The Diminished Cycle and ii-V-Is

In the previous ii-V-I post I outlined what I take to be the most standard, well-known ways to play a ii-V-I. This is a quick note of a well-known fact about them.

Here are all the actual ii-Vs we looked at there, as opposed to extended phrases, individual chord subs etc. We'll do everything in the key of C again, i.e. the I we want to reach is CΔ:

It will not have escaped your notice that these four keys are not only one third of all the possibilities, but they spell out the notes of a diminished seventh chord: G-Bb-Db-E. That's not a coincidence.

Let's start with G7, which usually resolves to C. Going up a minor third gets us to E7, which usually resolves to A, the relative minor of C. While A and C are very different as major keys, the keys of Am and C contain all the same notes, and a ii-V can resolve to major or minor -- it's usual to play the iim with a b5 if we're going to a minor chord but if you're not playing fifths in your voicings, who can tell? Anyway, the tritone subs of these two then take us the rest of the way around the circle of minor thirds that makes up a diminished arpeggio.

Here's another perspective. Any functioning dominant chord -- i.e. one that's resolving -- can in theory have a b9 added to it. Let's look at the result of doing that with each of the dominant chords above:

• G7  becomes G-B-D-F-Ab
• Db7 becomes Db-F-Ab-B-D
• Bb7 becomes Bb-D-F-Ab-B
• E7  becomes E-Ab-B-D-F

You'll have to excuse the enharmonic spellings but I did it for a reason. Look again -- every one of these chords is just the diminished seventh chord B-D-F-Ab with one added note. You can usually substitute this "diminished up a semitone" for a dominant chord (the exception is when it's non-functioning, like the IV7 in a blues, but that's not relevant to our ii-V-I discussion).

So it's not a huge stretch to imagine that this diminished chord is the "real" dominant structure, and each of these seventh chords is just a mild alteration of it. So of course you can switch between them to get different but related sounds. You can stack them up to get tasty chords with lots of alterations, or you can play the arpeggio of one over another in a melodic line. Here's what you get, although personally I don't find this kind of breakdown especially useful:

• Db7 over G7 implies G7b5b9 (That's the tritone sub, which is a "pure altered" sound)
• Bb7 over G7 implies G7b9#9 (The third of Bb7 is D, the natural 5 of G)
• E7 over G7 implies G7b9 13 (The root of E7 is the natural 13 of G)

These sound best to my ears if you just play them with normal root-shell voicings, without the fifths.

There's a certain kind of jazz musician who sees mystical depths in the appearance of diminished and augmented cycles in the music. Since 3 and 4 are the only interesting factors of 12, relationships like this are bound to crop up simply by chance. And it's unclear (to me, anyway) how often the founding musicians of the music were thinking in systematic terms like this. Still, it can be useful as a way to mentally organise things.

In the next post I'll tackle the ii-Vs that are not on this list and we'll start our journey into more challenging territory.