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8 More Properties of Pitch Class Sets and Interval Vectors

8.1 Common Tones when Transposed

If two PC Sets contain the same pitches, these are called the "common tones".

For example, the common tones between [0, 3, 4, 7] and [1, 3, 4, 8] are 3 and 4.

The interval vector can tell you how many common tones you will have after transposing a pitch class set. Simply look up the transposing interval in the interval vector, and the number you find will be the number of common tones after transposing the pitch class set.
For example, if the interval vector of the PC Set is: <324222> (6-Z13), and if you transpose the PC Set by a minor third, you will have 4 common tones between the original PC Set and the transposed PC Set.
How does the interval vector help? Take any Pitch Class Set
Algorithm:

Step 1: Find the Prime Form of your PC Set. Suppose it is (0,3,4,7)
Step 2: Lookup the PC Set, (0,3,4,7), in the Prime Forms table and find its interval vector, <102210>.
Step 3: The elements of the PC Set will tell you how many common tones to expect as you transpose the PC Set.

For example, if [0,3,4,7] is transposed by a half-step it becomes [1,4,5,8]. The original PC Set and the transposed PC Set have one common tone (4 = E).
As a second example, if [0,3,4,7] is transposed by a major third it becomes [4,7,8,11], which has two common tones (4 and 7).

Based on the interval vector <102210>, here is a complete list of how many common tones to expect when the PC Set is transposed:

a half-step => 1 pitch class remains the same
a whole-step => All new pitch classes
a minor-third => 2 pitch classes remains the same
a major third => 2 pitch classes remains the same
a perfect 4^th => 1 pitch class remains the same
a tritone => All new pitch classes

Except: For tritones (it would figure). When transposing by a tritone, you get double the number of common pitches as specified in the interval vector. For example, if you transpose [0,1,6] by a tritone, you would get two common pitch class sets, rather than one (see above for an example).
You can use this fact for composition to either make transitions smoother or more abrupt. If two adjacent harmonies in your music have many common tones, they will transition smoothly from one to the other. If they have few common tones, then the transition will be less smooth. For example:

Use for common tone transposition / modulation: Transpose a PC Set around a common tone for smoother transitions.

Alternatively, transpose with all new notes to emphasize the difference.

Also, the interval vector can be used to help identify when a PC Set can be combined with itself to make larger PC Sets with all unique pitches.

For example, the PC Set 6-8 has 6 pitches and the interval vector <343230>. This PC Set can be combined with itself by transposing it a tritone to make up a complete twelve tone row.

8.1.1 More Examples and Transpositional Symmetry

Let us consider two interesting PC Sets: The diatonic scale and the whole tone scale. Both of these scales have some rather interesting properties when they are transposed.

Diatonic Scale, 7-35:(0,1,3,5,6,8,10) which has the interval vector <254361>

Transpose the scale by a fifth or fourth (i.e. modulate to the dominant or the sub-dominant) and there will be 6 common pitch classes, and 1 new pitch class.
Transpose the scale by a half step, and there will be only 2 common pitches and 5 new ones. For example: C Major to C Major, or C Major to B major.
This gives rise, in tonal music, to the notion of "near" and "distant" keys.

The whole tone scale: 6-35:(0,2,4,6,8,10) has interval vector <060603>

Transpose this scale by any interval and either 1) all the pitch classes will be new or 2) all the pitch classes will be different.
Remember to double the value of the tritones entry (from 3 to 6).

If any entry of the interval vector is equal to the number of pitch classes in the set, then the PC Set can be transposed to itself with all pitch classes in common. This is called "Transpositional Symmetry."
In the Pitch Class Set Table, any PC Set with a "Count" column smaller than 12 has some Transpositional Symmetry.

8.2 Inversional Symmetry

The best way to see if (and how) a PC Set contains common pitch classes when inverted is to visualize the PC set on a clock face, and then look for one or more axis of symmetry.
OR: when looking at the table of all prime forms in the Appendix, if a PC Set has no entry in the "inversion" column, then it is inversionally symmetric on at least one axis.
Otherwise, there is no special math involved to determine inversional symmetry.
A PC Set which can invert to itself (on some axis of inversion) is said to be "Inversionally Symmetric".

Looking for inversions and inversional symmetry is another way to manipulate PC Sets to get new sounds.

In the following examples, the first PC Set is inversionally symmetric, and the second is not.