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4 Interval Vectors

An "Interval Vector" is a list of six numbers which summarizes the interval content in a PC Set. With a little experience, you will be able to get a sense for how a PC Set sounds when you see its interval vector. Further, once you know the interval content of a PC Set, you will also be able to manipulate the sound of the PC Set by inversion and octave displacement of pitches to emphasize certain intervals over others.

4.1 Pitch Intervals

The distance between any two pitches is called a "pitch interval". This is the standard definition for an interval in music. For example:
Ordered Intervals:

A3 to D5 = Perfect 11^th ascending = +17 half steps
D5 to A3 = Perfect 11^th descending = -17 half steps

Un-ordered Intervals:

A3 to D5 = Perfect 11^th = 17 half steps
D5 to A3 = Perfect 11^th = 17 half steps

4.2 Interval Classes

In the same way that many pitches "sound alike" and are therefore put into the same Pitch Class, there are also many intervals which sound alike and so are put into the same Interval Class.
There are six different interval classes which are numbered from 1 to 6.

m2 / M7 => 1 (half-steps)
M2 / m7 => 2 (whole-steps)
m3 / M6 => 3 (minor thirds)
M3 / m6 => 4 (major thirds)
P4 / P5 => 5 (perfect intervals)
A4 / d5 => 6 (tritones)

The interval class number (1 to 6) is the count of half steps between two pitch classes. In other words, it is the minimum distance between two pitches ignoring the octave displacement of either pitch.
For example, in the case of A3 to D5, if you moved A3 up an octave to A4, then the distance between the two is a perfect 4^th. And so the interval class is a '5', for a perfect interval.

4.3 Interval Vectors

An Interval Vector is a summary of all of the intervals between all pairs of pitches in a pitch class set. It is, essentially, a histogram of all of the interval classes which can be found in a PC Set.
For the purposes of this tutorial, an interval vector will be represented with angle brackets as follows:

For example, a C major chord is represented by the PC Set: [047] and has the interval vector <001110>. This is because a C major chord contains one minor third (from E to G) one major third (from C to E) and one perfect interval from C to G). Since a major chord contains no half steps, whole steps, or tritones, these entries in the interval vector are all set to zero (0).
Note that there is no agreed upon standard punctuation for representing an interval vector. The angle brackets appear to be the most common, but there are many other representations being used.

How to compute an interval vector:

o Step 1: Go through all pairs of pitches in your PC Set.

        If your PC Set has:           It will contain:
                2 pitches                        1 interval
                3 pitches                       3 intervals
                4 pitches                       6 intervals
                5 pitches                      10 intervals
                6 pitches                      15 intervals

Overall, the formula for computing the number of intervals (Ni) from the number of pitchs (Pi) is:
Number of Intervals = ( N*(N-1) ) / 2

o Step 2: For each pair, subtract the smaller number
from the larger number.

o Step 3: Take the result of step 2 and increment
the appropriate slot in the interval vector
using the following chart:

Example: [0, 2, 7, 8]

Step 1: 4 pitches in the pitch class set = 6 intervals = 6 pairs of pitches:
[0, 2] [0, 7] [0, 8] [2, 7] [2, 8] [7, 8]
Step 2: For each pair, subtract the smaller number from the larger number:
[0, 2] = 2; [0, 7] = 7; [0, 8] = 8; [2, 7] = 5; [2, 8] = 6; [7, 8] = 1
Step 3: For each difference in Step 2, increment the appropriate slot from the chart above:

Therefore, for our example, the interval vector is: <110121>
This means that the Pitch Class Set [0, 2, 7, 8] contains the following interval classes:
1 half-step, 1 whole-step, 1 major third, 2 perfect intervals, and 1 tritone
When I listen to this PC Set [0278], what I hear is a triad based on perfect fifths (0,2,7) = <010020> with an additional pitch (8) that adds some significant 'bite', via the half-step and tritone dissonance.