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.
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 the 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 4th and 5th intervals. Why? Because the 5th - a consonant interval - becomes a 4th 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 4th interval (measure 4) taking care to uses it as an appoggiatura reached by contrary motion. When the parts are inverted, the 4th becomes a 5th (measure 8):
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 10th and the 12th one of the voices moves an octave while the other moves a 10th or a 12th. The following animation may help you visualize it better:
Writing music invertible at the 10th or the 12th presents new problems not found on the inversion at the octave.
The following table will help us see what becomes of each interval when we use the inversion at the 12th:
As this table shows, a 2nd becomes an 11th after inversion, a 3rd becomes a 10th, etc. The intervals shown in gray are those that can create problems after inversion. The problematic interval in the inversion at the 12th is the 6th interval that becomes a 7th. In other words, a consonant interval (a very useful one in counterpoint!) becomes a dissonance.
By looking at the 10th table we find that there will be a lot more of problems with this inversion:
The most useful intervals - 3rds and 6ths - become 8ves and 5ths when inverted! This means that parallel or direct 3rds and 6ths will become parallel or direct 8ves and 5ths.
Let's see how Bach uses the inversion at the 10th and the 12th in the following fragment (measure 44) of Contrapunctus X from the Art of the Fugue:
In measure 66 we see the inversion at the 10th, the higher voice is moved one octave down while the lower voice is moved a 10th up::
In measure 85 the same passage is inverted at the 12th (the upper voice moves one 12th up):