Now that you are familiar with beats and meters in simple and compound meters, we will explore less common rhythms and meters.

7.1   Syncopation

In Chapters 3 and 5, we learned that there can be two, three, or four beats per measure. Although note values differ, the pattern of strong and weak beats remains the same for simple and compound meters.

Example 7.1.1. Weighted beats in simple meters

Example 7.1.2. Weighted beats in compound meters

• In Examples 7.1.1 and 7.1.2, the strongest beats are called downbeats and are indicated by the blue Ss in boxes.
• The next strongest beats differ.
  • In triple meters, beat 2 is weaker than the downbeat but stronger than beat 3.
  • In quadruple meters, beat 3 is weaker than the downbeat but stronger than beats 2 and 4.
• In each example, the last beat is a weak beat.

Sometimes composers intentionally make naturally weaker beats stronger by emphasizing them more. When this happens, it is called syncopation. John Philip Sousa creates syncopations using two different methods in Example 7.1.3.

Example 7.1.3. Syncopation: Sousa,^[1] “Washington Post March”

• The blue beats in the boxes below Example 7.1.3. show syncopations.
• The earliest examples of syncopation come from accent markings. These accents make beat 2 stronger than the downbeat.
• The final example of syncopation involves a longer duration tied into the next downbeat. Since no new sound occurs on the following downbeat, the sound on beat 2 becomes more prominent.

Although Example 7.1.3 is not the beginning of the piece, a time signature has been added for clarity. Since this chapter covers less common rhythms and meters, every example will include a time signature.

Syncopations are not just a single accent. Instead, as in Examples 7.1.3, there must be accents on at least two regular beats in a row. Now look at Example 7.1.4, which shows an accent on a weak beat that is not syncopation.

Example 7.1.4. Not a syncopation: Haydn,^[2] Symphony No. 94, (“Surprise”), ii – Andante

In this piano reduction of Haydn’s “Surprise” Symphony (Example 7.1.4), a seemingly random fortissimo appears after nearly sixteen bars of piano and pianissimo. The music then humorously returns to piano. No matter how strong this beat is, it is not considered a syncopation because there is no sense of regular pattern.

Example 7.1.5 does include syncopations because accents return at least twice.

Example 7.1.5. Syncopations: Guerreiro,^[3] Bôas Festas, Op. 12

Several features contribute to the syncopations in Example 7.1.5.

• Accent marks are on beat 2 of the treble clef in measures 5, 6, and 8.
• The quarter note’s longer duration on the accented notes also emphasizes beat 2.
• The accents happen three times in the four measures of Example 7.1.5. If the accent only occurred once, it would not be considered an example of syncopation.

There can also be syncopations on a smaller scale. While the downbeat is the strongest beat in a measure, any note that falls on the beat is stronger than the other parts of the beat. Notes that do not fall on the beats are called offbeats.

Example 7.1.6. Weighted parts of beats

• Example 7.1.6A:
  • At the beat level, beat 1 (highlighted in a blue box) is strong, and beat 2 is weak.
• Example 7.1.6B: At the division level, beat 1 is stronger than beat 2.
  • However, beat 2 (in a blue circle) is stronger than the final eighth note of beat 2, which is an offbeat.
• Example 7.1.6C: At the subdivision level, beat 1 is stronger than beat 2.
  • However, beat 2 is stronger than the last three sixteenth notes of beat 2.
  • Additionally, the “&” of beat 2 is stronger than the final sixteenth note of beat 2.

Example 7.1.7 illustrates syncopations that occur on the offbeats (weak part of beats), rather than on weak beats.

Example 7.1.7. Offbeat syncopations: Suppé,^[4] Overture to Poet and Peasant

In Example 7.1.7, Suppé’s melody falls on the offbeats for five bars.

• Observe how the melody (shown by the blue arrows) occurs between the beats written above. Though there are no accent marks, the notes land on the offbeats and are sustained. This emphasizes the offbeats more.
• The regular (unsyncopated) rhythm resumes at measures 116 and 118, but it almost seems as if they are syncopated.

Many types of dance music utilize syncopation. In Example 7.1.8, Guerreiro makes use of syncopation in a tango.

Example 7.1.8. Guerreiro, “Saude e…dinheiro” (“Health and…Money”)

• Like Example 7.1.7, Example 7.1.8 does not contain accent markings. Instead, notice where the longer note values of eighth notes appear: in both hands, they often fall between beats.
• The two eighth notes in beat two of the left hand establish the meter and consistently reinforce the simple duple beat. This repeating pattern of offbeat longer notes creates syncopation.

Ragtime, an influencer of jazz, is also characterized by syncopated rhythms (Example 7.1.9).

