Tuplet

This article is about the note groupings. For mathematical grouping, see tuple.

Irrational rhythm (<phonos file="Irrational rhythm.mid">Play</phonos>): triplet above second beat features three rather than the usual two equal divisions of the beat, while the four sixteenth notes (semiquavers) above the third beat are rational, four being a multiple of two

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".

Terminology

Sextuplet (<phonos file="Sextuplet rhythm.mid">Play</phonos>), or six notes. As the extra brackets show: six notes in the time of four = three notes in the time of two X 2

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".

"True sextuplet": in order to contrast with the above "false sextuplet", the 1st, 3rd, and 5th notes of a sextuplet must be stressed rather than the 1st and 4th (Baker, Slonimsky, and Kuhn 1995, 208).

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).

Besides "triplet", the terms "duplet", "quadruplet", "quintuplet", "sextuplet", "septuplet", and "octuplet" are used frequently. The terms "nonuplet", "decuplet", "undecuplet", "dodecuplet", and "tredecuplet" had been suggested but up until 1925 had not caught on (Dunstan 1925,). By 1964 the terms "nonuplet" and "decuplet" were usual, while subdivisions by greater numbers were more commonly described as "group of eleven notes", "group of twelve notes", and so on (Read 1964, 189).

Triplets

The most common tuplet (Schonbrun 2007, 8) is the triplet (Ger. Triole, Fr. triolet, It. terzina or tripletta, Sp. tresillo), shown at right.

File:Tuplets.png

Tuplet: a standard triplet; a triplet denoted without a bracket; a tuplet denoted as a ratio

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).

Tuplet notation

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.

Septuplet rhythm: seven against four (more frequent) and seven against eight (sometimes found) (<phonos file="Septuplet rhythm.mid">Play</phonos>).

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). Thus, a septuplet lasting a whole note can be written with either quarter notes (7:4) or eighth notes (7:8). 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)—which is also called sestole, sestolet, sextole, 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).

Duplet and quadruplet notated in 6/8 <phonos file="Duplet and quadruplet.mid">Play</phonos>. Two duplets or four quadruplets equal three regular eighth notes or a dotted quarter note.

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 (or quartole) 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). A duplet in compound time is more often written as 2:3 (a dotted quarter note split into two duplet eighth notes) than 2:1.5 (a dotted quarter note split into two duplet quarter notes), even though the former is inconsistent with a quadruplet also being written as 4:3 (a dotted quarter note split into two quadruplet eighth notes) (Anon. 1997–2000).

File:Quadruplet.png

"Quadruplet" with each note on a different drum in a kit used as a fill (Peckman 2007, 129). <phonos file="Quadruplet.mid">play</phonos>

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.

Usage and purpose

Tuplets can produce rhythms such as the hemiola, or may be used as polyrhythms when played against the regular duration. They are extrametric rhythmic units.

Sextuplet in quintuple time: six against five (<phonos file="Sextuplet against five rhythm.mid">Play</phonos>).

Traditional music notation favors duple divisions of a steady beat or time unit. A whole note (semibreve) divides into two half notes, a half note into two quarters, etc. and other notes are made by tying these together.

An irrational rhythm (by definition) is one that uses exact time points or durations that lie outside the scope of the duple system.

The n-tuplet notation shows the proportional increase or decrease of tempo needed for the bracketed notes, relative to the prevailing tempo. For example, a bracket labeled "5:4" (read five in the space of four) could group together durations (notes or rests) with a total of five sixteenth notes. A tempo 5/4 faster than usual then compresses these events into the space of four sixteenth notes.

The actual duration can be found by dividing the notated duration by the indicated tempo increase ((5/16)/(5/4) = 1/4, in this example).

Normally, the total duration of the bracketed notes is chosen to be exactly equal to the duration of one of the duple divisions. For the example of a 5:4 bracket, this is possible if the total bracketed duration has a 5 in its numerator, 5/16 in the example.

Sometimes though that requirement is dropped to create total durations not exactly expressible in the duple system. For example, one might have only three of the usual five sixteenth notes grouped by a bracket marked "3 of 5:4".

Counting

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

Four eighth-note triplets = one half-note triplet.

To play a half-note (minim) triplet accurately in a bar of 4/4, count eighth-note triplets and tie them together in groups of four. With a stress on each target note, one would count:

1-2-3 / 1-2-3 / 1-2-3 / 1-2-3

The same principle can be applied to quintuplets, septuplets, and so on.

See also

• Cross-rhythm
• Metric modulation
• Schaffel
• List of musical works in unusual time signatures

References

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