====== On Double Counterpoint ======
===== First Section: Double Counterpoint in Two Parts =====
==== Inversion at the Thirteenth ====

Double counterpoint at the //thirteenth// or //sixth// is obtained by the same method as the other double counterpoints; that is to say, by the two series of figures. These are they which belong to the counterpoint in question:

^ Counterpoint  |  1  |  2  |  3  |  4  |  5  |  6  |  7  |  8  |  9  |  10  |  11  |  12  |  13  |
^ Inversion  |  13  |  12  |  11  |  10  |  9  |  8  |  7  |  6  |  5  |  4  |  3  |  2  |  1  |

It is easily seen that two sixths in succession must not be employed in this kind of counterpoint. Since the //seventh// cannot be resolved in a regular manner, it must be employed as a passing discord.

The //second//, //third//, //fourth//, //fifth//, and //ninth//, must be prepared by the //sixth// or by the //octave//, either above or below, and be saved by one of these intervals.

{{ :cherubini_counterpoint_and_fugue:ex175a-1.png?650 |}}

{{ :cherubini_counterpoint_and_fugue:ex175a.mp3 | :cherubini_counterpoint_and_fugue:ex175a.ogg |}}

{{ :cherubini_counterpoint_and_fugue:ex175b-1.png?650 |}}

{{ :cherubini_counterpoint_and_fugue:ex175b.mp3 | :cherubini_counterpoint_and_fugue:ex175b.ogg |}}

The range of the //thirteenth// serves as a limit to this counterpoint.

An extended example of double counterpoint at the //thirteenth//, or //sixth//, will now be given. This counterpoint is less frequently used than the counterpoints at the //octave//, at the //tenth//, and at the //twelfth//.

{{ :cherubini_counterpoint_and_fugue:ex176-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex176.mp3 | :cherubini_counterpoint_and_fugue:ex176.ogg |}}

This counterpoint is inverted by first transposing the upper part at the //thirteenth//, below the theme. Then the theme should be transposed a //sixth// higher, or a //third// lower, while the counterpoint does not stir; the theme may also be transposed a //third// lower, and the counterpoint a //third// higher, //etc//.