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Many composers in the twentieth century expanded the traditional rhythmic palette of Western art music. Below, nine less traditional rhythmic techniques are discussed. These rhythmic techniques were not necessarily “new;” many of them were found in non-Western music or even Western music before 1900. However, Western composers did incorporate them more frequently after this year.

Asymmetrical Meter

Music in asymmetrical meters contain measures which are divided into unequal groupings of beats or divisions, creating an uneven metrical pulse. It is important to note that the term “asymmetrical meter” is not universally adopted, and such meters are known by other names, such as “mixed meters,” “complex meters,” and “non-isochronous meters,” among others. Listen to Example 1, whose meter is [latex]\begin{smallmatrix}5\\8\end{smallmatrix}[/latex]. Notice the division of the two beats; one is grouped into three eighth notes, and the other into two eighth notes. This creates a feeling of a long pulse followed by a short pulse:

Example 1. Asymmetrical meter in Stephen Sondheim’s “Ladies In Their Sensitivities” (1979).

This song is found in the musical Sweeny Todd (1979) whose music is by Stephen Sondheim. The off-kilter feeling of the [latex]\begin{smallmatrix}5\\8\end{smallmatrix}[/latex] meter reinforces the anxious thought process of the singer; he is a servant, recommending his boss for a shave, and is hesitant to do so because of the difference in his social position.

Now listen to Example 2, which is also in [latex]\begin{smallmatrix}5\\8\end{smallmatrix}[/latex] meter. Notice how the division of the two beats is grouped in the opposite manner of the last example, with the first beat divided into two eighth notes, and the second beat into three eighth notes:

Example 2. Asymmetrical meter in Rolf Rudin’s “Aufbruch” (2008), measures 141–149.

The [latex]\begin{smallmatrix}5\\8\end{smallmatrix}[/latex] measures juxtapose the more metrically typical [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex] measures, creating a call and response texture between the woodwinds and brass. The character of the woodwinds is much more lively and playful as compared to the staunch and powerful brass.

There are other asymmetrical time signatures whose beat division unit is the eighth note, such as [latex]\begin{smallmatrix}7\\8\end{smallmatrix}[/latex], [latex]\begin{smallmatrix}11\\8\end{smallmatrix}[/latex], [latex]\begin{smallmatrix}13\\8\end{smallmatrix}[/latex], etc. Usually we can identify the groupings of the meter’s subdivisions by either how the notes are beamed, or via explicit directions written in the music (e.g. a [latex]\begin{smallmatrix}7\\8\end{smallmatrix}[/latex] meter signature may be written as [latex]\begin{smallmatrix}(2+2+3)\\8\end{smallmatrix}[/latex], [latex]\begin{smallmatrix}(3+2+2)\\8\end{smallmatrix}[/latex], or another way).

Now listen to Example 3, which is in the time signature [latex]\begin{smallmatrix}13\\8\end{smallmatrix}[/latex] (3+3+3+2+2), beginning at 0:17. Try counting along with the example!

Example 3. Asymmetrical meter in Andrew Lloyd Webber’s “Skinbleshanks: The Railway Cat.”

This song is found in the musical Cats (1981), with music written by Andrew Lloyd Webber. The [latex]\begin{smallmatrix}13\\8\end{smallmatrix}[/latex] section could have been written in a prototypical [latex]\begin{smallmatrix}12\\8\end{smallmatrix}[/latex] meter; however, adding an extra eighth note reinforces the lively nature of the character by not allowing the listener to fall into a regular rhythmic pattern.

Asymmetrical meters can have other beat units, such as the sixteenth note or quarter note. Some examples you may encounter are [latex]\begin{smallmatrix}5\\4\end{smallmatrix}[/latex], [latex]\begin{smallmatrix}7\\16\end{smallmatrix}[/latex], or [latex]\begin{smallmatrix}5\\2\end{smallmatrix}[/latex]. The beats of these time signatures are still divided or grouped unequally, though their beat unit changes. Observe Example 4, which is first written in [latex]\begin{smallmatrix}7\\4\end{smallmatrix}[/latex] and rewritten in [latex]\begin{smallmatrix}7\\8\end{smallmatrix}[/latex]:

Example 4. Asymmetrical meter in “Unsquare Dance” (1961) written by the Dave Brubeck Quartet.

