Symmetric Difference in the Musical Context
Symmetric Difference of sets A and B, denoted A △ B or A ? B, is the set of all objects that are a member of exactly one of A and B (elements which are in one of the sets, but not in both). For instance, for the sets {1,2,3} and {2,3,4}, the symmetric difference set is {1,4} . It is the set difference of the union and the intersection,(A ∪ B) \ (A ∩ B) or (A \ B) ∪ (B \ A).
In simpler terms, whatever item that is only in one of the sets is considered the symmetric difference.
For example,
Let’s say I have two sets of items; set A {cat, dog, fish} and set B {dog, fish, bird}.
If I put the set A over set B I will get the following:
{cat, dog, fish}/{dog, fish, bird}
The symmetric difference here is {cat} and {bird} because these items are exclusive to their respective groups.
In music, we can use the concept of symmetric difference to use notes that are outside of a key or a chord. The use of notes outside of a key or a chord can be referred to as passing tones, neighbor tones, etc. or at a simpler level they can be interpreted at ‘color’. The musical style that uses ‘color’ the most frequently is jazz where the 9’s b9’s #4’s etc. are included in chords to create a more colorful harmony and melody.
I’ll show an example of symmetric difference in music.
Let’s say I’m in the key of C major and I want to improvise over the key by focusing on notes which are outside of the key. We can isolate the notes outside of the key of C by finding the symmetric difference between the key of C and a chromatic scale.
For instance,
Set A = all available notes in C major
Set B = all available notes in a chromatic scale
{C, D, E, F, G, A, B} / {C, C#, D, D#, E, F, F#, G, G#, A, A#, B}
The symmetric difference is the set of notes that are only in one of the two sets.
Therefore, the symmetric difference is {C#, D#, F#, G#, A#}.
In western music this set of notes can be interpreted as a C# minor pentatonic scale. Thus, if one wants to use notes outside of the key of C major they can use the C# minor pentatonic scale.
Using the mathematical concept of Symmetric Difference can enable us to isolate groups of notes that we want to use.