A whole new way to visualize and systematize musical chords
I’ve been moving in new directions for understanding the foundations of music, and looking at chord progressions in some recent articles here, and in a Science Moment.
Set aside the deeper idea I’m working on that chords might indicate a mover’s direction of movement (the broad topic of my earlier book, HARNESSED), here I want to tell you about an amazing new — I think — way to systematize all the chords within a key. The space is a cousin of one I described recently, but it’s in fact crucially different. You see it above, and it’s beautiful!
But what does it mean?
The dark black chords around the circle are just the Circle of Thirds. …all within the key of C. That’s when you shift two whole notes to play the next chord. Like from C to Em, or from F to Am.
My motivation here for the Circle of Thirds is that each of these chords is adjacent to its nearest neighbor in terms of the minimum number of semitone changes needed to get to one from to the other (where we don’t care which variant or inversion of the chord it is).
For example, although to get from C to Em one might shift from C up to Em, each finger moving two whole notes rightward, we can get from C to Em faster by just moving the c note of the C chord down to the b note. That’s an Em, b-e-g, just an inversion form of it.
So, as one moves around the circle, one is moving in the least chord-changey way, so to speak. This space in fact is the metric — or distance space — for chords within a key. The farther away a chord is around the circle from another chord, the more note shifts that must be made to get from one to the other.
But, my trick here is to think of the space as a vector space, where when two chords add, it’s just the union of their notes, and the resultant direction is the appropriate vector direction.
So, between C and Am, say, we can ask, What is the chord resulting from their sum? Whatever it is, it will be directed in between C and Am in this space.
The sum of C and Am — that is, just using all their notes combined — is just Am7, shown in between.
The same story goes for all the light gray chords all around. All the “sevens.”
That means that, in a sense, in moving from C clockwise, Am7 is nearer to C than is Am, and then Am is next-nearest in that direction. And so on.
One sanity check is that, notice that each of these chords has an opposite.
For example, Am has on the opposite side G7. In what sense are they opposites? Note-wise, they happen to have no shared notes, and so they sum to all notes in the scale. The idea is that two chords are opposites if they sum to “zero”, where zero occurs when all the notes are a result. Intuitively, if all seven notes are being played, it indicates none of the directions in this space.
The sanity check is that, for each of black-or-gray chord around the circle, the pairs of chords opposite each other do indeed add to “zero” (ahem, add to all the notes on the scale).
And, they perceptually feel kind of like opposites, in the sense that they’re not a subtle shift from one to the next in the way, say, C to Am7 is.
Ok, NOW, among the chords we have so far, we can find a pair separated by two chords in between, and add them.
For example, what happens if we add C and Fmaj7? When you combine all those notes, you actually have Fmaj7,9 (or just Fmaj9). And it’s direction in this space is in between Am7 and Am.
And it turns out you get the same sum when you add Am7 and F.
So, that’s why Fmaj7,9 is shown in red extending from in between Am7 and Am all the way to in between Am and Fmaj7.
We do the same additions all around the circle, and we get the other red “nines”chords. (Seven in all.)
Now, remember that opposite any chord is the chord that is its negation, or all the notes not in the chord. For these “nines” chords, each has five notes, and so each has an opposite chord with just two notes (not much of a chord at all).
For example, across from Fmaj7,9 must be the “chord” having the two notes not used in Fmaj7,9, and that’s b and d. That is to say, it somehow is the case that the b-d pairing is an opposite in this space to Fmaj7,9.
It also means that Fmaj7,9 and Am point in about the same direction (although the former seems maybe less determinate?). And the b-d chord pair is somehow pointing in about the same direction as G7.
Ok, now what happens if we add each adjacent pair of these “nines” chords? We end up with the “elevens” chords as shown on blue, also with a range of directions.
Interestingly, each now has an opposite, or negative, that is just a single note (see the single blue notes within the circle). And, in this way even individual notes have a direction in this space.
I suspect that as we move away from chords having 3.5 notes (ahem, 3 or 4, or midway of seven, the size of the diatonic scale), that the chords of longer or shorter lengths means shorter vector magnitude in this space. Maybe.
Music is well known to have chord progressions that broadly tend to move clockwise in this space, but by no means always.
My hunch is, as I mentioned, that this space is leveraged to make music whose chord progressions describe the trajectory of a mover in our midst, the topic of my earlier book HARNESSED, although I didn’t get at chord progressions in that book.
But at the moment I just enjoy having a beautiful way to meaningfully arrange all the chords within a key.