Too good not to share. Anyone who has read Hofstadter's 'Godel, Escher, Bach' all the way through is probably hip to this already, but it makes the point visually for people who've never had coursework in harmony or species counterpoint.
This is the famous "crab canon" from J.S. Bach's Musical Offering, which contains a highly-chromatic theme which is difficult to harmonize according to the species counterpoint rules that formed the foundation of much Baroque and Classical music, and which is still typically required of undergraduate music students to this day. The theme is given in C minor, with three flats, among them Bb. Interestingly enough, Bb is the seventh degree of the minor scale employed, but it is the only pitch class (10) that does not appear in the theme's original statement. That means the theme stops just short of being one of those twelve-tone row forms invented by Schoenberg:
More than a little bizarre to see this in the 18th Century, but what Bach does with this jewel is a real treat for the mind as well as the ear. According to some sources, this theme was either composed or collected by Frederick the Great, who was a great admirer of "Old Bach". The king played the theme for Bach in 1747, when the giant of Baroque music was 62 years old. Bach was not only old at the time, but was actually considerably less popular than some of his sons as a composer, because the public's tastes had moved away from contrapuntal music of a religious nature. The "new music" was, more often than not, forms derived from popular dance and subject matter derived from classical Greece and Rome. The Enlightenment was underway, and "enlightened men" no doubt thought all those canons and fugues passe.
It is interesting to think about Frederick the Great's motives (sorry) in this affair. He was a decent amateur musician, but by all accounts preferred the new music. Yet it was clear to the king that, fashionable or no, that "Old Bach" had mad skills that went far beyond what was being attempted in the new music. In providing this Thema Regium to the old master, the king was essentially challenging Bach to apply his skills to a piece of music that was to some degree artificial, like the "Wissenschaft" motif in Also Sprach Zaruthustra. While still tonal, just attempting a simple three-part harmonization that sounds sensible is actually quite a challenge. To actually realize it as Bach intended from the minimalist single-line score is a significant feat of transcription, and to really effectively play the thing almost essential. As with some of the Viennese serialists, one gets the impression that there is no meaningful distinction between composition and analysis for some of the "puzzle canons" that Bach derives from the king's theme.
Bach's motives (again, sorry) are a little obscure, as well. It is doubtful that Bach actually expected either the King or his courtiers to really understand the work, or even be able to play much of it from the score he provided. Playing the darn thing requires analysis, no button-pushers need apply. In physical decline, Bach surely knew that he could've satisfied the King's request with some simple two or three-part inventions, or perhaps a chaccone using the King's theme as a ground bass. That would've met the challenge in and of itself. Instead, Bach invested great energy and thought into what became the Musical Offering. The canons and fugues contained are some of the deepest and most intellectually challenging pieces to analyze in the entire (ahem) canon of western music.
It's a bit much to take in just by listening. How wonderful, then, that someone has done the work of providing visuals that help convey some of the intricacies of Bach's genius:
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