Dominant and Diminished Chords

Dominant and diminished chords are a special bunch. They share so many similarities that are not immediately obvious and allow for very swift modulation.

When you move the root of a dominant 7 up a semi-tone, you have a diminished chord.

Allow me to repeat myself ;
When you move the root of a dominant 7 up a semi-tone, you get a diminished chord.
This makes modulations to other keys extremely easy, first off, let me explain the above sentence before we talk about modulation.

Let's take our simple V in C (G7)

It's chord tones are

G7

G B D F
1 3 5 b7

by moving the root of this G7 (G) up a semi-tone, we would get an Ab diminished chord

Abdim

Ab B D F
1 b3 b5 bb7

Since we can now form a diminished 7 from somewhere we couldn't before, it also opens up more options for modulation.

A diminished chord has the same notes wherever you move it if you move it by a minor third. That means that no damage is done by going up 50000 minor thirds, you're still playing the same notes!

Abdim

Ab B D F
1 b3 b5 bb7

Bdim

B D F Ab
1 b3 b5 bb7

Ddim

D F Ab B
1 b3 b5 bb7

Fdim

F Ab B D
1 b3 b5 bb7

Okay, so we know that;

1) If we raise a dominant 7th's root up a semi-tone we get a diminished 7. Therefore, if we move a diminished 7th's root DOWN a semi-tone, we'll get a dominant.

2) We also know that we can move this diminished chord wherever we want in minor thirds.

Knowing this, I'm pretty sure we can pull off some modulations. These may not sound pretty (after all, who modulates by a minor third? and, who modulates to a distant tonality in one bar?). Generally harsh modulations like this (ones going to distant keys) should be done across more than one bar to accustom the listener to the changes, but for the sake of brevity, we will do a short and quick modulation.

First, I'll show you the sequence, and then I'll explain it

C G7 Ddim Bdim E7 A

We start off on our I chord (C).

And then we move to our V chord (G7). We play this G7 in its second inversion (with a D in the bass, this sets us up for our next move).

We then raise the G in our G7 up a semi-tone.

From

D G B F

To

D Ab B F

Knowing that it is safe to move up or down by a minor third, we do just that and move down to a B diminished chord

To

B F G# D

Now that we're on this chord, we simply move the F in the B diminished chord down a semi-tone and would you look at that, an E7!*

B E G# D

We're now in A major's territory, we play the E7 (The Vth of A) and then proceed to play the A.
Simples!

*Diminished chords are actually much more flexible than the dominant chord, any chord tone can be flattened to reveal a dominant chord (how exciting!).

On the B diminished, we could have gone to many different places other than the E7 down to the A. For example, if we flattened the B rather than the F, we could have had a Bb7 chord, there were quite a few options so let's look at a few...

Here's the original chord

B dim

B D F G

And here're a few options

Bb7

Bb D F Ab

E7

B D E# G

C#7

B F G# C#

G7

B F G D

Conclusion

We've only really scraped the surface of the dominant/diminished relationships but hopefully this will give you a few ideas the next time you encounter either of those chords.