Metric modulation

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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 tempo modulation.

Figure 1-a shows an example of a tempo modulation where the sixteenth-note serves as the pivotal value. This operation transforms the tactus speed and all its subdivisions by a factor of 4/7; expressed in ratios, the tempo relationship from old to new is 7:4. The same modulation can be renotated as in Figure 1-b. In this case, the use of the septuplet is avoided in favor of a time signature change. That is, the tempo change is expressed as a metric modulation.

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:

• 

(DeLone et. al. (Eds.), 1975, chap. 3)

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:

• 

(ibid, 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. For instance, in Figure 1-a above, the relationship between the sixteenth-note and the eighth-note is at first 2:1. After the modulation, this relationship becomes 7:2. Even though the old sixteenth-note is now notated as a septuplet, both values proceed at the same speed, which means that they are perceived as identical. Benadon (2004) has explored some compositional uses of tempo modulations, such as tempo networks and beat subdivision spaces.

[edit] 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)
• DeLone et. al. (Eds.) (1975). Aspects of Twentieth-Century Music. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5.