What is a secondary dominant chord? What's the theory behind them? How are they used in composition?
In common-practice theory, secondary dominant chords are chromatic harmonies used to approach a non-tonic chord with greater urgency. Let's use C major for examples:
I might want to approach the V chord (G) with a secondary dominant to give greater direction or "color" to the approach. I construct the secondary dominant by going to the V chord of the V chord. In other words, I briefly pretend I'm in G Major, and borrow its V, which is a D major chord. The standard Roman numeral notation is V/V, and the chord is chromatic to the original key of C major due to the third (F#). Just as often, I might borrow the V7 from the target chord's key, and this would be labeled V7/V and would be a D dominant 7th chord. Generally only major or minor non-tonic diatonic chords are used as targets, but obviously one could extend this to other qualities and to chromatic chords.
In some cases, particularly V/IV, the secondary dominant would not be chromatic. In C major, the target chord IV would be F Major, and its key's dominant would be C major, which isn't chromatic to the original key. In such cases, composers will almost always use the V7/IV instead, which in C major would have the chromatic note Bb.
The concept is easily extended to viio chords, since they also have dominant function. In C major viio/V would be F#dim for example. Most theorists distinguish between the two by calling second vii chords secondary leading tone chords instead of secondary dominants, but the concept is precisely the same.
In all cases, their purpose is to highlight the target chord, and create a sharper resolution to it.
As the other answers have said, a secondary dominant is a 7th chord that resolves to a chord other than the tonic. The classic example is a D7 chord in the key of C: the D7 resolves to G, which itself resolves to C.
Dominants create tension
To understand the theory and practice behind secondary dominants, you have to understand that all dominant chords serve a functional purpose, namely to drive the harmony forward by creating tension. That tension demands release and resolution, and so dominant chords provide the music with momentum and forward drive.
Sustaining tension is a tricky business. Obviously, music with no tension (and subsequent release) is dull and lacks momentum. But you can't just play the same V chord over and over hoping to draw out the tension as long as possible, because eventually the excitement dissipates and the listener is left with an unsatisfied, unresolved feeling.
So secondary dominants are used to extend the harmonic drive. They create some tension and offer the composer the opportunity to resolve it, but because they resolve to a non-tonic chord, the resolution itself creates new tension that can then be itself resolved later.
Suppose (again) you're in the key of C. You could write a progression that goes something like C-C-G7-G7-C. The first two chords establish a key center (C), the second two chords establish tension relative to that key center, and the final chord resolves that tension.
Now consider this progression: C-C-D7-D7-G7-G7-C. The same basic principle applies: establish a key center, create tension, resolve it—but now the portion of the music that creates tension is twice as long. Playing C-C-G7-G7-G7-G7-C is just as long, but not really effective (try it!), but using the secondary dominant allows the tension to extend without losing its power.
And you can keep this idea going: C-C-A7-A7-D7-D7-G7-G7-C. Now the harmonic tension is three times as long as in the original chord progression, but by using secondary dominants, it manages to sustain interest without dissipating. This is an extremely popular chord progression; in the key of Bb, it goes Bb-Bb-G7-G7-C7-C7-F7-F7-Bb, which you may recognize as the progression behind "I Got Rhythm" and countless other tunes.
And that's the idea. Dominants create tension and make the listener anticipate the coming resolution. Secondary dominants allow the composer to extend that tension, thus raising the emotional stakes of the music.
Secondary Dominants are chords that are the Dominant (V) chord of a certain key other than the Tonic (I) key.
For example, let the current key be C Major. An F Major chord is the Subdominant (IV) of the current key (C Major). How about a C Dominant 7th chord? That is not in the current key, but you know it's the Dominant 7th (V7) of F Major, and F Major is in C Major.
That C Dominant 7th chord is therefore the a Secondary Dominant.
Please note that Secondary Dominants can work in all valid forms (including the Root form and its 2 inversions, and the Dominant 7th form and its 3 inversions).
