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In music a tuplet (also irrational rhythm or groupings, artificial division or groupings, abnormal divisions, irregular rhythm, gruppetto, extra-metric groupings, or, rarely, contrametric rhythm) is "any rhythm that involves dividing the beat into a different number of equal subdivisions from that usually permitted by the time-signature (e.g., triplets, duplets, etc.)" (Humphries 2002, 266). This is indicated by a number (or sometimes two), indicating the fraction involved. The notes involved are also often grouped with a bracket or (in older notation) a slur. The most common type is the "triplet".
The modern term 'tuplet' comes from a mistaken splitting of the suffixes of words like quintu(s)-(u)plet and sextu(s)-(u)plet, and from related mathematical terms such as "tuple", "-uplet" and "-plet", which are used to form terms denoting multiplets (Oxford English Dictionary, entries "multiplet", "-plet, comb. form", "-let, suffix", and "et, suffix1"). An alternative modern term, "irrational rhythm", was originally borrowed from Greek prosody where it referred to "a syllable having a metrical value not corresponding to its actual time-value, or ... a metrical foot containing such a syllable" (Oxford English Dictionary, entry "irrational"). The term would be incorrect if used in the mathematical sense (because the note-values are rational fractions) or in the more general sense of "unreasonable, utterly illogical, absurd".
Alternative terms found occasionally are "artificial division" (Jones 1974, 19), "abnormal divisions" (Donato 1963, 34), "irregular rhythm" (Read 1964, 181), and "irregular rhythmic groupings" (Kennedy 1994). The term "polyrhythm" (or "polymeter"), sometimes incorrectly used of "tuplets", actually refers to the simultaneous use of opposing time signatures. (Read 1964, 167)
Whereas normally two quarter notes (crotchets) are the same duration as a half note (minim), three triplet quarter notes total that same duration, so the duration of a triplet quarter note is 2/3 the duration of a standard quarter note. Similarly, three triplet eighth notes (quavers) are equal in duration to one quarter note. If several note values appear under the triplet bracket, they are all affected the same way, reduced to 2/3 their original duration. The triplet indication may also apply to notes of different values, for example a quarter note followed by one eighth note, in which case the quarter note may be regarded as two triplet eighths tied together (Gherkens 1921, 19).
If the notes of the tuplet are beamed together, the bracket (or slur) may be omitted and the number written next to the beam, as shown in the second illustration.
For other tuplets, the number indicates a ratio to the next lower normal value in the prevailing meter. So a quintuplet (quintolet or pentuplet (Cunningham 2007, 111)) indicated with the numeral 5 means that five of the indicated note value total the duration normally occupied by four (or, as a division of a dotted note in compound time, three), equivalent to the second higher note value; for example, five quintuplet eighth notes total the same duration as a half note (or, in 3/8 or compound meters such as 6/8, 9/8, etc. time, a dotted quarter note). Some numbers are used inconsistently: for example septuplets (septolets or septimoles) usually indicate 7 notes in the duration of 4—or in compound meter 7 for 6—but may sometimes be used to mean 7 notes in the duration of 8 (Read 1964, 183–84). To avoid ambiguity, composers sometimes write the ratio explicitly instead of just a single number, as shown in the third illustration; this is also done for cases like 7:11, where the validity of this practice is established by the complexity of the figure. A French alternative is to write pour ("for") or de ("of") in place of the colon, or above the bracketed "irregular" number (Read 1964, 219–21). This reflects the French usage of, for example, "six-pour-quatre" as an alternative name for the sextolet (Damour, Burnett, and Elwart 1838, 79; Hubbard 1924, 480).
