VIII. 20th- and 21st-Century Techniques
Interval-Class Vectors
Bryn Hughes and Megan Lavengood
Key Takeaways
- The overall sound of a set class is characterized in part by the intervals between the pitch classes.
- An interval class vector is a tally of each interval class inside the set, written as a six-digit number enclosed in angle brackets (<xxxxxx>).
- Each digit in the interval class vector corresponds to the count of a certain interval class: the first digit is the number of IC 1s, the second is the number of IC 2s, and so on. For example, in a C major triad, there are 0 of IC 1, 0 of IC 2, 1 of IC 3, 1 of IC 4, 1 of IC 5, and 0 of IC 6; so, our interval-class vector is <001110>.
- Interval class vectors may be helpful for recognizing set classes as they occur in pieces of music: if you know what the possible interval classes are within a set class, you might look for those interval classes in the piece.
- Sets containing a greater number of pitch classes have, correspondingly, more and/or higher numbers in their interval-class vector. 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.
The quality of any sonority can be roughly quantified by summarizing all the intervals it contains. Because all of the intervals contained in a sonority contribute to its overall sound, we must find the interval class (IC) formed by each pitch class in a set, not just those that are next to each other.
Let’s try a simple example first: a C major triad. There are three possible interval classes to be made using the pitch classes of this triad: between its root and third, third and fifth, and root and fifth. C–E is a major third (IC 4), E–G is a minor third (IC 3), and C–G is a perfect fifth (IC 5).
Interval class vectors summarize this as a six-digit number enclosed in angled brackets; each digit corresponds to the count of a certain interval class, as shown in
. The IC vector for the C major triad is <001110>, which shows that the C major triad has:<img class="wp-image-748 size-medium" src="https://viva.pressbooks.pub/app/uploads/sites/78/2019/05/tallying.002-e1662408328441-300×96.png" alt="the IC vector , with arrows pointing to each digit in the vector and labelling them IC 1, IC 2, IC 3, IC 4, IC 5, and IC 6.” width=”300″ height=”96″> Each digit in an interval-class vector corresponds to an interval class: the first digit is for IC 1, the second is for IC 2, etc. The number shown in the vector indicates how many of that interval class are present in the set.
- 0 of IC 1
- 0 of IC 2
- 1 of IC 3
- 1 of IC 4
- 1 of IC 5
- 0 of IC 6
Method
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. Here is a three-step process for creating an interval-class vector:
- Write the set in normal form.
- Find the interval class for every combinati