In music, the term “modulation” is most frequently used to refer to a shift in keyality. One speaks of modulating from the tonic to the dominant, and, at least in traditional harmony, using a pivot chord to achieve the modulation; that is, a chord that assumes a duel role of one function in the current key, and a different function in the new key. For example, a C major chord can be used as the subdominant in G major, and the tonic in C major. When considering non-pitched concepts such as meter and tempo and rhythm, a beat value can serve as the pivot duration between two tempos in the same way a chord can serve as the pivot chord between two keys.
A metric modulation is a change in tempo resulting from a regrouping of divisions of a beat, so that the ictus formed by the new group is divided into more or fewer divisions, resulting in a slower or faster beat, respectively. For example, if the music is proceeding in paired eighth notes with a quarter note beat, and then proceeds in groups of three, making a dotted quarter beat, the tempo of the eighth notes remains unchanged, but the tempo of the ictus modulates from quarter note to dotted quarter note, essentially creating a slower tempo, though the subdivisions have not slowed.
This can easily be demonstrated with the third movement from Suite No. 2 For Military Band in F by Holst. The movement begins at 8:28 in six-eight meter, with an ictus of dotted quarter note. At 9:22, a second, slower beat, accomplished through a metric modulation, begins with the dotted half note as the new ictus, but with the eighth note tempo remaining constant. This is a good example to understand metric modulation, because both the “old” and “new” tempos are present simultaneously. This is not requisite for a metric modulation, but it does make a clear example. What is requisite for a metric modulation is that the new tempo is derived from a subdivision of the old tempo. This can be through a re-grouping within the same meter, or across a change in meter. Examples of the former can be shifting from duplets to triplets, and of the latter from going from two-four meter to three-eight meter, as Stravinsky was fond of doing. Listen to 0:00-1:38 of “Soldiers March” from L’Histoire du Soldat.
Confusion sometimes arises over whether or not tempo change wherein the tempo is altered by speeding up or slowing down the ictus beat, as one does in a standard accelerando or ritardando, is a metric shift. For example, consider the first movement from symphony “from the New World” by Dvorak. The introduction is in four-eight meter throughout, but the conductor shifts between conducting eighth notes and subdividing sixteenth notes. Does the conductor’s action make this a metric change? Is a new tempo being performed because the conductor is beating twice as fast for sixteenth notes and then half as fast for eighth notes? Is this a metric shift?
This is an interesting question. If you couldn’t see the conductor, would you intuitively perceive the ictus to shift back and forth between eighth and sixteenth notes, or do you intuitively hear the whole introduction in measures of four eighth note beats? Rhythmic structure is hierarchical; there are multiple levels of beats always present in music. That is what makes subdivision possible. All at once, a listener can perceive an elongation, the ictus, and divisions as nested beats. Shifting one’s perception from one level to the other does not in itself create a metric shift. If it did, metric shifts could happen anywhere, anytime and would be entirely left to the whim of the listener, essentially vanishing all meaning of the term. In order to be an instance of a metric shift, the music must bring you to the conclusion that the most compelling beat, the one that makes the most sense as the ictus, has changed, but is still related to a division of the previous ictus. The back and forth shifting evidenced by the conductor is not a regrouping of divisions, but a shift to a different rhythmic level that does not change the most compelling ictus, which remains the eighth note.
At 1:47, the first movement proper begins. It is Allegro and in two-four meter. The ictus is the quarter note, and this quarter note beat is equal to the sixteenth note subdivision from the introduction. Is this a metric shift? Here is a different situation. The division of sixteenth note equals the division of the quarter note. This fulfills our definition of a metric shift. But it can be one only if the new quarter note tempo selected by the conductor equals the old sixteenth note tempo. If the conductor selected a different new tempo, then that moment in the music would not be a metric shift.
This brings us to an important point; one that has been made elsewhere in this blog. Music is an aural art form. Though notated music is essential in Western culture art music, it is a means of transmission, and not the music itself. The music itself is what is performed and heard. If, when the music is played properly, it sounds like a metric modulation, then it is. If, regardless of what a composer wrote down, it does not sound like a metric modulation, then it is not. The ultimate analysis must come from what is heard.
There is one more point to made on this subject. In tonal analysis, temporary instances of non-diatonic tonalities as results from chromaticism forming secondary dominants, Neopolitan second chords, augmented sixth chords, and so forth, are often not considered modulations, because their resolutions do not bring the music to a new key, but only circle round to the same key as before. We must take the same approach with metric modulations. Though irregular meters such as five-eight and seven-eight do regroup subdivisions into faster and slower beats, these frequent shifts within a single measure do not initiate a new overall tempo for the music, but rather establish a regular pattern of irregular beat durations. As such, they are no more modulations than a melodic minor passage is considered one key in its ascending form and another key in its descending form. Any alteration is so short-lived that it cannot be construed as a modulation. For further inquiry, listen to Eliot Carter’s first string quartet, from which the concept of metric modulation gained traction.