Musica Mathematica: Traditions and Innovations in Contemporary Music

290 Pages·2017·7.859 MB·English

by  Rima Povilioniene

9
Methodology of Music Research
The concept of musica mathematica seeks to accurately examine the intersec-
tion of two seemingly radically different subject areas. From the perspective of
a European perception, the definition of the science of music was a result of
the Pythagorean concept of universal harmony. The Pythagoreans were the 9
first in European culture to raise the issue of uniting music and mathematics,
sound and number.
In the three parts of the monograph, versatile cases of the intersection of mu-
sic and mathematics are displayed, moving from philosophical and aesthetic
considerations about mathesis to practical studies, discussing the interaction
between music and other kinds of art (architecture, painting, poetry and litera-
Rima Povilioniene˙
ture), and providing a practical research of contemporary music compositions.
Musica Mathematica
Traditions and Innovations
in Contemporary Music
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Rima Povilioniene˙ holds a PhD in Musicology. She is a researcher at the Inter- P

national Semiotics Institute (ISI) at Kaunas University of Technology and an a
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associate professor in the Department of Musicology of the Lithuanian Academy Ri
of Music and Theatre.
www.peterlang.com ISBN 978-3-631-71381-5
9
Methodology of Music Research
The concept of musica mathematica seeks to accurately examine the intersec-
tion of two seemingly radically different subject areas. From the perspective of
a European perception, the definition of the science of music was a result of
the Pythagorean concept of universal harmony. The Pythagoreans were the 9
first in European culture to raise the issue of uniting music and mathematics,
sound and number.
In the three parts of the monograph, versatile cases of the intersection of mu-
sic and mathematics are displayed, moving from philosophical and aesthetic
considerations about mathesis to practical studies, discussing the interaction
between music and other kinds of art (architecture, painting, poetry and litera-
Rima Povilioniene˙
ture), and providing a practical research of contemporary music compositions.
Musica Mathematica
Traditions and Innovations
in Contemporary Music
a
c
ti
a
m
e
h
t
a
M

a
c
si
u
M

·
e˙
n
e
ni
o
vili
o
Rima Povilioniene˙ holds a PhD in Musicology. She is a researcher at the Inter- P

national Semiotics Institute (ISI) at Kaunas University of Technology and an a
m
associate professor in the Department of Musicology of the Lithuanian Academy Ri
of Music and Theatre.
www.peterlang.com
Musica Mathematica
Methodology of Music Research
Methodologie der Musikforschung
Edited by / Herausgegeben von Nico Schüler

Vol. / Bd. 9
Rima Povilionienė

Musica Mathematica
Traditions and Innovations in Contemporary Music
Bibliographic Information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the internet at http://dnb.d-nb.de.

Library of Congress Cataloging-in-Publication Data
Names: Povilionienė, Rima.
Title: Musica mathematica : traditions and innovations in contemporary music /
Rima Povilionienė.
Other titles: Musica mathematica. English
Description: Frankfurt am Main ; New York : PL Academic Research, 2016. | Series:
Methodology of music research ; Vol. 9 | Translated from Lithuanian. | Includes
bibliographical references and index.
Identifiers: LCCN 2016048131 | ISBN 9783631713815
Subjects: LCSH: Music—Mathematics. | Music—Philosophy and aesthetics. | Symbolism of
numbers in music.
Classification: LCC ML3800 .P81513 2016 | DDC 780/.051—dc23 LC record available at
https://lccn.loc.gov/2016048131

This book was originally published in Lithuanian under the title Musica mathematica. Tradicijos ir
inovacijos šiuolaikinėje muzikoje by the Lithuanian Academy of Music and Theatre (Vilnius: Lietuvos
muzikos ir teatro akademija, 2013, ISBN 978-609-8071-10-8)

Supported by the Lithuanian Council for Culture, the Research Council of Lithuania,
and the Lithuanian Academy of Music and Theatre

Mathematical editing by Dr. GIEDRIUS ALKAUSKAS
Translated by LAIMA VINCĖ SRUOGINIS
Language editing by KENNETH BASFORD

ISSN 1618-842X
ISBN 978-3-631-71381-5 (Print)
E-ISBN 978-3-631-71382-2 (E-PDF)
E-ISBN 978-3-631-71383-9 (EPUB)
E-ISBN 978-3-631-71384-6 (MOBI)
DOI 10.3726/b10528
© Peter Lang GmbH
Internationaler Verlag der Wissenschaften
Frankfurt am Main 2016
All rights reserved.
PL Academic Research is an Imprint of Peter Lang GmbH.

