FOR THE CURIOUS CELLIST

Bach: Canon In Hypodiatesseron Al Rovescio

Here is a very curious work, the exact title of which is “Canon in Hypodiatesseron al Rovescio e per Augmentionem Perpetuus”, from Bach’s masterwork of counterpoint “The Art of Fugue”.

By definition, in a canon, the second voice plays exactly the same notes as the first voice. In this canon the second voice does exactly that, with the “only” difference being that this second voice  ……. starts one fourth lower (hypodiatesseron) ….. is in the mirror image of the first voice (al rovescio – where the top voice goes up, the bottom voice goes down by the same interval, and vice versa where the top voice goes down) ……… and goes at half speed (per augmentionem perpetuus)!

Such a piece is, of course, very easy to compose …… but only if we don’t care about our creation sounding horribly dissonant (dodecaphonic music follows many of the same principles). The astonishing achievement of Bach is that somehow he makes this puzzle sound like a piece of music, and not like a dodecaphonic aural nightmare. Even if we know nothing about all the mathematical juggling involved, this is still “music for the ears”: it sounds good, with harmony, cadences, modulations etc. It is difficult to imagine how a human being could do this. Did he just go at it randomly, correcting any notes that created dissonances when played by the second voice? He uses the same four-bar beginning theme in another canon of identical characteristics called “Canon in Augmentation in Contrary Motion”, which could be considered as the non-identical twin of the one that we offer.

So here it is, in various different versions, for cello and violin duo but also for two cellos. In Bach’s original (keyboard) version the canon is repeated with the voices inverted between the hands (the leading voice is given on the repeat to the left hand). This gives an “equal” duo, in which both players (or hands) get to play both voices. We also offer the “unequal” duo, in which the second player stays always on the somewhat easier lower voice: this is our “Easier Version”. When the cello plays the first voice in the “Equal” Duo for violin and cello, we have taken the violin part at times down an octave. In all versions except the “Unequal” Violin/Cello Duo, the first three notes of the Finale have been lowered an octave for the first voice.

         “Equal” Duo for Violin and Cello     “Unequal” Duo for Violin and Cello

   “Equal” Duo for Two Cellos    “Unequal” Duo for Two Cellos

“Equal” Duo for Viola and Cello       “Unequal” Duo for Viola and Cello