So far we’ve only talked above ascending intervals, or intervals whose “base” or “root” note is the lower one. But what if the root note is the higher of the two notes? Then you get something called an “inverted” interval.
The formula to figure out an inversion is: 9 – (interval) = (inverted interval). However, once you’ve found the new number value, you must flip the major/minor and diminished/augmented quality of the interval. Perfect intervals stay perfect when inverted.
Ex: A major 3rd is an inverted minor 6th. 9 – 3 = 6, then switch the “major” to “minor.” A perfect fourth is an inverted perfect fifth because 9 – 4 = 5, and the quality (perfect) stays the same.
Here are some more:
|Minor 2nd||Major 7th|
|Major 2nd||Minor 7th|
|Minor 3rd||Major 6th|
|Major 3rd||Minor 6th|
|Perfect 4th||Perfect 5th|
|Augmented 5th||Diminished 4th|
|Perfect 5th||Perfect 4th|
|Minor 6th||Major 3rd|
|Major 6th||Minor 3rd|
|Minor 7th||Major 2nd|
|Major 7th||Minor 2nd|
The formula for inverting intervals works in the reverse manner, too. You can “un-invert” an inverted interval.
Ex: An inverted 4th is a regular 5th, and an inverted minor 7th is a major 2nd.
TL;DR: inverted intervals are intervals whose root note is the top note, rather than the bottom one. The formula for converting regular intervals to inverted ones, and vice versa, is: 9 – (interval) = (inverted interval). If the interval is major/minor, or augmented/diminished, you must flip its quality. Major becomes minor, augmented becomes diminished, perfects stay perfect.