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difference between diminished fifth and augmented fourth

They are the same note (as in, frequency of the sound wave), however they differentiate in where they land in the scale you are working with. An augmented 4th will still be the 4th note of the scale, whereas the diminished 5th would be the 5th note in the scale.

Please correct me if I'm wrong, but that's how I understand it.
That is the way I see it.
 
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They are the same note (as in, frequency of the sound wave), however they differentiate in where they land in the scale you are working with. An augmented 4th will still be the 4th note of the scale, whereas the diminished 5th would be the 5th note in the scale.

Please correct me if I'm wrong, but that's how I understand it.

That's getting at the idea. Consider the tritone between the 3rd and 7th notes of the dominant scale/chord. The 7th of C7 is Bb, which is a diminished 5th above E, the 3rd. But the tritone ascending between the 4th and 7th in a major scale is an augmented 4th (F to B in C maj).
 
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An incredibly simplified take on this (that obviously doesn't cover a lot of the practical reasons) would be to just look at the alphabet:

1 2 3 4 5
C D E F G

C to F# would be an augmented 4th (F is the fourth letter up from C)
C to Gb would be a diminished 5th (G is the 5th letter up from C)
Same Pitch. Different analysis.

Yes, I know that doesn't tell the whole story but it's one way of explaining it in a very simple way that might provoke a click moment and further interest.
 
An incredibly simplified take on this (that obviously doesn't cover a lot of the practical reasons) would be to just look at the alphabet:

1 2 3 4 5
C D E F G

C to F# would be an augmented 4th (F is the fourth letter up from C)
C to Gb would be a diminished 5th (G is the 5th letter up from C)
Same Pitch. Different analysis.

Yes, I know that doesn't tell the whole story but it's one way of explaining it in a very simple way that might provoke a click moment and further interest.

Yes.

This.

Now begins the 7 pages of nonsense. Please let it be known that this was the sanest way of understanding it.
 
The way that I see it, beyond just if its the 5th or 4th scale degree, is in which direction it wants to resolve. The b5 finds stability resolving down, whereas the #4/#11 wants to go up.

There it is. Simple. The only real reason that the same sound has different names. But, this is something that works in the world of written music. If you are dealing with this only in sound, you have little need to know the difference, but you are missing a lot of the picture of how and why music works as it does.

Same kind of argument that crops up here now and again like why is there a B#? (answer is directly above).
 
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As just mentioned above, you only have to look at the basic degree patterns of a bunch of scales to see the different enharmonic spellings:
Major - 1 2 3 4 5 6 7
Lydian - 1 2 3 #4 5 6 7
Locrian - 1 b2 b3 4 b5 b6 b7

It's no different to looking at why you might use a Minor 6th interval instead of an Augmented 5th and if you want to really stoke up the argument just ask about the diminished 7 interval ;)
 
As everyone said, same pitch relative to the root. I'd assume the difference is important when building harmonies and chord progressions. If you build a 1/3/5 triad, then the augmented status of the 4th is inconsequential. However, it becomes relevant if the 5th is diminished.

Chances are you won't have a scale with an augmented 4 and a diminished 5, so there is a melodic importance when ascending and descending. You can augment a 4 and still have a natural 5. You can have a natural 4 and a diminished 5. You can have all sorts of other combination too.
 
C to F# would be an augmented 4th (F is the fourth letter up from C)
C to Gb would be a diminished 5th (G is the 5th letter up from C)

It is this simple. It is not a matter of theory; it's a matter of notation. If it's written C to F#, we call it an augmented fourth; If it's written C to G flat, we call it a diminished fifth. It is the same interval.
 
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It is this simple. It is not a matter of theory; it's a matter of notation. If it's written C to F#, we call it an augmented fourth; If it's written C to G flat, we call it a diminished fifth. It is the same interval.

Yeah, it's the same pitch--but we did have a lengthy discussion on another thread about whether or how much it mattered how it was notated. I'm in the camp that holds the notation should follow function, in general.

For instance, if you had a maj7#11 chord (think of it as a B minor triad stacked on a C major triad), it would have an F#. But a C dim7 chord would have a Gb. If you arpeggiate those chords on your bass, the difference in tone color is noticeable, and you can easily imagine how they would function differently in chord progressions.