Editing Tuplet

{{Short description|Musical note value}}
{{About|the musical term|mathematical concept|Tuple}}
{{Image frame|content=<score sound="1">
\new RhythmicStaff {
   \clef percussion
   \time 4/4
   \set Score.tempoHideNote = ##t \tempo 4 = 80
   c4 \tuplet 3/2 { c8 c c }
   c4 \tuplet 5/4 { c16 c c c c }
   c4
}
</score>|width=300|caption=Rhythm with tuplets: a triplet on the second beat and a quintuplet on the fourth}}

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 (music)|beat]] into a different number of equal [[divisive rhythm|subdivisions]] from that usually permitted by the [[time signature|time-signature]] (e.g., triplets, duplets, etc.)"{{sfn|Humphries|2002|loc=266}} This is indicated by a number, or sometimes two indicating the fraction involved. The [[Musical note|notes]] involved are also often grouped with a bracket or (in older notation) a [[slur (music)|slur]].

The most common type of tuplet is the triplet.

==Terminology==
The modern term 'tuplet' comes from a [[rebracketing]] of compound 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 [[multiplet]]s (''Oxford English Dictionary'', entries "multiplet", "-plet, ''comb. form''", "-let, ''suffix''", and "-et, ''suffix''<sup>1</sup>"). An alternative modern term, "irrational rhythm", was originally borrowed from Greek [[Meter (poetry)|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 [[irrational number|mathematical sense]] (because the note-values are [[rational number|rational]] fractions) or in the more general sense of "unreasonable, utterly illogical, absurd".

Alternative terms found occasionally are "artificial division",{{sfn|Jones|1974|loc=19}} "abnormal divisions",{{sfn|Donato|1963|loc=34}} "irregular rhythm",{{sfn|Read|1964|loc=181}} and "irregular rhythmic groupings".{{sfn|Kennedy|1994}} The term "[[polyrhythm]]" (or "polymeter"), sometimes incorrectly used instead of "tuplets", actually refers to the simultaneous use of opposing time signatures.{{sfn|Read|1964|loc=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.{{sfn|Dunstan|1925|p={{Page needed|date=September 2013}}}} 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.{{sfn|Read|1964|loc=189}}

== Triplet ==

The most common tuplet{{sfn|Schonbrun|2007|loc=8}} is the '''triplet''' (German ''Triole'', French ''triolet'', Italian ''terzina'' or ''tripletta'', Spanish ''tresillo''). Whereas normally two [[quarter note]]s (crotchets) are the same [[duration (music)|duration]] as a [[half note]] (minim), three triplet quarter notes have that same duration, so the duration of a triplet quarter note is {{2/3}} the [[duration (music)|duration]] of a standard quarter note.

:<score sound="1"> \new RhythmicStaff {
   \clef percussion
   \time 2/4
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   c4 c
   \tuplet 3/2 { c4 c c }
}
</score>

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.

:<score sound="1"> \new RhythmicStaff {
   \clef percussion
   \time 2/4
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   c8 c c c
   \tuplet 3/2 { c8 c c } \tuplet 3/2 { c8 c c }
}
</score>

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.{{sfn|Gehrkens|1921|loc=19}}

:<score sound="1"> \new RhythmicStaff {
   \clef percussion
   \time 2/4
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   \tuplet 3/2 { c4 c8 } \tuplet 3/2 { c8 c4 }
}
</score>

In some older scores, rhythms like this would be notated as a [[Dotted note|dotted]] eighth note and a sixteenth note as a kind of shorthand{{sfn|Troeger|2003|loc=172}} presumably so that the beaming more clearly shows the beats.

== Tuplet notation ==

=== Notation ===
Tuplets are typically notated either with a bracket or with a number above or below the [[Beam (music)|beam]] if the notes are beamed together. Sometimes, the tuplet is notated with a [[ratio]] (instead of just a number) — with the first number in the ratio indicating the number of notes in the tuplet and the second number indicating the number of normal notes they have the same duration as — or with a ratio and a note value.

:<score> {
\override Score.TimeSignature #'stencil = ##f
\new RhythmicStaff {
   \clef percussion
   \time 5/4
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   \tuplet 3/2 { c8 c c }
   \once \override TupletBracket.bracket-visibility = ##t \tuplet 3/2 { c8 c c }
   \once \override TupletNumber.text = "3:2" \tuplet 3/2 { c8 c c }
   \once \override TupletNumber.text = "3:2♪" \tuplet 3/2 { c8 c c }
} }
</score>

=== Rhythm ===

==== Simple meter ====
For other tuplets, the number indicates a [[ratio]] to the next ''lower'' normal value in the prevailing meter (a [[Power of two|power of 2]] in [[simple meter]]). So a '''quintuplet''' ('''quintolet''' or '''pentuplet'''{{sfn|Cunningham|2007|loc=111}}) indicated with the numeral 5 means that five of the indicated note value total the [[duration (music)|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 {{music|time|3|8}} or compound meters such as {{music|time|6|8}}, {{music|time|9|8}}, etc. time, a dotted quarter note).

