XI. Rhythm and Meter
Metrical Dissonance
Mark Gotham and Megan Lavengood
Key Takeaways
- Metrical dissonance refers to the coexistence of two or more unaligned metrical layers in a single passage of music.
- Metrical dissonance can be divided into two types:
- Displacement dissonance, in which two or more layers with the same meter (or grouping structure) are displaced against each other
- Grouping dissonance, in which two or more layers with unique grouping structures (that are not simple multiples/factors of each other) can be heard simultaneously.
- Grouping dissonance often occurs with an ostinato—a repeating musical pattern within a context that is not similarly repeating.
“Dissonance” is one of those multi-purpose terms with a range of uses both within and beyond musical analysis. Outside of music, terms such as cultural or cognitive dissonance usually have negative connotations, but music has a way of making a virtue of the “wrong” or “ill-fitting.” Musical dissonance is not “bad” in the same sense as some of these other contexts (though stylistic norms may require proper handling and resolution, at least in the case of tonal counterpoint).
These same notions of consonance and dissonance have been applied to meter.[1] The modern conception of metrical dissonance was galvanized by Harald Krebs (1999).[2] This chapter is based on Krebs’s theory, and categorizes metrical dissonance as either displacement dissonance or grouping dissonance, each of which is discussed further below.
Displacement Dissonance
Displacement dissonance occurs when one layer of the music seems to be out-of-sync with the surrounding context. displacement dissonance: when two layers imply the same meter, but one is offset from the other by a consistent rhythmic value. This rhythmic value is called the displacement index; so, the displacement index of Example 1b is +1♪.
uses a familiar tune to illustrate: in , the tune is in sync with the bass line; in , it is an eighth note “late.” then is an example of displacement dissonance. Note that the final note of arrives properly in sync with the bass; this kind of adjustment is common in real pieces as well, especially to create a sense of conclusion or arrival (e.g., at cadences). This is a simple example of
Listening to
and , one would likely default to beginning these passages on beat 1 of a [latex]\mathbf{^3_4}[/latex] measure, creating four bars of [latex]\mathbf{^3_4}[/latex]. In the actual piece these lines are drawn from—Marie Jaëll’s Impromptu—the two lines occur simultaneously, but the melody is offset from the bass line, moved forward by one eighth note (again, a displacement index of +1♪).
B section (at 1:45 and 2:36, to the lyrics “you just call on me, brother, when you need a hand”) of Bill Withers’s iconic “Lean on me” (1972). In the A section of this song, the meter and grouping are clear and set the listener’s metric expectations. In the B section, the hand claps create a strong sense of displacement dissonance. Hand claps are typically a snare-drum substitute in pop drumbeats, playing the backbeat on beats 2 and 4 of a quadruple meter. But here, the handclaps are on the 2nd and 6th eighth notes of the bar, suggesting a backbeat that occurs an eighth note early (−1♪).[3] The song’s overarching meter may never be in doubt, but the rhythmic-metrical richness of this passage can be helpfully understood in terms of the systematic syncopations that potentially point to alternative displacements of the meter.
provides another example from the
Grouping dissonance
Grouping dissonance is the combination of two layers with different internal grouping organization; more specifically, the two layers articulate groupings with different numbers of pulses (pulse cardinalities), and these cardinalities are not multiples/factors of one another. These layers can be referred to in terms of their cardinalities to distinguish them: for example, a “3-layer” and a “4-layer.” provides simple illustrations of grouping dissonance.
The juxtaposition of 2- and 3-layers as in hemiola. The hemiola is a grouping dissonance found even in early Western concert music, such as G.F. Handel’s Water Music Suite No. 2, mvt. II (“Alla hornpipe”), shown in . This sort of hemiola is often found at important cadences of pieces in triple meter, where two measures of triple meter (a 3-layer) can be interpreted as three groups of two beats (a 2-layer) instead. This dissonance heightens excitement and reinforces the sense of tension-and-release created by the familiar cadential harmonic progression.
is common enough to be known by a special name: the
Three-against-four (3:4) is another widespread rhythmic-metrical pattern that can be readily heard in terms of grouping dissonance. tresillo rhythm can be viewed as a partial cycle of the 3-against-4 dissonance ( ).
