So now we know wherefrom these names come. But where do they go
In music of the Common Practice Period, the semitone is actually one of the most powerful of intervals you’ll find. Dissonances arise frequently in such music; and it is often (but by no means always, in music, rules are made to be broken!) by moving the distance of a semitone that such dissonances are resolved. Thus, in tuning and in the expectations of the listener, semitone relationships yield some of the most exciting notes in phrases; this excitement is often defined and ruled by the augmented fourth and the diminished fifth, which are much, much more than semantically different.
The Augmented Fourth
Since we already know what an augmented fourth is, we can speak to its more secretive, insidious nature. You see, the augmented fourth has no intention of remaining an augmented fourth.
It really wants to be a minor sixth. Or, barring that, a Perfect fifth.
In music of the Common Practice Period, there is a substantial difference in spelling a note as, say, an F# as opposed to a Gb. That’s because the F# is intended to move upward, the Gb downward. A great example of this is the Augmented sixth chord.
If an augmented sixth chord from the audience would kindly volunteer:
The augmented sixth chord derives its name from its characteristic interval, D-B#, which is...an augmented sixth, consisting of ten semitones. In music where it appears, it is usually based on the minor sixth degree of the tonal area either preceding or following, depending on its function. In this case, we’ll say that the chord is a Ger+6 in F# Major. But the augmented sixth chord derives its real power from the augmented fourth buried within, in this case, F#-B#.
Because it’s in that interval where the tension really lies. You see, to the ear, the Ger+6 chord really sounds like a V7/N. Which is to say, the B# might sound more like a C. But, by spelling the tritone as an augmented fourth, the composer is indicating that the B#’s destiny lies upward. So the Ger+6 resolves thusly:
D -> C#
F# -> E#
A -> G#
B# -> C#
Or, really, to satisfy my old theory professor’s persnicketiness:
D -> C#
F# -> F#
A -> A#
B# -> C#
A resolution which avoids the dreaded parallel fifths of which the first example is guilty.
In F# Major, then, the Ger+6 chord leads to either a V or a I6-4 (although it should be noted that the cadential six-four chord, spelled I6-4 or V6-4, depending on your expertise and the context, isn’t really a chord in its own right but a couple of suspensions used to obscure an otherwise ugly-sounding voice-leading quirk). A fairly typical progression would be:
Ger+6 -> I6-4 -> V(7) -> I
This is all because the augmented fourth forces the B# to move upward a semitone and the F#, eventually, downward to the E#. Thus, the augmented fourth is destined to become the minor sixth.
But what if the augmented fourth were re-spelled as a diminished fifth?
The Diminished Fifth
Let’s take our old Ger+6, apply some makeup and shoe-shine:
This chord is enharmonic to the Ger+6, above, but it is functionally and substantially different. It is now a dominant seventh chord, in particular, of the Neapolitan, which is a chord based on the flat second degree of the major scale.
Everything is enharmonically the same, but different in name. The augmented sixth, D-B#, is now a minor seventh, D-C, which wants to resolve inward because of a diminished fifth between F#-C, formerly known as F#-B#. Whereas the augmented fourth wants to become a minor sixth, the diminished fifth is similarly restless, wishing only to become a major or minor third. The typical resolution:
D -> G
F# -> G
A -> G
C -> B
One might note here that the root, G, of the resolution chord is repeated three times and the fifth, D, not a once. This is due to voice-leading principles and can be resolved by doubling the D in the first chord or changing the inversion of the first chord so that D is not in the bass – Trivialities all. What is relevant here is that the F# must resolve upward by semitone, and the C must resolve downward by semitone. This is all because the unsettling dissonance that is the tritone has been spelled as a diminished fifth.
In F# Major, this chord would probably be best described as V7/N, N symbolizing the Neapolitan. A typical progression might be:
V7/N -> N -> V7 -> I
It’s a totally different-sounding progression than above, and would be even more so, probably, because inversions of most of the chords would be necessary for good voice-leading. But I won’t go into that.
Enough of the theory, where can I hear it?
Well, the diminished fifth goes way back. Its importance in the dominant seventh brings it to bat in cadences all throughout the Common Practice Period. It also shows up in the diminished II chord of the minor mode and in the diminished VII chords of both major and minor, and also in the half- and fully-diminished seventh chords of either mode; although, again, this usually happens at cadences, where a proscribed direction is most appropriate.
The augmented fourth, in contrast, is rarer and more spectacular. It shows up most famously in Beethoven’s Fifth Symphony, second movement, where the entire orchestra gears up for a huge augmented sixth chord. It is really through the augmented sixth chord that the augmented fourth finds its greatest use. Such chords are responsible for special kinds of modulations and novel colors in the short term. But, again, I won’t go into that.