Is Pitch Class Multiplication Just Nonsense?

There's a technique that occasionally pops up in discussions of post-war serial music that seems to exemplify the idea of "paper music" -- music written to be studied an analyzed but not actually listened to. Let's see if we can make any sense of it.

To kick us off, here's something by American composer Charles Wuorinen, whose very nice book Simple Composition is where I learned about pitch class multiplication for the first time:

The Basic Idea

OK, so in simple terms the multiplication we're interested in is a procedure that works like this:

1. Find a collection of notes you're interested in, with no repetitions (e.g. a dominant 7 chord). If there's a concept of "root note" or "tonic", we'll assume you've transposed the notes so this is C (e.g. C-E-G-Bb).
2. Write the notes as numbers, starting with C=0, C#=1 and so on. That C7 chord becomes 0-4-7-10.
3. Multiply all these numbers by another number of your choice. Multiplying C7 by 3 gives us 0-12-21-30
4. Divide the resulting numbers by 12 and take just the remainder. Doing this to our example gives 0-0-9-6. (We do this because there are only 12 notes in the octave; 30 is 30 semitones above C, but that's just two octaves plus a tritone and the octaves don't mean much here so we "factor them out".)
5. Remove any duplicates and see what you've got. In our case we remove the extra zero and get 0-9-6, which is C-A-Gb. This is a Gb diminished triad.

It sounds complicated but if you do it a few times you'll find it's not too bad (if you know your twelve times table, anyway) and of course a computer makes it very easy. But what does it mean to say that C7 x 3 = Gb dim? It looks like we just shuffled numbers around without ever thinking about music, and that's not guaranteed to yield anything more interesting than doing the same thing randomly. Let's see what we can spot.

We can multiply by any number between 0 and 11 and get different results but they're not all equally interesting. Here's a table from Wikipedia:

In this post I want to focus on multiplication by 5 and 7 (M5 and M7 for short). These are the only numbers in the range that don't share a divisor with 12, apart from 1 and 11 of course. Multiplication by 1 doesn't change anything (as usual!) and by 11 has a simple interpretation we'll see in a moment. Apart from 5 and 7, the others can only ever give us a subset of the possible notes, so we quickly end up always getting things like diminished triads no matter what we started with.

Multiplication by 5 and 7

A closer look at the row for M5 reveals that it "fixes" the diminished seventh chord, in the sense that applying M5 to 0-3-6-9 just gives you 0-3-6-9 back. But the other eight notes get shuffled around in some kind of interesting way. M7 is similar but if "fixes" the whole tone scale instead, so only six notes get rearranged.

A rather closer look reveals that all these multiplications are "self-inverse" in the sense that applying one to some set of notes X and then applying it again to the result gives us X again.

Let's begin by understanding the M3 and M5 transformations a bit more clearly. To be honest they looked pretty chaotic until I tried this:

I'm not always a big fan of clockface diagrams but when they work, they work! Here we can see that:

• M11 is nothing but inversion about the 0 axis (we noticed this last time just by inspection)
• M5 inverts two pairs of notes -- [2, 10] and [4, 8] -- about 0 and another two pairs -- [1, 5] and [7, 11] -- about 3, leaving the rest fixed
• M7 inverts [3, 9] about 0, [1, 7] about 4 and [5, 11] about 8, leaving the rest fixed

Perhaps, therefore, we should think of M5 and M7 as follows:

• M5 is a sort of "inversion around the diminished seventh" (or "inversion around minor thirds")
• M7 is a sort of "inversion around the augmented triad" (or "inversion around major thirds")

A stray thought makes me wonder why we can't also have these, and perhaps lots of others:

There might be a whole universe here, or it might all end up being trivial. This is a distraction for now but maybe I'll come back to it.

