Kirnberger's most puzzling canon

The title page of the first volume of Johann Philipp Kirnberger's treatise "Die Kunst des reinen Satzes in der Musik" (1774) bears a humorously notated puzzle canon packed with subtleties and esoteric knowledge, but its solution nonetheless produces a pleasant enough result. The canon appears just below the dedication to Kirnberger's princess student, Anna Amalia, Abbess of Quedlinburg (1723-1787).

The text "Wir irren allesamt nur jeder irret anderst," a quotation of Georg Christoph Lichtenberg (Göttinger Taschenbuch 5), perhaps means, "We all make mistakes, only some make them differently." And this is a fitting text to this perplexing puzzle. [An aside: Beethoven also used this text for a 2-voice canon (WoO 198).]

The solution makes a 5-voice piece including the continuo part below the top line which is itself the 4-in-1 canon. Let's examine the clues to see how the puzzle is solved.

Key: The canon is given with a signature of six flats, but the continuo is given with the signature to the enharmonic key: six sharps. This should not be overlooked. Any first-year theory student today would be familiar with the idea that G-flat major and F-sharp major are enharmonic keys, i.e. that they are the same, but until equal temperament was accepted the idea that these two keys were the same was not coherent. I will present a solution in sharps.

Time: The canon is given in four measures of common time (4/4) as indicated by the "C", but the continuo is given in eight measures of cut time (2/4, in the modern baroque), indicated with "₵". Simple enough, once one realizes that either the continuo must be diminuted (twice as fast), or the canon must be augmented (half as fast).

Imitation: The mysterious numbers (60, 45, 48, 64) appearing below the music indicate the pitch intervals of imitation. The last two of these numbers indicate through their inversion on the page that the last two voices are to follow in contrary motion. If taken in order, the four numbers show the relative frequencies of the four canonic parts. If the dux (leader) begins on A# (symbolized by '60'), then the first follower must begin on a lower note bearing the ratio 45:60 with the entrance note of the dux. Since this ratio reduces to 3:4, that of the perfect fourth, the answer must fall on E#, the note whose frequency is a perfect fourth below A#. Likewise, the inverted '48' indicates the first note of the next voice is to begin with the note given by the ratio 48:60 with respect to the dux's entrance note. This fraction reduces to 4:5, and indicates that the third voice should enter on F#, continuing in contrary motion. Lastly, the fourth voice's ratio is 60:64, which reduces to the semitone 15:16, yielding the final voice's entry on B-natural. Since this number is inverted, the fourth voice continues in contrary motion as well.

While these enigmatic devices seem to indicate an intentional obfuscation that is not musically justified, one must remember that the principles, relations and concepts behind the solutions to the mysteries of this puzzle are to be learned in Kirnberger's treatise. The enigmatic frontispiece therefore presents some intrigue, both jocular and perplexing, perhaps even intimidating to the would-be student or to Princess Amalia.

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