Those Confounding Triplets (and such)
During a recent master class, an old performance problem reared its ugly head: the proper execution of slow (tempo or duration) triplets, which no one in the class could play accurately.
Quotable: “Equidistant! Please! Five equal notes!”-Leonard Bernstein, 1981 (referring to some quarter-note quintuplets during a rehearsal)
At the risk of sounding overly academic or theoretical, triplets can be defined as three equidistant notes played in place of other notes, most often two of the same denomination (i.e. three quarter- notes in place of two quarter-notes, or 3:2 (example1). At “normal” tempi triplets are routinely played; however, at slow tempi or with larger units (i.e. half-notes), accurate note placement can be problematic. This may not be particularly critical in all musical venues, like solo playing, but such rhythmic accuracy is considered as routine in ensemble contexts, especially at the highest levels of the music profession.
The time-honored approach to performing slow triplets is to (mentally) subdivide by two, via multiplication (3:2 = 3 x 2) while playing, as in example 2 (the x notes represent the “thinking” subdivision and the lower line the playing). A wonderful example of this performance problem is the half-note triplets in the next-to-last measure of the contrabass fugato episode in Richard Strauss’ Also Sprach Zarathustra (example 3), which, if not carefully executed, can sound like a rhythmic motif of a beguine. In this Strauss example, the preceding measure consists of two sets of quarter-note triplets (3 x 2 = 6); so, a de facto 6:4 ratio is already set, presumably rendering the the half-note triplets almost impossible to play incorrectly (editorial note: conductors and members of symphony orchestras are not allowed to react to this assertion).
Musicians familiar with New Music performance practices are cognizant of the fact such subdivisions can apply beyond simple triplets, as in example 4, which also involves simple multiplication. The measure calls for a half-note triplet to be played in 5/4 meter (3:5). When one multiplies the two figures, 3 x 5, it equals 15. There are fifteen eighth-note triplets in 5/4 meter; so, the duration of each half-note triplet equals five eighth-note triplets.
A further example of the multiplication/subdivision approach to dealing with rhythmic configurations is shown in example 4. In this instance, four equal quarters are written within 3/4 meter (4:3). 3 x 4 = 12, and there are twelve sixteenths in 3/4 meter. 12 ÷ 4 = 3; therefore, each note in the quadruplet has the durational equivalent of three sixteenth notes.
The mathematical approach looks ridiculous and overly academic when put in writing as in this post, and it doesn't always compute in actual practice; however, when it does, it can be extremely useful since it takes the guesswork (what one famous musician calls the "swindlesmanship") out of the rhythmic placement equation. In this regard, during the master class in question, we took the necessary time to insure that all the participants could properly execute the triplets before we moved on to other issues.
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