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.
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:
| newtempo | |
| oldtempo |
=
| numberofpivotnotevaluesinnewmeasure | |
| numberofpivotnotevaluesinoldmeasure |
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:
\begin{align}
| x | |
| 84 |
&=
| 3 | |
| 2 |
,\\ {84} ⋅
| x | |
| 84 |
&=
| 3 | |
| 2 |
⋅ {84},\\ {x}&=
| 3 ⋅ 84 | |
| 2 |
,\\ {x}&={126}\\ \end{align}
Note that this tempo, quarter note = 126, is equal to dotted-quarter note = 84 ((=) = (=)).
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 has explored some compositional uses of tempo modulations, such as tempo networks and beat subdivision spaces.
Three challenges arise when performing metric modulations:
Examples of the use of metric modulation include Carter's Cello Sonata (1948), A Symphony of Three Orchestras (1976), and Björk's "Desired Constellation" (=).Beethoven used metric modulation in his Trio for 2 oboes & English horn, Op. 87, 1794.
Metric modulations are generally notated as 'note value' = 'note value'. For example,
=
This notation is also normally followed by the new tempo in parentheses.
Before the modern concept and notation of metric modulations composers used the terms doppio piu mosso and doppio piu lento for double and half-speed, and later markings such as:
(Adagio)=(Allegro)
indicating double speed, which would now be marked (=).
The phrase l'istesso tempo was used for what may now be notated with metric modulation markings. For example: to, will be marked l'istesso tempo, indicating the beat is the same speed.
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