Tuplet

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

Irrational rhythm ( Play (help·info)): triplet above second beat features three rather than the usual two equal divisions of the beat, while the four sixteenth notes 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 ( Play (help·info)), 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 to of "tuplets", actually refers to the simultaneous use of opposing time signatures. (Read 1964, 167)

Triplets

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

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

Whereas normally two quarter notes are the same duration as a half note, 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 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) ( Play (help·info)).

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).

Duplet and quadruplet notated in 6/8  Play (help·info). 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 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).

"Quadruplet" with each note on a different drum in a kit used as a fill (Peckman 2007, 129).  play (help·info)

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 ( Play (help·info)).

Traditional music notation privileges duple divisions of a steady beat or prevailing time unit. A whole note 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.

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

• Schaffel
• List of musical works in unusual time signatures

References

• Baker, Theodore (ed.). 1895. A Dictionary of Musical Terms. New York: G. Schirmer.
• Baker, Theodore, Nicolas Slonimsky, and Laura Dine Kuhn. 1995. Schirmer Pronouncing Pocket Manual of Musical Terms. New York: Schirmer Books. ISBN 0825672236.
• Cooper, Paul. 1973. Perspectives in Music Theory: An Historical-Analytical Approach.. New York: Dodd, Mead. ISBN 0-396-06752-2.
• Cunningham, Michael G. 2007. Technique for Composers. Bloomington, Indiana: AuthorHouse. ISBN 1425996183
• Damour, Antoine, Aimable Burnett, and Élie Elwart. 1838. Études élémentaires de la musique: depuis ses premières notions jusqu'à celles de la composition: divisées en trois parties: Connaissances préliminaires. Méthode de chant. Méthode d’harmonie. Paris: Bureau des Études élémentaires de la musique.
• Donato, Anthony. 1963. Preparing Music Manuscript. Englewood Cliffs, NJ: Prentice-Hall, Inc. Unaltered reprint, Westport, Conn.: Greenwood Press, 1977 ISBN 083719587X.
• Gehrkens, Karl W. 1921. Music Notation and Terminology. New York and Chicago: The A. S. Barnes Company.
• Hubbard, William Lines. 1924. Musical Dictionary, revised and enlarged edition. Toledo: Squire Cooley Co. Reprinted as The American History and Encyclopedia of Music. Whitefish, Montana: Kessinger Publishing, 2005. ISBN 1417902000.
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• Lobe, Johann Christian. 1881. Catechism of Music, new and improved edition, edited and revised from the 20th German edition by John Henry Cornell, translated by Fanny Raymond Ritter. New York: G. Schirmer. (First edition of English translation by Fanny Raymond Ritter. New York: J. Schuberth 1867.)
• Kennedy, Michael. 1994. "Irregular Rhythmic Groupings. (Duplets, Triplets, Quadruplets)". Oxford Dictionary of Music, second edition, associate editor, Joyce Bourne. Oxford and New York: Oxford University Press. ISBN 0198691629.
• Köhler, Louis. 1858. Systematische Lehrmethode für Clavierspiel und Musik: Theoretisch und praktisch, 2 vols.. Leipzig: Breitkopf und Härtel.
• Latham, Alison (ed.). 2002. "Sextuplet [sextolet]". The Oxford Companion to Music. Oxford and New York: Oxford University Press. ISBN 0198662122.
• Marx, Adolf Bernhard. 1853. Universal School of Music, translated from the fifth edition of the original German by August Heinrich Wehrhan. London.
• Peckman, Jon. 2007. Picture Yourself Drumming: Step-by-Step Instruction for Drum Kit Setup, Reading Music, Learning from the Pros, and More. Boston, MA: Thomson Course Technology. ISBN 1598633309
• Read, Gardner. 1964. Music Notation: A Manual of Modern Practice. Boston: Alleyn and Bacon, Inc. Second edition, Boston: Alleyn and Bacon, Inc., 1969., reprinted as A Crescendo Book, New York: Taplinger Pub. Co., 1979. ISBN 0800854594 (cloth), ISBN 0800854535 (pbk).
• Riemann, Hugo. 1884. Musikalische Dynamik und Agogik: Lehrbuch der musikalischen Phrasirung auf Grund einer Revision der Lehre von der musikalischen Metrik und Rhythmik. Hamburg: D. Rahter; St. Petersburg: A. Büttner; Leipzig: Fr. Kistnet.
• Schonbrun, Marc. 2007. The Everything Music Theory Book: A Complete Guide to Taking Your Understanding of Music to the Next Level. The Everything Series. Avon, Mass.: Adams Media. ISBN 1593376529.
• Sembos, Evangelos C. 2006. Principles of Music Theory: A Practical Guide, second edition. Morrisville, NC: Lulu Press, Inc. ISBN 1430309555.
• Shedlock, Emma L. 1876. A Trip to Music-Land: An Allegorical and Pictorial Exposition of the Elements of Music. London, Glasgow, and Edinburgh: Blackie & Son.
• Stainer, John, and William Alexander Barrett. 1876. A Dictionary of Musical Terms. London: Novello, Ewer and Co.
• Taylor, Franklin. 1879–89. "Sextolet". A Dictionary of Music and Musicians (A.D. 1450–1883) by Eminent Writers, English and Foreign, 4 vols, edited by Sir George Grove, 3:478. London: Macmillan and Co.
• Taylor, Franklin. 2001. "Sextolet, Sextuplet." The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.