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One option if you're primarily interested in representing the individual digits of pi is to use a representation in a base other than 10. For example pi base 12 would have an individual digital for each chromatic note. Here's a website that might help get you started: http://www.virtuescience.com/pi-in-other-bases.html
The number 10 doesn't necessarily map well to values in traditional musical theory. (For instance, there are 12 chromatic pitches per octave, using conventional divisions of the octave; diatonic scales have seven pitches; note durations are related as powers or negative powers of 2). So, for this reason, the world is your oyster! I guess you can choose any ...
One Idea I haven't seen mentioned is rhythm. Perhaps you can use some of the spare digits as a change in pace (f.e. switch from eights to quavers). Or you could map the spare digits to pre-conceived rhythmic motives. Another idea would be to use the digits that are not mapped to a note to switch instrument. HTH.
Why use base 10? You have to make some compromise somewhere, and since π is already transcendental, there is no rational radix that will accurately represent π. If you use heptary, π ≈ 3.0663651432036134110263402244652226643520650240155443215426431025161154565220002622436103301443233631. These digits map perfectly to the seven pitches in an octave. Using ...
An option which no one has really mentioned is to use those extra digits for special purposes (ie Change tempo, another instrument). If the primary instrument is a piano, I'd imagine that simply assigning a digit to the snare, bass and cymbal would add a lot of flair to your final music. In fact, adding new instruments will open you up to a bunch of new ...
Pi can also be expressed through various infinite series. I like François Viète series discovered in 1593: Square root from 2 is half octave distance. Maybe it is possible to represent the series as some sequence of sounds? Or maybe some other series would fit better? This might reproduce the spirit of Pi even better than replaying its decimal ...
Other answers have suggested using different bases. For an event in the Physics department, I did play pi in quintal, and there is a video. The sheet was generated using a script and Lilypond. Bonus: also in octal, but this one is not annotated.
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