In music, a difference of three whole tones between two notes can get all of the above names. This might have you asking one of the following three questions:
  1. Good point, Footprints, why are there so many names for one interval, and what is the difference between them?
  2. What the hell are you talking about?
  3. Who cares?
If you asked question number 3, then go read something else, maybe something like Grandma Got Run Over By a Reindeer.
If you asked question 1, skip the introduction and go straight to the explanation.
If you asked question 2 exclusively (i.e. without adding question 3), read on:

Introduction

The chromatic scale has twelve notes, and they are all separated by half a tone or a semitone. And from this, we can build all the other intervals. The scale is:

C (B#), C# (Db), D, D# (Eb), E (Fb), F (E#), F# (Gb), G, G# (Ab), A, A# (Bb), B (Cb)


In brackets are the enharmonic names of the notes. So between C and C# there is half a tone. (Between C and Db there is the same half tone). Between C and D there is a whole tone. Between C and E there are two whole tones. Between C and F# (or Gb) there are three whole tones. Also, between A and Eb there are three whole tones, as it is in fact a circle (after B comes C). Three tones are called a tritone. This interval is sometimes referred to as an augmented fourth, a diminished fifth, a flat fifth (b5) or a sharp eleventh.

Explanation

Tritone
Tritone means three tones. So when there is a difference of three tones between two notes, this is a tritone. This terms is used when the interval is out of musical context. For example, if in ear training, if I play C and Gb and ask you what interval I played, you would (should) answer tritone (this is true, as you have no idea whether I'm playing C and Gb, C and F#. (Or, for that matter, B# and F#)). As we shall see, there is difference between them when they are in a musical context.
Diminished fifth and augmented fourth
If the tritone is in context, we shall see it is either an augmented fourth or a diminshed fifth. A perfect fourth is a distance of four notes on a major scale. For example F is a perfect fourth above C. Using a quick calculation, we find that F is in fact two and a half tones above C. A perfect fifth is a distance of five notes along a major scale. So G is a perfect fifth above C. By the same calculation, G is three and a half tones above C. Augmented, in music, means adding a half tone, and diminished means lowering a half tone. So as you can see, adding a half tone to a perfect fourth (making an augmented fourth), we'd get three tones, and lowering a half tone from a perfect fifth we get a diminished fifth, also three tones. Let's apply this to the scale. An augmented fourth from C is F#. A dimished fifth from C is Gb. (# stands for + half a tone, b for - half a tone). Now F# and Gb are enharmonic (they sound the same). But they are not technically the same.

Gb is NOT an augmented fourth from C. It is a fifth (G) that has been diminished. I'll give some more (more difficult examples) - a perfect fourth from B is E. An augmented fourth from B is not F. F is a diminished fifth, as F# is the fifth. An augmented fourth would be E#. The fifth of Gb is Db. What would it's diminished fifth be? Not C, but Dbb. Simple, see?

#11 and b5
When dealing with intervals, we use the terms "tritone", "diminished fifth" and "augmented fourth". However, when dealing with chords and scales we use the terms b5 (flat fifth) and #11 (sharp eleventh).

For example, let's take a look at the C mixolydian scale: C, D, E, F, G, A, Bb - C is the first note of the scale, D is the second, E the third, etc. But chords are built using thirds, so take C, add the third, E, and the next third, G. We get C, E, G - C major. Adding the next third we get C,E,G,Bb - a C7 chord. Add the next third - C,E,G,Bb,D - C7,9 (often called C9). (Notice d is the ninth. It is also the second, but it was added after the seventh, so it is the ninth). Continue if you wish - it's easy. F is the eleventh of the scale. What if we raise F half a tone? We get a sharp 11th (#11). This would give us: C,D,E,F#,G,A,Bb. This gives us a mixolydian #11 scale. (This is actually known as the lydian b7 scale, but that's just a name). In it, we still find the major triad (CEG) and the seventh chord (C7), but the fourth (eleventh) is raised half a tone from the mixolydian.
What if we take C mixolydian and lower the fifth? The fifth is a G, so we get a Gb. We get C,D,E,F,Gb,A,Bb. As you can see, this is a totally different scale. It can be called mixolydian b5. It is completely different from the previous (lydian b7), as you can see. In fact, this scale is not used at all in jazz or classical music, whereas the lydian b7 is constantly used (in jazz at least).

When talking about chords, a C7b5 chord has C, E, Gb, and Bb. A C7#11 has C, E, G, Bb and F#. Play these on the keyboard. They are very different, and are used in different contexts. Actually, we don't often get a b5 in a major chord. It is most commonly found in minor chords (thus making them diminished, or half diminished. This is actually where the diminished comes from - it's a diminished fifth) Also, when improvising over them, you would use a lydian b7 over the C7#11, but (most probably) a whole tone scale over the C7b5.

So as you can see, this little semantic difference is not so little after all!

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:

D
F#
A
B#

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:

D
F#
A
C

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.

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