Metric modulation

In music, metric modulation is a change in pulse rate (tempo) and/or pulse grouping (subdivision) which is derived from a note value or grouping heard before the change. Examples of metric modulation may include changes in time signature across an unchanging tempo, but the concept applies more specifically to shifts from one time signature/tempo (metre) to another, wherein a note value from the first is made equivalent to a note value in the second, like a pivot or bridge. The term "modulation" invokes the analogous and more familiar term in analyses of tonal harmony, wherein a pitch or pitch interval serves as a bridge between two keys. In both terms, the pivoting value functions differently before and after the change, but sounds the same, and acts as an audible common element between them. Metric modulation was first described by Richard Franko Goldman while reviewing the Cello Sonata of Elliott Carter, who prefers to call it tempo modulation. Another synonymous term is proportional tempi. A technique in which a rhythmic pattern is superposed on another, heterometrically, and then supersedes it and becomes the basic metre. Usually, such time signatures are mutually prime, e.g., and , and so have no common divisors. Thus the change of the basic metre decisively alters the numerical content of the beat, but the minimal denominator ( when changes to ; when, e.g., changes to , etc.) remains constant in duration. The following formula illustrates how to determine the tempo before or after a metric modulation, or, alternatively, how many of the associated note values will be in each measure before or after the modulation: Thus if the two half notes in time at a tempo of quarter note = 84 are made equivalent with three half notes at a new tempo, that tempo will be: Example taken from Carter's Eight Etudes and a Fantasy for woodwind quartet (1950), Fantasy, mm. 16-17. Note that this tempo, quarter note = 126, is equal to dotted-quarter note = 84 (( = ) = ( = )).