The Connection Between Math And Music: Pattern Making With The Circle Of Nths

Circle Of 5ths

My son is learning about counting by a certain number to help him with multiplication. They call this method count-bys.

Counting by 5s means you keep adding 5 to the sum: 0,5,10,15 etc.

Music composers are familiar with this, but refer to these jumps on a note wheel called The Circle Of Fifths.

I've mentioned in earlier posts, that the 5th note in a scale creates a great deal of tension and makes you want to return to the root. You could even say that the 5th makes you homesick for the root chord.

Knowing what the 5th is in any scale becomes very important if a composer plans on changing keys in the middle of a song, or wants to write a composition using the 5th of the 5th chord as John Coltrane did in Giant Steps.

This circle serves as the breadcrumbs that always lead you back home.

Going around the Circle Of 5ths clockwise gives you 5ths. If you move counter-clockwise, you'll get 4ths.

Circle of 5ths (notes): C, G, D, A, E, B, F#, C#, G#, D#, A#, F, C
Circle of 5ths (numbers, C scale): 1, 5, 2, 6, 3, 7, 4.5, 1.5, 5.5, 2.5, 6.5, 4, 1

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Circle Of 3rds
A nice alternate pattern to try is the Circle of 3rds. It forces you to play an augmented chord (raise the 5th by a half step). If you start at C, you will always end up playing C, E, G# regardless of the direction you travel. It looks less like a circle and more like a triangle of 3rds.

Circle of 3rds (notes): C, E, G#, C
Circle of 3rds (numbers, C scale): 1, 3, 5.5, 1

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Circle of 3rds and 5ths

The circle doesn't have to jump by the same interval. It can jump by alternating intervals like 3 then 5.

Circle of 3rds and 5ths (notes): C, E, B, D#, A#, D, A, C#, G#, C
Circle of 3rds and 5ths  (numbers, C scale): 1, 3, 7, 2.5, 6.5, 2, 6, 1.5, 5.5, 1

The circle of 3rds and 5ths makes for a great composition tool. And although the progression is not something you hear often, it returns back home to the root completing the circle.