yan Pang, William Heinze, Kyriaki Gkoudina, Gian Afrisando Pujakusuma, Jacob Grunklee, Zilu Chen, Tianxue Hu, Yiqing Ma

Research output: Chapter in Book/Report/Conference proceedingChapter


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.

Original languageEnglish (US)
Title of host publicationComputational Music Science
PublisherSpringer Nature
Number of pages11
StatePublished - 2018

Publication series

NameComputational Music Science
ISSN (Print)1868-0305
ISSN (Electronic)1868-0313

Bibliographical note

Publisher Copyright:
© 2018, Springer Nature Switzerland AG.


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