VIII. 20th- and 21st-Century Techniques

# Interval Class Vectors

Bryn Hughes

The quality of any sonority can be roughly quantified by summarizing all the intervals it contains.  For simplicity’s sake, we will only account for interval classes when attempting to perform this assessment.  Because all of the intervals contained in a sonority contribute to its overall sound, we must find the interval class formed by each pitch class in a set, not just those that are next to each other.  To make this task easier, and less susceptible to error, it is best to make a tally chart to keep track of the interval classes involved.  Start with one pitch class, and measure the interval class between it and every other pitch class in the set.  Record each of these on your tally chart.  Repeat this process with every other pitch class until they are all accounted for.  Finally, total the number of interval-classes in each column, and copy these numbers into your interval class vector.

The Interval Class Vector tells us what intervals a composer may exploit from a particular set, which gives us the set’s particular sound and/or associations. Some sets contain a more or less even distribution of all the Interval Classes and are especially prized by atonal composers for these properties, while others are especially heavy on one IC or another and may contain none of a particular IC.
Interval Class Vectors also help us to make aural-analytical connections between two or more sets that may not appear to be immediately related. If they share a similar distribution of interval classes, they will also sound similar in a free-atonal context.
Sets containing a greater number of pitch classes will have more and/or higher numbers in their ICVs, naturally, because there are a lot more intervals at play. One way of checking your work with an Interval Class Vector is to add the numbers up. Each trichord has 3 available Interval-classes (and so the vector should always total to 3), each tetrachord has 6 (and so should sum to 6), each pentachord has 10, and so forth.

# How to find the IC Vector of a pc set

1. Write the set in Normal Form
2. Find the Interval Class for every 2 pitch-class combination in the set. Record your results on a tally-chart.
3. Total the number of tallies in each column (including zeros), and enclose these totals in angle brackets < >.

To figure out the interval-class vector, we’re going to simply figure out the interval class created by each combination of two pitches in the sonority. Let’s try a simple example first: A C-major triad. This is much easier to do if we use the pitch-class integer clockface, so the first thing we need to do is re-write our C-major triad as pitch-class integers: C is 0, E is 4, and G is 7. Next, we are going to look at the interval class for each pair of integers. Remember, the interval class is the shortest distance between two pitch classes on the clockface. So, 0 to 4 is interval class 4. 0 to 7 is interval class 5. Finally, 4 to 7 is interval class 3. When writing out the interval-class vector, we write out a “scoreboard” that tallies the number of each interval class in the sonority. There are six spaces in the interval-class vector, each representing one of the six interval classes. In our C-major triad example, there are 0 interval class 1s, 0 interval class 2s, 1 interval class 3, 1 interval class 4, 1 interval class 5, and 0 interval class 6s. So our interval-class vector is . We contain the vector in square brackets to differentiate it from other strings of integers that might appear in our analysis.

For our second example, let’s look at a really dissonant, cluster chord: F, F#, and G. In integers, this would be 5, 6, and 7. Next, we’re going to figure out the interval class between each pair of notes in the sonority. 5 to 6 is interval class 1. 5 to 7 is interval class 2. Lastly, 6 to 7 is interval class 1. All together, then, we have 2 interval class 1s, 1 interval class 2, and no other interval classes. So the interval-class vector is 

For the more visually inclined, here is a video on Interval Class Vectors.

Assignments
1. Worksheet on Interval Class Vectors (.pdf, .docx). Asks students to calculate interval class vectors for several sets. 