Visualizing secondary dominants

In my MusicRadar column honoring Roberta Flack, I thought of a new analogy for secondary dominant chords, and I figured that I should work it into a new explainer with some new graphics. So, if you are having trouble understanding how these chords work, read on.

Secondary dominant chords solve a specific problem: how to make functional tonal harmony more interesting while still keeping things functionl. The first thing you need to know is that every major and minor key has a chord built on its fifth scale degree, the V7 chord. In Western tonal theory, these are the dominant chords, so named because they dominate the key: they create tension and suspense, which is released when you resolve to the tonic chord. The circle of fifths is useful visualization scheme for the keys and their dominant chords.

Now let’s consider the key of C major. It comprises the notes C, D, E, F, G, A, and B. You can combine these notes to form seven chords: C, Dm, E, F, G, Am, and B°. These chords sound good together in pretty much any order and any combination, but they are bland and lacking in drama. How can you introduce some more color into your C major harmony without throwing random notes and chords around?

One good method for enriching the key of C major is to think of each of its chords as a temporary key center unto itself. You can then introduce chords from those other keys, especially their dominant chords. So rather than just think about the chords C, D, Em, F, G, Am and B°, we can think about the keys of C major, D minor, E minor, F major, G major, A minor, and B… something. People don’t usually treat Em or B° as key centers, actually, so we can ignore those.

Let’s focus on the key of A minor. The strongest way to define a key in Western tonal music is with a V7-I cadence. To make A minor really sound like the center of harmonic gravity, you want to set it up with its V7 chord, E7, and then have that resolve to Am. The important voice leading motion is the leading tone G-sharp, the third of the E7 chord, resolving to the tonic A, the root of the Am chord.

Now let’s say you have a song that moves from C to Am. It sounds fine. But now try moving from C to E7 to Am. That G-sharp in the E7 chord is from outside the key of C major, and it sounds “wrong.” However, when the E7 resolves to Am, it retroactively makes sense. Your ear goes, oh, I get it, that chord was from A minor, not C, cool! This is a more intellectually satisfying chord movement.

Now let’s say you have a song that goes from C to Dm. Again, it sounds fine. But now try inserting the V7 chord from the key of D minor, so you go C to A7 to Dm. The main event here is the leading tone C-sharp, the third of the A7 chord, resolving to the tonic D in the Dm chord. That C-sharp is the most dissonant note you could possibly hear without leaving the piano keyboard entirely, so it’s disruptive. Having that disruption explained retroactively with the resolution to Dm comes as a relief.

Using an A7 chord in C major is an especially cool thing to do because A7 is so similar to Am. Hearing an Am chord is about as surprising as finding jelly with peanut butter in a sandwich. But what if you’re expecting jelly and you get anchovy paste instead? That’s the effect of swapping Am for A7: it’s superficially similar to what you were expecting, but also terribly wrong. When the A7 resolves to Dm, though, it’s like your peanut butter sandwich has suddenly turned into a Caesar salad. Anchovy paste may have been awful with peanut butter, but with romaine lettuce and croutons, it’s delicious.

So those are the related minor keys. How about the related major keys? Say you want to get from C to G. You can get there via the V7 chord in G major, a D7 chord. This chord’s third, F-sharp, is from outside the key of C major, but it’s only mildly disruptive. In fact, the disruption is so mild that you don’t even have to resolve the D7 to G at all; you could go to some other chord, like F, or back to C. Rock musicians do this routinely. My analogy here is that expecting F and hearing F-sharp is like expecting strawberry jam and getting marmalade: different, but not distressing.

Finally, say you want to get from C to F. You get there via the V7 chord in F major, which is C7. The leading tone is E, which is inside the key of C. The B-flat in the C7 chord is from outside the key, technically, but you may not even hear it as dissonant. Flat sevenths are so ubiquitous in major-key music that my students hear it as being solidly within the key, just another flavor of C chord. So they just have to take it on faith that C7 is a secondary dominant that tonicizes F.

So, in summary, here’s a list of the keys most closely related to C major  with their V7 chords, the secondary dominants:

• C major – G7
• D minor – A7
• F major – C7
• G major – D7
• A minor – E7

Fun fact: the secondary dominant chords’ roots spell out a C major pentatonic scale. 

You can also use diminished chords to set up your related key centers. Let’s think about the home key of C major first. There is a very close relationship between the V7 chord, G7, and the vii chord, B°. The notes in G7 are G, B, D and F. The notes in B° are B, D and F; that’s just G7 minus its root. In the context of C major, G7 and B° are pretty much interchangeable.

Now let’s think about B°7. It’s the notes B, D, F and A-flat. If you put a G underneath it, you get G7(b9). That’s more of a minor-key flavor, but it resolves nicely to C anyway. You can build diminished seventh chords on all the leading tones of the keys in the C major neighborhood to create the feeling of V7b9 chords resolving.

• C major – B°7
• D minor – C#°7
• F major – E°7
• G major – F#°7
• A minor – G#°7

The roots of the °7 chords spell out an E major pentatonic scale.

The last and hippest way to define key centers is with tritone substitutions, dominant seventh chords whose roots are a half step higher than their destination notes. This is a complicated idea, but in practice, all you have to do is precede your destination chord with a dominant seventh a half-step higher:

• C major – Db7
• D minor – Eb7
• F major – Gb7
• G major – Ab7
• A minor – Bb7

The substitute dominants’ roots form a Gb major pentatonic scale.

Hear all of the secondary dominants in action:

Understanding Secondary Dominants

Here is the above information in graphical form.

 

Here’s my preferred view, wrapped in a circle.

You can find examples of songs that use secondary dominants here. Here’s a fuller explanation of diminished chords. Happy harmonizing!