Mathematics In Music

I've heard a lot of people say that music is based on mathematical patterns, but I've always found that a bit of a laugh: Nobody determines what notes to put in music by pulling out a calculator. Well, I was listening to one of my favorite pieces of piano music today and subsequently found an article written on the mathematical formula behind it.

Linus Kesson wrote this excellent article deconstructing Arvo Pärt's "Fratres". I hope he will excuse me for borrowing liberally his work, but this is such a great example of mathematical patterns in music, and his illustrations show so well the inherent patterns, that I really can't explain this without using them.

Fratres (the ear-wrenching minute-long violin solo at the beginning notwithstanding) is constructed of 9 "phrases" with a repeating and unchanging "bridge" separating each one. Although each phrase seems quite distinct from the others, it is only a matter of expression: structurally... mathematically they are all equal.

Each phrase is divided into 2 halves. Each half phrase has 3 measures building to a final measure of 8 chords. All chords are of 3 notes each. The first measure is in 7/4 time, and always contains 4 chords: 1 and 2, plus 7 and 8 of the final 8-chord set. The second measure is in 9/4 time, and sticks chords 3 and 6 in the middle to make 6 chords: 1-2-3 and 6-7-8. The third measure is in 11/4 time, and puts in chords 4 and 5, to create the full 1 through 8 chord set. That's the first half of the complete phrase.

For the second half of the phrase, there are 3 more measures in 7/4, 9/4, and 11/4 time. The chords are played backwards... but not as "chords 8 through 1", but instead played mathematically backwards. Keep that in mind.

So playing the 2 halves (the 3 plus 3 measures) as described above makes one of the 9 phrases. (A violin is added as a counterpoint — a narration, I like to say — on top of the phrases, but the mathematical action is always in the chords.)

Now, as mentioned, each of the chords is comprised of 3 notes. The middle note is restricted to the A-minor triad that the piece was written in. The top and the bottom notes work their way through a D-minor scale. (Stay with me on this.)

The circle to the left is the 7 notes of the D-minor scale. What Fratres did was start the bottom note of each phrase at one point of the D-minor scale and work stepwise around the circle 8 steps (arriving back at the original note), while the top note follows it two steps behind. (Halfway through the phrase, the progression skips up or down an octave, depending on which direction it is moving.)

This circle is the middle note of the 3-note chord phrases. It only plays from the triad of the A-minor chord. The mathematical reasoning behind the particular progression of tone choices between "A", "C", and "E" is explained in Linus Kesson's article, but it is bit too detailed to explain on my non-musical blog to my non-musical audience. Suffice it to say that the middle voice of the 3-note chords is always either A, C, or E.

So what you have finally are the 2 circles (the top and bottom notes on the inside, the middle note on the outside) rotating together through each phrase. Again, it is a little complicated as to how Arvo Pärt choose the starting points and the rotational direction for each of the 9 phrases, but the mathematical progression is absolutely precise and does not deviate through the whole piece.

Remember that I said that the first half (measures 1 through 3) of each phrase was chords 1 through 8, and that the second half of each phrase (measures 4 through 6) was mathematically backwards? Now you understand it when I say that the second half of the phrase is created by playing through the circles in the reverse order. Notice that there are 14 notes in the outer circle? When the second half of the phrase begins from its starting point, the middle note plays backwards a different progression of notes than it did going forward.)

Finally, under each phrase, you place a constant left hand "A+E" foundation, and you have the entire piece.

For those of you who can read music, take a look at the first half of the first phrase and the chord pattern as described becomes instantly recognizable:

Remember, the final phrase is in the third measure. The first two are the incomplete phrase being filled.

Now we have established that, indeed, composers can create music based on very strict mathematical progressions. But, let's face it: That's an easy thing to do. I can cook dinner on mathematical progressions. The question is: will it taste any good? I can paint a picture with a mathematical progression of colors. The question is: will it look okay? The question with mathematical music is thus: How does it sound?

Well, if you ask me, I don't think a prettier piece of music has ever been written.