# Union

“Union” in set theory is a very simple concept. If you had one set of items, let’s say a {cat, dog} and another set of items {fish, bird} the union of these two sets would be: {cat, dog, fish, bird} The symbol used for Union is “∪,” thus in set theory our example would be written {cat, dog} ∪ {fish, bird} In other words the union of {cat, dog}, {fish, bird} is: {cat, dog, fish, bird}. If we apply this to numbers the union of {1, 2, 3} and {4, 5, 6} is the set {1, 2, 3, 4, 5, 6}. Here’s just one more level to understand Union. If we had two sets that had some of the same items in both sets the union would only include the unique items and not the repeated items; for instance the Union of: {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.

There are many ways to apply the idea of Union to music. Here is one way often used to find the Passing Diminished Scales that are commonly used in jazz improvisation.

It is a common mistake to believe that a Diminished Scale i.e. 1, 2, b3, 4, b5, b6, 6 and 7 is the correct scale to use over all of these Passing Diminished Chords. The proper scale to use is a Union of the the Passing Diminished Chords’ Chord Tones and the Chord Tone of the chord that the Passing Diminished resolves to. Therefore if we had C#°7 to D minor 7 we would combine the notes of these two chords, i.e. “The Union,” to find the Passing Diminished Scale:

C#°7 Chord Tones: C#, E, G, Bb
D-7 Chord Tones: D, F, A C

The Union of these two chords creates this scale:

C, C#, D, E, F, G, A, Bb

Passing Diminished are always used in a key center, so it is best to think of these scales in relationship to the key. Below is the Passing Diminished Scales thought of in relation to the key center of C.

The following is a list of the passing diminished
chords and their relationship to the key.
#1 Diminished 1, b2, 2, 3, 4, 5, 6, b7
b2 Diminished 1, b2, 3, 5, b7, 7: 2 and 6 are often added to this scale to remove larger intervals.
#2 Diminished 1, 2, b3, 3, #4, 5, 6, 7
b3 Diminished 1, 2, b3, 4, #4, 5, 6, 7
#3 Diminished 1,#1, 2, 3, 4, 5, 6, b7
#4 Diminished 1, 2, b3, 4, #4, 5, 6, 7
b5 Diminished 1, b3, 3, 4, b5, 5, 6: 2 and 7 are often added to this scale to remove larger intervals. 3 is sometimes omitted by some musicians.
#5 Diminished 1, 2, 3, 4, 5, #5, 6, 7
b6 Diminished 2, 4, 5, b6, 7: 1,3 and 6 are often added to this scale to remove larger intervals.
#6 Diminished 1, #1, 2, 3, 4, 5, 6, b7, 7
b7 Diminished 1, b2, 3, 5, 6, b7: 2 and 4 are often added to this scale to remove larger intervals.

There are many more examples of the use of Union in music. Over time we will explore more of these –or try to come up with your own!