How to differentiate between a diminished fifth and an augmented fourth interval?

Question

Both are a tritone appart, both sound the same. How can we differentiate when an interval is an augmented fourth or a diminished fifth?

Context:

In voice leading we learn that when the spelling of the tendency tones (7th and 4th degrees) forms a diminished fifth they are going to resolve inwards (towards each other), and when the spelling of the tendency tones forms an augmented fourth they will tend to resolve outwards (a way from each other).

This means that in V7 resolving in I in C major (G7 to C) the tendency tones are B and F. Depending on the voicing B and F can be either an augmented fourth or a diminished fifth.

What's the difference?

Answer 1

The difference is in the spelling. The tritone (augmented 4th (A4)/ diminish 5th(d5)) is named in the context it is analysed in.

The notes of G7 in order are G, B, D, and F. G to B is a Major 3rd (M3), G to D is a Perfect 5th (P5), and G to F is a minor 7th (m7). We analyse the G as the root note everything is based on the distance from G to the other notes. We could look at B to G as a minor 6th(m6), or D to G as a Perfect 4th (P4), or F to G as a Major 2nd (M2). The reason we don't is it isn't useful to analyse this way because we stack chords in 3rds. We analyse from the bottom (root) up. Because of this the distance from B to D is a minor 3rd (m3) and the distance from B to F is a diminished 5th (d5). Again it can be inverted, but this is not useful for analysis.

In the context of scales, there are two modes that demonstrate how the a diminished 5th and an augmented 4th are different. The modes are Lydian and Locrian. Because the 4th degree of the Lydian scale is greater than perfect, the interval is an Augment 4th and because the 5th degree of the Locrian scale is less than perfect the interval is a diminished 5th. Below I put the two modes based on C major.

F Lydian:
F  G  A  B  C  D  E  F
P1 M2 M3 A4 P5 M6 M7 P8

B Locrian:
B  C  D  E  F  G  A  B
P1 m2 m3 P4 d5 m6 m7 P8

It is the same internal, but the context lets you know what to call it. If you need more clarification let me know this is a rather broad topic.

Here is a source on interval notation and their inversion.

Answer 2

In just intonation, the dim. fifth and aug. fourth have slightly different pitches. In 5-limit just intonation, it is the case that the interval (frequency ratio) between a higher F and a lower B is a diminished fifth ( 610 cents) and the interval between a higher B and a lower F is an augmented fourth (590 cents). This difference in pitch can be achieved by continuous pitch instruments, e.g. non-fretted strings and especially voice; so they're different because they (can) sound different. The notation reflects this difference in terms of the "measuring intervals by counting letters" approach.

For voice-leading, I learned this rule as these two voices move to the available notes that are a 1/2 step away, rather than as a rule describing the contraction/expansion of an interval. However, the latter approach does make sense. As an example consider a voicing (low to high) B F B (I've dropped any other notes from the G7 chord) typically these voices would go to C E C in a G7-C cadence. The dim. fifth has contracted, and the aug. fourth has expanded, just as your rule would indicate.

Answer 3

We can't, unless it's written on the stave. An aug. 2nd sounds just like a min.3rd; a min 6th sounds just like an aug 5th.A dim 7th sounds just like a maj.6th etc.etc. It's the context musically that will denote which it needs to be called. Just listening to an interval cannot tell us how it should be written or named. If, for example, the B to higher F is in the key of say G, then it'll be a b5.To make it a #4, it'll technically be B to E#, which puts it into another key, but then it ceases to be an F anyway.Using the note names, as you did, of B and F, it can only be called b5. If you quoted, e.g. C and F#, this would only be a #5.(As would Cb and F).

EDIT : Yes, it does all depend on the voicing. Intervals are traditionally counted from the lower to the higher note. Therefore - B>F is a diminished 5, as B>F# is P5, whereas F>B is an augmented 4, as F>Bb is P5. That's the proof on paper, at least!

After some more thought - B-F is always a b5, and F-B is always a #4. It has to be, as B-F# is the P5, and F-Bb the P4. There is no other correct answer. Unless you're asking just about any two notes which are a tritone apart, giving, obviously, the same sound. As I said earlier, just from the sound, an interval cannot necessarily be given a correct name.

Answer 4

This means that in V7 resolving in I in C major (G7 to C) the tendency tones are B and F. Depending on the voicing B and F can be either an augmented fourth or a diminished fifth.

On a G7 chord (G-B-D-F) the B - F is a diminished 5th always. However in an inversion of the chord (say B-D-F-G) it may became F - B which is an augmented 4th. As you see in any case, the resolution remains the same, B goes to C and F goes to E.

So this is true:

when the spelling of the tendency tones (7th and 4th degrees) forms a diminished fifth they are going to resolve inwards (towards each other), and when the spelling of the tendency tones forms an augmented fourth they will tend to resolve outwards (a way from each other)

Since chords are built upon 3rds, its only natural to call it a diminished 5th all the times. However, if we are doing a harmonic analysis we have to consider whether this V7 is in root position or in an inversion.

Answer 5

How can we differentiate when an interval is an augmented fourth or a diminished fifth?

It is all very simply. Assuming you where born with at least five fingers and you can count to five you can do it.

Lets take C as an example. You take your thumb you make your thumb represent this C. When we get to a fourth finger (ring) We have a fourth. So C = 1 = Thumb. D = 2 = Index E = 3 = Middle and then we get F = 4 (Fourth).

So C to whatever type of F is a fourth. If you count to five with this patented 8 finger pedagogue methodology you can also get a fifth. C to whatever type of G is a fifth.

When you have a perfect interval like fourths and fifths and you take the interval closer to each other by either raising the bottom note or lowering the top note then this perfect interval becomes an diminished interval.

If this interval is taken away from each other by either raising the top note or lowering the bottom note then the perfect interval becomes an augmented interval

C- F# and C - Gb may be enharmonic equivalents of each other BUT THEY ARE NOT THE SAME INTERVAL!. The one is a fourth and the other is a fifth or to be more specific the on is an Augmented Fourth and the other is a Diminished Fifth.

Hope that helps.