Cartesian Product

The Cartesian Product of A and B, denoted A × B, is the set whose members are all possible ordered pairs (a,b) where a is a member of A and b is a member of B. The cartesian product of {1, 2} and {red, white} is {(1, red), (1, white), (2, red), (2, white)}.

In more simpler terms the Cartesian Product takes the items and pairs them into all the possible combinations that they could be distributed in. These pars are then placed as subsets within a much larger set.

Another example can be displayed using sets {maple, mahogany} and {table, chair}

Using the Cartesian Product of these sets would be notated as: {maple, mahogany} × {table, chair}

This would result in {(maple, table), (maple, chair), (mahogany, table), (mahogany, chair)} as the Cartesian Product.

In the musical context, we can apply these mathematical methods to the direction that a C major scale may move. By doing this we could create many different types of melodies or practice patterns.

With a C major scale: {C, D, E, G, A, B} there are two possible diatonic movements {up, down}. Answers for example would be {C, up}, {C, down} ….etc… You could then calculate different directions in a sequenced scale. Taking all the possible combinations you could create different types of melodies or practice patterns.

{C, D, E, G, A, B} × {up, down}

C up to D up to E

D up to E up to F

E up to F up to G

F up to G up to A

G up to A up to B

A up to B up to C

These are just a few that can be named. The possibilities and sequences that could be made out of this are really extensive and useful for making scales or melodies. The use of the Cartesian Product can be found throughout improvising in Music with the choice of moving up and down in a scale.