Inversion of INTERVALS

When we take the lower note of an interval and transpose it (move it) up an octave, we get the inversion of an interval. 

As an example, let us invert the interval C4 – G4, which is a perfect fifth. The next figure shows that when we transpose C4 up an octave to C5, we get the interval G4 – C5, which is a perfect fourth. Thus, the inversion of a perfect fifth produces a perfect fourth.

Figure: The inversion of a perfect fifth produces a perfect fourth

If we repeat this procedure and invert the interval G4 – C5, we get the interval C5 – G5 which is the same as the starting interval, only an octave higher. This shows that only one inversion of intervals exists. In chapter Chord inversions in close position, we shall use the same procedure to invert chords and see that chords have two or three inversions.   

The next table shows the relationship between the number of the interval and the number of its inverted interval. From the table, we see that wider intervals have narrower inverted intervals (and vice versa).

A similar relationship exists between the quality of the interval and the quality of its inverted interval. This is shown in the next table.

Using the tables, we can easily see that, for instance, the inversion of a minor third produces a major sixth. Another interesting fact is that the inversion of an augmented fourth produces a diminished fifth. That is, the inversion of a tritone produces also a tritone. The reason for this is that a tritone divides an octave into two equal halves since an octave spans 6 wholes steps while a tritone spans 3 whole steps. 

Let us explain the practical usage of the inversion of intervals. Usually, it is easier to find a narrower than a wider interval. For instance, let’s find the ascending major sixth from F#4. The inversion of a major sixth is a minor third. If we apply the descending minor third from F#4, we get D#4. Thus, the ascending major sixth from F#4 is D#5. Also, it is sometimes easier to find an ascending interval than a descending interval. For instance, instead of finding the descending perfect fifth from C4, we can find the ascending perfect fourth from C4 which is F4. Thus, the descending perfect fifth from C4 is F3. In chapter Circle of fifths, we shall use the ascending and descending perfect fifth. Herein, instead of using the descending perfect fifth, we can also use the ascending perfect fourth and get the same results.