At times, music notation can be confusing. Even for highly trained musicians, certain aspects of music notation requires them to pause and investigate how a particular rhythmic grouping should be performed. This is particularly true with rhythmic groupings known as tuplets. All tuplets are mathematical ratios. The first number in the ratio is the number of notes in a bracketed group. That number also often (but not always) appears over the bracket.  The second number in the ratio is the number of the same kind of notes which represents the time span over which the tuplet should be completed. Triplets (usually just referred to as triplets, not tuplets), provide an easy example. An eighth note triplet consists of three eight notes bracketed together. Those three eighth notes in the triplet are to be played in the normal time it takes to play two eighth notes that are not part of a triplet. In this case, the ratio would be 3:2. Three eighth notes played in the time span usually taken up by two, that is played in one beat of common time. All triplets have the same ratio—only the note values change. For a quarter note triplet, the three quarter notes in the triplet are played over the time span otherwise occupied by two quarter notes, that is, played in two beats of common time. The kind of note (quarter, half, etc.) stays the same on both sides of the ratio. 

Quintuplets (five notes bracketed together), sextuplets (six notes) and septuplets (seven notes) are all played in the time span of four like notes in common time. The ratios are 5:4 for the quintuplet, 6:4 for the sextuplet and 7:4 for the septuplet. This is the conventional notation for tuples in modern music notation. These ratios can be seen in the example below.

Unfortunately, composers do not always use conventional notation. In these cases, performers must interpret the notation that encounter and adjust their performance to the closest match. For example, an eighth note septuplet should be played over two beats according to its ratio (7:4) because four eighth notes occupy a time span of two beats. But if that eighth note septuplet occupies an entire measure in common time, clearly it must be played over four beats, not two. According to conventional notation, this would be incorrectly notated, but when it appears, the composer leaves no choice but to play it over four beats. A composer might also choose to notate a septuplet to be played over one beat in thirty-second notes because of the rapid speed at which such a septuplet must be played. Thirty-second notes are an accurate representation of how the music sounds, though it is not accurate according to conventional notation, in which it would be notated in sixteenth notes in accordance with its ration of 7:4. (Blatter, 2007) In practice, the musician must read what the allotted time-span for the tuple is in the notation, and evenly space the number of notes in the tuples across that time span.

So far, I have only discussed tuplets in common time. There are also tuplets in triple meter, and they have their own ratios. For example, in three-four meter, two quarter notes bracketed together and filling out the entire measure have a ratio of 2:3—that is, the two quarter notes of the duplet must be played over three beats. A quarter note quintuplet in three-four meter and filling out the entire measure would require five notes to be played over three quarter note beats, a ratio of 5:3. It is even possible to have an eighth note septuplet filling out an entire measure in three-four meter, requiring that seven eighth notes be played in the time span normally assigned to six, that is, in three beats. That ratio would be 7:6. 

With tuplets becoming more frequently used by recently active composers, they have often clarified their intent by including the ratio above the music notation. This leaves no doubt how the rhythmic figure is to be interpreted. In the case of duplets in duple compound meter, two eighth notes might be bracketed together with a “2” over the bracket, or two dotted eighth notes might be notated with no bracket or number above. Both are ways of notating the same thing. 

The occurrence of three durations over the time span normally occupied by two of the same durations is most often called a triplet. A triplet is sometimes called a hemiola usually when its effect is to disrupt the established metric feel. Hemiola originally described the ratio of one and a half to one. We can see how this applies to triplets if we take an eighth note triplet. Three eight notes is one and a half beats in common time, and the three eighth notes of the triplet are performed over one beat. Brahms was fond of using them, and Bernstein famously used hemiolas in his song “America” from West Side Story. “I like to be in A”- is sung with six notes, two beats in six-eight meter, then “-mer-i-ca!” Is also two beats, but with three notes. That’s the hemiola—three notes during two beats, 3:2. 

Regardless of which tuplet occurs, and in which meter, the function of a tuplet is to borrow a rhythm that does not naturally occur in the current meter and impose it on that meter, changing the grouping structure of the musical phrase. Triplets occur in duple meter, and duplets occur in triple meter. Because rhythms are already notated in groups of three in triple meter, triplets are not necessary there, and because rhythms are already notated in groups of two in duple meter, duplets are not necessary there. Tuplets temporarily disrupt the metric structure, and derive their effectiveness from that disruption. If a particular tuplet is repeated at length, it no longer sounds like a tuplet, but instead like a change in meter. 

Conventional notation helps us understand tuplets in many but not all cases. Extrapolating the correct interpretation of a tuplet where conventional notation is not used is done by fitting it into the allotted time span according to the metric context given in the notation. 

Blatter A. (2007). Revisiting music theory : a guide to the practice. Routledge.