Example 7.1.9. Ragtime syncopations: Joplin,^[5] “The Easy Winners”

In Example 7.1.9, there are two levels of syncopation. The syncopations covered in this section are marked with asterisks.

• In measure 5, the quarter note falls on the second eighth of beat one, which is the “and” of beat one.
• In measures 6 and 7, the first eighth note falls on the second sixteenth of the beat (i.e. that is, the “ee” of beat one).
• In measure 6, the second eighth note occurs on the second eighth of beat two.
• Across the bar line from measure 7 to measure 8, a note value equal to an entire measure plus a sixteenth note falls on the fourth sixteenth note of beat two (i.e. that is, on the “uh” of beat two).
  • There are two “voices” in this piano example. Notice that in the treble clef, one voice has stems that only point down, while the other voice has stems that only point up. This allows the first voice to be held for the entire measure in measure 8, while the second voice begins an anacrusis into the next measure.

Three types of syncopation occur in Example 7.1.9 (syncopations on “and,” “ee,” and “uh”). Joplin makes the syncopations sound natural and not rhythmically disjointed by maintaining constant eighth notes in the left hand, reinforcing the simple duple meter.

Syncopations can also happen in compound meters (Example 7.1.10).

Example 7.1.10. Offbeat syncopations: Röntgen-Maier,^[6] Violin Sonata in B Minor, i – Allegro

• Example 7.1.10 is in 6/8, which is a compound meter.
• The violin plays the rhythm of the division (i.e. that is, three eighth notes). However, instead of the beat being accented, the last (third) note of the division (an offbeat) is marked with a sforzando symbol, creating syncopations. These syncopations are labeled with asterisks.
• To support the syncopations in the violin, the piano also has sforzandi on the last eighth note offbeats.

Directions:

• Circle all syncopations in the following examples.
• How does the composer place emphasis on the syncopations?

1. de Gardéev,^[7] Rumänisches Charakterstück, Op. 44

2. Szymanowska,^[8] Prelude No. 1

3. Gabriel,^[9] “Only”

7.2   Borrowed Division: Triplets

We learned that simple meters have beats that are evenly divided into two, while compound meters have dotted beats that are evenly divided into three. Sometimes, a simple meter will divide its beat into three equal parts, and a compound meter will divide its beat into two even parts. When this occurs, it is called borrowed division (Example 7.2.1).

Example 7.2.1. Borrowed division

Borrowed division can occur in simple or compound meters. When a beat is not divided into its usual parts, it is called a tuplet.

When you divide a beat (without a dot) into three equal parts, the tuplet is called a triplet. The most important thing to remember when writing triplets is that you must include the number “3.” Otherwise, you are simply writing three notes.

Remember that a triplet is a type of borrowed division. This means the note value of the division remains the same. In simple meters, instead of the beat splitting evenly into two, a triplet uses the same note value as the division but features three notes instead of two (Example 7.2.2).

Example 7.2.2. Triplet at the beat level

• Example 7.2.2A:
  • The beat is a half note.
  • The division consists of two quarter notes. Therefore, the borrowed division is three quarter-note triplets.
  • When there is no beam, add a bracket with the number “3.” Write the “3” on the stem side (opposite the notehead).
• Example 7.2.2B:
  • The beat is a quarter note.
  • The division consists of two eighth notes. Therefore, the borrowed division is three eighth-note triplets.
  • The stems point downward, so the number “3” is added to the bottom (opposite the notehead). Since there is a beam, an extra bracket is not necessary.
• Example 7.2.2C:
  • The beat is an eighth note.
  • The division consists of two sixteenth notes. Therefore, the borrowed division is three sixteenth-note triplets.
  • Because sixteenth notes have two flags, they are connected by two beams. The number “3” is positioned on the stem side, opposite the notehead. A bracket is not needed.

Triplets are simple to recognize because they typically include the number “3” (Example 7.2.3).

Example 7.2.3. Triplets: Rossini,^[10] The Thieving Magpie, Overture

• In 3/4, the quarter note receives one beat, and the division is two eighth notes. Beats are marked with the blue square brackets.
• The division is labeled in the second violins and also appears in the cellos and double basses, with an eighth note and an eighth rest.
• Triplets appear in the first violins and violas, playing an octave apart. The division of three, usually seen in compound meters, is used in this simple meter as triplets.

You may come across triplets in your music that don’t have “3” written above or below. While this is acceptable in published scores, we will always include the “3” when writing out triplets (see Example 7.2.4).

Example 7.2.4. Implied triplets: Le Beau,^[11] Theme and Variations, Op. 3, No. 7

In Example 7.2.4, the first and third beats of each measure have triplets, even though the “3” is not written. Musicians should understand that because of the beaming, triplets (three eighth notes beamed together) occur on the first and third beats, while the division (two eighth notes beamed together) occurs on the second and fourth beats.

