# Why is Eb to A# not a perfect 5th?

I read a mock question that asked if Eb to A# was a perfect 5th.

I counted up 7 semitones, and concluded it was.

The answer says that it’s not, but not why.

Is it simply because the enharmonic equivalent Bb should have been used, as that’s in the Eb scale?

I find it an odd one if so, because if played you’d identify it as a perfect fifth.

• Did this "mock question" also mention a piece where this double augmented 4th appears? I know Mikrokosmos Nr. by Bartok where he notates an augmented 3rd (that sounds like a 4th) but in context it makes sense to write it as a >3rd. While I would be interested to see a progression of chords or voices where this double augmented 4th would make sense. – Albrecht Hügli Feb 23 '20 at 9:09
• No it didn’t have any context - it was simply a list of intervals I had to say were perfects fifths or not. – Marcos Scriven Feb 23 '20 at 9:19
• Anyway - it makes sense to ask this question to test if a student has understood the theory of intervals and the principle of labeling them. – Albrecht Hügli Feb 23 '20 at 9:21
• @AlbrechtHügli - asking the question as it was in the exam makes little sense. See my answer. – Tim Feb 24 '20 at 8:33

'If played, you'd identify it as a perfect fifth'. Maybe, maybe not.

The sound of any interval is only half the story. Every interval has at least two different names but only one sound.

The other half of the story is what actual notes are being played. Their names, and where they live on the stave.

E♭ > A♯. Any E>A is going to be a 4th - of some sort. E-F-G-A - 4 letter names. Now, let's do some juggling. E♭ > A♭ is a P4. Count up the scale notes if you like, or state that P4 uses 5 semitones.

Keep the bottom note of E♭, but take that A♭ up to A ♮. Now the interval is an augmented 4th. Going up by another semitone to A♯ makes it a double augmented 4th.

Not a common interval at all, but an interval nevertheless! If intervals were recognisable purely by their sound, it would be a lot easier. But the problem then is how would we write them. Half the time (or more!) they'd be technically wrong, meaning a lot of the time they'd be harder to read. I understand that some would say that doesn't matter, but isn't it better to get things right? Like in key E, there's rarely a D♭m chord - should be C♯m...

And, as a question, that mock is flawed. 50:50 chance of getting it right. Doesn't reveal much. Better, perhaps, to ask why that isn't a P5..? Like you did?

• Thank you - I was not familiar with double augmentation. In one way it’s a bit easier than I expected. The question was presented as written on a stave - just counting there were four steps would immediately tell me it was not a fifth - regardless of key or the number of semitones. – Marcos Scriven Feb 23 '20 at 7:41
• Yes, but just hearing those two notes, most folk would automatically say P5, I guess. – Tim Feb 23 '20 at 7:58
• Not only the most people. Everybody would. Or they wanted to give a mock answer ;) – Albrecht Hügli Feb 23 '20 at 9:10

Because it wasn't spell it with a `Bb`.

E to A is a fourth of some type. Lower the `E` with a flat and it becomes an augmented fourth. Add the sharp to the `A` and it is augmented (made larger by a half step) again and so it's called double augmented, a double augmented fourth.

...Is it simply because the enharmonic equivalent Bb should have been used, as that’s in the Eb scale?

Which is it? `Eb` and `A#`or `Eb` and `Bb`?

Yes, they are enharmonically equal, but the whole point of asking about `Eb` and `A#` is to see if someone understands complex interval spelling in writting notation.

...I counted up 7 semitones...if played you’d identify it as a perfect fifth...

That's beside the point. The interval was spelled out with sharps and flats.

We need to take a step back and think about terminology.

Interval in the generic sense is just a distance of time or space, or some other measurable thing. If there are 7 steps between two things, it's just an interval of 7. (Not a musical seventh, I mean the distance is just 7 units, frets, piano keys, etc.) If the two things were frequencies you could describe the interval with a ratio like 3:2.

But, those are merely "physical" measurements. They are not musical notation.

7 half steps or 3:2 is not a perfect fifth. They are possibly a perfect fifth, double augmented fourth, diminished sixth, etc. You cannot use terms like perfect, augmented, diminished, etc. until the interval is notated or spelled with the letters and sharps and flats of notation.)

If someone played the two tones would they identify a perfect fifth? Yes. But that would be a kind of ear training test.