There are disagreements about the sextuplet (pronounced with stress on the first syllable, according to Baker 1895, 177) or sestole or sestolet (Baker 1895, 177) or sextole (Baker 1895, 177) or sextolet (Baker 1895, 177; Cooper 1973, 32; Latham 2002; Shedlock 1876, 62, 68, 87, 93; Stainer and Barrett 1876, 395; Taylor 1879–89; Taylor 2001). This six-part division may be regarded either as a triplet with each note divided in half (2 + 2 + 2)—therefore with an accent on the first, third, and fifth notes—or else as an ordinary duple pattern with each note subdivided into triplets (3 + 3) and accented on both the first and fourth notes. Some authorities treat both groupings as equally valid forms (Damour, Burnett, and Elwart 1838, 80; Köhler 1858, 2:52–53; Latham 2002; Marx 1853, 114; Read 1964, 215), while others dispute this, holding the first type to be the "true" (or "real") sextuplet, and the second type to be properly a "double triplet", which should always be written and named as such (Kastner 1838, 94; Riemann 1884, 134–35; Taylor 1879–89, 3:478). Some go so far as to call the latter, when written with a numeral 6, a "false" sextuplet (Baker 1895, 177; Lobe 1881, 36; Shedlock 1876, 62). Still others, on the contrary, define the sextuplet precisely and solely as the double triplet (Stainer and Barrett 1876, 395; Sembos 2006, 86), and a few more, while accepting the distinction, contend that the true sextuplet has no internal subdivisions—only the first note of the group should be accented (Riemann 1884, 134; Taylor 1879–89, 3:478; Taylor 2001).
In compound meter, even-numbered tuplets can indicate that a note value is changed in relation to the dotted version of the next higher note value. Thus, two duplet eighth notes (most often used in 6/8 meter) take the time normally totaled by three eighth notes, equal to a dotted quarter note. Four quadruplet eighth notes would also equal a dotted quarter note. The duplet eighth note is thus exactly the same duration as a dotted eighth note, but the duplet notation is far more common in compound meters (Jones 1974, 20).
In drumming, "quadruplet" refers to one group of three sixteenth-note triplets "with an extra [non-tuplet eighth] note added on to the end", thus filling one beat in 4/4 time (Peckman 2007, 127–28), with four notes of unequal value.
Traditional music notation privileges duple divisions of a steady beat or prevailing time unit. A whole note (semibreve) divides into two half notes, a half note into two quarters, etc. Up to any given tolerance, by tying together sufficiently many notes, purely duple notation can express any time point or duration.
An irrational rhythm occurs when a musical score indicates an exact time point or duration that lies outside the scope of the duple system.
Rather than specifying the new tempo by means of a metronome marking, the prevailing notation indicates the proportional increase or decrease relative to the prevailing tempo. For example, a bracket labeled 5:4 (read five in the space of four) might group together durations (occurring as a sequence of notes and rests) that total to the equivalent of, say, five sixteenth notes. A tempo 5/4 faster than usual then compresses these events into the space of four sixteenth notes. While, in principle, one can increase the pace of any sequence of rhythmic events by 20%, the completion of an irrational rhythm will usually return the count to the duple system. For this to occur with a 5:4 bracket, the total bracketed duration must have a 5 in its numerator, 5/16 in the example. Note that one obtains the actual duration of the bracketed events by dividing two fractions, the notated duration and the indicated tempo increase, (5/16)/(5/4) = 1/4, in this example.
Another variant involves a tempo increase that does not return to the original duple rhythm framework. For example, one might have merely three sixteenth notes grouped by a bracket marked 3 of 5:4.
Tuplets may be counted, most often at extremely slow tempos, using the lowest common multiple (LCM) between the original and tuplet divisions. For example, with a 3-against-2 tuplet (triplets) the LCM is 6. Since 6/2=3 and 6/3=2 the quarter notes fall every three counts (overlined) and the triplets every two (underlined):
1 2 3 4 5 6
This is fairly easily brought up to tempo, and depending on the music may be counted in tempo, while 7-against-4, having an LCM of 28, may be counted at extremely slow tempos but must be played intuitively ("felt out") at tempo:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
The same principle can be applied to quintuplets, septuplets and so on.
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