Any note (not a dotted note) that divides evenly into three parts forms a triplet. This includes notes that are part of the division (not the beat) (Example 7.2.5).

Example 7.2.5. Triplets at the division level

• Example 7.2.7A: Each eighth note in the division is equally divided into two sixteenth notes in the subdivision.
• Example 7.2.7B: The second eighth note in the division is equally divided into three sixteenth notes, resulting in a triplet.
  • This triplet is not equal to the beat; instead, it is equal to one of the divisions.

Example 7.2.6 illustrates a triplet dividing a division (rather than the beat).

Example 7.2.6. Triplets: Chopin,^[12] Polonaise, Op. 40, No. 1

• In 3/4, the quarter note gets one beat, and the division is two eighth notes. Beats are marked with the blue square brackets.
• In both examples with square brackets, the second eighth note is divided into a sixteenth-note triplet.
  • The sixteenth-note triplet is equivalent to two sixteenth notes, which equals one eighth note.
  • Combined with the first eighth note, the eighth note and sixteenth-note triplets complete one beat.

Triplets can also appear in compound meters, but never to divide the beat since the beat is a dotted note and already divides evenly into three. In compound meters, triplets will take place within the division of the beat (Example 7.2.7).^[13]

Example 7.2.7. Triplets: de la Hye,^[14] Le corsaire rouge

• In 6/8, the dotted quarter note receives one beat, and the division is three eighth notes.
• On the first beat of measure 13, the division of three eighth notes is clear.
• On the second beat of measure 13, the first eighth note is divided into sixteenth-note triplets.

Although 6/8 is a compound meter, we see triplets in Example 7.2.7. However, the triplets are not divisions of the beat (which are dotted notes), but rather divisions of the division (which are not dotted notes).

Triplets can also be worth more than one beat (Example 7.2.8).

Example 7.2.8. Triplets of longer values than one beat.

In Example 7.2.8, the quarter note equals one beat, meaning the half note equals two beats. When the half note is divided into three equal quarter notes instead of two, it becomes a quarter-note triplet worth two beats. The triplet from Example 7.2.8 is shown within the context of Example 7.2.9.

Example 7.2.9. Triplet: Smyth,^[15] Fête Galante: A Dance-Dream in One Act

• In 2/4, a half note fills an entire measure and can be divided into two equal quarter notes, as shown in measure 2.
• Just as two quarter notes fill a measure, three quarter-note triplets also fill a measure (measure 3).

Example 7.2.10 illustrates another triplet that spans an entire measure.

Ex. 7.2.10. Triplet: Villa-Lobos,^[16] A prólo do bébé, “Moreninha”

• In 4/4, a whole note fills an entire measure and can be divided into two equal half notes.
• Because two half notes fill a measure, three half-note triplets also fill an entire measure.

Whether the triplet is equal to a beat, half a beat, or multiple beats, always use the same method when writing triplets: Find what would divide the value into two and apply that note value to a triplet.

Calculating Triplets

Students are often intimidated by triplets. However, as long as you understand beat divisions, we can approach it step by step.

For example, if a quarter note equals one beat, how many beats are the following triplets in Example 7.2.11 worth?

Example 7.2.11. Calculating triplets

Let’s start by breaking down a quarter note beat. If a quarter note equals one, its division is two eighth notes (Example 7.2.12).

Example 7.2.12. Quarter note beat

If a quarter note is divided into two eighth notes, then the borrowed division consists of three eighth notes, with the number “3” added (Example 7.2.13).

Example 7.2.13. Borrowed division

Since three eighth note triplets equal one beat, the following statements are also true:

• A quarter note is twice the duration of an eighth note. Therefore, quarter note triplets are worth two beats.
• A sixteenth note is half the duration of an eighth note. Therefore, sixteenth note triplets are worth half a beat.

Example 7.2.14 shows the solution for Example 7.2.11.

Example 7.2.14. Calculation of Triplets

When calculating the value of triplets, always begin with the basic division of the beat. For beats that correspond to other note values, such as an eighth note or a half note, follow the same steps.

What if you are asked to write triplets? Examples 7.2.15 and 7.2.16 demonstrate how to add triplets in different time signatures.

Example 7.2.15. Adding a triplet where a quarter note equals one beat

• Example 7.2.15A: The instructions ask to add a triplet in beat two.
• Example 7.2.15B: First, identify the beat’s division. A quarter note equals two eighth notes.
• Example 7.2.15C: Because the division is two eighth notes, the triplet would consist of three eighth notes. Since the three eighth notes are already beamed together, you only need to add the number “3” on the stem side.
• Example 7.2.15D: This example is incorrect because the number “3” is missing. As a result, there are too many beats: There are 3½ beats in 3/4.

