Invertible Counterpoint

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Invertible counterpoint is a way of composing two or more voices so that their registral positions can be reversed, i.e. so that the higher voice can become the lower and vice versa, without breaking the 18th-century conventions of dissonance treatment. Sometimes, the technique is called "double counterpoint" because each of the two voice serves a dual role, either as upper or as lower part. Three or four voices can also be written so that they are texturally invertible, in which case we speak of triple and quadruple counterpoint.

The technique goes back to the 1500s. Nicola Vicentino discusses the topic in his 1555 treatise L'antica musica ridotta alla moderna prattica. A classic description of invertible counterpoint and guidelines for applying the technique compositionally appear in Gioseffo Zarlino's treatise Le Istitutione harmoniche (1558), Part 3 (The Art of Counterpoint), chapter 56.

Along with other contrapuntal techniques, such as melodic inversion, retrograde, stretto, diminution, and augmentation, invertible counterpoint allows composers to maximize material already presented in a piece--in a sense, "recycling" it. Having written a passage with voices in one registral disposition, composers can reverse the positions of the voices and instantly generate another passage. The technique thus serves the age-old artistic principle of "unity in variety," presenting the same material (unity) in a new way (variety). Finally, invertible counterpoint came to be seen as a sign of compositional mastery; with it a composer could demonstrate his accomplishment and worthiness as an artist.

When inverting parts, the intervals formed between them are, of course, inverted. In the familiar case of inversion "at the octave," for instance, where one of the two voices is transferred above or below the other voice by transposing it up or down an octave, a third inverts into a sixth, a unison into an octave, second into a seventh, augmented fourth into a diminished fifth.[1] In those cases, consonance inverts into consonance, dissonance into dissonance. However, the interval of fifth, one of the quintessential consonances, inverts into a fourth, which is a dissonance in two-voice counterpoint. Fifths in two-voice invertible counterpoint must therefore be written as though they were dissonances, that is, approached and left by step, as in unaccented and accented passing and neighbor notes (including the double neighbor and escape tone), or in a suspension figure. In the last case (suspension), a fifth must be prepared by a consonance, and the lower note of the fifth must be resolved downward by step, as though it were a dissonance. When the parts are inverted, the lower note of that fifth will become the upper note of a dissonant fourth, which will resolve correctly: downward by step.

Beginning with Angelo Berardi's treatise, Raggionamente musicali (1687), the intervallic transformations involved in invertible counterpoint were summarized in a chart, like the following:

1 2 3 4 5 6 7 8
8 7 6 5 4 3 2 1

As we can see by examining these transformations, writing a 5th (consonance) between the voices is problematic, especially on a strong beat, because when the voices exchange places, it will become a dissonant 4th, requiring resolution. In writing a strong-beat (metrically accented) 5th, it will be necessary, therefore, for the lower voice to descend by step so that when the lower voice becomes the upper, and the 5th becomes a 4th, the dissonant 4th will resolve down by step, as 18th-century contrapuntal practice demands. Metrically and rhythmically unaccented 5ths are unproblematic so long as they continue by step.

Less commonly, we find invertible counterpoint at the 12th, which has the following interval transformations when the voice positions are reversed:

1 2 3 4 5 6 7 8 9 10 11 12
12 11 10 9 8 7 6 5 4 3 2 1

Here again we face the problem of a consonance transforming into a dissonance. This time a 6th becomes a 7th when the voices are reversed. That transformation is most inconvenient because it means that all 6ths, which are common in 18th-century counterpoint, will become dissonant 7ths and require resolution! The lower voice of a metrically accented 6th would have to descend by step so that the 7th resulting from reversing the voices would resolve down by step. Metrically and rhythmically unaccented 6ths are unproblematic so long as they continue by step.

Invertible counterpoint at the 10th, the hardest to write and thus infrequent, is even more problematic, as a glance at the intervallic transformations shows:

1 2 3 4 5 6 7 8 9 10
10 9 8 7 6 5 4 3 2 1

A series of parallel 3rds or 6ths, frequent in 18th-century music, will become, respectively, forbidden parallel 8ves and 5ths!

Examples in the Inventions
A few brief examples of invertible counterpoint occur in the F-major Inventio (no. 8). Compare m. 3 with m. 14. The 4th on the third 16th of beat 1 (E-A), m. 3, becomes a 5th (E-B) on the third 16th, beat 1, m. 14. In m. 3, the 6th (A-F) on beat 3 becomes a 3rd (C-E) in m. 14 on beat 3. Compare also m. 15 with m. 19.

Another example appears in the C-major Inventio (no. 1). Compare mm. 3 and 11. The right-hand E at the beginning of m. 3 is transposed down a 7th, to the left-hand F in m. 9. The left-hand C at the beginning of m. 3 is transposed up a 2nd (+ 8ve) in m. 9. The transposition intervals (7th and 2nd) sum to 9, indicating double counterpoint at the 8ve.

In Inventio 1, we can see Bach's awareness of the one pitfall when writing invertible counterpoint at the 8ve, namely that a 5th becomes a 4th when the voices are reversed. Compare mm. 1-2 (up to the first note in m. 2) with mm. 7-8 (up to the first note in m. 8). The voices are reversed: right-part of m. 1 is transposed down a 4th, plus 8ve, to become the left-hand part in m. 7; and the left-hand part of m. 1 is transposed up a 5th, plus 8ve, to become the right-hand part in m. 7. 5+4= 9, hence double counterpoint at the 8ve.

Between mm. 1-2, the left hand leaps up a 5th, from C to G, and on the downbeat of m. 2 we find a 5th (G-D) formed by the two voices. Between mm. 7-8, the right-hand part does not copy the left-hand part from m. 1; it does not leap up a 5th. Instead, it moves downward by step to F#. Had it copied the left-hand part from m. 1 and leaped up a 5th (to D), a dissonant 4th (A-D) would have resulted on the downbeat of m. 8. Bach foresaw that and clearly wanted to avoid it, so he altered the upper voice.

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1. Alternatively, both voices may be transposed such that the sum of their intervallic movements equals an 8ve; that is, one voice is transposed a 5th downward (or 8ve + 5th), while the other is transposed a 4th upward (5th + 4th = 8ve). When invertible counterpoint at the 8ve is applied to two voices, all 3rds in the original arrangement of the voices become 6ths when the voices exchange places. All 2nds become 7ths, and all 4ths become 5ths. Note well: in invertible counterpoint at the octave, the sum of the interval transpositions is always 9.
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