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Music and the Making of Modern Science$
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Peter Pesic

Print publication date: 2014

Print ISBN-13: 9780262027274

Published to MIT Press Scholarship Online: May 2017

DOI: 10.7551/mitpress/9780262027274.001.0001

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Euler: The Mathematics of Musical Sadness

Euler: The Mathematics of Musical Sadness

Chapter:
(p.133) 9 Euler: The Mathematics of Musical Sadness
Source:
Music and the Making of Modern Science
Author(s):

Peter Pesic

Publisher:
The MIT Press
DOI:10.7551/mitpress/9780262027274.003.0010

Throughout his life, the great mathematician Leonhard Euler spent most of his free time on music, to which he devoted his first book. This chapter discusses how he reformulated the ordering of musical intervals on a new mathematical basis. For this purpose, Euler devised a “degree of agreeableness” that numerically indexed musical intervals and chords, replacing ancient canons of numerical simplicity with a new criterion based on pleasure. Euler applied this criterion (and Aristotle’s teachings about the pleasure of tragedy) to argue that minor intervals and chords evoke sadness through their greater numerical complexity, hence lower degree of agreeableness than the major. This work involved extensive attention to ratios and numerical factorization immediately preceding his subsequent interest in continued fractions and number theory. Having devised a new kind of index, Euler was prepared to put forward indices that would address novel problems like the Königsberg bridge problem and the construction of polyhedra, basic concepts of what we now call topology. Throughout the book where various sound examples are referenced, please see http://mitpress.mit.edu/musicandmodernscience (please note that the sound examples should be viewed in Chrome or Safari Web browsers).

Keywords:   Leonhard Euler, Musical Theory, Pleasure, Aristotelian theory of tragic pleasure, Continued fractions, Number theory, Königsberg bridge problem, Topology

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