Let’s try an exercise where a quarter note does not equal one beat (Example 7.2.16).

Example 7.2.16. Adding a triplet where a half note equals one beat

• Example 7.2.16A: The instructions ask to add a triplet in beat two.
• Example 7.2.16B: First, identify the beat’s division. A half note equals two quarter notes.
• Example 7.2.16C: Because the division is two quarter notes, the triplet would be three quarter notes. Since the three quarter notes are not beamed together, you need to add a bracket on the stem side and add the number “3.”
• Example 7.2.16D: This example is incorrect because the number “3” is missing. As a result, there are too many beats: 2½ beats in cut time.

Whether you are interpreting or writing triplets, always start with the beat’s division. From there, you can figure out how to interpret and write triplets.

Directions:

• Write the borrowed division.

Directions:

• Based on the given note that equals one beat, write how many beats or how much of one beat the following triplets are worth.

Directions:

• Identify all triplets and write how many beats or how much of a beat each triplet is worth.
  • Remember that for published scores, the number “3” can be left out for triplets.

1. Macfarren,^[17] “Bonnie Scotland”

2. Mayer,^[18] Valse, Op. 32

3. Virgil,^[19] Barchetta, Op. 23, No. 1

7.3   Other Divisions of the Beat

In the last section, we learned about the triplet, which is when a note (not a dotted note) divides equally into three parts. As we will see in this section, beats can be divided into any number of notes.

Duplets

We learned that dotted notes divide evenly into three. When you split a dotted note into two equal parts, this tuplet is called a duplet. The key point when writing duplets is to include the number “2.” Otherwise, you are simply writing two notes (see Example 7.3.1).

Example 7.3.1. Duplet

• Example 7.3.1A: The dotted half note receives the beat.
  • The division is three quarter notes. Therefore, the borrowed division is two quarter-note duplets.
  • When there is no beam, add a bracket with the number “2.” Write the “2” on the stem side (opposite the notehead).
• Example 7.3.1B: The dotted quarter note receives the beat.
  • The division is three eighth notes. Therefore, the borrowed division is two eighth-note duplets.
  • The stems point upward, so the number “2” is placed at the top (opposite the notehead). Since there is a beam, an additional bracket is not needed.
• Example 7.3.1C: The dotted eighth note receives the beat.
  • The division is three sixteenth notes. Therefore, the borrowed division is two sixteenth-note duplets.
  • Because sixteenth notes have two flags, they are connected with two beams. The number “2” appears on the stem side opposite the notehead. An extra bracket is not needed.

Duplets are less common than triplets but still appear in the repertoire (Example 7.3.2).

Example 7.3.2. Duplets: Debussy,^[20], Suite bergamasque “Clair de lune”

• In 9/8, the dotted quarter note receives one beat, and the division is three eighth notes. Beats are labeled with the blue square brackets.
• The first beat of measure 3 clearly shows the division of three eighth notes.
• The next two beats of measure 3 are duplets, with two eighth notes evenly dividing the dotted quarter note beat.

Now, let’s figure out how to calculate and write duplets.

Calculating Duplets

Students are often more intimidated by duplets than triplets. Hopefully, after practicing a few exercises step by step, you won’t feel intimidated.

If a dotted half note equals one beat, how many beats are the following duplets worth (Example 7.3.3)?

Example 7.3.3. Calculating duplets

Let’s begin by breaking down a dotted half note beat. If a dotted half note equals one, its division is three quarter notes (Example 7.3.4).

Example 7.3.4. Dotted half note beat

If a dotted half note is divided into three quarter notes, then the borrowed division consists of two quarter notes with the number “2” added (Example 7.3.5).

Example 7.3.5. Borrowed division

Since two quarter note duplets equal one beat, the following statements are also true:

• An eighth note is half the duration of a quarter note. Therefore, eighth note duplets are worth half a beat.
• A half note is twice the duration of a quarter note. Therefore, half note duplets are worth two beats.

Example 7.3.6 shows the solution for Example 7.3.3.

Example 7.3.6. Calculation of duplets

Just like when you calculated the value of triplets, always begin with the basic division of the beat for duplets.

Writing duplets is quite straightforward if you remember two steps:

1. Write the division of the beat (i.e., the three notes that equally divide the beat).
2. Change the division of three to a division of two, and add the number “2” on the stem side. If there is no beam, add a bracket.

Let’s practice an exercise that requires adding a duplet (Example 7.3.7).

Example 7.3.7. Adding a duplet where a dotted quarter note equals one beat

• Example 7.3.7A: Directions ask to add a duplet in beat 2.
• Example 7.3.7B: First, identify the beat’s division. A dotted quarter note equals three eighth notes.
• Example 7.3.7C: Because the division is three eighth notes, the duplet consists of two eighth notes. Since the two eighth notes are already beamed together, you only need to add the number “2” on the stem side.
• Example 7.3.7D: This example is incorrect because the number “2” is missing. As a result, there are not enough beats because the measure is missing one eighth note.

