When a polyphonic passage is written so that each voice can be used as lower, upper or middle voice we are using invertible counterpoint. It is called double counterpoint when it involves two voices, triple and quadruple when three or four voices are used.

#### Inversion at the octave

Take a look at the following excerpt from Bach's BWV 780 Invention. The lower voice in measure 5 plays what the upper voice played in measures 1 to 4 (red on the score) while the upper voice in measure 5 plays what was played by the lower voice in measures 1 to 4 (blue on the score). When a piece is written so that each voice can be used as lower or upper voice we are using invertible counterpoint:

The inversion used by Bach in this piece is the most common and simple one and is known as inversion at the octave. Counterpoint books give us tables that show how intervals change when inverted. Although we may not need it for the inversion at the octave, we will show the table to help understand tables related to other types of inversions (see Inversion of Intervals for more information).

 intervals in this line: 2 3 4 5 6 7 become the corresponding interval in this line: 7 6 5 4 3 2

These tables can help us find the intervals that may be problematic when inverted. In the case of inversion at the octave the only problem is related to the fourth and fifth intervals. Why? Because the fifth - a consonant interval - becomes a fourth that is considered in counterpoint a dissonant interval.

With this in mind let's take another look at Bach's Invention. As you can see below, he uses only one fourth interval (measure 4) taking care to uses it as an appoggiatura reached by contrary motion. When the parts are inverted, the fourth becomes a fifth (measure 8):

#### Inversion at the tenth and the twelfth

In the inversion at the octave the lower voice moves one or more octaves up while the upper voice moves one or more octaves down. In the inversion at the tenth and the twelfth one of the voices moves an octave while the other moves a tenth or a twelfth. The following examples may help you visualize it better:

Inversion at the octave: upper voice moves down one or more octave and lower voice moves up one or more octave:

Inversion at the tenth: upper voice moves down one octave and lower voice moves up a tenth:

Inversion at the twelfth: upper voice moves down one octave and lower voice moves up a twelfth:

Writing music invertible at the tenth or the twelfth presents new problems not found on the inversion at the octave.

#### Inversion at the twelfth

The following table will help us see what becomes of each interval when we use the inversion at the twelfth:

 2 3 4 5 6 7 8 9 10 11 11 10 9 8 7 6 5 4 3 2

As this table shows, a second becomes an eleventh after inversion, a third becomes a tenth, etc. The intervals shown in red are those that can create problems after inversion. The problematic interval in the inversion at the twelfth is the sixth interval that becomes a seventh. In other words, a consonant interval (a very useful one in counterpoint!) becomes a dissonance.

#### Inversion at the tenth

By looking at the tenth table we find that there will be a lot more of problems with this inversion:

 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2

The most useful intervals - thirds and sixths - become octaves and fifths when inverted! This means that parallel or direct thirds and sixths will become parallel or direct octaves and fifths.

Let's see how Bach uses the inversion at the tenth and the twelfth in the following fragment (measure 44) of Contrapunctus X from the Art of the Fugue:

In measure 66 we see the inversion at the tenth, the higher voice is moved one octave down while the lower voice is moved a tenth up:

In measure 85 the same passage is inverted at the twelfth (the upper voice moves one twelfth up):

Read the the analysis of Contrapuntus X, Canon at the tenth and Canon at the twelfth from the Art of the Fugue and Fugue BWV 885 for some examples of this contrapuntal technique.