Editing Metric modulation (section)

==Determination of the new tempo==
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:
:<math>\frac{\text{new tempo}}{\text{old tempo}} = \frac{\text{number of pivot note values in new measure}}{\text{number of pivot note values in old measure}}</math>{{sfn|Winold|1975|loc=230-231}}

[[File:Metric modulation 2=3.png|thumb|upright=1.4|Metric modulation: 2 half notes = 3 half notes[[File:Metric modulation 2{{=}}3.mid|center]] or [[File:Metric modulation 2{{=}}3 with eighth notes.mid|thumb|center|Play with eighth note subdivision for tempo/metre comparison]]]]

Thus if the two half notes in {{time signature|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:

:<math>
\begin{align}
\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{align}
</math>
Example taken from Carter's ''Eight Etudes and a Fantasy'' for woodwind quartet (1950), Fantasy, mm.&nbsp;16-17.{{sfn|Winold|1975|loc=230}}

Note that this tempo, quarter note = 126, is equal to dotted-quarter note = 84 (({{music|halfnote}} = {{music|dottedhalf}}) = ({{music|quarternote}} = {{music|dottedquarter}})).

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.{{sfn|Benadon|2004}}

Three challenges arise when performing metric modulations:{{sfn|Weisberg|1996|loc=54}}
#Grouping notes of the same speed differently on each side of the barline, ex: (quintuplet {{music|sixteenthnote}}=sextuplet {{music|sixteenthnote}}) with sixteenth notes before and after the barline
#Subdivision used on one side of the barline and not the other, ex: (triplet {{music|eighthnote}}={{music|sixteenthnote}}) with triplets before and quarter notes after the barline
#Subdivision used on neither side of the barline but used to establish the modulation, ex: (quintuplet {{music|quarternote}}={{music|quarternote}}) with quarter notes before and after the barline

Examples of the use of metric modulation include Carter's Cello Sonata (1948),{{sfn|Cunningham|2007|loc=113}} ''[[A Symphony of Three Orchestras]]'' (1976),{{sfn|Farberman|1997|loc=158}} and [[Björk]]'s "[[Medúlla|Desired Constellation]]" ({{music|dottedquarter}}={{music|halfnote}}).{{sfn|Malawey|2007|loc=142-144}}
Beethoven used metric modulation in his Trio for 2 oboes & English horn, Op. 87, 1794.