A couple years ago I created a piece of music for "Pi Day" from the first 256 digits of Pi. I won't bother to go into the details for that experiment, save to say that I simply took the numbers from Pi and added those to a root note of a major scale and let the notes fall where they may. This was a pretty simple exercise, and I'd been kicking around an idea for a much better exercise ever since.
With that in mind, given the proximity to St. Patrick's day, I decided to create a new piece with an Irish feel.
Here's what I did for this experiment:
I chose to use a 5-note pentatonic scale instead of a 7-note major scale, and I did so because there are 10 numbers in our base 10 numbering system, and 2 x 5 = 10. With that in mind, in my first draft of this experiment, all of the notes in the piece were derived by using a pentatonic scale with a 2-octave range, and mapping the numbers 0 to 9 from the first 252 places of Pi to the 10 notes of the 2-octave scale. (I'll explain why I used 252 places of Pi later.) This first draft placed the piece within the range of an Irish Tin Whistle, and I chose the key of D Major since that's the predominant key for that instrument.
However, while I was entering the notes and listening to the playback, many of the notes were often too far apart from their surrounding notes, with very strange octave jumps, which made the whole piece sound random. With that in mind, I decided to use modulus division to cut the range in half, thereby forcing all of the notes into a 1-octave pentatonic scale. In other words, if a number from Pi was over 5, then I subtracted 5.
This change for my second draft of this experiment resulted in a much smaller scale of "D E F# A B" to work with, and the 1-octave scale fell within range of the bagpipes, so I added drones for "D A D" beneath the melody to add to the illusion of a piper playing. However, during playback with a bagpipe sample, something sounded weird: every time there were two notes of the same pitch next to each other, it sounded odd. I quickly realized that was because an Irish musician won't hold a note for two beats - they'll use ornamentation to separate the identical pitches so it doesn't sound like one continuous note.
My good friends Randy Clepper (www.randyclepper.com) and Mark Wade (www.markalanwade.com) have taught a lot of classes about Irish ornamentation. I leveraged some of the things that I learned from them, and I added "cuts" to each of the sections where there were two notes that needed to be separated. By way of explanation, a "cut" is when you play a quick grace note above the note that is in the melody line. So if you have a A followed by an A in the melody, you would play the first A of the melody, then jump up quickly and play a B before returning to the second A of the melody, making sure to land the second A of the melody on the beat where it belongs. (Depending on the instrument that you are playing, you would play a cut by playing the first A of the melody, then hit a grace note A before jumping to the grace note B, and returning to the second A of the melody. It's like a really fast triplet.) Once I added the Irish ornamentation throughout the piece, it contributed significantly to the Celtic feel.
The drum beat was another exercise in self-indulgence that was fun to do. Because this entire experiment is about math, I chose to create a "Slip Jig," because they're in a 9/8 time signature. Hardly anyone uses that time signature, but it added a lot of possibilities. The accents that I chose were based on the steps that Irish dancers would use for a Slip Jig, which are beats 1 3 4 6 7, which creates a | X - X X - X X - - | beat. Since I play bodhran, I added rolls where I might use them if I were playing in a session.
Lest I forget, the 9/8 time signature is the reason for using the first 252 places of Pi. In my previous Pi Day experiment, I used the first 256 places of Pi, because 256 is one of those golden geek numbers. Since I already had those numbers lying around, I divided 256 by the 9 from the time signature, which resulted in 28.4. I rounded that down to 28, which gave me the number of measures that I would create. So 28 measures of 9 notes each meant that I only needed 252 places for this experiment. (See? It's all so simple, isn't it?)
And last but not least, the 157 bpm tempo that I chose to use was derived from taking 314 (e.g. "3.14") and dividing by 2. ('Cause, you know - more math.)