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Invertible counterpoint makes a good Bach canon better

August 24, 2016

What is it?

Invertible counterpoint is a concept arising from music that centers around consonance. Its employment is an important technique in the art of counterpoint. The challenge of invertible counterpoint is to preserve the consonant sense of two or more voices (melodies or parts) even when upon repetition one of those voice's parts is transposed beyond the others, specifically, when a higher voice assumes the role of bass, that being a part of particular aural importance in establishing the consonance upon which the entire enterprise rests. Simply put, what was once the treble melody now becomes the bass melody, and vice versa.


Invertible counterpoint is furthermore dependent upon the idea of varied repetition. One might say that two parts in counterpoint have the potential for inversion, but that latent capacity is only realized once the two (or more) parts repeat, albeit in different registers.


Take as an example the principal subject and countersubject from Bach's Invention in C minor (BWV 723).


In measure 3, the subject is heard below the countersubject, but in measure 13 these roles are exchanged. The casual listener might assume that any two melodies inverted this way would produce a pleasing result. Au contraire! The two melodies must be invented with care in order to allow for such a pleasing result.


The Technical Details

For intervals between voices to be consonant, they must be octaves (8), perfect fifths (5), major or minor sixths (6) or major or minor thirds (3) or unisons (U). When a note in either voice changes, the interval between those voices changes. When the two melodies are written together the sequence of intervals between them is formed, but when the two melodies are swapped (inverted), those intervals change. What was once a pleasant and aurally sensible ordering of intervals may change. in baroque music and later, some dissonances can stand alone as sensible sonorities (d5, A4, m7), but each must resolve in a specific way.


When the voices are inverted by an octave, such as in our example,

a unison (U) or octave (8) inverts to an octave (8) and vice versa

a second (2) inverts to a seventh (7)

a third (3) inverts to a sixth (6)

a perfect fourth (4) inverts to a perfect fifth (P5)


and vice versa.  Also…


a diminished fifth (d5) inverts to an augmented fourth (A4)


Here are the two passages again, with the important intervals labeled:


Notice that in both passages, the consonant intervals U, 3, 6 and 8 are the ones relied upon the most, while the interval P5 is avoided, since it inverts to a (dissonant) perfect 4th.


Notice that the interval d5 appears halfway through on the downbeat. this meaningful dissonance must resolve inward to a third, each voice moving by a step. This happens on beat 3. Likewise, on beat 4 a different diminished fifth (d5) appears, and it resolves on the following downbeat. Notice that in measure 14, the dissonant intervals are now augmented fourths (A4), and they resolve outward to sixths (6) as required.


The necessary resolution of those exceptionally tense intervals (harmonies) may begin to give a glimpse of what might be termed the 'sense' of the harmonic motion which underlies the relationship between two or more voices as they move through time, in contrast to the immediate consonance or decorative dissonance that might appear at any given instance.


The Benefits of Invertible Counterpoint in Fugue

Invertible counterpoint is indispensable to Bach's conservation of motivic materials — that sense with which his fugal pieces is imbued that every measure is woven from the same cloth, that despite the vast diversity of the character of his fugues, each follows its own program and explores its own motivic materials. While literal repetition is eschewed, motivic ideas are explored, combined and recombined, rarely developed in a Beethovenian sense but rather arranged in new combinations. Invertible counterpoint offers tremendous resources for the accomplishment of this exploration.


Examine for instance Bach's C minor fugue from the first book of the Well-Tempered Clavier. You will find that the expositions and episodes are drawn from the same motives, and moreover, both the episodes and expositions are given fresh life as the direct result of inversion.


Invertible Counterpoint in Canon

It may not be quite as clear what the value of invertible counterpoint might be in the construction of a canon. Certain types of canon benefit greatly.


Bach's 'Canon a4' from the Musical Offering shows how quadruple counterpoint (4-voice invertible counterpoint) can make an otherwise repetitive piece into a rich embroidery of sound.



The enigmatic form of the canon, not the realization, is given in the manuscript and printed editions of the piece. A solution by Bach's student Kirnberger is included with the Bach Gesellschaft edition printed in the late nineteenth century, but I give a different one here. Bach gives two clefs in his enigma: the rare French violin clef and the bass clef, but there is no need to interpret these clefs as the only ones permitted. The complete invertiblity allows for other octaves, despite the fact that no clef used in Bach's time permit for single-octave transposition. The operative device here is clearly invertibility, not specific register. Indeed the canon sounds better with a viola rather than two cellos, as the result of the additional registral differentiation, and the new bass lines that result as higher bass notes are not obscured by the repetition of lower strands. The finished canon is quite attractive, chromatic and distinctive.


This is a bit of an aside, but Beethoven seems to have been intrigued by the twisting melody in the first several measures as he patterned part of the second theme of his 8th Symphony after it.




Romberg's canon for four voices, featured last October draws much of its interest in the registral combinations offered by its invertible counterpoint as well.





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