Why "Secondary Dominant"? It's because it's the Dominant of a key other than the original key (as stated in the first sentence). Please note that if a chord that appears to be secondary chord leads to a key change, that chord is NOT considered a secondary chord.
The Secondary Dominant can be the Dominant of any key other than the Tonic. If the current key is C Major (again), then:
Remember that Secondary Dominants can also be used in Dominant 7th form. For the 3rd condition (Secondary Dominant of Subdominant), the C Major chord can be C Dominant 7th (and in this case, should be).
Secondary Dominants add harmonic color to the music.
Please note that each Secondary Dominant usually is followed by the Tonic (I) chord of the Secondary Dominant's key. If the Secondary Dominant chord is F Dominant 7th, and the current key is F Major, then the Tonic of the chord is B♭, so the following chord should be a B♭ chord.
Let's do an exercise!
Given that the current key is B Major, and there is an A# Major chord, what chord will most likely follow that A# Major chord?
Well A# Major is clearly not in B Major, and we can see that it's the Dominant key of D# Major and d# minor, and we can see that d# minor is in B Major (the current key), so the d# minor chord will most likely follow that chord.
Since Secondary Dominants are the Dominant of a certain key, Secondary Dominants are written as V/?, where ? is a Roman Numeral such that the Secondary Dominant is the Dominant of the ? of the original key.
For example, V/V means that the Secondary Dominant is the Dominant of the Dominant of the original key (we'll use C Major again; it's easy to work with). That translates to the Dominant of the Dominant of C Major. The Dominant of C Major is G Major, so V/V translates to the Dominant of G Major, which is D Major.
Let's do another one, where the current key is D Major this time. V/ii of D Major means that the Secondary Dominant is the Dominant of the Supertonic (ii) of D Major, which is the Dominant of e minor, which is B Major. Get it?
In real notation, you don't need to go "V/IV of E Major". Instead, you only need to go "V/IV", because the key is already given in the music score.
Let's do an exercise!
What is the V/VI of a minor?
Well the VI of a minor is F Major, and the Dominant of that is C Major, so the V/VI of a minor is C Major.
That's it! I hope you learned what Secondary Dominants are! Feel free to comment if you have any questions or comments (duh)!
A secondary dominant chord is when you turn a minor or major 7th chord into a dominant chord, in order to make another chord the tonic chord. I can't tell you exactly what's the theory behind this, but I can tell you a bit about how they're used, not so much for composition but for arrangements:
Take for example a typical chord progression I-VI-II-V. If we use the C scale, then it's CM7-Am7-Dm7-G7
First of all, any degree can turn into a secondary dominant, except the IV.
Now, in this progression, if we take the third chord, D, and imagine for a moment that it's the tonic, then the previous chord, A, happens to be the V grade of D, so we can turn Am7 into A7. This will give the sensation of resolution that happens with the V-I so commonly used in jazz and other genres, but right in the middle of the progression.
But, D also happens to be the V grade of G (if G were the tonic), so we can turn Dm7 into D7 and have yet another feeling of resolution. So we turn the progression into CM7-A7-D7-G7. This is known as extended secondary dominants.
But, that's not all... any dominant can be accompanied by the II degree that precedes it in a full cadence (II-V-I). Using the same example progression (the original one), if we once again turn Am7 into A7 because it's the V degree of D, then the II degree of D would be Em7, so we can squeeze Em7 before A7 and play those 2 chords in the same timeframe in which A7 was played before.
So if you're playing quarter notes, then CM7 for 1/4, then Em7 for 1/8, A7 for 1/8, Dm7 for 1/4 and finally G7 for 1/4. The same can be done when turning Dm7 into D7: squeeze in the II degree of G which happens to be Am7. So now we have:
CM7, Em7+A7, Am7+D7, G7
There are other things you can do with secondary dominants, such as replacing a dominant with the dominant born from its tritone; this means that instead of playing G7, which is G-B-D-F (the tritone is B-F), you can switch the order of B-F to F-B and that would be the tritone of Db7.