# How to enharmonically change the bottom note of the interval A# to E#

A# to E# is a perfect fifth, and an exercise book I am working through says to enharmonically change the bottom note and identify the resulting interval.

I cannot see any possibility except rewriting it as Bb to E#, which would be a double augmented 4th. The book has not introduced doubly augmented or double diminished intervals, and I can't even find those terms in the glossary.

Is there another possibility? It specifically says to enharmonically change the bottom note.

Sure there are other possible enharmonic equivalents you can convert the interval to C♭♭ to E♯, but it makes the interval uglier (a tripy augmented 3rd). Even if the book misprinted this and did not intend for you to have any doubly augmented or doubly diminished it's still an acceptable exercise for you.

Intervals of a 4th and 5th are the most likely place you'll ever see a doubly diminished or doubly augmented interval due to the nature of these intervals. In fact, there may be chords you play that have these intervals.

Let's look at a C7b5#9 and a C7#5b9

```C7b5#9 - C  E  Gb  Bb  D#

C7#5b9 - C  E  G#  Bb  Db
```

In the case of a C7b5#9, the interval between Gb and D# is a doubly augmented 5th and the Gb wants to go to an F(tonic) and the D# wants to go to an E(tonic's 7th aka leading tone). In the case of a C7#5b9, the interval between G# and Db is a doubly diminished 5th and the G# wants to go to an A(the 3rd of tonic) and the Db wants to go to a C(the 5th of tonic).

The point of these excesses is more to just drill and tax your knowledge of the topic and taking it to extremes to think most likely is the goal of that specific example. In fact in collage in my Music Theory 101 course my professor gave us interval worksheets that were very rigorous with the last interval being a triply augmented 5th (G♭♭ to D♯!) which was meant to stretch our understanding of the concept of intervals and is something that you'll almost certainty never see. It also wasn't in the book we were using and he even said as a joke "

If you ever see a G♭♭ to D♯ in the wild, drop what you are doing and contact me.

I have yet to contact him due to this.

• I've been using aug 4 to mean augmented 4th. Is there a standard way to adapt this notation to a doubly augmented 4th?
– user45413
Nov 17, 2017 at 1:11
• @KyleSchlitt typically you write the augmented part twice. For example and Augmented 4th I would typically write A4 and for a doubly augmented 4th I would write AA4.
– Dom
Nov 19, 2017 at 20:24

Yes, if the requirement is to change only the lower note, the only sensible alternative is Bb. Which results in a not-very-sensible double-augmented 4th. If this interval did occur in a piece of music, calling it a double-augmented 4th is most unlikely to be useful. It will actually be a perfect 5th coinciding with an enharmonic shift. Something rather different to the dual-function nature of e.g. the tritone B (Cb) - F as part of alternatively G7 or Db7.

The author of this book may be throwing in an extreme case just to test your logic! Or he may not have thought the question through carefully enough. What's the book?