We are introduced to Frederick the Great (King of Prussia) and J.S. Bach’s historical friendship. The King acquires fifteen of Gottfried Silbermann’s piano-fortes. Meaning “soft-loud”, this piano is different from the usual piano of that time when the keys of the piano could only be played at uniform loudness. There was no way to strike a note more loudly than its neighbours. Bach, upon invitation by the King to try out his new piano-fortes by improvising, improvises a 6-part fugue in his “Musical Offering” for the King known as Ricercar. This astonishing feat is compared to playing sixty simultaneous blindfold games of chess and winning them all. The inscription written on the page preceding the music reads “At the King’s Command, the Song and the Remainder Resolved with Canonic Art.” is written in Latin. Canonic appears to have a double meaning: “with canons” and “in the best possible way”. The Latin initials “RICERCAR also means “to seek” in Italian.
Douglas informs us of what a “canon” means: a melody is transformed and added to the original piece, not unlike a copy. It could be transformed by moving the time the copy plays, playing the copy in a higher or lower tone, quickening (diminution) or slowing down (augmentation) of the copy, inverting, and even playing in reverse. Here is an example of a canon, “Good King Wenceslas“. All of these are found in Bach’s Musical Offering. This leads him to introducing the first abstract concept of what an isomorphism is: an information-preserving transformation.
A fugue, on the other hand, is like a canon with all its transformative properties intact. A fugue is less rigid than that of a canon however, which allows much more artistic expression. The main sign which indicates whether a piece is a fugue is by observing the start of the piece: one voice is introduced, then another voice transformed five scale notes up or four scale notes down. The second voice is the “countersubject” to the first voice (subject). Other voices are introduced as subjects and their respective countersubjects, until all the voices arrive. At this point, apart from basic musical rules, there truly are no rules which governs how the fugue will develop. Hence, a fugue by formula apparently cannot be composed.
Each of the canons in the Musical Offering is a different variant of the King’s Theme, and all the transformative properties described above are fully exploited to give maximal sophistication. One of the canons is cryptically labelled “Quarendo invenietis” (“By seeking, you will discover”), one of the puzzles Bach gives to the King. A solution is found by one of Bach’s pupils Johann Philipp Kirnberger, but one can still wonder whether there are other solutions one might find.
One three-voiced canon in the Musical Offering is simply called “Canon per Tonos” and has a very interesting property. As the piece goes on, the pitch goes higher and higher until it seemingly concludes, but in fact the keys are changed (modulate) without the listener realizing it. Tonalities are changed increasingly remote from the original tone of C minor until it actually returns to it. Hence, the piece is very much does not ends; it is endless.
This is given as our first example to Douglas’s notion of Strange Loops. By constantly moving upwards (or downwards) a hierarchical system we somehow find ourselves back to where we started. He advises us with Bach’s message to the King: “Quarendo invenietis”.