sets out a complete cycle of this grouping dissonance, but partial cycles are more common in musical practice, especially in pop music—from classic jazz to electronic dance music (EDM). The Afro-Cubanfour-on-the-floor kick drum and handclap on the backbeat, but the synth and bass riffs creates metrical dissonance out a conflicting pattern based on groupings of three sixteenth notes. Note that the 3-layer breaks off with a 2-pulse-long grouping so that the riff has a downbeat arrival on every other measure.
shows a partial 3:4 grouping dissonance (sixteenth-note pulse) from Calvin Harris’s “I’m Not Alone” (2009). The song’s overall simple quadruple meter is clear from the
Grouping dissonances and polyrhythms can often be easier to understand in box notation rather than traditional Western notation, which prioritizes metrical hierarchies in a way that might not be helpful when dealing with grouping dissonance.
is a box notation diagram illustrating “I’m Not Alone.” In Western notation, we are forced to choose between unusual dotted rhythms or very frequent ties, both of which have advantages and disadvantages, but in box notation, this issue is entirely circumvented. The box notation also clarifies the mathematical properties of grouping dissonances. Any grouping dissonance will have a point of realignment, where both layers have a simultaneous attack point. This happens after x number of pulses, where x is the lowest common multiple between the cardinalities of the two layers. 12 is the lowest common multiple between 3 and 4, so a 3:4 grouping dissonance realigns every twelve pulses.Grouping dissonance via ostinati
A common musical device in many sorts of music is the ostinato: a rhythmic motive that repeats within a context that is not similarly repeating. If the length of the ostinato does not align with the prevailing meter, the ostinato can create a sense of grouping dissonance. In these cases, the rhythms on their own may not indicate a grouping dissonance, but combined with the pitch content of the ostinato, a grouping dissonance becomes audible.
Igor Stravinsky, a particular lover of metrical dissonance and other kinds of rhythmic surprise, creates grouping dissonance with an ostinato in the “Alleluia, laudate dominum” from Symphony of Psalms (conduct the [latex]\mathbf{^3_2}[/latex] meter of the melody; the second, conduct [latex]\mathbf{^4_2}[/latex] with the bass line.
). While the half notes that form the bass line of this excerpt are not themselves in conflict with the [latex]\mathbf{^3_2}[/latex] meter, the repeating pattern of four notes (E♭–B♭–F–B♭) suggests a four-beat grouping that does not align with the three-beat grouping of the melody and other lines. Listen to the example twice: the first time,
Changing Dissonance
Some music gets into a groove and sticks with it, but many do not: a piece may use metrical dissonance in some places and not others, creating a sense of tension-and-release. Consider the Bill Withers example above: the metrical dissonance in the B section contrasts with the A section that rhythmically very straight and without dissonance.
Let’s close with an example of this technique of changing dissonance in Brahms well-known “Lullaby” (“Wiegenlied”, Op. 49, No. 4).[4] The rhythm of the vocal melody is highly regular in the notated [latex]\mathbf{^3_4}[/latex]. Now have a look at the piano accompaniment, shown in . The left hand (bass) is every bit as regular as the voice, but the right hand (treble) goes its own way:
-
- We start with a rhythmic pattern suggesting three-eighth-note-long groupings that might fit better in [latex]\mathbf{^6_8}[/latex] rather than [latex]\mathbf{^3_4}[/latex], displaced by an eighth. (So before the voice even enters we arguably have both displacement and grouping dissonance!)
- The grouping dissonance disappears almost immediately and shifts to displacement dissonance: we settle on a two eighth note grouping structure, but displacement by an eighth. This remains for the majority of the verse.
- The opening double-dissonance briefly returns, before giving way to…
- …the right hand (finally) playing a note on the downbeat! The dotted-quarter on beat 2 may still suggests displacement dissonance, but of a milder kind: displacement forward by a quarter note, to emphasize the second beat instead of the first (like a sarabande).
- Finally, a measure later, we have all 6 eighth notes in the bar in a strong-weak alternation—no sense of dissonance at all.
- The last four-bar pattern repeats.
This reading (which you may, of course, disagree with) describes a process of (slight) tension to release that is entirely in keeping with the mood of this lullaby.
- Gotham, Mark. 2022. “Towards a Cognitively Based Quantification of Metrical Dissonance.” In The Oxford Handbook of Time in Music, edited by Mark Doffman, Emily Payne, and Toby Young. Oxford University Press. https://doi.org/10.1093/oxfordhb/9780190947279.013.12.
