Music

Pi(s)ano.

The Fibonacci sequence is defined by starting with 0, 1, and computing each subsequent term by adding the two terms before it. This gives us the first several terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. When taken modulo n for various natural numbers n, the Fibonacci sequence becomes periodic. For example, taken modulo 4, 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... --> 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 3, 1, ...

The melodies of this little piece come straight from the Pisano periods for various choices of n, by assigning 0 = middle C, and then taking each integer as the appropriate number of steps up the Phrygian dominant scale.  In the sheet music, each section is labeled with which n it is based on. For example, the Fibonacci numbers modulo 6 go "0, 1, 1, 2, 3, 5, 2, 1, 3, 4, ..." and the melody line labeled "mod 6" goes "C, Db, Db, E, F, Ab, E, Db, F, G, ...". The duration of the notes and the accompaniments in the left hand are determined according to whatever I felt like doing at the time. When a line says "mod n over mod m," it just means I added in another sequence as harmony underneath the melody line.

Cosine Fugue.

To construct the "subject" of this fugue, I took a graph of cosine. I treated every step of π/6 along the x-axis as representing a quarter note, and divided the y-axis into levels such that the range [-1, 1] would correspond to a full octave or 12 tones. Then, for each subinterval of length π/6, I looked at how and when the graph of cosine crossed these tone levels. As the "steepness" of cosine changes, the duration of notes adjusts to reflect it, so that some are quarter notes, some are eighth notes, and some are triplets. After this subject was constructed, I loosely built a fugue around it, featuring three voices, transposing (which can be thought of as a vertical shift of the graph) and overlapping so as to produce the contrapuntal effect of a fugue.