The Connection Between Math And Music: The Measurable Efficiency In Inversions

rubenalexander (69)in #mathematics • 8 years ago (edited)

In my last post, @allgoodthings said his friend had told him to memorize the Circle of Fifths. This brute force method is somewhat tricky over fast chord changes and requires a lot of movement on the piano. @allgoodthings learned inversions to help with this and we'll talk about why they call inversions a game changer.

The notes passed while playing these chords turns out to be very redundant, because the notes you are moving towards are already underneath your fingers. They are just in a different form.

A chord inversion can be understood like a PB & J sandwich. The main sandwich has bread, jelly, and peanut butter. This order is arbitrary and you can start with any ingrediant first. Yes, you could lay down peanut butter on a cutting board, bread, then jelly, if you were feeling a little odd one day. Or you could start with the bread, add jelly, then add peanut butter. Some versions are more messy than others, but they are all still a PB & J sandwich.

The same goes for chords. Take the C Major chord for example. It has a C, E, and a G. These can be rearranged, but will still be a C Major chord. [E, G, C], [G, C, E], and [C, E, G] are all still C Major.

C Major: C, E, G
C Major 1st Inversion: E, G, C
C Major 2nd Inversion: G, C, E

This may seem trivial in a one chord song, but as you add chords or increase the tempo the delays start adding up.

If we look at moving from a C Major to an F Major chord in the standard positions, this movement will take a total of 15 half steps.

Sum of C Major to F Major Half Steps C to F = 5 half steps E to A = 5 half steps G to C = 5 half steps

Total: 15 half steps

But if we move from C Major to the 2nd inversion of F Major it only takes a total of 3 half steps.

 
Sum of C Major to F Major (2nd Inversion) Half Steps
C to C = 0 half steps
E to F = 1 half steps
G to A = 2 half steps 
Total: 3 half steps

That is an 80% reduction when inversions are used!

If we consider all the half step changes required to play 4 chords in the Circle of 5ths starting from C Major it takes a total of 82 half steps

C Major to G Major to D Major to A Major Half Steps C to G to D to A: 28 half steps E to B to F# to C#: 28 half steps G to D to A to E: 26 half steps

Total: 82 half steps

This same progression could be reduced to 17 half steps if you invert half of the chords.

C Major to G Major (2nd Inversion) to D Major to A Major (Second Invesion) Half Steps
C to D to D to A: 2 + 0 + 2 = 2 half steps
E to G to F# to C#: 3 + 0.5 + 1.5 = 5 half steps
G to B to A to E: 4 + 2 + 4 = 10 half steps
Total: 17 half steps

That's a 79% reduction!

You can get a natural sound with less moving notes. Notice how the D key stays the same between the G Major and the D Major.

Here's an example of how inversions can be used to transition through chords in a song.