This is an excerpt of the “Unsquare Dance” (1961) by the Dave Brubeck Quartet. A typical square dance is in a quadruple meter; however, this “unsquare dance” is in an asymmetrical meter, which diverts the listener’s expectations. Notice how the rhythm can be written in both [latex]\begin{smallmatrix}7\\4\end{smallmatrix}[/latex] and [latex]\begin{smallmatrix}7\\8\end{smallmatrix}[/latex] meters, while still sounding the same.

Ametric Music

Music written ametrically does not have any perceivable meter. Ametric music may or may not have an explicit meter signature, and is not necessarily played or sung in a strict metrical style. In other words, the notes may still have rhythmic values, but rhythmic durations might not result in a perceivable beat pattern. This technique may allow for a performer’s own improvisation and phrasing, and is employed in some cadenzas, passages of free improvisation, Gregorian chant, and post-tonal music. Listen to Example 5, a post-tonal song that does not have a time signature:

Example 5. Ameter in Charles Ives’s “The Cage” (c. 1904).

“The Cage” (circa 1904) is a song written by American composer Charles Ives. The ametric nature of the song supports the anxious behavior of the leopard who is pacing his enclosure. Without a strict metrical pulse and explicit meter signature, the rhythm of the piano is erratic, which showcases the leopard’s temperament from his long-term confinement.

Perceived vs. Notated meter

In some cases, the written meter may not be the meter that the listener perceives; furthermore, different listeners could hear different meters in a given passage of music. Musicians call this is an issue of perceived versus notated meter. Sometimes, this results in a perception of ameter, or music without a perceivable beat, such as in example 6:

Example 6. Perceived versus notated meter in measures 1–8 of Edgard Varèse’s “Density 21.5” (1936).

Though “Density 21.5” (1936) uses a written common time signature, the listener perceives ameter due to various syncopations, tuplet rhythms, and obscured bar lines.

However, in some cases, the notated meter is different than the aurally perceived meter, as seen in Example 7:

Example 7. Notated meter different than the aural perception of a work.

In this excerpt, the time signature of [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex] is implied, as seen by the dotted barlines. This is not written however, as the excerpt is notated in [latex]\begin{smallmatrix}4\\4\end{smallmatrix}[/latex]. Additionally, the pattern of accents and the harmonic resolutions of the phrase further imply a [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex] meter.

Changing Meter

Changing meter is a method of composition which incorporates any change of meter. There are no limits or rules to changing the meter, and it can happen multiple times within a piece, even measure to measure. Listen to Example 8, which features several metric changes from [latex]\begin{smallmatrix}2\\4\end{smallmatrix}[/latex] to [latex]\begin{smallmatrix}5\\8\end{smallmatrix}[/latex] to [latex]\begin{smallmatrix}6\\8\end{smallmatrix}[/latex] to [latex]\begin{smallmatrix}5\\8\end{smallmatrix}[/latex] to [latex]\begin{smallmatrix}6\\8\end{smallmatrix}[/latex] to [latex]\begin{smallmatrix}3\\8\end{smallmatrix}[/latex]:

Example 8. Changing meter in measures 54–60 of Tui St. George Tucker’s “Libera Me” from Requiem (1995).

This is measures 54–60 of Tui St. George Tucker’s “Libera Me” from her Requiem (1995). The “Libera Me” text fearfully depicts judgement day: “Deliver me, O Lord, from death eternal in that awful day,” proceeds the line of text from this excerpt “When the heavens are shaken…” Tucker paints this chaos musically in part through changing meter, showing how the “heavens are shaken” through the this metric instability.

