Music, Mathematics and Computer
Science
By Peyman Nasehpour
Music and
mathematics have always been considered two fields of research to be related
very closely to each other and fortunately computer science has come to help the
progress of the research recently. The history of investigating of the
relationship of music and mathematics goes back to the ancient times and many
mathematicians and musicians from Greece, Persia, India and Arabic States have worked on music and mathematics.
Today new aspects of the relationship between music and mathematics have arisen
and some researchers have started working on them, though it seems the
relationships of musical rhythms and drums with mathematics have not been
worked very broadly, though some have started to work over this field too. In this short note, we will discuss about the survey
of
how music, mathematics and computer science relate to each other.
MIDI Files
MIDI is the abbreviation of “musical
instruments of digital interface” and apparently since 1982, the MIDI files have been a standard for
communication and storage between electronic musical instruments. My friend,
Jeff Senn (AKA Jas) has worked on MIDI Files of
Arabic rhythm cycles to transfer them into electronic sounds to be played by a
computer machine with a certain tempo on the WWW through a Web browser. The
extension of the theory and practice of MIDI Files are necessary to use MIDI
Files to be able to extract a more natural sound from MIDI Files and transfer
the musical rhythms of other cultures such as Persia and India.
More info
can be found here:
http://it.stlawu.edu/~math/activities/FOS/2000/
MIDI Files and Prime Numbers:
http://www.2357.a-tu.net
Jas’ Middle
Eastern Rhythms FAQ:
http://www.khafif.com/rhy
Computer
Music Composition:
For the theory of computer
music composition the fields of mathematics, computer science and music are
used. There is a book written on music and computers by the following people:
Phil
Burk, SoftSynth.com
Larry Polansky, Dartmouth College
Mary Roberts, Princeton University
Dan Rockmore, Dartmouth College
Douglas Repetto, Columbia University
More info can be found
here:
http://www.keycollege.com/catalog/titles/music_and_computers.html
The Relationships of
Music and Algebra
The consideration of the
relationships between music and elementary arithmetic has an ancient history
going back to Pythagoreans. For example Fibonacci Numbers and Golden Section in
Art, Architecture and Music has been discussed here:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html
Then more advanced topics
such as logarithm and differential equation has been used (particularly in
acoustics and sound engineering).
Another interesting topic
is the application of combinatorics in music (melody and rhythm):
A good example is the book
“Adventures of Musical Combinatorics”:
http://members.aol.com/s6sj7gt/poly.htm
Recently some
mathematicians have tried to research about the relationships between music and
algebra. Some of the references are:
G-systems
An algebraic theory of
chord structures is being presented in this paper. Every tone grouping is
depicted as an instance of the so-called G-system. The aim is to provide a
simple algorithm for a generation of musical structures. It should be
useful for programmers of computer music as well as for those interested in
musical analysis. The theory of G-systems gives some known mathematical results
in a simple and clearly organized way. Therefore it might be inspiring for
mathematicians studying methods of enumeration, theory of groups, algebraic
solutions of combinatorial problems and other areas (Fermat's theorems, Gauss's
theory of equation classes, Polya's enumeration, Sylowov's groups...).
http://www.sweb.cz/vladimir_ladma/english/music/articles/dide99.htm
Another article is:
http://graham.main.nc.us/~bhammel/MUSIC/compose.html
(Finite groups and so on…)
Can One Hear the Shape
of a Drum?
The sounds of different
types of drums in a marching band are easy to distinguish, even without seeing
the instruments.
What makes these sounds so
readily identifiable is that each drum vibrates at characteristic frequencies,
depending mainly on the size, shape, tension, and composition of its
sound-generating drumhead. This spectrum of frequencies -- the set of pure
tones, or normal modes, produced by a vibrating membrane stretched across a
frame -- gives a drum's sound its particular color.
Physicists and
mathematicians have long recognized that the shape of the boundary enclosing a
membrane plays a crucial role in determining the membrane's spectrum of
normal-mode vibrations. In 1966, mathematician Mark Kac, then at Rockefeller University in New York City, focused attention on the opposite
question.
Kac asked whether knowledge
of a drum's normal-mode vibrations is sufficient for unambiguously inferring
its geometric shape. His paper, which proved remarkably influential, bore the
playful title "Can One Hear the Shape of a Drum?"
Reference:
http://www.maa.org/mathland/mathland_4_14.html
Another interesting book on
this topic is the book that will be released very soon, as Dave Benson reports
in his website: http://www.math.uga.edu/~djb/html/math-music.html
Prof. Dr. Habil Guerino
Mazzola is the Zurich-based specialist of MusicMedia Science and has many books
and articles on music and mathematics. His university page is:
http://www.ifi.unizh.ch/staff/mazzola/
And his website is:
http://www.encyclospace.org/
And another mathematician
is the late Iannis Xenakis (1922-2001) that has worked on music, mathematics
and computer science broadly. There is a website for his remembrance here:
http://www.iannis-xenakis.org/english/