- Krebs, Harald. 1999. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. New York: Oxford University Press.
- Grouping Dissonance Composition Exercise (.pdf, .mscz). Asks students to compose two mini pieces with repeating motives of different lengths, creating grouping dissonance. References the pentatonic and hexatonic collections.
Key Takeaways
- Metrical dissonance refers to the coexistence of two or more unaligned metrical layers in a single passage of music.
- Metrical dissonance can be divided into two types:
- Displacement dissonance, in which two or more layers with the same meter (or grouping structure) are displaced against each other
- Grouping dissonance, in which two or more layers with unique grouping structures (that are not simple multiples/factors of each other) can be heard simultaneously.
- Grouping dissonance often occurs with an ostinato—a repeating musical pattern within a context that is not similarly repeating.
“Dissonance” is one of those multi-purpose terms with a range of uses both within and beyond musical analysis. Outside of music, terms such as cultural or cognitive dissonance usually have negative connotations, but music has a way of making a virtue of the “wrong” or “ill-fitting.” Musical dissonance is not “bad” in the same sense as some of these other contexts (though stylistic norms may require proper handling and resolution, at least in the case of tonal counterpoint).
These same notions of consonance and dissonance have been applied to meter.[5] The modern conception of metrical dissonance was galvanized by Harald Krebs (1999).[6] This chapter is based on Krebs’s theory, and categorizes metrical dissonance as either displacement dissonance or grouping dissonance, each of which is discussed further below.
Displacement Dissonance
Displacement dissonance occurs when one layer of the music seems to be out-of-sync with the surrounding context. displacement dissonance: when two layers imply the same meter, but one is offset from the other by a consistent rhythmic value. This rhythmic value is called the displacement index; so, the displacement index of Example 1b is +1♪.
uses a familiar tune to illustrate: in , the tune is in sync with the bass line; in , it is an eighth note “late.” then is an example of displacement dissonance. Note that the final note of arrives properly in sync with the bass; this kind of adjustment is common in real pieces as well, especially to create a sense of conclusion or arrival (e.g., at cadences). This is a simple example of
Listening to
and , one would likely default to beginning these passages on beat 1 of a [latex]\mathbf{^3_4}[/latex] measure, creating four bars of [latex]\mathbf{^3_4}[/latex]. In the actual piece these lines are drawn from—Marie Jaëll’s Impromptu—the two lines occur simultaneously, but the melody is offset from the bass line, moved forward by one eighth note (again, a displacement index of +1♪).
B section (at 1:45 and 2:36, to the lyrics “you just call on me, brother, when you need a hand”) of Bill Withers’s iconic “Lean on me” (1972). In the A section of this song, the meter and grouping are clear and set the listener’s metric expectations. In the B section, the hand claps create a strong sense of displacement dissonance. Hand claps are typically a snare-drum substitute in pop drumbeats, playing the backbeat on beats 2 and 4 of a quadruple meter. But here, the handclaps are on the 2nd and 6th eighth notes of the bar, suggesting a backbeat that occurs an eighth note early (−1♪).[7] The song’s overarching meter may never be in doubt, but the rhythmic-metrical richness of this passage can be helpfully understood in terms of the systematic syncopations that potentially point to alternative displacements of the meter.
provides another example from the
Grouping dissonance
Grouping dissonance is the combination of two layers with different internal grouping organization; more specifically, the two layers articulate groupings with different numbers of pulses (pulse cardinalities), and these cardinalities are not multiples/factors of one another. These layers can be referred to in terms of their cardinalities to distinguish them: for example, a “3-layer” and a “4-layer.” provides simple illustrations of grouping dissonance.
