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

Double counterpoint is a musical-composition technique that comes in various types. By far the most important type in terms of tonal and structural enrichment of the music is double counterpoint in (or at) the twelfth.

Counterpoint
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 Twelfth
Double counterpoint in the twelfth involves one of two scenarios: (1) the transposing of the lower part up an octave and the transposing of the upper part down a fifth, or (2) the transposing of the lower part up a fifth and the transposing of the upper part down an octave.

(The simple transposing of the lower part up a twelfth produces the same result as the second scenario but at a higher octave. The simple transposing of the upper part down a twelfth produces the same result as the first scenario but at a lower octave.)

These transpositions cause 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.

The alteration of intervals in double counterpoint in the twelfth is as follows:

1 (unison) becomes 12
2 becomes 11
3 becomes 10
4 becomes 9
5 becomes 8
6 becomes 7
7 becomes 6
8 becomes 5
9 becomes 4
10 becomes 3
11 becomes 2
12 becomes 1

For example, the first inversion, from a unison (1) to a twelfth (12), will result from either one of the double-counterpoint scenarios.

Consider the first scenario. Suppose the original interval is a unison (1) on C. If the C representing the lower part is transposed up an octave to another C, and the C representing the upper part is transposed down a fifth to F, the result is an F-C twelfth (12, that is, a fifth plus an octave).

Consider the second scenario, again starting with unison C's. If the C representing the lower part is transposed up a fifth to G, and the C representing the upper part is transposed down an octave to C, the result is a C-G twelfth (12, that is, a fifth plus an octave).

In both scenarios, the unison (1) inverts to a twelfth (12).

The next inversion, from a second (2) to an eleventh (11) operates the same way. Consider the first scenario. Suppose the original interval is C-D, a second (2). If the lower part, C, is transposed up an octave to another C, and the upper part, D, is transposed down a fifth to G, the result is a G-C eleventh (11, that is, a fourth plus an octave).

Consider the second scenario, again starting with a C-D second (2). If the lower part, C, is transposed up a fifth to G, and the upper part, D, is transposed down an octave to another D, the result is a D-G eleventh (11, that is, a fourth plus an octave).

In both scenarios, the second (2) inverts to an eleventh (11).

The other intervals (3 to 10, 4 to 9, and so on) follow the same pattern.

The reason the two intervals add up to 13 instead of 12 (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 the C-D (2) and G-C (11) inversion discussed above, the C is counted twice: first as the lower part of C-D and then as the upper part of G-C.

Here is a simpler example, involving only one transposed part instead of two: the original interval is a C-C octave (8), the lower part is transposed up a twelfth to G, and the result is a C-G fifth (5). The C-C octave (8) and the C-G fifth (5) add up to 13 because one of the C's is counted twice: first as the upper part of C-C and then as the lower part of C-G.

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

(A) Perfect intervals. After the inversion of the parts in double counterpoint in the twelfth, most perfect intervals remain perfect. For example, a unison (1) becomes a twelfth (12), which is simply a fifth (5) plus an octave (8). The other inversions: fifth (5) to octave (8), octave (8) to fifth (5), and twelfth (12) to unison (1). Therefore, if the traditionally forbidden parallel unisons, fifths, and octaves are avoided in the original writing, they will automatically be avoided in the inversion.

(B) Dissonances. After the inversion of the parts in double counterpoint in the twelfth, most dissonances remain dissonant: second (2) to eleventh (11, that is, a fourth plus an octave), fourth (4, perfect and augmented) to ninth (9), ninth (9) to fourth (4), and eleventh (11) to second (2). The exception is a seventh (7), which inverts to a sixth (6), a consonance. In all these cases, if the original dissonant interval is properly treated as a dissonance, the voice leading in the inverted form will also be correct.

(C) Third. After the inversion of the parts in double counterpoint in the twelfth, a third (3) becomes a tenth (10), and a tenth (10) becomes a third (3). A tenth, of course, is simply a third (3) plus an octave (8). In other words, for contrapuntal purposes, a third (3) remains a third (3).

(D) Sixth. This is the problem interval in double counterpoint in the twelfth. After the inversion of the parts in double counterpoint in the twelfth, a consonant sixth (6) becomes a dissonant seventh (7). Therefore, in the original writing, a sixth (6) must be treated as a dissonance so that, after the inversion, its inverted form, a seventh (7), will be so treated.

Because of the unavailability of the sixth for consonant purposes, double counterpoint in the twelfth is often characterized by the prominent use of thirds between the parts to make up for the loss of sixths.

Double counterpoint in the twelfth is one of the musically richest forms of counterpoint. It is richer, for example, than double counterpoint in the octave.

In double counterpoint in the octave, the parts invert, but the pitch classes remain the same, thus creating little or no harmonic change. For example, the original interval progression C-E to D-F implies a chordal progression of C major to D minor. Inverted in the octave, the progression is E-C to F-D, which still implies the same chords.

However, double counterpoint in the twelfth generates new pitches and new chordal implications. For example, an original C-E to D-F interval progression becomes, in the first scenario, A-C to B-D, with the new pitches A and B and with the new implied chords A minor and B diminished. In the second scenario, the inverted intervals are E-G and F-A, with the new pitches G and A and with the new implied chords E minor and F major.

The pitch and chord permutations of double counterpoint in the twelfth give the affected musical passage a sense of compellingly fruitful variety within the basic unity of the same melodies.
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The New Grove Dictionary of Music and Musicians. 2nd ed. London: Macmillan, 2001.

Merriam-Webster's Collegiate Dictionary. 11th ed. Springfield, Mass.: Merriam-Webster, 2006.