A Geometric Approach to Music.

I wanted to share a cool geometric frame1 on music that I came across while trying to coax nicer sounds out of my guitar.

Some basics

Some people say that music is the space between notes.
They might have a point, but frustratingly that doesn’t help me sound more interesting ( I went 7 years between notes once - and I would not reccommend it ).
Let’s talk about chords for a bit.

Turns out sounding two notes the same time is all you need to do to call yourself a chord these days. That being said, the distance between these notes makes a difference.

The distance between two notes is called an interval. Because us guitarists are a highly industrious bunch, we’ve used numbers to describe these intervals and can describe chords using formulae.

First things first. The way I’ve understood it, a piece of music is in a key. This usually clues us in on what note sounds most stable in this context - our Root note. Let’s consider the key of C and the C major scale, as there are no accidentals ( sharps or flats ). This scale nonsense that I’ve just introduced is a subset of all available notes. It follows the following intervals - W W H W W W H - where W is a whole step (the 2nd in interval-speak) and H is a half step (the flat 2nd b2).

Now our notes are sitting between these intervals. So you’d have the following notes and intervals on the C major scale:

R 2 3 4 5 6 7 R

Now any good guitar instructor will tell you that the spelling of a major chord is R 3 5
Armed with this triad, we can jump into this fancy new perspective that I’ve been enjoying lately:

Enter Shapes and Sounds:

It starts with imagining that all the chords and sounds that you’re hearing exist on an non-euclidean space.
Consider each note to be a point in this sound-space if you will. When we play a chord, and (let’s talk about three note chords for now, a triad) we’re forming a shape in this space for the duration of that chord.
Each note exists in a fixed distance from one another. That distance is the interval.
So if we take our R 3 5 we get a neat little triangle. The triangle moves around every time we play a different major chord ( different roots ). Cool huh?
Well, what would you do with a physical object? If you’re anything like me, you’d spin it around, or stretch the sides about. Turns out we can do the same thing in our musical space!
Rotate the triangle on a vertex, so that a different vertex, side or face hits you first. You’ve just done something that we call inversions in music. Inversions of the same chord sound different, and evoke different feelings in the listener. (Think 5 R 3 with the 5 in the bass or 3 5 R) This could be rather interesting to the discerning composer.
You can also compose similar shapes with notes with the same intervals from different octaves.

Now what happens when we take a minor chord? The formula for that triad is R b3 5 which means we play the flat third. This sounds VERY different. Our triangle is still a triangle, but the lenght of it’s sides have changed. Different triangles sound different huh?

What happens when we add other points? 7th Chords sound ‘Jazzy’ because of the different interval we’ve added to this basic triad. It’s now a different shape in our space. A polytope of sorts.

This was a neat frame - different shapes have different sounds, music can be ‘seen’ or ‘felt’ as a succession of different shapes.
To the improvising guitarist, remember that intervals exist close together and far apart on your fretboard.
Finding these different shapes and exploring spread intervals has been rather enjoyable, and I’m seeing shapes and colors in sounds that I didn’t know existed before.

This plays nicely with harmony as well. Imagine the rest of the band filling out different parts of your polytope at that point in time. You get to pick and choose the point you write, in turn expanding or shrinking or dramatically altering the shape and sound of the composition washing over any ears nearby.

Fun stuff!

Write to me on twitter @HollaAtJosh if you have any comments, can educate me further, or want to share something on the topic.


  1. My fascination with this interpretation may or may not have anything to do with the fact that I came across it while also taking graduate mathematics classes.