The juxtaposition of 2- and 3-layers as in hemiola. The hemiola is a grouping dissonance found even in early Western concert music, such as G.F. Handel’s Water Music Suite No. 2, mvt. II (“Alla hornpipe”), shown in . This sort of hemiola is often found at important cadences of pieces in triple meter, where two measures of triple meter (a 3-layer) can be interpreted as three groups of two beats (a 2-layer) instead. This dissonance heightens excitement and reinforces the sense of tension-and-release created by the familiar cadential harmonic progression.
is common enough to be known by a special name: the
Three-against-four (3:4) is another widespread rhythmic-metrical pattern that can be readily heard in terms of grouping dissonance. tresillo rhythm can be viewed as a partial cycle of the 3-against-4 dissonance ( ).
sets out a complete cycle of this grouping dissonance, but partial cycles are more common in musical practice, especially in pop music—from classic jazz to electronic dance music (EDM). The Afro-Cubanfour-on-the-floor kick drum and handclap on the backbeat, but the synth and bass riffs creates metrical dissonance out a conflicting pattern based on groupings of three sixteenth notes. Note that the 3-layer breaks off with a 2-pulse-long grouping so that the riff has a downbeat arrival on every other measure.
shows a partial 3:4 grouping dissonance (sixteenth-note pulse) from Calvin Harris’s “I’m Not Alone” (2009). The song’s overall simple quadruple meter is clear from the
Grouping dissonances and polyrhythms can often be easier to understand in box notation rather than traditional Western notation, which prioritizes metrical hierarchies in a way that might not be helpful when dealing with grouping dissonance.
is a box notation diagram illustrating “I’m Not Alone.” In Western notation, we are forced to choose between unusual dotted rhythms or very frequent ties, both of which have advantages and disadvantages, but in box notation, this issue is entirely circumvented. The box notation also clarifies the mathematical properties of grouping dissonances. Any grouping dissonance will have a point of realignment, where both layers have a simultaneous attack point. This happens after x number of pulses, where x is the lowest common multiple between the cardinalities of the two layers. 12 is the lowest common multiple between 3 and 4, so a 3:4 grouping dissonance realigns every twelve pulses.Grouping dissonance via ostinati
A common musical device in many sorts of music is the ostinato: a rhythmic motive that repeats within a context that is not similarly repeating. If the length of the ostinato does not align with the prevailing meter, the ostinato can create a sense of grouping dissonance. In these cases, the rhythms on their own may not indicate a grouping dissonance, but combined with the pitch content of the ostinato, a grouping dissonance becomes audible.
Igor Stravinsky, a particular lover of metrical dissonance and other kinds of rhythmic surprise, creates grouping dissonance with an ostinato in the “Alleluia, laudate dominum” from Symphony of Psalms (conduct the [latex]\mathbf{^3_2}[/latex] meter of the melody; the second, conduct [latex]\mathbf{^4_2}[/latex] with the bass line.
). While the half notes that form the bass line of this excerpt are not themselves in conflict with the [latex]\mathbf{^3_2}[/latex] meter, the repeating pattern of four notes (E♭–B♭–F–B♭) suggests a four-beat grouping that does not align with the three-beat grouping of the melody and other lines. Listen to the example twice: the first time,
Changing Dissonance
Some music gets into a groove and sticks with it, but many do not: a piece may use metrical dissonance in some places and not others, creating a sense of tension-and-release. Consider the Bill Withers example above: the metrical dissonance in the B section contrasts with the A section that rhythmically very straight and without dissonance.
Let’s close with an example of this technique of changing dissonance in Brahms well-known “Lullaby” (“Wiegenlied”, Op. 49, No. 4).[8] The rhythm of the vocal melody is highly regular in the notated [latex]\mathbf{^3_4}[/latex]. Now have a look at the piano accompaniment, shown in . The left hand (bass) is every bit as regular as the voice, but the right hand (treble) goes its own way:
-
- We start with a rhythmic pattern suggesting three-eighth-note-long groupings that might fit better in [latex]\mathbf{^6_8}[/latex] rather than [latex]\mathbf{^3_4}[/latex], displaced by an eighth. (So before the voice even enters we arguably have both displacement and grouping dissonance!)
- The grouping dissonance disappears almost immediately and shifts to displacement dissonance: we settle on a two eighth note grouping structure, but displacement by an eighth. This remains for the majority of the verse.
- The opening double-dissonance briefly returns, before giving way to…
- …the right hand (finally) playing a note on the downbeat! The dotted-quarter on beat 2 may still suggests displacement dissonance, but of a milder kind: displacement forward by a quarter note, to emphasize the second beat instead of the first (like a sarabande).
- Finally, a measure later, we have all 6 eighth notes in the bar in a strong-weak alternation—no sense of dissonance at all.
- The last four-bar pattern repeats.
This reading (which you may, of course, disagree with) describes a process of (slight) tension to release that is entirely in keeping with the mood of this lullaby.