:<score sound="1"> \new RhythmicStaff {
   \clef percussion
   \time 2/4
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   c8 c c c
   \tuplet 5/4 { c8 c c c c }
   c2
}
</score>

Some numbers are used inconsistently: for example '''septuplets''' ('''septolets''' or '''septimoles''') usually indicate 7 notes in the [[duration (music)|duration]] of 4—or in compound meter 7 for 6—but may sometimes be used to mean 7 notes in the duration of 8.{{sfn|Read|1964|loc=183–184}} Thus, a septuplet lasting a whole note can be written with either quarter notes (7:4) or eighth notes (7:8).

:<score> \new RhythmicStaff {
   \clef percussion
   \time 4/4
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   \tuplet 7/4 { c4 c c c c c c }
   \tuplet 7/8 { c8 c c c c c c }
}
</score>

To avoid ambiguity, composers sometimes write the ratio explicitly instead of just a single number. 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.{{sfn|Read|1964|loc=219–221}} This reflects the French usage of, for example, "six-pour-quatre" as an alternative name for the ''sextolet''.{{sfn|Damour, Burnett, and Elwart|1838|loc=79}}{{sfn|Hubbard|1924|loc=480}}

There are disagreements about the '''sextuplet''' (pronounced with stress on the first syllable, according to Baker{{sfn|Baker|1895|loc=177}}—which is also called '''sestole''', '''sestolet''', '''sextole''', or '''sextolet'''.{{sfn|Baker|1895|loc=177}}{{sfn|Cooper|1973|loc=32}}{{sfn|Latham|2002}}{{sfn|Shedlock|1876|loc=62, 68, 87, 93}}{{sfn|Stainer and Barrett|1876|loc=395}}{{sfn|Taylor|1879–1889}}{{sfn|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. This is indicated by the beaming in the example below.

:<score> \new RhythmicStaff {
   \clef percussion
   \time 2/4
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   \tuplet 6/4 { c16 \set stemRightBeamCount = #1 c \set stemLeftBeamCount = #1 c \set stemRightBeamCount = #1 c \set stemLeftBeamCount = #1 c c }
   \tuplet 6/4 { c16 c \set stemRightBeamCount = #1 c \set stemLeftBeamCount = #1 c c c }
}
</score>

Some authorities treat both groupings as equally valid forms,{{sfn|Damour, Burnett, and Elwart|1838|loc=80}}{{sfn|Köhler|1858|loc=2:52–53}}{{sfn|Latham|2002}}{{sfn|Marx|1853|loc=114}}{{sfn|Read|1964|loc=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.{{sfn|Kastner|1838|loc=94}}{{sfn|Riemann|1884|loc=134–135}}{{sfn|Taylor|1879–1889|loc=3:478}} Some go so far as to call the latter, when written with a numeral 6, a "false" sextuplet.{{sfn|Baker|1895|loc=177}}{{sfn|Lobe|1881|loc=36}}{{sfn|Shedlock|1876|loc=62}} Still others, on the contrary, define the sextuplet precisely and solely as the double triplet,{{sfn|Stainer and Barrett|1876|loc=395}}{{sfn|Sembos|2006|loc=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.{{sfn|Riemann|1884|loc=134}}{{sfn|Taylor|1879–1889|loc=3:478}}{{sfn|Taylor|2001}})

==== Compound meter ====
In [[compound meter (music)|compound meter]], even-numbered tuplets can indicate that a note value is changed in relation to the [[dotted note|dotted]] version of the next higher [[note value]]. Thus, two '''duplet''' [[eighth note]]s (most often used in {{music|time|6|8}} [[Metre (music)|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.{{sfn|Jones|1974|loc=20}}

:<score sound="1"> \new RhythmicStaff {
   \clef percussion
   \time 6/8
   \set Score.tempoHideNote = ##t \tempo 4 = 100
   c8 c c c c c
   \tuplet 2/3 { c8 c } c8. c
   \tuplet 4/3 { c8 c c c } c16. c c c
   c2.
}
</score>

A duplet in compound time is more often written as 2:3 (a dotted quarter note split into two duplet eighth notes) than 2:{{frac|1|1|2}} (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 four quadruplet eighth notes).{{sfn|Anon.|1997–2000}}

=== Nested tuplets ===
On occasion, tuplets are used "inside" tuplets. These are referred to as ''nested tuplets''.

:<score sound="1"> \new RhythmicStaff {
   \clef percussion
   \time 2/4
   \set Score.tempoHideNote = ##t \tempo 4 = 68
   \once \override TupletBracket.bracket-visibility = ##t \tuplet 5/4 {c8[ \tuplet 3/2 { c16 c c] } c4 \tuplet 7/4 { c32[ c c c c c c] } } c2
}
</score>

== Counting ==
Tuplets can produce rhythms such as the [[hemiola]] or may be used as [[polyrhythm]]s when played against the regular duration. They are [[extrametric]] [[rhythmic unit]]s. The example below shows sextuplets in quintuplet time.

