For this project I have Fourier transformed a drum sound to find component frequencies. As I understand it, a “noisy” instrument like a drum does not have a fundamental frequency or harmonics whereas a melodic instrument like a piano does. I’ve taken some images from a website on audio signal processing for a visual example. The first two images below are the signal and frequency spectrum of a voice, following that is a piano note, and the last two images are the signal and frequency spectrum of a snare drum.

I have found the drum sounds which produce a Fourier transform with the most clearly defined component frequencies. I then found the melodic sounds which produced Fourier transforms with a fundamental frequency matching the shape of the frequency components of the drum sound. I call these melodic sounds the drum component keys. Using the drum component keys, I have constructed a song in which the keys gradually start to overlap. Once all the keys overlap, a melody is no longer heard. What is heard is a drum beat.

One way to write a melody where each drum component key eventually overlaps is to use group theory to design appropriate arpeggio timing. Suppose for instance, that we end up with five drum component keys. We play each key in regular intervals, the length of those intervals each corresponding to an element of some cyclic group. If we have a cyclic group with at least five generators we can construct an arpeggio with five keys each ringing out separately initially and then coming together eventually.

As an aside, for reference, I will briefly describe the points of group theory which I am employing. For more basic reading, the reader should use search words such as “abstract algebra”, “group”, and “group theory”.

A group G is cyclic if there is an element a in G that can generate every element in the group.

The order of a group G is the number of elements in a group.

The order of an element a in G is the smallest positive integer n such that a^n = e, where e is the identity element of the group.

Let denote the group generated by a (if a is a generator of G then = G). If a has finite order n then = {e, a, a^2,…,a^(n-1)} and a^j = a^i if and only if n divides i-j.

An integer k in Z_n is a generator of Z_n if and only if gcd(n,k) = 1.

Since we are dealing with some musical scale, our drum components keys will be isomorphic to some Z_n. If we have five drum component keys, our musical scale is isomorphic to Z_5. These five notes will not likely be found on the Western scale, and so I must use as my drum component keys, melodic tones which can be shifted outside of the Western scale to have any fundamental frequency I choose.

Given that our drum component scale consists of five notes, our scale is isomorphic to the group Z_5 = {1,2,3,4,5} = {0,1,2,3,4}. What components of Z_5 generate Z_5? We require gcd(5,k) = 1. This applies to all the elements of Z_5. For example, <2> = {2^1 = 2, 2^2 = 4, 2^3 = 6 = 1, 2^4 = 8 = 3, 2^5 = 10 = 0} = {2,4,1,3,0}.

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