Metric modulation

In music a metric modulation is a change (modulation) from one time signature/tempo (meter) to another, wherein a note value from the first is made equivalent to a note value in the second, like a pivot. The term was invented to describe the practice of Elliott Carter, who prefers to call it temporal modulation.

For metric modulation to exist 3 things have to occur:

1. There has to be an exact relationship between two different tempi
2. A common pulse must exist between these two tempi
3. The name and function of the pulse changes.

The following formula illustrates how to determine the tempo before or after a metric modulation, or, alternately, how many of the associated note values will be in each measure before or after the modulation:

• ${\frac {new\ tempo}{old\ tempo}}={\frac {number\ of\ pivot\ note\ values\ in\ new\ measure}{number\ of\ pivot\ note\ values\ in\ old\ measure}}$

(Winold 1975, 230-31)

Thus if the two half notes in 4/4 time at a tempo of quarter note = 84 are made equivalent with three half notes at a new tempo, that tempo will be:

${\begin{aligned}\qquad {\frac {x}{84}}&={\frac {3}{2}},\\{84}\cdot {\frac {x}{84}}&={\frac {3}{2}}\cdot {84},\\\qquad {x}&={\frac {3\cdot 84}{2}},\\\qquad {x}&={126}\\\end{aligned}}$

(Winold 1975, p.230, example taken from Carter's Eight Etudes and a Fantasy for woodwind quartet (1950), Fantasy, mm. 16-17.)

A tempo (or metric) modulation causes a change in the hierarchical relationship between the perceived beat subdivision and all potential subdivisions belonging to the new tempo. Benadon (2004) has explored some compositional uses of tempo modulations, such as tempo networks and beat subdivision spaces.

Score notation

Metric modulations are generally notated as 'note value' = 'note value'.
For example,

This notation is also normally followed by the new tempo in parentheses.
This is analogous with the assignment in imperative computer languages:
{x = f(x);}  ≡  {x_new = f(x_old);}

References

• Benadon, Fernando (2004). Towards a Theory of Tempo Modulation. Proceedings of the 8th International Conference on Music Perception and Cognition. Evanston, IL. (563 - 566)
• Winold, Allen (1975). "Rhythm in Twentieth-Century Music". In Aspects of Twentieth-Century Music, edited by Gary Wittlich, . Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5.