Polyrhythm

Sometimes composers use a tuplet in one part while maintaining a regular division (or subdivision) in another part. This phenomenon is called polyrhythm or cross rhythm (Example 7.3.8).

Example 7.3.8. Polyrhythm: Rogers,^[21] Violin Sonata in D Minor, op. 25, iii – Allegro giojoso

• In 6/4, the dotted half note receives one beat, and the division is three quarter notes. Beats are labeled with blue square brackets.
• The piano performs the typical division of three quarter notes per beat.
• At the same time, in measure 90, the violin plays duplets (two quarter notes equal one beat).
• Since these rhythms happen simultaneously, it creates a polyrhythm.
• Notice that both the violin and the piano parts still have two beats per measure. Only the divisions differ.

Polyrhythms can extend beyond just one or two measures. In fact, when the polyrhythm lasts throughout an entire movement, composers might simply change the time signature, as Bach does in Example 7.3.9.

Example 7.3.9. Polyrhythm with different time signatures: Bach,^[22] Herz und Mund und Tat und Leben, (Heart and Mouth and Deed and Life), BWV 147, “Jesus bleibet meine Freunde” (“Jesus Shall Remain My Joy”)

• The melody appears in the oboes and first violins in a compound triple meter (9/8).
• At the same time, the other instruments have a time signature of a simple triple meter (3/4).
• Instead of writing triplets each time, Bach simply changed the time signature of the oboes and first violins to 9/8.
• This is an example of polyrhythm because the compound triple and simple triple time signatures happen at the same time. Both meters have three beats per measure; however, their divisions differ.

In Example 7.3.10, Smyth uses polyrhythm and combines two duplets to form a quadruplet, or a division of four.

Example 7.3.6. Duplets and quadruplets: Smyth, String Quartet in E Minor, i – Allegretto lirico

• The second violin and viola firmly uphold the simple duple meter of ¢ƒ. In addition to the division of three, each beat is accented.
• On the third eighth note where the second violin and viola have rests, the cello enters with a syncopated eighth note tied to a quarter note to fill the meter’s division of three.
• The first violin plays a duplet in measures 226 and 229. The two eighth notes evenly divide the beat of the dotted quarter note.
• The quadruplet in Example 7.3.6 was not a subdivision of the beat; rather, the quadruplet in bar 228 merged two duplets in a row, where each duplet is a borrowed division of the beat. However, music can also use borrowed subdivisions (Example 7.3.7).

The quadruplet in Example 7.3.10 was not a subdivision of the beat; rather, the quadruplet in bar 228 merged two duplets in a row, where each duplet is a borrowed division of the beat. However, music can also use borrowed subdivisions (Example 7.3.11).

Example 7.3.11. Sextuplet: Schubert,^[23] “Ave Maria”

• • In common time, the quarter note gets one beat, and the subdivision consists of four sixteenth notes.
  • The treble clef of the piano accompaniment is divided equally into six sixteenth notes, creating sextuplets. Since the subdivision of six resembles the subdivision of a dotted quarter note beat, the sextuplet is an example of borrowed subdivision.
  • In the voice part, the sixteenth notes suggest a consistent subdivision of four, which leads to the occurrence of polyrhythm.

Alternative Divisions of the Beat

In addition to borrowed divisions and subdivisions, beats can be divided into any number (within reason). In Example 7.3.12, Massenet divides the quarter note beat into three and five parts.

Example 7.3.12. Quintuplet: Massanet,^[24] Méditation from Thaïs

• In common time, the quarter note gets one beat, the division is two eighth notes, and the subdivision is four sixteenth notes.
• In the first bar, the violin plays an eighth-note triplet, where three eighth notes make up one beat.
• In measure 3, five sixteenth notes equal one beat: this is a quintuplet.

A quintuplet is not a type of borrowed division, like three eighth notes, or a type of borrowed subdivision, like six sixteenth notes, but is instead a kind of tuplet. Placing five sixteenth notes into one beat represents a different division than the usual division (two eighth notes) or subdivision (four sixteenth notes). Since a quintuplet of five is closer to the four-sixteenth-note division, the note values used for the quintuplet are sixteenth notes instead of eighth notes.

A beat can be divided into any number of notes. Although each type of tuplet has a specific name (for example, a beat divided into nine is called a nonuplet), we will only use the most common divisions.

• Triplet: borrowed division in simple meters
• Duplet: borrowed division in compound meters
• Sextuplet: borrowed subdivision in simple meters
• Quadruplet: borrowed subdivision in compound meters
• Quintuplet: beat divided equally into five parts
• Septuplet: beat divided equally into seven parts

What type of note value does Le Beau use in Example 7.3.13? As a division of seven, these tuplets are septuplets. 

