## Geometric Chord System November 15, 2009

Posted by eric22222 in General, Math.

No, not that kind of chord.

When I was in high school, I had a bulletin board in my room. Strung across three thumb tacks was a rubber band. It may have appeared to be no more than a triangle, but I had placed the tacks precisely so that strumming the three legs of the triangle created a major chord.

Yesterday, while playing with a rubber band, a thought struck me. You can change the pitch of a plucked note by changing the length of the string. So, you could represent any chord imaginable with a polygon! So here we go: Dobbs’s Geometric Chord System.

Firstly, a line. This represents our root note. For all the examples in this post, we’ll be in the key of C, so C is our root note (260 Hz).

Next, we need to figure out what the lengths of our other notes are. They will be connected to the endpoints of our root. Let’s make a C major chord by adding E and G. First, here’s how you find the new length, where x is the number of half-steps up the note is from the root. In this case, well use 1 for our length and 4 for our x.

$2^{-x/12}$

So our new length is 2^(-4/12) = 0.7937. For G, the length is 2^(-7/12) = 0.6674. If you already know the exact frequency of both notes, you’ll get the same results from their ratio (C/G = 260/390 = 0.667). So let’s map those on our root!

The colored circles our of radius 0.7937 (red) and 0.6674 (blue). The point where they intersect is what we wanted to find.

Tada! This triangle represents the major triad. Before you make the same assumption I almost did, it is not a right triangle. But it would’ve been cool if right triangles exhibited some kind of interesting musical properties. Moving on: the points where our three-note chords can meet up are bounded. That is, if C is our lowest note, the other two will never escape a radius equal to C’s length. Furthermore, if those other notes are lower than the next C up, they’ll be bounded on the other side by C/2 (a string of half length produces a note one octave higher). So, here’s what we’ve got:

The deep green center is where all three note chords will wind up. Notice the symmetry. Any chord on the left side of the area has a twin on the right side. So let’s map some actual chords to our plane:

Vaguely diamond shaped, yeah? No. But it was my first guess. It’s actually a zig-zag. Check it out:

And because that last one looked pretty cool, here’s a full grid of chords, with half-note steps. Notice the logarithmic behavior of the grid.