When I was 7 and had to learn English for the first time, I regressed to mathematics for comfort. I listened to English through my Chinese ears and the only thing that made sense was to resort to math.
“What Chinese animal year were you born in?” I motioned with my hands.
Instead of asking my American teachers for their age, I figured it out through the Chinese zodiac system. It was the logic of Modulo 12.
The Circle of Fifths works by the same principle. There are 12 notes in an octave, each corresponding to a key — hence 12 major keys. Each key is related to the next by a perfect 5th interval.
Such mathematics gives us an appreciation for the beauty of music. When I was studying composition at the conservatory, I learned that mathematics could reduce and ease the decision making required when composing.
One of these techniques was deciding on the interval to use. At each step you choose the next note to be the interval above or below. At some point, you switch to another interval. [Listen to Interval Scherzo by Anne Ku 2 min 47 sec, live recording of pianist Elwin Hendrianto's world premiere in Utrecht, The Netherlands, 22 March 2005]
Serial music is an example of a predefined decision rule. To learn about twelve tone music composition, I created a spreadsheet to compute the different rows, inversions, etc. I proudly showed it to my teacher. I found a way to use Excel to compose music!
I like objectives and constraints to be preset to help bound a problem. A commission such as “write a piece for my new born baby to reflect the our French and Japanese heritage” is better than “write a piece for my new born baby.” A commission that has a goal and set of criteria or constraints help make the job of a composer much easier.
Most of the time, we as performers ask composers to write a piece for piano and guitar with no criteria or constraints. Implicitly we want the composition to be playable, interesting to listen to, and have a longevity beyond the amount of time it takes to write and learn to play it. Every composer has his or her own ideas. They are not always explicit at the outset.
Understanding the mathematics of music helps to determine decision rules for composing.