Here’s
something you may have wondered at some point in your musical 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?
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.
1 2 3 4 5 6 7 + 1 2
= 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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