I. Fundamentals

# 4.1 Introduction

Chapter 2 outlines the different ways in which the underlying pulse of a piece of music (the beat) can be regularly divided. Chapter 3 provides an introductory discussion of the nature of simple meters, those in which the beat is regularly divided into two equal notes. This chapter continues that discussion and explores compound meters, those in which the beat is regularly divided into three equal notes.

This chapter will also provide a brief discussion of tuplets—a type of beat division that strays from the norm for a given meter.

# 4.2 Compound meters

As discussed in Chapter 2, compound meters are characterized by how the beat is regularly divided into three equal durations. Consider the following excerpt. This piece is in a compound duple meter. There are two beats per measure (each equal in length to a dotted quarter note) and each beat is divided into three eighth notes:

Compound meters, like simple meters, are indicated with time signatures. Understanding compound time signatures, however, is not quite as straightforward as the examples seen in Chapter 3. For simple meters, the time signature conveys information about the beat: the top number indicates the number of beats per measure and the bottom number the note value of each beat. With compound meters, on the other hand, time signatures convey information about the beat division.

The feeling of subdividing the beat into three equal durations is the defining characteristic of a compound meter. Try listening to Example 4–2 while quickly counting “1 2 3 4 5 6” with each measure. You may find it difficult to keep up with the music, especially at this brisk tempo. Now try listening again while counting “1 da da 2 da da” with each measure. Notice how musical stresses in each measure line up so naturally with “1” and “2” in this second hearing. This difference explains why this piece would be best described as a duple meter with compound subdivision.

The following example also uses a compound meter:

Note: Chapter 2 discusses how beams can be used to group notes in a way that clarifies the meter. While the right hand of the piano part in Example 4–1 does just this (organizing the eighth notes into groups of three), the notated vocal part in the top staff does not. This is typical of vocal music, where beams are used primarily to group notes that belong to a single syllable of the text.

Take the time signature from the examples above: 6/8. As explained above, this is a compound duple meter. Each beat is equal to a dotted quarter note and each beat is regularly divided into three eighth notes:

As you can see from Example 4–2, the numbers in the time signature refer to the beat division. Each measure of 6/8 has six eighth notes. In other words, to find the number of beats per measure, one must divide the top number by three. If the top number is 6, the meter is duple. If the top number is 9 or 12, the meter is respectively triple or quadruple. These are the most common top numbers for compound-meter time signatures: 6, 9, and 12.

Since the bottom number indicates the duration of the beat division, one must add three of these note values together to get the beat unit. In the case of 6/8, the lower number indicates that the beat division is equal in duration to an eighth note. Three eighth notes add up to one dotted quarter note. The beat unit of a compound meter will always be a dotted note.

Note: A good rule of thumb to follow is that if the top number of a time signature is 2, 3, or 4, the meter is simple. If the top number is 6, 9, or 12 (any multiple of three, greater than three), the meter is compound.

Activity 4-1

Activity 4–1

Identify each of the following time signatures as simple or compound and as duple, triple, or quadruple. Then identify the note value of the beat.

### Question

Does the following time signature represent a simple or compound meter?

#### Hint

Remember, if the top number is 6, 9, or 12, the meter is compound.

compound

### Follow-up question

Is this meter duple, triple, or quadruple?

#### Hint

For compound meters, divide the top number by three to determine the number of beats per measure.

triple

### Follow-up question

What is note value of a single beat?

#### Hint

For compound meters, the beat unit is three times the length of the note value indicated by the bottom number of the time signature.

dotted quarter note

### Question

Does the following time signature represent a simple or compound meter?

#### Hint

Remember, if the top number is 6, 9, or 12, the meter is compound.

simple

### Follow-up question

Is this meter duple, triple, or quadruple?

duple

### Follow-up question

What is note value of a single beat?

half note

### Question

Does the following time signature represent a simple or compound meter?

#### Hint

Remember, if the top number is 6, 9, or 12, the meter is compound.

simple

### Follow-up question

Is this meter duple, triple, or quadruple?

### Follow-up question

What is note value of a single beat?

eighth note

### Question

Does the following time signature represent a simple or compound meter?

