Power Set

Power set of a set A is the set whose members are all possible subsets of A. For example, the power set of {1, 2} is {{}, {1}, {2}, {1,2} } .

A larger more complex example can be shown using the set {1, 2, 3, 4, 5, 6, 7}:

Since the power set of {1, 2, 3, 4, 5, 6, 7} is all the possible subsets of {1, 2, 3, 4} we would end up with:

{ {}, {1}, {2}, {3}, {4}, {1, 2}, {2, 3}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4} }

The power set is a pretty simple concept that can really start to have a large number of subsets when there are more and more items in the set used.

In music, we can use power sets to show how some scales are present in other scales.

For example, the first five notes of a C major Ionian scale {C, D, E, F, G} we can find a C major chord {C, E, G}.

Lets take the power set of {C, D, E, F, G}

We will end up with:

{ {}, {C}, {D}, {E}, {F}, {G}, {C, D} {C, E}, {C, F}, {C, G}, {D, E}, {D, F}, {D, G}, {E, F}, {E, G}, {F, G}, {C, D, E}, {C, D, F}, {C, D, G}, {C, D, A}, {C, E, F}, {C, E, G}, {C, F, G}, {D, E, F}, {D, E, G}, {D, F, G}, {E, F, G}, {C, D, E, F}, {C, D, E, G}, {C, D, F, G}, {C, E, F, G}, {C, D, E, F, G}}

In all of these subsets, one can find the sub set {C, E, G} which is a C major triad!

The capabilities of using the power set with scales could allow us to come up with every possible chord within a given key in music.