Updated Version at New Location.
An updated version of this article on interval vectors is published at my new website of music theory resources at music-theory-practice.com.
Click here to go to the updated Interval Class Vector website at https://music-theory-practice.com/post-tonal/interval-vector-calculator.html
An interval class vector (ICV) is a series of six numbers that represents the intervalic content of a collection of pitch classes. ICVs may seem like a complex topic (as evident by the tweets below), but hopefully the following tutorial will help alleviate any apprehension.
An ICV isn't a single number, such as "032140," but instead a series of six separate single digits. Each digit represents the amount of times an interval occurs within a specified collection or sonority. It may be helpful to think of an ICV as a series of six columns where each column represents an interval and contains only a single digit. The first column represents the number of occurrences of half-steps in the collection/sonority, the second column represents the major seconds, and so on through the sixth column which represents the tritones. These "columns" are referred to as "Interval Classes." Traditionally, ICVs are notated with within carrots, "<" and ">," although this practice seems to be becoming less common.
Put simply: Inversions.
The largest interval represented by ICVs is a tritone. This is because any interval greater than a tritone can be inverted and thus becomes in an interval smaller than a tritone. This is why the tritone is always the farthest distance from any given pitch-class.
For example, the pitches C and A may at first appear to represent an interval of a major sixth, however, this assumption implies specific pitches rather than pitch-classes. A pitch, such as A440, is the A above middle C, also known as A4. A440 is not the same pitch as an A below middle C. On the other hand, the pitch-class "A" represents all As, from A0 through A7 and beyond. Therefore, the pitch-classes C and A ultimately boil down to a minor third rather than a major sixth.
Here is a chart that shows the inversions of intervals within an octave.
For more information on inversions consider viewing my page on musical inversions.
There are many ways to calculate the ICV of a collection, including just doing it in your head (assuming that the collection is small enough). What I've illustrated below is the way I was first taught to calculate an ICV by my post-tonal theory professor at the University of Southern California. The process may seem long as it's presented here, but it actually doesn't take much time at all. The demonstration below uses the following pitch classes: C, Eb, E-natural, and G. Here they are notated, just remember that we're treating these pitches as pitch classes -- not (octave dependent) pitches.
First, draw a circle and write the pitches of the chromatic scale around it in the same way you would for the circle of 4ths or 5ths.
Alternatively, you can replace the pitch names with their integer equivalent, i.e. C = 0, C# = 2, D = 3, etc. Just remember to use "T" for Bb/10 and "E" for B/11.
For demonstration purposes, I'm going to combine the two chromatic circles so that the pitch names are on the outside of the circle and the integers are on the inside. However, it's important to remember that this is actually redundant information -- they literally mean the same thing. In most introductory classes to basic post-tonal theory, you will be expected to be as comfortable with integer-notation as pitch/letter-name notation.
This table will provide the scaffolding for keeping track of a collection's intervals. Take note of the columns and rows, as the numbers of rows and columns will change depending on the size of your collection. The number of columns (Cs) will equal the size of your collection (SoC), and the number of rows (Rs) will equal the size of the collection minus one (SoC - 1).
In the first column of the table write the letter names of your collection of pitches. Here, I preceded each pitch name with "from," which is short for "Intervals from *insert pitch name*."
I suggest first learning this by counting the half-steps between each combination of pitches. To start, let's look at the first two pitch-classes in our collection: C and Eb. Using the chromatic circle, count clockwise from the first note, C, and count each note between and including the next note, Eb. Your final number should be three, which is, of course, the interval of a minor third. Clearly you can circumvent the use of the chromatic circle, but this method is useful for beginners and will help prevent any mistakes. Once you're comfortable enough with integer-notation and intervals you won't need the aid of a chromatic circle. If you want to speed up your knowledge of intervals, check out my Scale Degree Game.
Now that we have the number "3," write three in the first box on our table, directly next to "From C." Next, do this for the remaining intervals from C: C-E and C-G. Play the GIF to see how I like to visualize the process.
If you followed my process, you will have found seven half-steps from C to G. However, seven is too high and has no place on our ICV. But why? Well, "7" is too high because it represents a perfect fifth, an interval greater than a tritone, and ICVs "top out" at tritones. ICVs top out at tritone because tthey're greatest distance from any one note once inversions are taken into account. Therefore, any time that you get an interval higher than a tritone, 6, it needs to be inverted. A perfect fifth, or a "7," now becomes a perfect fourth, a "5." For more help with inversions, check out my music inversion calculator.
The benefit of using the chromatic cirlce is that you can easily invert an interval by just counting counter-clockwise. Check out the GIF again to see how couting from C-G clockwise gets the number (interval) 7 (a perfect fifth), but counter-clockwise reveals the number 5 (a perfect fourth) -- the inversion of a perfect 5th.
Now that you've finished counting the intervals "From C," try finishing the rest of the table. Play the GIF to check your results.
Now that the table is completed (right), we need to count the number of times that each number (read: interval) appears in the table. We know that there won't be any intervala larger than a 6 (see step 3.1.2), and we also know that there will not be any zeros on the table. Therefore, let's count in order, from 1s to 6s.
Once you've counted the number of times an interval occurs, place that number in the appropriate column (interval class) within the ICV. Play the GIF below to see how I do it:
And that is how you find the ICV for a collection/sonority!
I hope that helps -- leave a comment below if you have any thoughts or questions. Thanks!!