- Gotham, Mark. 2022. “Towards a Cognitively Based Quantification of Metrical Dissonance.” In The Oxford Handbook of Time in Music, edited by Mark Doffman, Emily Payne, and Toby Young. Oxford University Press. https://doi.org/10.1093/oxfordhb/9780190947279.013.12.
- Krebs, Harald. 1999. Fantasy Pieces: Metrical Dissonance in the Music of Robert Schumann. New York: Oxford University Press.
- Grouping Dissonance Composition Exercise (.pdf, .mscz). Asks students to compose two mini pieces with repeating motives of different lengths, creating grouping dissonance. References the pentatonic and hexatonic collections.
Media Attributions
- Tresillo as grouping dissonance © Megan Lavengood is licensed under a CC BY-NC-SA (Attribution NonCommercial ShareAlike) license
- Box chart for “I’m Not Alone” © Megan Lavengood is licensed under a CC BY-NC-SA (Attribution NonCommercial ShareAlike) license
- Notwithstanding notable differences, there are sufficiently strong commonalities between tonal and metrical dissonance to set them out in comparable ways, based on the simplicity of the proportional relationships between the constituent parts. ↵
- Krebs, in turn, cites an 1837 article by composer Hector Berlioz ("Feuilleton du Journal des Débats, November 10, 1837") as the earliest explicit comparison using these terms. ↵
- Other instrumental parts could also potentially be heard as further layers of displacement dissonances. Starting with the bass guitar, while the anacrustic pattern (G–A–C) in the low register reinforces the main meter, there is also a prominent note in a higher register on beat 3 that asserts itself as a different voice. This potentially represents an alternative [latex]\mathbf{^4_4}[/latex] meter with a displacement index of one half note. The backbeat articulated with open hi-hat notes, in turn, could also be heard as a displacement dissonance (index of one quarter note), but note that many listeners will find this backbeat pattern so familiar that it actually reinforces the original meter. ↵
- View a full score transcription. ↵
- Notwithstanding notable differences, there are sufficiently strong commonalities between tonal and metrical dissonance to set them out in comparable ways, based on the simplicity of the proportional relationships between the constituent parts. ↵
- Krebs, in turn, cites an 1837 article by composer Hector Berlioz ("Feuilleton du Journal des Débats, November 10, 1837") as the earliest explicit comparison using these terms. ↵
- Other instrumental parts could also potentially be heard as further layers of displacement dissonances. Starting with the bass guitar, while the anacrustic pattern (G–A–C) in the low register reinforces the main meter, there is also a prominent note in a higher register on beat 3 that asserts itself as a different voice. This potentially represents an alternative [latex]\mathbf{^4_4}[/latex] meter with a displacement index of one half note. The backbeat articulated with open hi-hat notes, in turn, could also be heard as a displacement dissonance (index of one quarter note), but note that many listeners will find this backbeat pattern so familiar that it actually reinforces the original meter. ↵
- View a full score transcription. ↵
two or more different ways of hearing the music's metrical structure presented at once, for example with a consistent pattern of conflicting rhythms
A type of metrical dissonance in which two or more musical layers imply the same meter, but one is offset from the other by a consistent rhythmic value (the "displacement index").
A type of metrical dissonance in which two or more musical layers have a different number of beats/pulses which are not multiples/factors of one another, as in a polyrhythm.
A rhythmic/pitch motive that repeats within a context that is not similarly repeating.
A recurring pattern of accents that occur over time. Meters are indicated in music notation with a time signature.
The rhythmic value by which two musical layers are offset from one another in a displacement dissonance.
An accent on beats 2 and 4 of a quadruple meter. Backbeats are common in jazz and pop styles.
The number of elements in a set or other grouping.
A grouping dissonance between a 2-layer and a 3-layer. Hemiolas commonly occur in triple-meter pieces by grouping six beats (e.g., two measures of 3/4) into three groups of two beats, rather than the expected two groups of three beats.
A drumbeat in which the kick drum is sounded on every beat.
A pitch collection built with the interval pattern ma2–ma2–mi3–ma2–mi3. This collection can also be generated by using scale degrees 1̂, 2̂, 3̂, 5̂, and 6̂ only of the major scale.
A six-note collection that alternates between half steps and minor thirds, such as C–C♯–E–F–G♯–A.