Example 7.3.3. Septuplets: Le Beau, Prelude, Op. 12, No. 1

Le Beau uses sixty-fourth notes. We can approach it step by step to understand why this note value is correct. The septuplet fills in half a beat.

• There is one quarter note per beat or one eighth note every half beat.
• Two sixteenth notes equal one eighth note, or half a beat.
• Four thirty-second notes are equivalent to two sixteenth notes, or half a beat.
• Eight sixty-fourth notes equal four thirty-second notes, or half a beat.

Since eight sixty-fourth notes are closest to the division of seven, the note values used are sixty-fourth notes.

The next example (Example 7.3.14) illustrates how extreme other divisions can be.

Example 7.3.14. Other tuplets: Szymanowska, Fantasie for the Piano

• In measure 2, the value of a sixteenth note is divided equally into twelve parts.
• In measure 3, the value of a sixteenth note is divided equally into eleven parts.

Although a division of 11 can be called an “undecuplet” and a division of 12 can be called a “dodecuplet,” for numbers greater than eight, it is more common to simply refer to these larger tuplets as the number of divisions followed by the word “tuplet.” In Example 7.3.14, the first tuplet is more commonly called an “11-tuplet,” and the second tuplet is called a “12-tuplet.”

As you can see, beats (or even part of a beat) can be evenly divided into any number of parts (within reason).

Directions:

• Write the borrowed division.

Directions:

• Based on the given note that equals one beat, write how many beats or how much of one beat the following tuplets (duplets and others) are worth.

Directions:

• Identify all tuplets. Write how many beats or how much of a beat each tuplet is worth.

1. Rothschild,^[25] “Les Papillons”

2. Savage,^[26] Six Easy Lessons for the Harpsichord or Pianoforte, Sonata No. 5, i – Moderato

3. Milanollo-Parmentier,^[27] Lamento, Op. 7

4. Grandval^[28], Mazeppa, Act 2, Scene 6

7.4   Metric Events

Although syncopations and polyrhythm deviated from the norm, the pulse remained steady. In this section, we encounter metric phenomena where accents do not occur as expected.

Hemiola

The time signature of 3/4 is an example of a triple meter: there are three beats per measure, and the downbeat has the strongest accent (Example 7.4.1).

Example 7.4.1. Typical triple meter

Example 7.4.1 shows that there are three beats in each measure, with the accented downbeats enclosed in boxes.

Sometimes composers temporarily shift from three beats per measure to two beats per measure, creating a hemiola. Although the accents change, it is important that the underlying pulse of the beat remains consistent (Example 7.4.2).

Example 7.4.2 Hemiola

• In measure 1, there are three beats per measure, and the downbeat (in blue boxes) receives the strongest accent.
• In measures 2-3, accents occur every two beats (shown in blue) instead of every three beats. This creates a hemiola.
• In measure 4, the usual three-beat accent pattern resumes.

Example 7.4.3 shows how Example 7.4.2 sounds.

Example 7.4.3. How a hemiola sounds

Examples 7.4.2 and 7.4.3 sound the same. The quarter note beat remains steady throughout. The hemiola becomes evident when the time signature and bar lines change. Mozart writes a hemiola in Example 7.4.4.

Example 7.4.4. Hemiola in Mozart,^[29] Piano Sonata No. 12 in F Major, K. 332, i – Allegro

• In measures 62-63, the triple meter is clear, with Mozart marking the accented downbeats as forte. Beats two and three then immediately drop to piano.
• On the third beat of measure 64, the pattern is interrupted by a sudden forte on beat three.
• Two beats later, on the second beat of measure 65, the next forte occurs.
• In measure 66, the original triple meter returns.

In Example 7.4.4, the triple meter briefly shifts to ôµ, producing a hemiola. The contrast in dynamics emphasizes the hemiola even further.

When a hemiola occurs alongside the written meter, it creates a polymeter. While a polyrhythm maintains the beat with different divisions happening simultaneously, a polymeter occurs when two meters happen simultaneously. Look at the following melody by Johann Strauss in Example 7.4.5.

Example 7.4.5. Two beats per measure: Strauss,^[30] Frülingsstimmen Waltz, Op. 410

The melody in Example 7.4.5 clearly shows two beats per measure with the downbeat accented. However, this is not what Strauss notates. Strauss composed this duple melody over a triple-meter accompaniment (Example 7.4.6).

Example 7.4.6. Polymeter with hemiola: Strauss, Frülingsstimmen (Voices of Spring) Waltz, op. 410

• The right hand plays the melody from Example 7.4.5, which implies a duple meter (accents every two beats).
• At the same time, the left hand plays an accompaniment in triple meter, with accents every three beats.

