Borrowed Divisions

Typically, a meter is defined by the presence of a consistent beat division: division by two in simple meter, and by three in compound meter. Sometimes composers will use a triple division of the beat in a simple meter, or a duple division of the beat in a compound meter; these rhythms are called tuplets. Triplets are a type of tuplet in which a beat (or subdivision, or multiple beats) in simple meter is divided into three parts. Triplets are sometimes thought of as “borrowed” from compound meter, because the beat in compound meter is usually divided into three parts. A triplet is notated by writing the number “3” on top of an appropriate rhythm, as seen in Example 1:

Example 1. Notate a triplet by writing a “3” on top of the appropriate rhythm.

Triplets may occur at both the beat and division levels, as seen below in the second measure of Example 2:

Example 2. A triplet at the subdivision level on the first beat of measure 2.

In the second measure of Example 2 (beat 1), the sixteenth-note triplet is equivalent to one eighth note (the division in this time signature).

Triplets also may occur across multiple beats, as seen in Example 3. In this example, the triplet (notated with three quarter notes) takes up the space of one half note:

Example 3. A triplet which spans two beats.

A duplet is a tuplet in which a beat (or subdivision) in compound meter is divided into two parts. Duplets are notated by writing the number “2” on top of an appropriate rhythm, as seen in Example 4:

Example 4. Notate a duplet by writing a “2” on top of the rhythm.

Like triplets, duplets can occur at the subdivision level, as seen in Example 5:

Example 5. A duplet at the subdivision level on the second beat.

In Example 5 the sixteenth-note duplet is equivalent to one and a half eighth notes (an eighth note is the division in this time signature).

Counting for these rhythms is usually “borrowed” as well. For example, triplets are usually counted 1-la-li, while duplets are usually counted 1-and, 2-and etc.

Meter Beyond Measure (Hypermeter)

We have seen that beats are either accented or non-accented which was observed in the discussion of conducting patterns in the previous two chapters (see Simple Meter and Time Signatures and Compound Meter and Time Signatures). Hypermeter refers to groups of measures that form patterns of accentuation, especially at faster tempos. In order to identify patterns of accentuation across multiple measures, one can place hypermetrical numbers above measures (centered). Example 6 shows an example from the “Scherzo” of Ludwig van Beethoven’s Symphony No. 9 (1824), played on a keyboard:

Example 6. Eight measures of the “Scherzo” of Ludwig van Beethoven’s Symphony No. 9 (1824).

While listening to Example 6, try conducting along to the hypermetrical numbers, in a quadruple pattern. By doing this you will be able to feel (and hear) which measures are more accented (1 and 3) and which are less accented (2 and 4).

Syncopation

Often rhythm and meter work together to create and reinforce a “grid,” which you can feel by conducting along to a work. Syncopation happens when there are off-beat rhythmic accents, which can be created by ties, dots, rests, and/or dynamics. Example 7 shows several different syncopated rhythms with proper counts:

Example 7. Different examples of syncopated rhythms.

In measure 1, the rhythm is syncopated due to ties, while in measure 2 it is syncopated because of both a dot and a tie. In measure 3 the rests create a sense of syncopation, while in measure 4 the syncopation is the result of dynamics (accents). See Rhythm and Meter for more information about syncopation.

• Triplets and Duplets (YouTube)
• Triplets (Hello Music Theory)
• Duplets (Hello Music Theory)
• Hypermeter (John Buccheri)
• Syncopation (Music Theory Academy)
• Syncopated Rhythm Practice (YouTube)

1. Counting Triplets (.pdf), and pp. 32–33 of .pdf)
2. Counting Duplets, p. 13 (.pdf)
3. Hypermetrical Numbers (.pdf)
4. Counting Syncopation (.pdf, .pdf, website, .pdf)

1. Triplets and Duplets, Hypermeter, Syncopation (.pdf, .docx) Worksheet playlist