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/