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Of the fourteen canons in this work, twelve were unknown prior to 1974. The full set was discovered in France with a copy of the Goldberg Variations, and they are a brilliant set of explorations of the first eight notes of the Goldberg ground:
You'll hear this ground throughout the work. Now, here is the first of the canons, in Bach's own hand. As you can see, it's just the ground itself.
But see how Bach notates it. If we straighten everything out and get rid of a few ink blots, we get this.
Note the mirror image of the clef and key and time signature at right -- a clear indication that this figure is to be played backwards, or in retrograde, as it is also played forward, to yield the first of the fourteen canons. If we write this out explicitly on two staves, we get this (the ground is in the treble clef, the retrograde in the bass clef).
Each of the canons is scored like this, with notation that is both efficient and playfully cryptic. There is no indication of what instrument should be used, or how many repetitions of this little fragment of notes one should play before going on to the next canon, which reminds me of another, considerably more modern canon I have realized.
Now, we must admit the canon in the grand staff above is rather elementary, as are canons II through IV in this cycle. But that's just the logician Bach at work, and I encourage you to follow the program below as you listen, to see what he's up to.
Bach leaves the question of orchestration wide open, so I have used an orchestra of organs, two guitars, three winds, and strings trading off the various parts. The idea was to provide enough different timbres to present the canons in various registers and locations in the stereo field, hopefully bringing out the multitude of contrapuntal lines. Below is a brief guide to the program.
Just to make sure we're perfectly clear about the ground, here it is.
I: Canon simplex [0:11]
This is just what's shown in the grand staff above, repeated once.
II: All' roverscio [0:27]
Take Canon I and turn it upside down, and that's what you get here: the inversion of Canon I and its retrograde.
III: Motu recto e contrario [0:37]
Now Bach takes the Goldberg ground and combines it with its inversion, rather than with its retrograde. Listen for the ground, and then, after four notes, the inversion comes in. It's repeated twice, so you'll have time to pick it out. (Incidentally, the canon that sounds first is the leader; the one that follows it is, well, the follower.)
IV: Motu contrario e recto [0:58]
Finally, we get the inversion of the ground and its inversion, also following the ground by four beats. You'll hear much of the orchestra I've used for this realization trading off the parts. Listen for the inversion, and the inversion of the inversion, following each other around the various instruments.
What do I mean, "finally," after just four canons? Well, if your brain is generating an electrical smell after trying to sort out the inversion of the inversion from the inversion, you might want to pause, or even restart the MP3! More to the point, look at what Bach has done. We have a ground, and we can do two things to it: reverse it (retrograde) or turn it upside down (inversion). That leads to 2 x 2 = 4 permutations of the ground plus variants, and the ever-methodical Bach has given us just that. It all sounds a bit academic so far, but so what? I've always admired the game of baseball for requiring pitchers to actually throw four pitchouts to issue an intentional walk, rather than just motioning the batter to first. It's delightfully methodical. Likewise, Bach here intentionally walks us through the flavors of his ground. And now that he's done that, he lets it fly. Hang on!
V. Canon duplex a 4 voci [1:51]
Now we start cooking with gas. You'll hear the Goldberg ground and its inversion a la Canon III, but on top of that, Bach adds a sprightly new canon, thus, Canon duplex. The new canon appears in sixteenth notes against the quarters of the ground, and, yes, it also appears with its inversion. So now you have (Goldberg + inversion) + (new canon + inversion), woven together in an irresistible tapestry.
VI. Canon simplex uber besagtes Fundament a 3 voci [2:22]
Over the Goldberg ground in the bass, you will now hear a "mirror" canon in the other two voices. Here the two canons are the precise inversions of each other; the intervals used in the leader are exactly the same as those in the follower, but they all move in the opposite direction.
VII. Canon simplex uber besagtes Fundament a 3 voci [3:24]
Here is a second canon and its inversion against the Goldberg ground.
VIII. Canon simplex a 3 voci il soggetto in Alto [4:06]
Now Bach moves the ground from the bass voice to the alto voice, with the canons tracing a variant of the ground in contrary motion around it. Like Canon VI, this is a mirror canon, with the intervals precisely inverted between leader and follower.
IX. Canon in unisono post semifusam a 3 voci [5:12]
The querulous character of Canon VIII is transformed into a joyful bubbling in this canon. Both the leader and the follower here are on the tonic G (thus "canon in unisono"), and they trail each other by a mere sixteenth note.
X. Alio modo per sincopationes et per ligaturas a 2 voci [5:48]
When one sets these works up in an audio sequencer like Sonar, you get a MIDI map of the notes, their durations, pitches, velocities, controller commands, and so forth. Here's the MIDI map of this canon, featuring two lines each moving against their inversions.
I laughed out loud after playing this one in and scrolling the Sonar piano roll window out. That's the mind of a gentleman who loved symmetry!
XI. Canon duplex uber Fundament a 5 voci [6:16]
This fragment is the crux of the work -- literally. For a thorough discussion of the symbolism in this canon, upon which Bach inscribed Symbol: Christ will crown those who carry His cross, I refer you to the paper That Crown of Thorns, by my friend Dr. Tim Smith of Northern Arizona University. This 5-voice double mirror canon condenses into a few bars of musical notation a magnificent artistic achievement with multiple layers of meaning. While you read Tim's comments on the "chi" figure, consider the MIDI map for this canon.
XII. Canon duplex uber Fundamental-Noten a 5 voci [6:51]
Where Canon V presented four voices in two lines, each moving against their inversions, this canon is in five voices, with two canons plus inversions superimposed on the ground.
XIII. Canon triplex a 6 voci [7:19]
If a double canon, with each canon having a leader and follower, has 4 voices, then naturally a triple canon must have 6. This miraculous canon has a timeless sound to it, a sense of eternity that invites the listener to ponder, reflect, and remember Bach's signature notation: Soli Deo gloria.
XIV. Canon a 4 voci per Augmentationem et Diminutionem [7:48]
Bach concludes this cycle in glorious fashion. Here he demonstrates a figure that can be combined with itself not only in canon, and in inversion, but with the rhythmic values appearing in various multiples of the original. One could set this canon in any number of ways, but to bring out the progressive augmentation, I've scored it so the voices enter in order (soprano, alto, tenor, bass), and in increasingly augmented incarnations (1X, 2X, 4X, and 8X). The alto and bass appear in inversion, in which case the 8X augmented inverted bass recovers the Goldberg ground. Put together, they create a joyful noise that is a most satisfying conclusion to this set of small musical gems.
The Bach Number
Have you wondered why there are 14 canons in this work, and not some other number? Try this -- assign a number to each letter of the alphabet, with A=1, B=2, and so forth, one number per letter except for I and J both being assigned 9. This yields a common "number alphabet" of Bach's day. You can calculate easily that B+A+C+H = 14. Moreover, J+S+B+A+C+H = 41, which by an amusing circumstance is the inversion (symbolically, at least) of 14. Did Bach know about this? Absolutely, and letter-number mappings and meaningful numbers of patterns and repetitions, collectively termed gematrics, appear throughout his music.
Now, in my own research area of solar variations and Sun-climate interactions, the road to scholarly ruin is littered with the bones of astronomers who have gone too far looking for correlations that don't exist. In similar fashion, some musicologists have, in my estimation, gone quite overboard finding numerology in the music of Bach. But some amount of symbolism is unquestionably present. One has only to look at Bach's seal to see his love of puzzles, and he strikes one as having had a good sense of humor.
So in that spirit, all you good listeners, can you find the gematric tricks embedded in this realization? Old BACH can be found all over in it, and I humbly hope he might have been amused.
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