Overview
A secondary dominant is a chord which is used to stabilise a diatonic chord other than the tonic. For example, in the key of C major, a D minor seventh (or D minor) chord functions as a subdominant chord. But what if you (the composer / arranger) want the chord to function as a tonic center?

The best ways to stabilize a chord is by a cadence, and the most basic is the V7 I cadence (or V I cadence). The dominant chord stabilizes the tonic. So in order to make the D minor feel like it is the tonic center, we can put an A7 chord before it.

Definition
A chord must satisfy certain conditions in order to be a secondary dominant:
  1. It must be a dominant chord. This is usually a V, V7, viim7b5 or viio7, but I'll talk mostly about the V7 as it's the most common (in jazz, anyway).
  2. It's root must be diatonic to the parent scale. So in C major, it can be C7, D7, E7, F7, G7, A7, or B7. However, G7 is the dominant in C major, so it's not considered a secondary dominant. F7 is also not a secondary dominant (see below).
  3. It must be a dominant chord that resolves to one of the degrees of the parent scale that can function as a tonic. This means it has to resolve to a major or a minor, not to a diminished or augmented chord. So of the above chords, C7 resolves to Fmajor7, D7 resolves to G7, E7 resolves to Am7, F7 doesn't resolve to any chord diatonic to C major, so is not a secondary dominant. G7 resolves to C major7, so is not a secondary dominant. A7 resolves to Dm7, and B7 resolves to Em7.
Labelling
A secondary dominant is labelled as follows: X/Y, where X is the secondary dominant, and Y is the chord it is resolving to. So in Cmajor, an A7 chord would be V7/iim7. (It resolves to Dm7, which is the second degree of C major).
Use
I've actually said this, but it's important, so I'll repeat: a secondary dominant is used to stabilize a chord other than the tonic, and make it the tonic center. If we have the progression:
|| C | C | Am7 | Em7 | Dm7 | G7 | C ||,
and we want to stabilize the degrees other than 1, we can change it to
|| C | C E7 | Am7 B7 | Em7 A7 | Dm7 D7 | G7 | C ||
Secondary dominant scales
The secondary dominant chord is made up of the usual dominant arpeggio: 1, 3, 5, b7. (2 tones, 1.5 tones, 1.5 tones). But what about the other notes in the scale? What if you want to improvise on the tune? What if you are arranging a tune, and want to put a B7 chord before an Em7, but there is a D note in the melody? Will it work?

The secondary dominant scale is constructed by taking the chord arpeggio and inserting notes from the parent scale to fill in the blanks. So if we're in C major, and we have a D7, we have: D, F#, A and C, and we'll insert notes:
Between D and F# we have an E in the C major scale (we also have an F, but we only want one note between each two, and taking F would mean we have 1.5 tones between the first and second notes of the scale).
Between F# and A we have G.
Between A and C we have B.
So here's our D7 scale: D E F# G A B C - a mixolydian scale.

Let's try another: B7:
Between B and D# we have C and D. Let's take C. Don't worry about D for now.
Between D# and F# we have E and F. Let's take E (There should only be one note with each name in a scale. Of course, we could call F - E#, but we won't.)
Between F# and A we have G.
So we have B C D# E F# G A. This is a B mixolydian b9 b13 scale.

So if you have a D7 chord, you can improvise over it with a D mixolydian scale, and if you have a B7 chord, you can use a B mixolydian b9b13.

If you really want to get the hang of it, I suggest building all the secondary dominant scales, but if you're too lazy, here they are (for the major scale):
V7/iim7 - mixolydian b13 (natural 9)
V7/iiim7 - mixolydian b9b13
V7/IVmaj7 - mixolydian
V7/V7 - mixolydian
V7/vim7 - mixolydian b9 b13

And if you want a quick way to remember them - all dominants that resolve to a major chord are mixolydian, all that resolve to a minor chord are b13. Of the ones that resolve to a minor chord, only the V7/iim7 is natural 9, the others are b9.

Interchageability of b9 and #9
As you can see from the above, when constructing the scale for B7, I could choose either C or D. In B7, the natural 9 is a C#, so C is b9 and D (or CX) is a #9. You can construct the other dominant that has a b9 and you will find that it too, can have a #9. So ever b9 dominant is actually a #9 and every #9 is a b9.

Whew!

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