There is polymeter in Example 7.4.6 because the right hand has accents every two beats in measures 3-4, while the left hand has accents every three beats.

Johannes Brahms is well known for his use of hemiolas, polyrhythm, and polymeter. It is important to establish a strong sense of meter before applying any of these elements. In Example 7.4.7, a regular compound duple meter is set at the beginning.

Example 7.4.7. Established meter: Brahms,^[31] Intermezzo, Op. 119, No. 3

The time signature of 6/8 is a compound duple meter: there are two beats per measure, and each beat is divided into three equal parts. The meter is clearly established at the start. Later in the Intermezzo, Brahms shifts the meter (Example 7.4.8).

Example 7.4.8. Shift of meter: Brahms, Intermezzo, Op. 119, No. 3

• Measure 45 resembles the beginning of the piece, implying a compound duple meter.
• On the second beat of measure 47, the meter suddenly shifts to what sounds like 3/4. While the eighth note division remains unchanged, they are now grouped into two instead of three.
• The second half of measure 48 features a sixteenth-note quintuplet and an eighth note.

The hemiola at measure 47 sounds especially abrupt because the “downbeat” falls on beat 2. The change from compound to simple begins on the second beat of 6/8, which is technically an offbeat (the “and” of beat two) in 3/4. However, closer examination shows that Brahms prepares us for this metric shift (Example 7.4.9).

Example 7.4.9. Hemiola preparation: Brahms, Intermezzo, Op. 119, No. 3

Focus on the second bar of both Example 7.4.9A and B. In Example 7.4.9A, several factors strongly establish the compound duple meter.

• The melody, which is in the lower part of the piano’s right hand (see the notes with stems pointed down in the treble clef), is felt in groups of three. These groups are either sets of three or a quarter note paired with an eighth note, creating the swinging feel often associated with compound meters.
• The left hand’s accompaniment clearly divides into two beats per measure. In the first three bars, the two beats are separated melodically by a low note and the same note two octaves higher. For example, C₂ falls on the downbeat of the opening measure, and C₄ marks the second beat. The music changes direction at that point. The compound duple division becomes more apparent in the last two bars, as three-note ascending patterns are distinguished by beaming.

In Example 7.4.9B, the first two bars appear to be the same as Example 7.4.9A, but a closer look reveals an important change in the left hand.

• In measure 46, the left-hand pattern is now broken into clear groups of two:
  • B♭B flat1 to B♭B flat2
  • C3 to C4
  • C3 to B♭B flat2
• Pairing the left hand in groups of two now makes us reconsider the right hand’s melody. Is it divided into two, as in measure 2, or into three? Compare the two interpretations in Example 7.4.10, where the beaming has been changed to illustrate the patterns.

Example 7.4.10. Simple or compound?: Brahms, Intermezzo, Op. 119, No. 3

With the repetitions on G, it is likely that one could see measure 46 as Example 7.4.10B. Now, the left-hand pairings of two are more convincing. Moreover, the hemiola in measures 47-48 is not as surprising, since Brahms has prepared us for it.

In the Brahms intermezzo, metrical accents are uncertain. Our sense of meter is temporarily lost. However, this is best achieved by first preparing the listener with a clear pattern of accented beats from a regular meter.

Alternative Meters

In addition to the traditional time signatures we have learned, there are also irregular meters. Irregular meters include any time signatures that are not duple, triple, or quadruple meters, such as 5/4 (Example 7.4.11).

Example 7.4.11. Irregular meter: Holst,^[32] The Planets, Op. 32, “Mars, the Bringer of War”

• Holst begins this movement from The Planets in 5/4, an irregular meter.
• The time signature is more closely related to a simple meter because the beat is a quarter note (not a dotted note) that divides evenly into two. When the quarter note beat is divided into three, the number “3” is used to indicate borrowed division.

Irregular meters feature time signatures where the top number is not typical, such as “5” or “7.” Additionally, irregular meters can have time signatures that indicate how beats are divided within each measure, known as composite meters or asymmetrical meters. Composite meters combine simple divisions of two and compound divisions of three, while the divisions remain constant. This results in beats that are not symmetrical, as shown in Example 7.4.12.

Example 7.4.12. Composite meter: Reicha,^[33] Fugue, Op. 36, No. 20

• The time signature is a composite meter. It shows that each measure is divided into three eighth notes, followed by two eighth notes.
• Notice how the beaming matches the time signature. In measures 2 and 4, the first three eighth notes are beamed together, while in measures 1 and 3, the last two eighth notes are beamed together. The beaming significantly influences how one interprets the music.

