J.S. Bach: A Study in Musical Symmetry

Bach's music has often been described as "mathematical" or "pure." This is due in part to the intricate structures and symmetries present in his music. Symmetrical arrangements and repetitions were typical of compositions in Bach's time, but no one else approached his innovation and mastery of these forms. While symmetrical elements can be found throughout Bach's body of work, these elements are most apparent in his later pieces, particularly his canons.

The Canon

The canon, derived from the Greek word for rule, is perhaps the strictest form of counterpoint. It is made up of two separate voices, one of which is an exact repetition or contrapuntal derivation of the other. One usually follows the other, and the second voice is said to be "strictly generated" by the first. The two voices go by many names - leader/follower, dux/comes, antecedens/consequens, proposta/risposta - and are a stricter version of the subject/countersubject in a fugue. The canon is the darling of musical theory, for its two voices can be performed simultaneously with either melodic line as the bass.

Though the strict guidelines of a normal canon already generate symmetrical forms, there are a few special types of canon which are even more symmetrical. The crab canon calls for a single melodic line which is played forward and backward simultaneously. If you try to picture this visually, the two voices would consist of the first voice and its mirror image; the challenge is in constructing a melody which will fit perfectly with its inverse. The canon in contrary motion has bilateral symmetry: the two voices progress by the same intervals, but move in opposite directions. This technique of contrary motion is also known as melodic inversion, and the canon is sometimes called an inverted canon. An even more rigorous sub-category of the inverted canon is the mirror canon. In this case, the second voice mimics the precise quality of intervals invoked by the first voice; this is different from the normal inverted canon because composers will often be forced to inflect interval qualities in order to stay in key.

Canon Perpetuus: an example of a mirror canon>
Click title for RealAudio sample

Seek and Ye Shall Find

Bach became increasingly obsessed with the canon toward the end of his life (for a full biography click here), fascinated by the challenge they presented even the most adept composer. Canons were also often presented in the form of puzzles, and so became a favorite form of exchange for Bach and many of his students and friends. By puzzle, I mean that the canon was often left incomplete, with some sort of cryptic devi ce to indicate how to fill in the rest. For example, Bach would usually only provide the first voice of a crab canon, completed with a backward clef at its end to indicate its nature. Hence, the inscription Quarendo Invenietis, Seek and Ye Shall Find appeared in Bach's Musical Offering. The Offering itself is full of symmetry and follows the form:

I Ricercar
II Five Canons
III Trio Sonata
IV Five Canons
V Ricercar

Another interesting thing about the Musical Offering - from a letter Bach wrote to Meinrad Spiess we know that he intended to present this piece to the Mizler society. The society was sort of a forum for exploring the importance of mathematics and philosophy in music, and its influence on Bach has been hotly debated. Though some argue that Bach only joined the society to please his old colleague Christoph Mizler, Bach's later work clearly reflects some of the society's interests. His famous Canonic Variations for organ were written specifically for the society, and the intricate structures of canons and fugues dominated his last years of composition.

Malcolm Boyd, Bach (Oxford: Oxford University Press, 1983)
Istvan Hargittai, Symmetry (New York: Pergamon Press, 1986)
Marjorie Senechal and George Fleck, eds., Patterns of Symmetry (Amherst: University of Massachusetts Press, 1977)
Related Links
J.S. Bach Homepage

Bach Canons and Fugues

Dave's J.S. Bach Page and Midi Files

Page author: Sasie Sealy