The Euler Space

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

doi:10.1007/978-3-030-00982-3_6

The Euler Space

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

Music (Twin Cities)

Research output: Chapter in Book/Report/Conference proceeding › Chapter

Abstract

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 language	English (US)
Title of host publication	Computational Music Science
Publisher	Springer Nature
Pages	51-61
Number of pages	11
DOIs	https://doi.org/10.1007/978-3-030-00982-3_6
State	Published - 2018

Publication series

Name	Computational Music Science
ISSN (Print)	1868-0305
ISSN (Electronic)	1868-0313

Bibliographical note

Publisher Copyright:
© 2018, Springer Nature Switzerland AG.

Publisher link

10.1007/978-3-030-00982-3_6

Find It

Find It at U of M Libraries

Cite this

@inbook{2eb5125c598b4584a68963cc825b96c4,

title = "The Euler Space",

abstract = "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{\textquoteright}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{\textregistered} music software.",

author = "Guerino Mazzola and Yan Pang and William Heinze and Kyriaki Gkoudina and Pujakusuma, \{Gian Afrisando\} and Jacob Grunklee and Zilu Chen and Tianxue Hu and Yiqing Ma",

note = "Publisher Copyright: {\textcopyright} 2018, Springer Nature Switzerland AG.",

year = "2018",

doi = "10.1007/978-3-030-00982-3\_6",

language = "English (US)",

series = "Computational Music Science",

publisher = "Springer Nature",

pages = "51--61",

booktitle = "Computational Music Science",

address = "United States",

}

TY - CHAP

T1 - The Euler Space

AU - Mazzola, Guerino

AU - Pang, Yan

AU - Heinze, William

AU - Gkoudina, Kyriaki

AU - Pujakusuma, Gian Afrisando

AU - Grunklee, Jacob

AU - Chen, Zilu

AU - Hu, Tianxue

AU - Ma, Yiqing

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

UR - https://www.scopus.com/pages/publications/85136626564

UR - https://www.scopus.com/pages/publications/85136626564#tab=citedBy

U2 - 10.1007/978-3-030-00982-3_6

DO - 10.1007/978-3-030-00982-3_6

M3 - Chapter

AN - SCOPUS:85136626564

T3 - Computational Music Science

SP - 51

EP - 61

BT - Computational Music Science

PB - Springer Nature

ER -

The Euler Space

Abstract

Publication series

Bibliographical note

Publisher link

Find It

Other files and links

Fingerprint

Cite this