Editing Polyrhythm (section)

== In western art music ==
In some [[European art music]], polyrhythm periodically contradicts the prevailing meter. For example, in [[Wolfgang Amadeus Mozart|Mozart]]'s opera ''[[Don Giovanni]]'', two orchestras are heard playing together in different metres ({{music|time|3|4}} and {{music|time|2|4}}):

[[File:Mozart Don Giovanni 2 dances together.wav|thumb|Mozart ''Don Giovanni'' 2 dances together]]
[[File:Mozart Don Giovanni 2 dances together.png|thumb|center|400px|Mozart, ''Don Giovanni'' dances from act 1, scene 5]]

They are later joined by a third band, playing in {{music|time|3|8}} time.

Polyrhythm is heard near the opening of [[Beethoven]]'s [[Symphony No. 3 (Beethoven)|Symphony No. 3]]. (See also [[syncopation]].)

[[Chopin]] often explored the rhythmic possibilities inherent in the independence of a pianist's two hands.  A spectacular example may be found in his [[Étude Op. 10, No. 10 (Chopin)|Étude, Op, 10 No. 10]].  [[Alan Walker (musicologist)|Alan Walker]] comments that while this piece is straightforward for listeners, "From the player’s point of view, however, nothing is straightforward. Chopin has placed him inside a veritable hornets’ nest of cross-rhythms and syncopations. The melody first emerges from a background of triplets, then of duplets. Accents are changed without warning, shifting the balance of the phrase sideways, so to speak, together with the place of each note within it."<ref>Walker, A. (2018, p. 325), Fryderyk Chopin, a Life and Times. London, Faber.</ref>

Polyrhythm is a particularly common feature of the music of [[Brahms]]. Writing about the [[Violin Sonata No. 1 (Brahms)|Violin Sonata in G major, Op. 78]], [[Jan Swafford]] (1997, p.&nbsp;456) says "In the first movement Brahms plays elaborate games with the phrasing, switching the stresses of the {{music|time|6|4}} meter back and forth between {{serif|'''3+3'''}} and {{serif|'''2+2+2'''}}, or superimposing both in violin and piano. These ideas gather at the climax at measure 235, with the layering of phrases making an effect that perhaps during the 19th century only Brahms could have conceived."<ref>[[Jan Swafford|Swafford, J.]] (1997) ''Johannes Brahms, a Biography''. London, Macmillan.</ref>
[[File:Brahms Violin Sonata in G, 1, bars 235ff.wav|thumb|Brahms Violin Sonata in G, 1, bars 235ff]]
[[File:Brahms Violin Sonata in G, 1, bars 235ff.png|thumb|center|500px|Brahms Violin Sonata in G, 1, bars 235ff]]

In "The Snow Is Dancing" from his ''[[Children's Corner]]'' suite, [[Claude Debussy|Debussy]] introduces a melody "on a static, repeated B-flat, cast in triplet-division cross rhythms which offset this stratum independently of the sixteenth notes comprising the two dancing-snowflake lines below it."<ref>{{cite journal |last1=Oravitz |first1=M. |title=The human and the physical in Debussy's depictions of snow |journal=Les Cahiers de la Société québécoise de recherche en musique |date=2013 |volume=14 |issue=1 |pages=49–54 |doi=10.7202/1016198ar|doi-access=free }}</ref> "In this section great attention to the exactitude of rhythms is demanded by the polyrhythmic superposition of pedals, [[ostinato]], and melody."<ref>Schmitz, E.R. (1966, 124) The Piano Works of Claude Debussy.  New York, Dover.</ref>
[[File:Debussy, the Snow is Dancing, Bars 34-38.wav|thumb|Debussy, "The Snow Is Dancing", bars 34–38]]
[[File:Debussy, the Snow is Dancing, Bars 34-38.png|thumb|center|500px|Debussy, "The Snow Is Dancing", bars 34–38]]

