Double Canon on a Möbius Strip
Double Canon on a Möbius Strip was composed as the final culminating project for the course Musical Puzzles and Games, taught by Dr. Mark Janello. After analyzing and realizing numerous examples of cryptic, clever, and sometimes bizarre “enigma”/“puzzle” compositions (mostly from the Renaissance and Baroque periods), our final task as students was to produce one of our own to try stumping our peers and the general Peabody student population.
Among the canons of Johann Sebastian Bach, BWV 1087, there is one which makes use of the melody’s reflection across the center line of the staff as a canonic voice. Thus, one means of conceptualizing this canon is as one statement of the melody on a musical staff made into a Möbius strip, or Möbius "loop", after German mathematician August Ferdinand Möbius. This served as a jumping-off point for my own canon, which I wanted to further complicate with the addition of second canonic voice (at a different pitch level and, subsequently, mode) and its own reflection along the same axis. To the right, presented in a crude combination of faux-Renaissance engraving and obtuse prose, is my own original melody which, once solved, will produce a canon with those stipulations that largely follows the conventions of Renaissance counterpoint.
If someone should attempt to transcribe a realization of the four voices of this canon, a few things will be worthwhile for them to note:
The original melody, to be written correctly in a single orientation, needs to be considered in terms of the topology of a Möbius strip. At the outset, the single continuous surface of the Möbius strip could be considered as the two sides of the original paper strip set side-by-side, if “reading” it from the seam by travelling along the surface. This will also result in the backside of the original being oriented as upside-down and backwards relative to the front-facing side when formed into the Möbius Strip. For this reason, I’ve drawn extra attention to how the material on the backside of the original strip of manuscript paper is presented; the second C clef is not necessary to the final presentation, but does aid in alluding to the fact that the second half of the melody is presented in both inversion (along the center line) AND retrograde.
Also inherent to the properties of a Möbius strip is embedded a clue about the relationship between placing the entries of canonic voices relative to one another. Each line ‘meets’ its reflection on the Möbius strip halfway along its relative side, meaning that the reflected canonic voices can correctly be placed as entering halfway through their original lines. An intentionally omitted detail, however, is when the transposed canonic voice will enter relative to the original - to my knowledge, there exists only the one solution that will obey contrapuntal rules [proper modal treatment will also only necessitate musica ficta on one note]. Two clues I will offer in regard to this aspect of the puzzle are that 1) the offset is a natural number of whole-note values and 2) the order of the voices entering is original, transposed, reflected, reflected-transposed.
A conventionally-notated version of this canon with all four voices written out can be produced upon request.