# How to correctly invert non-compound greater-than-octave intervals?

(A.S.: Assuming that "non-compound" means "contains eight or less degrees.")

TL;DR: How to invert doubly augmented seventh?

I've ran into inversion problem with greater-than-octave intervals (not only compound ones).

Simple inversion is to invert something like major sixth (i.e. `C4 to A4`), – obviously, we will get minor third (`A4 to C5`).

A bit more complicated inversion is to invert compound intervals: you should first bring it to non-compound view; so, major third over octave `E♮4 to G♯5` becomes minor sixth – `G♯5 to E♮6` (or `G♯4 to E♮5`, or `G♯3 to E♮4`).

Other weird thing is to invert augmented octave: it considered as compound interval and becomes diminished octave (not diminished unison, as one might think, since diminished unison is incorrect interval).

So, if there is a doubly augmented seventh (`C♭4 to B♯4`), what interval will we got if invert it? Doubly diminished second (`B♯4 to C♭5`)? Is this correct? Is it OK that lower pitch is higher than upper one?

• How has the interval come up for you? Typically choices of notes are based on a function they serve, so I would expect that a double augmented 7 (x7) would have a function different than the b9, which it is enharmonically equivalent to. – Basstickler Apr 19 '16 at 13:01
• @Basstickler: Try not to think tonally, like in serial technique, — there could be anything, up to the four times diminished unisons. – Dima Parzhitsky Apr 19 '16 at 18:26
• Sounds interesting. I'm curious what the benefit of writing things out this way would be. Function aside, would it actually sound different and therefore have an advantage as being referred to as such? Serialism is essentially trying to break free from tonality, so ultimately requires another means of explanation/description, so it sounds like you're in a similar realm with this. Would you perhaps be able to provide something to listen to or read on the subject (for me, not necessarily SE)? – Basstickler Apr 19 '16 at 20:35
• @Basstickler: I'm not sure I got your point (especially, what exactly could sound different? after which action?). Anyway, the serial technique was mentioned here to appoint that tonal relationship of notes may not be taken into account (because of the specific of serialism) and can be safely omitted when inverting intervals. Schönberg's music can serve the example of serialism (but beware of its extreme expressionism). – Dima Parzhitsky Apr 19 '16 at 21:24
• @DmitryParzhitsky this interval wouldn't come up in serialism or at least the 12 tone flavor as you would stick with the naturally named notes and either all sharps or flats for the notes that are not naturally named. And even then you are more look at just the distance in semitones. – Dom Apr 20 '16 at 16:05

You are dealing with extremes of intervals so the result may not be straightforward and I can almost guarantee you will never see anything like that in practice. In general doubly diminished and augmented intervals are only seen in 4ths and 5ths due to them being perfect and are extremely rare in practical music as seen in and explained this question.

In general for inverting intervals you want the sum of the degrees to equal 9 and the qualities to balance out. For example, a P5 inverts to a P4, a M3 inverts to a m6, and an A4 inverts to a d5.

In your example of C♭ to B♯, you have the interval of a doubly augmented 7th written as AA7. The inverse interval B♯ to C♭ is a doubly diminished 2nd written as dd2 in accordance to what was talked about in the last paragraph.

The fact the pitches sound over an octave or reversed is irrelevant when it comes to naming as an interval is defined as how the pitch is written which is why the interval C to F♯ is not the same interval as C to G♭ even though in equal temperament they will sound exactly the same. If the interval looks like a second it needs to be denoted as such.

I've included a quick mock up of what both intervals look like in standard notion and anyone looking at them would realize pretty fast that the first is some kind of 7th and the second is some kind of 2nd. They may need to think about what the qualities are since this is very atypical, but the distinction is extremely important which this question goes more into.

This can be answered fairly simply.

The inversion of a doubly augmented seventh is a doubly diminished second. A doubly diminished second is -1 semitone, so it's more than "OK" that the (nominally) "lower" pitch is higher; it follows naturally from the definition. If you descend by -1 semitones, you go up in pitch.

You mention inverting an augmented 8ve. You can't invert an aug 8ve. If you try, the higher note still ends up being the higher note. That would make it an augmented unison. Essentially, it becomes reduced and not inverted.