Tuplets
A tuplet is a rhythm whose basis is a value that is not a multiple of a power of 2, or a rhythm that subdivides a beat in a non-standard way with respect to the time signature. Because western rhythmic notation only has dedicated symbols for powers of 2 (whole notes, half notes, quarter notes, eighth notes, etc) tuplets are necessary to express values that are not multiples of powers of 2. For example: 1/12 of a whole note is not a multiple of a power of two, and so a "1/12th note" cannot be expressed using the standard notes with dots or ties. Instead it is notated and named as an "eighth-note triplet". The most common usage of tuplets is as non-standard divisions of beats. For example, in 4/4, 4 and 2 are inherently felt subdivisions, but groupings of 3 and 5 would be considered tuplets. Similarly in 6/8, dotted-quarter-note beats are subdivided in 3 by default, so a grouping of 2 notes per beat could be written as a tuplet. In this case such a grouping could also be written with dotted notes. Tuplets whose numbers are powers of two (duplets, quadruplets, etc) can always be written as dotted notes, but the tuplet form is sometimes preferred in sheet music for readability.
Tuplets generally follow the naming scheme of tuples at large: a tuplet of 5 is often called a quintuplet or a 5let, while a grouping of seven is called a septuplet or 7let. Additionally, some tuplets have their own unique names (which coincide with their polyrhythmic counterparts), such as the hemiola (3:2). There are numerous ways to label tuplets; a more comprehensive list of names can be found on the Wikipedia tuplet page.
A triplet in 4/4 means that 3 beats are subdivided equally in the space of 4 (or 2). A duplet in 3/4 means that 2 beats are subdivided equally in the space of 3. Sometimes, however, the tuplet doesn't fit the inherent subdivision. When this happens, ratio tuplets are used, where the subdivision is presented as a ratio between two numbers. A ratio tuplet of 5:3 in 4/4 would mean 5 beats are played in the space of 3 beats in 4/4. This could also be considered "half" of a 5:3 polyrhythm, where only the 5 is present. Musicians often use polyrhythms to learn how to play complex tuplets effectively because they are fundamentally the same thing.
It should be noted that a tuplet can exist without every beat in the tuplet being played. For example, a drummer could play a quintuplet in 4/4 but only play the 1 and 4. Alternatively, the drummer could play all the beats in the quintuplet but accent the 1 and 4.
Counting Tuplets
Tuplets can be counted at lower tempos by finding the LCM (least common multiple) of the original note grouping and the tuplet subdivision, then dividing. So, for example, a 5:4 tuplet has an LCM of 20. This means subdividing the tuplet into 20 beats will ensure that each note in the tuplet lands on exactly one beat. 20 ÷ 5 = 4, and 20 ÷ 4 = 5, so the quintuplet notes will fall on every 4th beat, while the quarter notes fall of every 5th.
If we mark the quintuplet notes as bold numbers and the quarter notes as in italics, we can map out this tuplet as such (notice that the beats coincide):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Although the LCM method helps find the ideal subdivisions of a tuplet, the math isn't necessary to count simpler tuplets since it's pretty intuitive: just reverse the numbers (5:4 = quintuplets on every 4th beat, quarter notes on every 5th beat). However, the LCM method is still useful for more complex tuplets and nested tuplets.
Notating Tuplets
Tuplets are usually notated by bracketing the group of notes and signifying the tuplet type with a number above or below. Sometimes, in beamed notes, only the number is present and the notes aren't bracketed.
This applies to ratio tuplets as well. However, there are some differences: often, the note value is displayed next to the ratio above the grouping for clarity:
On the right, three triplets are played with varying note lengths: eighth notes, quarter notes, and sixteenth notes. We can also see that the beamed notes aren't bracketed in this case.
Nested Tuplets
When a tuplet is present inside of another tuplet, it is called a nested tuplet.
Here, two eighth-note triplets and a 64th-note quintuplet make up a larger triplet. This can be counted by finding the LCM of the three nested tuplets (the LCM of 3, 3, and 5 is 15), and dividing that by each of the nested tuplets (so 15 ÷ 3 = 5, meaning both of the triplets will fall on every 5th beat, and 15 ÷ 5 = 3, meaning the notes of the quintuplet will fall on every 3rd beat, starting at the one). Let's have the eighth-note triplets be bold, the 64th-note quintuplets be in italics, and the larger triplet notes be in parentheses. When mapped out across 15 subdivisions, that looks like this:
(1) 2 3 4 5 (6) 7 8 9 10 (11) 12 13 14 15
Other Types of Tuplets
Tremolo Tuplets
Sometimes, a tremolo is written as a tuplet to indicate the number of oscillations. Since bracketing is not needed, only the tuplet number is placed above or below.