Why Does An Interval & Its Inversion Add up To 9 Instead of 8?

Here’s something you may have wondered at some point in your musical studies or career: If there are only eight diatonic notes in an octave, why is it that when you invert any interval, the starting interval plus its inversion always add up to 9?
Shouldn’t it add up to 8? 🙂


Let me illustrate. Here’s an octave, with F as the root:




Eight staff positions are between the two F’s, therefore, the label “Octave” for “Eight.”


As shown in next example, when we omit the last note in the octave, we get a major seventh interval distance between the outer 2 notes; here shown relative to the note F:




Again, it is a seventh, so there are seven positions on the staff between F and its seventh.


Let’s now invert this interval by moving the F up an Octave:






Inverted, the major seventh has become a minor second. The second occupies two staff positions. The thing is, all we have done is move the F up one octave, so shouldn’t the interval plus its inversion be the same as an octave?


I’m sure you’ve figured it out: The reason that an interval plus its inversion adds up to nine instead of 8 is that one note is counted twice within the octave. In this example, the E is counted twice: once as the note 7 steps higher than F, and once as the note one step lower than F.


F          G          A          B          C          D          E          +          E          F  =  9 notes.


The Memory Trick


Interestingly, you can use this bit of information to your advantage. You can determine the inversion of any interval simply by hearing its name (intervals are named by numbers. That number is defined by the number of note names/letters involved) and deducting that from 9. Amaze your friends at parties! Win on Jeopardy! … Perhaps not…but it still can be useful to know what an interval’s inversion is WITHOUT having to see it on the staff. Here’s how:


First, determine the quality of the interval, that is, whether it’s perfect, major, minor, augmented or diminished. To find the quality of the inversion, simply use the opposite quality: minor inverted becomes major, and vice versa, perfect interval inverted becomes another perfect interval.


So if your beginning interval is a minor third, its inversion is going to be a major something. If your beginning interval is an augmented fourth, the inversion will be a diminished something. The exception would be the perfect intervals, because when perfect intervals are inverted, their inversions are also perfect.


Next, determine the number of staff positions of the interval. So, in the minor third example, the number of staff positions would be three. To find the inversion, simply subtract that number from nine. In this case 9-3=6. So, the inversion of a minor third will be a major sixth. The inversion of a perfect fourth is a perfect fifth, and so on.


Practice

Let’s try some difficult ones:

What is the inversion of an augmented sixth interval?


Well, since the interval is augmented, we know that its inversion must be the opposite of augmented, so it must be a diminished something. Since the number of staff positions it occupies is 6, we know to subtract that number from nine to determine the number of staff positions occupied by the inversion, so 9-6=3. Therefore, the inversion of the augmented sixth is the diminished third. Pretty simple, right?

How about this:

What is the inversion of the perfect octave?

Since this interval is perfect, we know that its inversion must also be perfect. We also know that since it is an octave, the number of staff positions it occupies is 8. So, to find the inversion, we will subtract that from 9: 9-8=1. The interval that occupies only one staff position is unison. Therefore, the inversion of the perfect octave is the perfect unison.

I hope you’ve learned why an interval plus its inversion equals nine and not eight, and I hope the simple logical formula helps you to determine the inversion of any interval when you need to remember it quickly. Keep practicing. Have fun.



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