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Geometric Chord System November 15, 2009

Posted by eric22222 in General, Math.
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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).

GeometricChordNotation 1

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!

GeometricChordNotation 2

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

GeometricChordNotation 3

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:

GeometricChordNotation 5

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:

GeometricChordNotation 6

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

GeometricChordNotation 7

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.

GeometricChordNotation 8

 

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Comments»

1. Site Admin - November 17, 2009

These are some awesome graphics for this very geometric chord system.


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