(You are listening to a MIDI performance of "Ma fin est mon commencement" by Guillaume de Machaut. This version is arranged for brass instruments. For a PDF file of the complete score, click eleven o'clock on Einstein's gold watch below.)

Although human beings tend to perceive time as an arrow that flies in just one direction, the laws of physics themselves are time-symmetric, and do not distinguish between forward and backward motion in time.

Although "going backwards in time" is not something we expect to observe at the macroscopic scale of everyday experience, composers have been fascinated by this possibility for centuries. Guillaume de Machaut's musico-poetic enigma, "Ma fin est mon commencement et mon commencement ma fin" ("My end is my beginning and my beginning my end"), composed before 1377, makes for a particularly instructive musical illustration of temporal symmetry on multiple levels. The cryptic lyrics of this three-voice rondeau reveal not only the composer's performance intentions but also disclose a mystery as elusive as any Zen koan:

As its name suggests, the conventional rondeau, with its ABaAabAB structure, is "round," its circularity suggested by the recurrence of each of two musical sections, either in their original form (A, B) or with a different text (a, b). Cyclical repetition of this kind, in which "The thing that hath been . . . is that which shall be," is common in music of all periods, but Machaut devised far more ingenious means of illustrating the meaning of his lyrics: the top two parts present the same melody, but in opposite temporal directions, and the lowest of the three parts (i.e., the "third voice") literally retraces its steps once it reaches the midpoint of the song. Since the two musical sections are repeated consecutively a total of three times, the third voice, true to the text, reverses itself exactly "trois fois" before the end. There is not a more apposite analogy for time-reversal symmetry in the entire history of music, although the composer was obviously not trying to advance any physical theory.

Nevertheless, we cannot discount the conceptual significance of Machaut's rondeau. Although retrograde motion per se is not unusual in fourteenth-century polyphony, Machaut's musical enigma is in a class by itself. Unlike his other rondeaux, whose sophisticated texts evoke the joys and sorrows of courtly love, "Ma fin est mon commencement" is singularly focussed on its central temporal enigma. As Gustave Reese noted, "Such close relation between music and text as we find here . . . is not characteristic of 14th-century composition."¹⁵ In this light, it is clear that the composer did not merely intend this rondeau to be a vehicle for displaying his contrapuntal skill but also saw it as a means of engaging the intellectual curiosity of both performers and listeners.

Machaut (c. 1300–77), who served as canon at Rheims Cathedral for nearly forty years, lived in an age when the mysteries of time fired the imagination of Christian mystics like Meister Eckhart (c. 1260–1326). An older contemporary who spent much of his career teaching in nearby Strassbourg and Paris, Eckhart held the position of vicar-general of Bohemia some twelve years before Machaut arrived there to join the household of King John as a secretary in 1323. If Machaut's rondeau were meant to suggest that the beginning and end of time are one and the same sub specie aeternatis, Eckhart made that view explicit when he wrote, "A day, whether six or seven ago, or more than six thousand years ago, is just as near to the present as yesterday. Why? Because all time is contained in the present Now-moment."¹⁶

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