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The Euler Space

  • Guerino Mazzola
  • Yan Pang
  • William Heinze
  • Kyriaki Gkoudina
  • Gian Afrisando Pujakusuma
  • Jacob Grunklee
  • Zilu Chen
  • Tianxue Hu
  • Yiqing Ma
Chapter
Part of the Computational Music Science book series (CMS)

Summary.

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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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Guerino Mazzola
    • 1
  • Yan Pang
    • 1
  • William Heinze
    • 1
  • Kyriaki Gkoudina
    • 1
  • Gian Afrisando Pujakusuma
    • 1
  • Jacob Grunklee
    • 1
  • Zilu Chen
    • 1
  • Tianxue Hu
    • 1
  • Yiqing Ma
    • 1
  1. 1.School of MusicUniversity of MinnesotaMinneapolisUSA

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