Tuplet

# Tuplet

Discussion

Encyclopedia
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

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
Rhythm
Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...

that involves dividing the beat
Beat (music)
The beat is the basic unit of time in music, the pulse of the mensural level . In popular use, the beat can refer to a variety of related concepts including: tempo, meter, rhythm and groove...

into a different number of equal subdivisions from that usually permitted by the time-signature
Time signature
The time signature is a notational convention used in Western musical notation to specify how many beats are in each measure and which note value constitutes one beat....

(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

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
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...

", "-uplet" and "-plet", which are used to form terms denoting multiplet
Multiplet
In group theory, a multiplet is formally a group representation of an algebra.In physics, multiplet can mean:* A group of related spectral lines* A group of related subatomic particles...

s (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
Meter (poetry)
In poetry, metre is the basic rhythmic structure of a verse or lines in verse. Many traditional verse forms prescribe a specific verse metre, or a certain set of metres alternating in a particular order. The study of metres and forms of versification is known as 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
Polyrhythm
Polyrhythm is the simultaneous sounding of two or more independent rhythms.Polyrhythm in general is a nonspecific term for the simultaneous occurrence of two or more conflicting rhythms, of which cross-rhythm is a specific and definable subset.—Novotney Polyrhythms can be distinguished from...

" (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.
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....

triole, Fr.
French language
French is a Romance language spoken as a first language in France, the Romandy region in Switzerland, Wallonia and Brussels in Belgium, Monaco, the regions of Quebec and Acadia in Canada, and by various communities elsewhere. Second-language speakers of French are distributed throughout many parts...

triolet, It.
Italian language
Italian is a Romance language spoken mainly in Europe: Italy, Switzerland, San Marino, Vatican City, by minorities in Malta, Monaco, Croatia, Slovenia, France, Libya, Eritrea, and Somalia, and by immigrant communities in the Americas and Australia...

terzina or tripletta, Sp.
Spanish language
Spanish , also known as Castilian , is a Romance language in the Ibero-Romance group that evolved from several languages and dialects in central-northern Iberia around the 9th century and gradually spread with the expansion of the Kingdom of Castile into central and southern Iberia during the...

tresillo), shown at right.
Whereas normally two quarter note
Quarter note
A quarter note or crotchet is a note played for one quarter of the duration of a whole note . Often people will say that a crotchet is one beat, however, this is not always correct, as the beat is indicated by the time signature of the music; a quarter note may or may not be the beat...

s are the same duration as a half note
Half note
In music, a half note or minim is a note played for half the duration of a whole note and twice the duration of a quarter 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
Beam (music)
A beam in musical notation is a thick line frequently used to connect multiple consecutive eighth notes , or notes of shorter value , and occasionally rests...

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
Ratio
In mathematics, a ratio is a relationship between two numbers of the same kind , usually expressed as "a to b" or a:b, sometimes expressed arithmetically as a dimensionless quotient of the two which explicitly indicates how many times the first number contains the second In mathematics, a ratio is...

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 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
Note value
In music notation, a note value indicates the relative duration of a note, using the color or shape of the note head, the presence or absence of a stem, and the presence or absence of flags/beams/hooks/tails....

. Thus, two duplet eighth note
Eighth note
thumb|180px|right|Figure 1. An eighth note with stem facing up, an eighth note with stem facing down, and an eighth rest.thumb|right|180px|Figure 2. Four eighth notes beamed together....

s (most often used in 6/8 meter
Metre (music)
Meter or metre is a term that music has inherited from the rhythmic element of poetry where it means the number of lines in a verse, the number of syllables in each line and the arrangement of those syllables as long or short, accented or unaccented...

) 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.

## Usage and purpose

Tuplets can produce rhythms such as the hemiola
Hemiola
In modern musical parlance, a hemiola is a metrical pattern in which two bars in simple triple time are articulated as if they were three bars in simple duple time...

, or may be used as polyrhythm
Polyrhythm
Polyrhythm is the simultaneous sounding of two or more independent rhythms.Polyrhythm in general is a nonspecific term for the simultaneous occurrence of two or more conflicting rhythms, of which cross-rhythm is a specific and definable subset.—Novotney Polyrhythms can be distinguished from...

s when played against the regular duration. They are extrametric rhythmic units.
Beat (music)
The beat is the basic unit of time in music, the pulse of the mensural level . In popular use, the beat can refer to a variety of related concepts including: tempo, meter, rhythm and groove...

or prevailing time unit. A whole note
Note
In music, the term note has two primary meanings:#A sign used in musical notation to represent the relative duration and pitch of a sound;#A pitched sound itself....

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
Counting (music)
In music, counting is a system of regularly occurring sounds that serve to assist with the performance or audition of music by allowing the easy identification of the beat. Commonly, this involves verbally counting the beats in each measure as they occur...

, 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
To play a half-note (minim) triplet accurately in a bar
Bar (music)
In musical notation, a bar is a segment of time defined by a given number of beats of a given duration. Typically, a piece consists of several bars of the same length, and in modern musical notation the number of beats in each bar is specified at the beginning of the score by the top number of a...

of 4/4, count eighth-note triplets and tie them together in groups of four. With a stress on each target note
Note
In music, the term note has two primary meanings:#A sign used in musical notation to represent the relative duration and pitch of a sound;#A pitched sound itself....

, 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.