Double Counterpoint in the Octave: A Concise Guide for Music Students and Listeners

Strictly speaking, counterpoint is any music consisting of two or more parts (also called lines or voices) that sound simultaneously. However, the term counterpoint is often used specifically to denote a musical texture in which each of the parts retains an individual character, as distinct from a texture in which one part clearly predominates (as the melody) and the remaining part or parts are clearly subordinate (as harmonic fillers).

The word counterpoint entered Middle English in the 15th century from Middle French contrepoint, from Medieval Latin contrapunctus. The Latin word itself is a condensed form of the expression punctus contra punctum ("note against note," or, by extension, "melody against melody").

The ways in which composers can and have applied counterpoint to musical composition are virtually limitless. However, certain structures and procedures have recurred often enough to be named and analyzed.

Invertible Counterpoint
One such procedure is called invertible counterpoint, that is, counterpoint that can be inverted. This type of counterpoint is designed so that it can also be performed with the lower part transposed to lie above the upper part or with the upper part transposed to lie below the lower part.

When applied to two parts, this method of composition is called double counterpoint. If applied to three or four parts, it is called respectively triple or quadruple counterpoint.

Double Counterpoint in the Octave
Double counterpoint in the octave involves the transposing of the lower part up an octave or the transposing of the upper part down an octave.

This inversion causes the harmonic (simultaneously sounded) intervals between the two parts to change. The student or composer must be aware of those changes so that, after inversion, the music will produce the desired effects.

When one part is transposed by an octave, either up or down, the following interval changes occur:

1 (unison) becomes 8 (octave)
2 becomes 7
3 becomes 6
4 becomes 5
5 becomes 4
6 becomes 3
7 becomes 2
8 becomes 1

For example, suppose the original interval is a unison (1) on C. If one of the C's is transposed up an octave, the result is a C-C octave (8). If one of the C's is transposed down an octave, the result is again a C-C octave (8) but an octave lower.

Suppose the original interval is C-D, a second (2). If the lower part, C, is transposed up an octave, the new interval becomes D-C, a seventh (7). If, instead, the upper part, D, is transposed down an octave, the new interval is still D-C , a seventh (7), but an octave lower. In either case, the original 2 becomes a 7.

The other intervals ( 3 to 6, 4 to 5, and so on) follow the same pattern.

The reason the two intervals add up to 9 instead of 8 (the interval of transposition) is that one of the tones is counted twice, once in the original interval and again in the inverted interval. For example, in C-D (2) and D-C (7), the D is counted twice: first as the upper part of C-D and then as the lower part of D-C.

"Rules"
The following observations pertain to a diatonic context (one that employs only the pitches of a single key).

(A) Consonances. After the inversion of the parts in double counterpoint in the octave, most consonances remain consonant: a unison (1), an octave (8), a major or minor third (3), and a major or minor sixth (6).

A unison (1) becomes an octave (8), and an octave (8) becomes a unison (1). Therefore, if the traditionally forbidden parallel unisons and parallel octaves are avoided in the original writing, they will automatically be avoided in the inversion.

A third (3) becomes a sixth (6), and a sixth (6) becomes a third (3). Both of these rich, useful consonances remain consonances in the inversion.

(B) Dissonances. After the inversion of the parts in double counterpoint in the octave, a second (2) inverts to become a seventh (7), and a seventh (7) to a second (2). Therefore, if these two dissonances are correctly treated (as passing tones, suspensions, and so on) in the original writing, they will automatically remain correct in the inversion.

(C) Fourths and fifths. After the inversion of the parts in double counterpoint in the octave, a fourth (4) becomes a fifth (5), and a fifth (5) becomes a fourth (4). This is the tricky area in double counterpoint in the octave. There are three situations to consider.

First, an augmented fourth (aug. 4) becomes a diminished fifth (dim. 5), and a diminished fifth (dim. 5) becomes an augmented fourth (aug. 4). Both intervals are dissonances. If correct voice leading governs these intervals in the original writing, that correct voice leading will also govern the inverted forms.

Second, a perfect fourth (4) becomes a perfect fifth (5). Therefore, care must be taken in the approach of the two parts to a fourth (4) because similar motion to a fourth (4) will lead, in the inversion, to similar motion to a fifth (5). Similar motion to a fifth (5) creates what is called a direct (or hidden, covered, exposed) fifth, which, especially if both parts leap to the fifth, is either forbidden or limited in various styles (two-part writing, three-part writing, 16th-century style, Bach style, and so on).

Third, a perfect fifth (5) becomes a perfect fourth (4). A fourth (4) is a dissonance in two-part writing. Therefore, the fifth (5), even though it is a consonance, must be treated as if it were a dissonance so that, after the inversion, the fourth (4) will be so treated.