Intervals that span a distance greater than an octave are called “compound intervals.” You can find out their number by adding 7 to the interval regular. Ex.: a “10^{th}” is the distance of a 3^{rd} an octave higher (3 + 7 = 10). We add 7 because there are 7 notes in a scale. This may seem confusing because a scale is usually thought of as 8 notes, but remember that the 8^{th} “note” is just a repeat of the first note; it’s just an octave higher. So we add the interval to 7 rather than 8.

Here are the most common compound intervals:

9^{th}, 10^{th}, 11^{th}, and 13^{th}.

Here’s what these four sound like:

Melodically:

Harmonically:

(Note: 12ths and 14ths are possible, too, but less common in popular music).

If we plug them into our “interval + 7” formula, we get:

A 9^{th} is a 2^{nd}, because 7 + 2 = 9

A 10^{th }is a 3^{rd} because 3 + 7 = 10

A 11^{th} is a 4^{th} because 4 + 7 = 11

A 13^{th}is a 6^{th} because 7 + 6 = 13

Compound intervals, like regular intervals, can be sharped or flatted. For example, jazz chords use a lot of “b9” (“flat-nine”) intervals in chords. Compound intervals come into play a lot in extended chords, but we’ll get to those soon.

TL;DR: Compound intervals are intervals greater than an octave. The most common ones are 9ths, 10ths, 11ths, and 13ths. The formula for a compound interval’s value is: (compound interval) – 7 = (interval value) or (interval) + 7 = (compound interval).