It is also important to note that the subdivision of the meter can change from simple to compound or vice versa, as seen in Example 9:

Example 9. Changing meter in measures 10–19 of Andreas Markis’s “Agean Festival Overture” (1967).

This is a condensed score of measures 10–19 of Andreas Markis’ Aegean Festival Overture (1967). It is common for Greek folk music to incorporate changing meter, and Markis evokes this folk style by including this compositional technique.

Changing meter doesn’t always include a constant subdivision. Instead, it can have a constant beat and a changing subdivision. This is explored in more detail below (see Metric Modulation).

Polymeter

Polymeter occurs when two or more meters are performed simultaneously. This technique can be heard in many modernist works—those created after ~1900—and the meters can be explicit or implicit. Explicit metrical notation means that two or more meters are actually written, while implicit polymeter is only implied. Example 10 demonstrates this concept explicitly:

Example 10. Polymeter in Béla Bartók’s “Ara táskor” (Harvest Song) (1931).

Example 10 shows “Ara táskor” (“Harvest Song”) (1931) written by Béla Bartók for two violins. Notice that Bartók uses more than one meter simultaneously beginning in m. 11. Bartók was heavily inspired by folk music, and he chose to write in polymeter in order to imitate this style.

Now observe Example 11, which shows implicit polymeter:

Example 11. Polymeter in Maurice Ravel’s “String Quartet in F Major,” Second Movement (1902).

In the second movement his of “String Quartet in F Major” (1902), Ravel utilizes a [latex]\begin{smallmatrix}6\\8\end{smallmatrix}[/latex] time signature, while also implying a melody in [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex]. In the first seven measures of this piece, the first violin and cello are implied to be in a [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex] time signature, whereas the second violin and viola switch every other measure between [latex]\begin{smallmatrix}6\\8\end{smallmatrix}[/latex] and an implied [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex]. Beginning in m. 8, the lower two instruments play in the written time signature, whereas the upper two instruments imply [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex].

Another way of thinking about ” implicit polymeter” is as a type of grouping dissonance. For a more in-depth discussion, read the chapter on Metrical Dissonance.

Metric Modulation

Metric modulation is a means of smoothing out abrupt tempo changes by introducing subdivisions or groups of beats in the first tempo that match durations in the new tempo.^[1] In this way, the change is near indeterminable by the listener, and is only recognized in retrospect. Metric modulations are most commonly notated with a “note value = note value” (for example, 𝅘𝅥  = 𝅘𝅥  ) indication above the music. Observe Example 12 and the metric modulation which occurs in m. 4:

Example 12. Metric modulation in which the division of the beat is kept the same.

Notice how the eighth note division stays the same between the [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex] and [latex]\begin{smallmatrix}9\\8\end{smallmatrix}[/latex] meters. Keeping the subdivision the same is one technique of metric modulation.

Now observe Example 13:

Example 13. Metric modulation in which the beat is kept the same.

Notice how the quarter note beat of the first meter becomes the dotted quarter note beat of the new meter. In this case, the eighth note division becomes faster.

Now observe metric modulations in the context of the composition “Canaries” written by Elliott Carter in example 14:

Example 14. Metric modulation in measures 1–24 of Elliott Carter’s “Canaries” from “8 Pieces for 4 Timpani” (1949).

Elliott Carter is well-known for his metric modulations. In this example, six measures from the end, Carter sets up the transition to the [latex]\begin{smallmatrix}3\\8\end{smallmatrix}[/latex] meter by adding triplets into the prior [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex] measure. This creates the effect of having the [latex]\begin{smallmatrix}3\\8\end{smallmatrix}[/latex] measure seem like an extension of the previous triplet rather than a meter change. By doing this, Carter creates a seamless transition between the two meters. Carter also uses the [latex]\begin{smallmatrix}3\\8\end{smallmatrix}[/latex] measure as a transition to the next [latex]\begin{smallmatrix}3\\4\end{smallmatrix}[/latex] measure by keeping the eighth note the same, again making the transition between the two meters quite seamless.