#### Hint

Remember, if the top number is 6, 9, or 12, the meter is compound.

simple

### Follow-up question

Is this meter duple, triple, or quadruple?

triple

### Follow-up question

What is note value of a single beat?

eighth note

### Question

Does the following time signature represent a simple or compound meter?

#### Hint

Remember, if the top number is 6, 9, or 12, the meter is compound.

compound

### Follow-up question

Is this meter duple, triple, or quadruple?

#### Hint

For compound meters, divide the top number by three to determine the number of beats per measure.

### Follow-up question

What is note value of a single beat?

#### Hint

For compound meters, the beat unit is three times the length of the note value indicated by the bottom number of the time signature.

dotted eighth note

### Question

Does the following time signature represent a simple or compound meter?

#### Hint

Remember, if the top number is 6, 9, or 12, the meter is compound.

simple

### Follow-up question

Is this meter duple, triple, or quadruple?

### Follow-up question

What is note value of a single beat?

quarter note

# 4.3 Tuplets

You will quite frequently encounter beat divisions that defy your expectations based on your observations about the meter. Consider the following example:

This excerpt is notated in 3/4, a simple triple meter. As the time signature indicates, one beat has a value equal to a quarter note. Since 3/4 is a simple meter, the quarter note beat normally divides into two eighth notes. But, on the second and third beats of m. 6 (and the first beat of m. 7), three eighth notes are squeezed into each beat.

This rhythmic figuration is known as a triplet and is normally notated as it appears in Example 4–4 with a small 3 written above or below a group of beamed notes. Triplets such as these represent a temporary shift to the corresponding compound meter. If 3/4 is a simple triple meter, 9/8 would be the corresponding compound triple meter:

In 3/4 each beat can be divided into two eighth notes. In 9/8 each beat can be divided into three eighth notes. A triplet, therefore, represents a kind of rhythmic borrowing from the corresponding compound meter. In other words, when we hear the triplets in mm. 6–7 of Example 4–3, it sounds as though Haydn has temporarily switched to a compound triple meter, where each beat is divided into three even eighth notes.

The generic term for this type of rhythmic alteration is a tuplet. In addition to triplets, another type of tuplet is a duplet. Duplets typically appear in pieces in compound meters, like the following:

As the name implies, a duplet alters the rhythm so that two notes take up the space that would normally accommodate three. In this sense, a duplet can be thought of as the opposite of a triplet. The example above is in 6/8, a compound duple meter. In this meter a typical beat—a dotted quarter note—would be divided into three eighth notes. In mm. 32–33, however, we see that each beat is divided into two eighth notes (indicated by the bracketed 2 above each group). Each of these duplets fills the space of an entire beat, or, three eighth notes.

Tuplets can be a very versatile tool with regards to the rhythmic expressiveness of a composition and many other types can be found. The two types discussed here, triplets and duplets, are by far the most common.

Activity 4-2

Activity 4–2

### Question

Triplets and duplets can be thought of as beat divisions borrowed from the corresponding simple or compound meter. The following example shows a rhythm in 3/4 (a simple triple meter), which contains some triplets borrowed from 9/8 (the corresponding compound meter). Convert the rhythm to 9/8 by adjusting each rhythm accordingly. The first measure has been done for you. (As you can see, eighth notes become eighth note duplets and triplets become straight eighth notes.)

# 4.4 Summary

Compound meters are those in which the beat regularly divides into three even durations (as opposed to simple meters, which divide into two durations). Like simple meters, compound meters are usually expressed with time signatures, though in such cases the numbers convey information about the beat division as opposed to the beat itself.

The bottom number of a compound-meter time signature indicates the durational value of the beat division and the top number—usually 6, 9, or 12—indicates how many beat divisions make up one full measure. Since the top number refers to the beat division, one need simply divide it by three to determine how many beats appear in a single measure: 6 indicates a duple meter, 9 indicates a triple meter, and 12 indicates a quadruple meter. To determine the note value of the beat, one would add three beat division units together. The beat in a compound meter will therefore always be a dotted note.

Composers are not limited to the normal beat division of a given meter. They may—and often do—borrow the beat division from the corresponding simple or compound meter. These rhythmic figurations are known generically as tuplets. Triplets provide a compound-meter beat division in a simple-meter context (three notes in the space of two). Duplets do just the opposite: they provide a simple-meter beat division in a compound-meter context (two notes in the space of three).