====== On Double Counterpoint ======
===== Second Section =====
==== Triple and Quadruple Counterpoint at the Octave ====

There are two ways of composing these counterpoints; the first and most easy, consists in adding a double counterpoint, one or two parts proceeding in //thirds//, either with the lower part, or with the upper part.

Double counterpoint, in order to be susceptible of receiving these two parts in //thirds//, or even a single one, must be constructed according to certain conditions:
  * Firstly, it should not contain -- throughout -- either two //thirds//, or two //sixths// in succession, and consequently, it should be written entirely in contrary movement or in oblique movement.
  * Secondly, it should contain no discord, except passing tones.

> Example of a double counterpoint at the octave

{{ :cherubini_counterpoint_and_fugue:ex179a-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex179a.mp3 | :cherubini_counterpoint_and_fugue:ex179a.ogg |}}

In order first to transform this double counterpoint into triple counterpoint, no more need be done, than to add a third part, either a third above the upper part, or a third above the lower part. ((**Editor's note**: Despite these instructions, in Example 179b, Cherubini adds a third //below// the upper part. In Example 179c, the upper part becomes this //added// third-below line.))

{{ :cherubini_counterpoint_and_fugue:ex179b-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex179b.mp3 | :cherubini_counterpoint_and_fugue:ex179b.ogg |}}

> Triple counterpoint: Adding a third above the lower part

{{ :cherubini_counterpoint_and_fugue:ex179c-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex179c.mp3 | :cherubini_counterpoint_and_fugue:ex179c.ogg |}}

To convert the same double counterpoint into quadruple counterpoint, there must be joined to the two principal parts, the two parts just added; the one, a third above the upper part, and the other, a third above the lower part.

{{ :cherubini_counterpoint_and_fugue:ex179d-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex179d.mp3 | :cherubini_counterpoint_and_fugue:ex179d.ogg |}}

Then the parts of this counterpoint may be inverted in several ways, as the following example shows:

{{ :cherubini_counterpoint_and_fugue:ex179e-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex179e.mp3 | :cherubini_counterpoint_and_fugue:ex179e.ogg |}}

The other way of introducing //triple// and //quadruple// counterpoint at the octave consists of combining the parts in such a manner, as that they may be inverted; that is to say, so that each part can be placed above or below, without changing the melody at all, and without there arising the least-objectionable point, or the least infringement of the strictest rules. It is indispensable, to ensure this, that the parts shall never form between each other either a //fourth// or a //fifth//; excepting in the case where the melody proceeds by conjunct movement, or in that where only discords prepared by the //second//, the //fourth//, and the //seventh// are employed. The discord of the prepared //ninth// is impracticable in this kind of counterpoint, as has already been said with regard to double counterpoint at the octave.

> Examples of triple counterpoint of this kind

{{ :cherubini_counterpoint_and_fugue:ex180-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex180.mp3 | :cherubini_counterpoint_and_fugue:ex180.ogg |}}

> Examples of quadruple counterpoint of the same kind (media player is above Example 181)

{{ :cherubini_counterpoint_and_fugue:ex181.mp3 | :cherubini_counterpoint_and_fugue:ex181.ogg |}}

{{ :cherubini_counterpoint_and_fugue:ex181-1.png?800 |}}

{{ :cherubini_counterpoint_and_fugue:ex181-2.png?800 |}}

This kind of counterpoint, by its nature and its regularity of inversions, may be applied to //counter-subjects// of a //fugue//; as will be seen when that sort of composition comes under discussion in the next chapter.