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 the triplet rhythm. 

Triplets may occur at both any metric level, as seen in Example 1. In Example 1a, the triplet is at the beat level—this is the most common use of a triplet. Example 1b shows a sixteenth-note triplet, which is equivalent to one eighth note (the division in this time signature). Triplets also may occur across multiple beats, as seen in Example 1c. This triplet (notated with three quarter notes) takes up the space of one half note.

Example 1. Triplets at (a) the beat level, (b) the subdivision level, and (c) the multi-beat level.

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 the duplet rhythm, as seen in Example 2.

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

Counting for tuplet 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 label patterns of accentuation across multiple measures, it can be helpful to annotate the score with hypermetrical counts as in Example 3, which shows a reduced excerpt from the “Scherzo” of Ludwig van Beethoven’s Symphony no. 9 (1824). While listening to Example 3, 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).

Example 3. Eight measures of the “Scherzo” of Ludwig van Beethoven’s Symphony no. 9 (1824) with hypermetric counts.

Syncopation

Syncopation happens when there are off-beat rhythmic accents (see Rhythm and Meter in Pop Music for more information), and as Example 4 shows, it can be created by ties, dots, rests, and/or dynamics. In measure 1, the ties create the syncopation, while in measure 2, the rhythm is syncopated because of both a dot and a tie. The sense of syncopation in measure 3 is created by the rests at the beginning of each beat. In measure 4, while there is an eighth note on each division, the off-beat accents (dynamics) produce syncopation.

Example 4. Different examples of syncopated rhythms.

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

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

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