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Oxford · Warszawa · Wien
All parts of this publication are protected by copyright.
Any utilisation outside the strict limits of the copyright law,
without the permission of the publisher, is forbidden and liable to
prosecution. This applies in particular to reproductions,
translations, microfilming, and storage and processing
in electronic retrieval systems.
This publication has been peer reviewed.
www.peterlang.com
Contents
List of Analyzed Music Scores ....................................................................... 9
Author’s Preface ................................................................................................ 11
Foreword: Mathesis as a Philosophy of the Beauty of Music ......... 15
Part 1
A Retrospective of the Traditions of Musica Mathematica ... 21
1. The Constructive Relationships between Music and
Mathematics: The Pythagorean Conception of Universal Music
and its Spread in the Worldview of Later Periods ......................... 23
1.1. Music and the Theory of the Quadrivium ............................................... 24
1.2. Review of the Harmony of Spheres .......................................................... 26
1.3. The Phenomenon of Mathesis Universalis ............................................... 35
1.4. Expression of Numerical Proportions and Progressions ....................... 37
1.5. Ars Combinatoria and the Constructivism of Music ............................. 44
2. Semantic Interpretation of the Interaction between Music and
Mathematics: Mystic Middle Ages and the Sacral Baroque ..... 51
2.1. Semantics of the Kabbalah in Music ........................................................ 52
2.2. Symbolic Thinking and Sacral Numerology ........................................... 55
2.3. Codes of the Numerical Alphabets in Music ........................................... 60
3. Constructive Aspects of the Interaction between Music and
other Arts ......................................................................................................... 69
3.1. “Frozen Music”: Dialogues between Music and Architecture .............. 69
3.2. Ut Pictura Musica: The Interaction between Music and Art ................. 75
3.3. Musical Cryptography as a Common Denominator of the Sound
and the Word ............................................................................................... 79
4. Musica Mathematica in Practice: Aspects of Analysis ................ 93
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Part 2
The Renewal of Mathematical Techniques in Musical
Compositions of the 20th and 21st Centuries ............................ 99
1. Constructive Aspects of Music Composition .................................103
1.1. The Implications of Numerical Proportions and Progressions in
Music ............................................................................................................103
1.1.1. The Number Proportions and Progressions of Antiquity ..............104
John Cage. First Construction (1939) .................................................116
1.1.2. Fibonacci and Number Sequences Derived from it .......................121
1.1.3. The Prime Numbers and other Mathematically Determined
Sequences ............................................................................................129
1.2. The Renewal of Polytempo, Polyrhythm, and Polymeter .....................137
1.3. Symmetrical Algorithms and the Confrontation between
Symmetries and Asymmetries .................................................................146
1.4. Transformational Elements (Combinatorics, Permutations,
Rotations) ....................................................................................................151
2. Semantic Aspects of Music Composition ........................................163
2.1. Cosmological Number Codes and Graphic Constructions .................163
2.2. The Symbols of Magic Number Squares .................................................168
2.3. The Implications of Sacred Numbers ......................................................175
2.4. Personalized Semantics: The Significance of Individual Numbers ......183
Part 3 Innovations of Mathematical Techniques in
20th and 21st Century Music .................................................................201
1. The Mathematized Musical Graph ......................................................205
1.1. A Geometric Prototype as an Algorithm for Musical Composition ...... 205
1.2. L- system Formalities in Music .................................................................213
1.3. The Numericalization of the Musical Score ...........................................218
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2. Implications of Modern Mathematical Theories ..........................223
2.1. The Practice of Algorithmic Music and Computer- Generated
Composition ...............................................................................................223
2.2. Tonal Adaptations of Complicated Mathematical Processes ...............228
2.3. Fractal Theory Analogies in Musical Compositions .............................239
Afterword ............................................................................................................253
Bibliography .......................................................................................................257
Index of Names .................................................................................................277
About the Author .............................................................................................285
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