=== Hemiola ===
Concerning the use of a two-over-three (2:3) [[hemiola]] in Beethoven's [[String Quartet No. 6 (Beethoven)|String Quartet No. 6]], Ernest Walker states, "The vigorously effective [[Scherzo]] is in {{music|time|3|4}} time, but with a curiously persistent [[cross-rhythm]] that does its best to persuade us that it is really in {{music|time|6|8}}."<ref>Walker, Ernest (1905: 79) ''The Music of the Masters; Beethoven''. New York: Brentano's Union Square.</ref>

[[File:Beethoven Scherzo from Op 18 No 6 Quartet version for audio.wav|thumb|Beethoven Scherzo from String Quartet, Op. 18, No. 6]]
[[File:Beethoven Scherzo from Op 18 No 6, violin and cello only.png|thumb|center|500px|Beethoven Scherzo from Op. 18, No. 6, violin and cello only]]

=== Polyrhythm, not polymeter ===
The illusion of simultaneous {{music|time|3|4}} and {{music|time|6|8}}, suggests [[polymeter]]: [[Meter (music)|triple meter]] combined with [[Meter (music)|compound duple meter]].

{{Block indent|<score sound="1">
    \new PianoStaff <<
       \new DrumStaff <<
           \drummode {
              \set Staff.timeSignatureFraction = 6/8
              \set Score.tempoHideNote = ##t \tempo 4 = 120
              \repeat volta 2 { tommh4. tommh }
           }
       >>
       \new DrumStaff <<
           \drummode {
              \set Staff.timeSignatureFraction = 3/4
              \repeat volta 2 { tomfh4 tomfh tomfh }
           }
       >>
   >>
</score>}}
However, the two beat schemes interact within a metric hierarchy (a single meter). The triple beats are primary and the duple beats are secondary; the duple beats are [[cross-beat]]s within a triple beat scheme.

{{Block indent|<score sound="1">
    \new DrumStaff <<
       \new voice \drummode { 
           \set Score.tempoHideNote = ##t \tempo 4 = 120
           \time 3/4
           \stemUp \repeat volta 2 { tommh4 r8 tommh r4 }
       }
       \new voice \drummode {
           \stemDown \repeat volta 2 { tomfh4 tomfh tomfh }
       }
   >>
</score>}}

=== Composite hemiola ===
The four-note [[ostinato]] pattern of [[Mykola Leontovych]]'s "[[Carol of the Bells]]" (the first measure below) is the composite of the two-against-three [[hemiola]] (the second measure).

{{Block indent|<score sound="1">
    \new Staff <<
       \mergeDifferentlyHeadedOn
       \mergeDifferentlyDottedOn
       \new voice \relative c'' {
          \set Score.tempoHideNote = ##t \tempo 4 = 136
          \key g \minor
          \time 3/4

          bes4 a8 bes g4 \bar "||"
          \stemDown bes4 a g \bar "||"
       }
       \new voice \relative c'' {
          s2.
          \stemUp bes4. bes
       }
    >>
</score>}}

Another example of polyrhythm can be found in measures 64 and 65 of the first movement of [[Wolfgang Amadeus Mozart|Mozart]]'s [[Piano Sonata No. 12 (Mozart)|Piano Sonata No. 12]]. Three evenly-spaced sets of three attack-points span two measures.

{{Block indent|<score sound="1">
    { \new PianoStaff <<
        \new Staff <<
            \set Score.currentBarNumber = #62
            \bar ""

            \relative c' {
                \set Score.tempoHideNote = ##t \tempo 4 = 160
                \clef treble
                \key f \major
                \time 3/4

                <d aes'>8 <d aes'> r <d aes'> r <d aes'>
                <d g> <d g> r <d g> r <d g>
                <c g'> <c g'> r <c g'> <c f>[ <c f>]
                r <c f> <b f'>[ <b f'>] r <b f'>
                }
            >>
        \new Staff <<
            \relative c, {
                \clef bass
                \key f \major
                \time 3/4

                <bes bes'>4 d' f
                <ees, ees'> g' bes
                <aes,, aes'> aes'' <d,, d'>
                d'' <g,,, g'> g''
                }
            >>
    >> }
</score>}}