Timeline Notation

Timeline notation is a contemporary metric technique that uses seconds as the measure of time, rather than traditional bar lines and meters. Composers indicate groups of seconds in uneven groupings or an even grid. This technique often results in a feeling of ameter. Timeline notation can also be accompanied by graphic notation, in which pitch and durations are specified by nonstandard symbols, as seen in Example 15:

Example 15. Timeline notation in Krzysztof Penderecki’s “Threnody for the Victims of Hiroshima” (1960).

Threnody for the Victims of Hiroshima (1960) was written by Krzystof Penderecki. It was originally called 8’37”; later, it was renamed to Threnody, and was eventually dedicated to the victims of the bombing of Hiroshima in 1964. This composition features timeline notation, graphic notation, and extended string techniques to create a jarring chaos that evokes programmatic sounds such as sirens or screams.

Example 16 shows the timeline notation from one of the systems on the ninth page of Threnody:

Example 16. Timeline notation from “Threnody for the Victims of Hiroshima,” page 9.

Notice how the number of seconds is split unevenly on this timeline. First there is a section that is 10 seconds in length, then one that is 7 seconds, followed by two more that are 10 seconds and 5 seconds respectively.

Now observe timeline notation in an even grid, as seen in Example 17:

Example 17. Timeline notation in John Cage’s “Water Walk for Solo Television Performer” (1959).

Water Walk for Solo Television Performer (1959) is a composition written by John Cage that was premiered on the television program “Lascia o Raddoppia” in Milan in 1959. The composition calls for 34 different objects, as well as a prerecorded single track tape. Most of the materials relate to water in some way; some objects include a bath tub, rubber duck, ice cubes, etc. Additionally, Cage also calls for five radios and a grand piano.

Example 18 shows the timeline notation from one of the systems on the first page of Water Walk:

Example 18. Timeline notation from “Water Walk for Solo Television Performer” (1959), page 1.

Notice how the number of seconds is evenly split on this timeline, increasing by five seconds in each interval.

Feathered Notes

Feathered beaming notates a gradual change in the speed of notes within a single beam. One can distinguish deceleration from acceleration based upon whether the value of the beam’s final note is longer or shorter than the note that the beam began with. Feathered notes are shown in Example 19:

Example 19. Feathered notes.

In Example 19, the first line shows an acceleration while the second line shows a deceleration. Feathered notes are considered to be an extended technique, most often heard in contemporary compositions, sometimes including cadenzas and ametric works. Due to the unpredictable nature of the acceleration or deceleration of the rhythm, feathered notes are not usually found in strictly metered compositions.

Ostinato

An ostinato is a repeated rhythmic or pitched musical idea. This repetition can be a single measure or multiple measures. Now listen to Example 20, one of the most well-known examples of an ostinato:

Example 20. Ostinati in measures 1–16 of Gustav Holst’s “Mars, the Bringer of War” (1918).

“Mars the Bringer of War” (1918) is the fourth movement of Gustav Holst’s suite The Planets, a programmatic work where each movement is named for a planet. The underlying ostinato in this movement creates both a foreboding and militaristic atmosphere, supporting the war-like nature of the of the Roman god Mars for whom the planet is named.

• Asymmetrical Meter (lcsproductions.net)
• Introduction to Asymmetrical Meters (Ash Stemke)
• Compound and Asymmetric Meter (learnmusictheory.net)
• Creating Interesting Motion with Time Signature Changes (David Bawiec)
• Advanced Rhythm and Meter/Percieved vs. Notated Meter (Frank Koonce)
• Songs that use Polymeter and Polyrhythm (David Bennett)
• Intro to Metric Modulation (Philippe Macnab-Seguin)
• Metric Modulation (Beyond Music Theory)
• Feathered Notes (Steinberg.org)
• Ostinato (Music Theory Academy)

• Advanced Rhythm and Meter Aural Skills (.pdf)

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