Some Simple Examples

Let's look at what it means for M5 to fix the dim 7 chord at the root. With C as the root, applying M5 to C dim 7 just gives us C dim 7 again. On the other hand, the Db dim 7 chord (1-4-7-10) becomes D-F-Ab-B (2-5-8-11), which is D dim 7. Applying it to D dim 7 gives Db dim 7 -- this is just a result of the self-inverse property of multiplication. So M5 fixes the diminished chord at the root and "swaps over" the other two: in musical terms, it converts Half-Whole language into Whole-Half language.

Turning now to M7, again with the root at C, this one fixes all the notes of the C Whole Tone scale. By necessity it also fixes the other Whole Tone scale, but it shuffles notes around within it. Specifically, it swaps over the augmented triads in it: Db aug becomes Eb aug and vice versa.

One way to think about this is that if you start with one mode of the Augmented Hexatonic (say, C dim 7 plus Db dim 7: C-Db-E-F-G#-A) applying M7 kicks you into the other one (C dim 7 plus Eb dim 7). I've long thought of this scale as an augmented analogue of Half-Whole and this fits with that view.

So, in an alternative slogan to the ones above: M5 permutes diminished language, M7 permutes augmented language. In each case, the diminished (or augmented) chords change but they remain diminished (or augmented). This is surely enough to say that M5 and M7 do something musically meaningful. Let's look at a more involved example.

The All-Interval Tetrachords

I like to think of an All-Interval Tetrachord (AIT) as a combination of a minor third with a tritone such that they don't overlap. There's a post about these here if you'd like to know more about them. There are four of them, and they fall into two categories -- with the minor third rooted at C they are:

• eAIT, Forte 4-z15 (the minor third is enclosed by the tritone):
• 4-z15A: C-Eb-F-B
• 4-z15B: C-Eb-E-Bb
• iAIT, Forte 4-z29 (the minor third is interleaved with the tritone):
• 4-z29A: C-Eb-Db-G
• 4-z29B: C-Eb-D-Ab

If we choose any one of these, M5 and M7 take us on a tour of all of them:

We could have predicted some of this. Since the minor third and the tritone belong to different diminished chords, we expect M5 to permute one of them but not the other, which turns one AIT into its inversion (i.e. swaps between A and B forms). It's more surprising to me that M7 switches between Z-partners in this case, though -- it should be noted that this doesn't always happen!

Additionally, notice that applying M5 followed by M7, or M7 followed by M5, is equivalent to inversion. That makes sense because 5 x 7 = 35, which is equal to 11 modulo 12, so doing these in either order is the same as doing M11. This will always happen, so if you start with any PC set the most variety you can get will be one other PC set and the two inversions.

It follows that M5 and M7 give us two ways to reach the "same" sound, but in its two possible inversions. If the inversions are actually the same, M5 and M7 produce the "same" results -- I've put "same" in scare quotes because this is only true from a PC set perspective; we usually care about inversion.

This means we can pair up PC sets / scales / chords / whatever by the relation "you can get from one to the other via M5 or M7, perhaps with an inversion included". In my searches I haven't found any very interesting examples of this besides the AITs we just looked at. But those searches were far from exhaustive and what's interesting or not is subjective anyway.

Tentative Conclusions

It certainly seems as if M5 and M7 are not complete nonsense. But I've struggled to find many examples of what you might call "happy surprises" when taking a PC set and applying the multiplications to it. The AITs certainly have that property but not much else seems to. So I'm yet to be convinced that these operations can generate rich new material for composers and improvisors, or that anybody could hear M5 or M7 in music they were listening to.

It has to be admitted that Wuorinen's book -- the only extensive source I know of on the subject -- makes no great claims about the applicability of these operations. It's looking at them in the context of serial composition, were M5 and M7 stand alongside T(ransposition), I(nversion) and R(etrograde) as ways to transform a series into another series in a more or less logical way, and I have to admit they fit better there than in the more ramshackle world this blog inhabits. Even so, Wuorinen explicitly warns the musician against mistaking these for musical procedures; for him they're merely preliminary ways to establish

I'll keep my eye out for interesting applications of M5 and M7 in future but I'm not expecting any grand revelations from them.