Set Theory’s Intersection in the Musical Context

The idea of “Intersection” in set theory is a fairly simple concept. Let’s use an example with two separate sets.

Example:

If you had one set of items, {apple, banana cherry}, and another set of items with {banana, cherry, date, eggplant} the intersection of these sets would be: {banana, cherry}. This is because they both contain these items.

The symbol used for Intersection is “ ∩,” thus in set theory this example would be written as: {apple, banana cherry} ∪ {banana, cherry, date, eggplant}.

In other words, the intersection of {apple, banana cherry} and {banana, cherry, date, eggplant} is: {banana, cherry}.

If we apply this to numbers the intersection of {1, 2, 3, 4} and {3, 4, 5, 6} is the set {3, 4}.

In layman’s terms, the intersection is simply the common item, or number, found in both sets.

Intersection will apply to more than two sets as well.

For example, if I had the sets {1, 2, 3,} {2, 3, 4} {4, 5, 6} and {5, 6, 7} I would have an Intersection of {2, 3, 4, 5, 6}. It is important to note that the concept of intersection applies to all items that intersect. This is the case even if the items are not found among all the sets in place. For instance, above the number 5 is only found in two sets but it is still in the Intersection since it intersects in two of the sets.

How does the set theory concept of Intersection apply to music?

We can apply Intersection in music just as we can apply the concept of “Union” to chords and scales.

To learn more about Union click here.

For example, we can make up a scale of notes by taking the intersection of chords in the key of C Major.

To learn more about chords click here.

Let’s take the chords C Major: {C, E, G}, A minor: {A, C, E}, G Major: {G, B, D}, B diminished: {B, D, F} and F Major: {F, A, C}.

These chords are all in the key of C. And they are all completely different chords with their own tonal functionality. Yet, when we calculate the intersection we get: {C, D, E, F, G, A, B}.

By using the set theory concept of Intersection, we have formed a C Major scale out of chords found in the key of C Major. This provides a mathematical explanation for the relationship that the notes in a key have with the harmonic properties of the key.

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