Whereas Example 7.4.11 in 5/4 was more like a simple meter, Example 7.4.12 resembles a compound meter more because the bottom number indicates the division rather than the beat. There are two beats per measure in Example 7.4.12, but the beats are uneven: the music sounds like it is limping, with the first beat longer than the second by an eighth note.

Example 7.4.13 illustrates another instance of a composite meter. Listen to how the beaming influences the accents of the beats.

Example 7.4.13. Composite meter: Bartók^[34], Six Dances in Bulgarian Rhythm, No. 2

• The composite meter maintains a steady eighth note, but the accents vary.
  • In the first two beats, every other eighth note is accented.
  • In the third beat, the first of the three eighth notes is accented.
• The beats (accents) are clear with the beaming: 2 + 2 + 3.

Time signatures can also vary throughout a work, called changing meters (Example 7.4.14).

Example 7.4.14. Changing meters: Mussorgsky,^[35] Pictures at an Exhibition, Promenade

In the opening of Mussorgsky’s Pictures at an Exhibition, he alternates between 5/4 and 6/4. The quarter note remains constant, and the time signature changes. The bottom number of a changing time signature does not need to stay the same, as in Example 7.4.15.

Example 7.4.15. Changing meters: Clarke,^[36] Sonata for Viola and Piano, iii – Adagio

In the Clarke viola sonata, two unexpected metric surprises emerge.

• In the first system, there are two bars of 1/4. We have not encountered a time signature with a top number of one. These bars are in simple meter because the quarter note gets one beat. Notice how Clarke uses these measures of simple single to repeat the patterns from the music in common time, but higher at each appearance.
• In the second system, the time signatures change from common time to 2/4 to 3/2. The bottom number of the time signature changes from 4 to 2. Often, when this happens, the composer will let the musician know whether the pulse or the beat stays the same. Although Clark does not specify, we can see from the violist’s part that the pulse remains unchanged. Indeed, the viola’s part in the entire second system repeats the same six notes six times.

If the violist’s music stays the same for three bars, why does Clarke change the time signature three times? Listen to the example—do you notice how the music builds tension and reaches a climax at measure 191? Clarke uses several techniques to show the increase in excitement.

• As previously mentioned, each occurrence of 1/4 results in a higher restatement of the music in common time. The higher pitch adds to the excitement. (Imagine how your voice sounds when you’re not impressed with something compared to when you’re extremely excited about it.)
• Additionally, each occurrence begins piano and then crescendos. The marking ‘sub. animando’ means to suddenly become more animated or lively.
• The final appearance in common time does not lead to 1/4, but to 2/4 instead. Whenever a pattern is broken, it contributes to more excitement.

Clarke uses multiple compositional techniques that highlight the climax at measure 191.

• The previous crescendos culminate in a grand forte.
• The pianist plays five octaves of As, from A₁ to A₅.
• The durational broadening of the piano’s part also enhances the climax. While faster rhythms suggest excitement and progress, broadening often signifies a moment of arrival.

Although the time signatures used may appear random, we can see that each one was carefully selected and contributed to the composer’s overall vision.

Directions:

• Describe what is occurring metrically in the following examples.

1. Brahms, Horn Trio, Op. 40, iv – Finale: Allegro con brio

2. Tchaikovsky^[37], Symphony No. 6, Op. 74, ii – Allegro con grazia

3. Villa-Lobos, A prólo do bébé, “O polichinelo”

7.5   Analysis: Brahms, Schicksalslied

As previously mentioned, Brahms is well known for his use of hemiolas. Observe the dramatic use of a hemiola in Example 7.5.

Example 7.5. Hemiola: Brahms, Schicksalslied, Op. 54

Listen to Example 7.5 while clapping the beat. Accent all the beats that are boxed.

• The first six measures (mm. 140-145) have three beats per measure. The straightforward triple meter is clearly established.
• In measures 146-152, a hemiola occurs as the vocalists’ parts alternate between a note and a rest, creating two-beat “measures.”
• In measure 154, the triple meter resumes after a full measure of rest in measure 153.

The words and the music go hand in hand in Example 7.5.

• When the music is in a triple meter (measures 140-145), the choir sings, “blindly from one brief hour to another.”
• Then, during the hemiola (measures 146-152), the choir sings, “Like water from boulder to boulder flung downward.” Listen to the example: Do you hear how the music sounds like water bouncing from one stone to the next?

Brahms uses word painting, where music tries to mimic the meaning of the words. He takes advantage of the hemiola to create a vivid musical picture of water skipping from rock to rock.

Terms

• asymmetrical meters
• augmentation
• borrowed division
• changing meter
• composite meter
• diminution
• duplet
• hemiola
• irregular meter
• offbeat
• polymeter
• polyrhythm (also cross rhythm)
• quadruplet
• quintuplet
• septuplet
• sextuplet
• syncopation
• triplet
• tuplet