:<score sound="1" override_midi="Sextuplet against five rhythm.mid">
<< \relative c' {
   \override Staff.StaffSymbol.line-positions = #'(-2 2)
   \clef percussion
   \time 5/4
   \override TupletNumber.text = #tuplet-number::calc-fraction-text
   \tuplet 6/5 { e4 e e e e e } e4
} \\ \relative c' {
   a4 a a a a a
}
>>
</score>

Tuplets may be [[Counting (music)|counted]], most often at extremely slow tempos, using the [[least common multiple]] (LCM) between the original and tuplet divisions. For example, with a 3-against-2 tuplet (triplets) the LCM is 6. Since {{nowrap|1=6 ÷ 2 = 3}} and {{nowrap|1=6 ÷ 3 = 2}} the quarter notes fall every three counts (overlined) and the triplets every two (underlined):

:{| style="width: 1.2in;"
|<span style="text-decoration: overline;"><u>1</u></span>
|2
|<u>3</u>
|<span style="text-decoration: overline;">4</span>
|<u>5</u>
|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:

:{| style="width: 8in;"
|<span style="text-decoration: overline;"><u>1</u></span>
|2
|3
|4
|<u>5</u>
|6
|7
|<span style="text-decoration: overline;">8</span>
|<u>9</u>
|10
|11
|12
|<u>13</u>
|14
|<span style="text-decoration: overline;">15</span>
|16
|<u>17</u>
|18
|19
|20
|<u>21</u>
|<span style="text-decoration: overline;">22</span>
|23
|24
|<u>25</u>
|26
|27
|28
|}
To play a [[half-note]] (minim) triplet accurately in a [[bar (music)|bar]] of {{music|time|4|4}}, count eighth-note triplets and tie them together in groups of four

:<score>
<< \relative c' {
   \override Staff.StaffSymbol.line-positions = #'(-2 2)
   \clef percussion
   \time 4/4
   \tuplet 3/2 { e4.~ e8 e4~ e e8~ e4. } e4
} \\ \relative c' {
   \tuplet 3/2 { a2 a a } a4
}
>>
</score>

With a stress on each target note, one would count: '''1''' – 2 – 3&nbsp;&nbsp;1 – '''2''' – 3&nbsp;&nbsp;1 – 2 – '''3'''&nbsp;&nbsp;1 – 2 – 3&nbsp;&nbsp;'''1'''
The same principle can be applied to quintuplets, septuplets, and so on.

== See also ==
{{Wiktionary|tuplet}}
* [[Composite rhythm]]
* [[Cross-beat]]
* [[Duple and quadruple metre]]
* [[Metre (hymn)]]
* [[Metre (poetry)]]
* [[Metric modulation]]
* [[List of musical works in unusual time signatures]]
* [[Schaffel]]
* [[Sextuple metre]]
* [[Triple metre]]

==References==
{{reflist|18em}}

'''Sources'''
{{div col|colwidth=45em}}
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* {{wikicite|ref={{harvid|Jones|1974}}|reference= Jones, George Thaddeus. 1974. ''Music Theory: The Fundamental Concepts of Tonal Music Including Notation, Terminology, and Harmony''. New York, Hagerstown, San Francisco, London: Barnes & Noble Books. {{ISBN|0-06-460137-4}}.}}
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* {{wikicite|ref={{harvid|Shedlock|1876}}|reference=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.}}
* {{wikicite|ref={{harvid|Stainer and Barrett|1876}}|reference=Stainer, John, and William Alexander Barrett. 1876. ''A Dictionary of Musical Terms''. London: Novello, Ewer and Co.}}
* {{wikicite|ref={{harvid|Taylor|1879–1889}}|reference=Taylor, Franklin. 1879–1889. "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.}}
* {{wikicite|ref={{harvid|Taylor|2001}}|reference=Taylor, Franklin. 2001. "Sextolet, Sextuplet." ''[[The New Grove Dictionary of Music and Musicians]]'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (musicologist)|John Tyrrell]]. London: Macmillan.}}
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{{div col end}}

==Further reading==
* [[Theodore Baker|Baker, Theodore]], [[Nicolas Slonimsky]], and Laura Dine Kuhn. 1995. ''Schirmer Pronouncing Pocket Manual of Musical Terms''. New York: Schirmer Books. {{ISBN|0-8256-7223-6}}.

==External links==
*{{Commons category-inline|Tuplet}}

{{Musical note values}}
{{Rhythm and meter}}

{{DEFAULTSORT:Irrational Rhythm}}
[[Category:Note values]]

[[de:Notenwert#Triole]]