Mathematical Forays into Musical Compositions
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In Swedish composer Daniel Cummerow’s musical works,
each musical fragment is determined by a mathematical
recipe through the use of a formula that links numerical
digits with musical notes. For example, the mathematical
constant p, has an intricate, vaguely medieval correlate,
whereas the decimal digits of the constant e, progress at a
relentless, suspenseful pace. 
A variety of different strategies are used by algorithmic
music composers to get such interesting results. One
method involves converting prime numbers directly into
their corresponding MIDI notes, at least up to 127, to get a
curiously rising scale. The MIDI specification assigns a
number to each note on a keyboard (e.g., middle C is 60,
C-sharp is 62, and so on, for a total of 128 tones) and is
used in computer programs to represent musical notes.
However, since there are an infinite number of primes, one
could continue by dividing each prime by a certain number, then use just the remainder to assign the musical
value—an elegant use of modular arithmetic. 
In some of his pi compositions, instead of mapping digits directly to their respective MIDI numbers, Cummerow
constructs the piece by assigning each digit from 1 to 8 to
a note in a specific scale; 0 signals a pause; and 9 means
either a pause or the repetition of the previous tone. 
In yet another experiment, Cummerow uses a unique
musical alphabet invented by French composer Olivier
Messiaen, which extended the German names of the notes
A to H by giving each letter of the alphabet its own pitch,
octave, and note value. Cummerow paired the first 255
digits of pi and applied the above formula to those that fell
below twenty-six. For the rest, he followed a different
recipe. 
Other techniques have yielded fascinating musical
pieces featuring the Fibonacci sequence, Pascal’s triangle,
and intriguing structures that have been associated with
chaotic dynamics, such as the Sierpinski triangle and
Lorenz’s butterfly.
Many other musicians have also delved into mathematical music, exploring ideas such as the fractal notion of
self-similarity in which each component is a miniature
replica of the overall structure.
Only rarely do composers enter the seemingly forbidding world of mathematicians and their abstruse concerns,
though the recent frenzy of forays into the field may be
just the start of a beautiful marriage between mathematics
and music. —F. Merali
Intriguing mathematical music compositions of the following artists can be found on the World Wide Web:
Cummerow (http://www.geocities.com/Vienna/9349/);
Chris K. Caldwell (http://www.utm.edu/research/primes/
programs/music/listen/); and José Oscar Marques
(http://www.midiworld.com/c/jmarques.htm)