Interval Permutations

Hello there, discerning and sophisticated music lovers!

I’ve decided it would be fun to use this blog to post information about various musical ideas and compositional tools that I’ve either used or plan to use when writing music. It could be useful to any of you who are interested in writing music yourself, or even for any of you who are just curious about some of the more obscure details of the music you listen to.

First up is a concept I came across about four years ago. I originally posted about it on an earlier blog that I used to write. I really didn’t say very much about it though, so I thought I’d revisit the idea to try to do it justice a little better.

Drew F. Nobile has written an article at Music Theory Online called “Interval Permutations” about how interval sequences can be permuted in order to produce different pitch class sets that are audibly related. I find the idea very interesting.

If you don’t know what an interval is, that is simply the distance between two pitches. Technically, it’s the difference between the wave frequencies of the two pitches, but it’s simpler if you think about it in terms of a piano keyboard. We’re just talking about how many white and black piano keys are in-between the two notes we’re dealing with. The number of semitones is the number of piano keys between the two notes plus one.

C and G are 7 semitones apart.

When Nobile refers to interval sequences, he’s talking about multiple intervals forming either a chord or a melody.

For example, a C major chord in root position (C, E, and G) can be formed by stacking a major third (four semitones) and a minor third (three semitones). C to E is the major third, and E to G is the minor third. A major third followed by a minor third is an interval sequence.

C major

Now we’re ready to understand interval permutations. If we change the order of the intervals in the sequence, we can produce a different chord. In our simple case, if we swap the order of the major and minor thirds, we get a minor chord instead of a major chord. From C, a minor third up is Eb. Then a major third up again is G. So we have a C minor chord.

C minor

Note also that there’s no reason the three notes must be played simultaneously. C, E, and G can produce a melody as well, and this also involves the very same sequence of intervals. So we can permute melodic interval sequences as well, thus transforming melodies in the same way that we transformed the C major chord into a C minor chord.

Now of course that was a really simple example, and while it is a different chord form and a different pitch set we haven’t actually arrived at a different set class. As different as they sound, major and minor chords are considered to be of the same set class because they are merely inversions of one another (as we just saw). I can’t get into the minute details of set classes right now. All you really need to know is that collections of pitches are called pitch sets and certain pitch sets are collected together into different set classes. And we need to know how to use interval permutation to transform chords and melodies. But if you are really curious you can read more about pitch class set theory and set classes and this tool is helpful for determining the set class to which any chord belongs.

You can apply the same kind of permutation to larger sets of intervals, and in some cases it will change not only the chord form but actually yield a different set class.

Of particular interest in Nobile’s paper is his example 18, a graph that depicts all of the possible relationships made by such intervallic permutation between tetrachordal set classes (that is, set classes of pitch sets containing four distinct pitch classes). After scrutinizing the graph, I noticed a natural symmetry in the relationships between the set classes, so I created my own version of the graph to emphasize that symmetry:

This looks cool, but how can the concept be used in music?

Well, it’s actually fairly simple. You can start with any set class and use interval permutation as a means to determine the musical material that follows. You don’t even necessarily have to strictly follow the rules. You can use it as a starting point and then tweak things until they work for you. (I mean, do you want your music to have weird mathematical relationships in it, or do you want it to sound good? Maybe it can be both, but if I had to choose one, I’d pick the latter.)

For instance, let’s say I begin with this chord:

This chord is in the set class with prime form 0247 (again, easily determined with this tool), and from the graph above I see that interval permutations are possible to set classes 0135, 0237, 0257, or 0358. Right now my interval sequence is a minor third (three semitones), followed by two major seconds (two semitones each). So let’s try moving the minor third into the middle of the sequence. That gives me this instead:

And this is 0257. Note that I can transpose this anywhere I like and move any pitch in the chord to a different octave and it’ll remain the same set class. So if I transpose down two semitones, I’ll get C-D-F-G, which is still 0257. And if I then transpose the C up to the top of the chord, it’s still 0257, but now I have this:

The interval sequence here is a minor third (three semitones), a major second (two semitones), and a perfect fourth (five semitones). Let’s move the fourth to the bottom so our sequence is a fourth, a minor third, and a major second, and let’s also transpose the result so the highest note is still C. This produces:

And this is once again 0247.

Now again, this is still a pretty simple example. You can probably come up with simple chord progressions without the aid of a tool like this. But you can do the same thing with larger set classes and chords with pitches clustered together more closely. For a musician who ordinarily sticks to basic chord progressions but wants to branch out into using more colorful and strange harmonies, it can be daunting finding sequences of chords that make sense.

This is where I have found interval permutation to come in very handy. If you want to find chord sequences that have an audible coherence to them, even when the chords in question have seven or eight distinct pitches in them, you can try transforming your chords using simple interval permutations to look for places to move to in the harmonic landscape. And remember that you don’t strictly need to keep the chords positioned on the same root. You can transpose them around after performing the interval permutation. Give it a try!

I’ll see you next time with more fun stuff about how I write music!