Editing Hemiola (section)

===Vertical hemiola: sesquialtera===
''The Oxford Dictionary of Music'' illustrates hemiola with a superimposition of three notes in the time of two and vice versa.<ref>[[Michael Kennedy (music critic)|Michael Kennedy]], "Hemiola, Hemiolia", ''The Oxford Dictionary of Music'' (Oxford and New York: Oxford University Press, 2002).</ref>

:<score>
\new Staff <<
 \new voice \relative c' {
  \clef percussion
  \time 6/8
  \set Score.tempoHideNote = ##t \tempo 4. = 80
  \stemDown \repeat volta 2 { g4. g }
  }
  \new voice \relative c' {
  \stemUp \repeat volta 2 { f4 f f }
  }
>>
</score> [[File:3 over 2.mid|frameless]]

One textbook states that, although the word "hemiola" is commonly used for both simultaneous and successive durational values, describing a simultaneous combination of three against two is less accurate than for successive values and the "preferred term for a vertical two against three … is ''sesquialtera''."<ref>[[Paul Cooper (composer)|Paul Cooper]], ''Perspectives in Music Theory; An Historical-Analytical Approach'' (New York: Dodd, Mead, 1973): 36.</ref> ''The New Harvard Dictionary of Music'' states that in some contexts, a sesquialtera is equivalent to a hemiola.{{sfn|Randel|1986|p=744}} ''Grove's Dictionary'', on the other hand, has maintained from the first edition of 1880 down to the most recent edition of 2001 that the Greek and Latin terms are equivalent and interchangeable, both in the realms of pitch and rhythm,<ref>[[W. S. Rockstro|W[illiam] S[myth] Rockstro]], "Hemiolia", ''A Dictionary of Music and Musicians (A.D. 1450–1880), by Eminent Writers, English and Foreign'', vol. 1, edited by [[George Grove]], D. C. L., (London: Macmillan and Co., 1880): 727; Rockstro, W[illiam] S[myth], ''Sesqui'', ''A Dictionary of Music and Musicians (A.D. 1450–1883), by Eminent Writers, English and Foreign'', vol. 3, edited by George Grove, D. C. L. (London: Macmillan and Co., 1883): 475</ref><ref name="Rushton" /> although [[David Hiley]], E. Thomas Stanford, and Paul R. Laird hold that, though similar in effect, hemiola properly applies to a momentary occurrence of three duple values in place of two triple ones, whereas sesquialtera represents a proportional metric change between successive sections.<ref>[[David Hiley]], E. Thomas Stanford, and [[Paul Laird|Paul R. Laird]], "Sesquialtera", ''[[The New Grove Dictionary of Music and Musicians]]'', 2nd edition, 29 vols., edited by [[Stanley Sadie]] and [[John Tyrrell (musicologist)|John Tyrrell]] (London: Macmillan, 2001): 23:157–159.</ref>

====Sub-Saharan African music====
A repeating vertical hemiola is known as [[polyrhythm]], or more specifically, [[cross-beat|cross-rhythm]]. The most basic [[rhythmic cell]] of sub-Saharan Africa is the 3:2 cross-rhythm. Novotney observes: "The 3:2 relationship (and [its] permutations) is the foundation of most typical polyrhythmic textures found in West African musics."<ref>{{cite thesis|author=Eugene Domenic Novotney|title="The 3:2 Relationship as the Foundation of Timelines in West African Musics|type=thesis|location=Urbana, Illinois|publisher=University of Illinois|year=1998|page=201|url=http://www.unlockingclave.com/free-download-32-thesis.html|url-status=dead|archive-url=https://web.archive.org/web/20160820170449/http://www.unlockingclave.com/free-download-32-thesis.html|archive-date=2016-08-20}} (blurb)</ref> [[V. Kofi Agawu|Agawu]] states: "[The] resultant [3:2] rhythm holds the key to understanding ... there is no independence here, because 2 and 3 belong to a single [[shape|Gestalt]]."<ref>[[V. Kofi Agawu]], ''Representing African Music: Postcolonial Notes, Queries, Positions'' (New York: Routledge, 2003): 92. {{ISBN|0-415-94390-6}}.</ref>

[[File:Gyil.JPG|thumb|Ghanaian gyil]]
In the following example, a Ghanaian [[gyil]] plays a hemiola as the basis of an [[ostinato]] melody. The left hand (lower notes) sounds the two main beats, while the right hand (upper notes) sounds the three cross-beats.<ref>David Peñalosa, ''The Clave Matrix; Afro-Cuban Rhythm: Its Principles and African Origins'' (Redway, California: Bembe Inc., 2009): 22. {{ISBN|1-886502-80-3}}.</ref>
:<score>
\new Staff <<
 \new voice \relative c' {
  \clef treble
  \time 6/8
  \set Score.tempoHideNote = ##t \tempo 4. = 80
  \stemDown \repeat volta 2 { b4. d }
  }
  \new voice \relative c' {
  \stemUp \repeat volta 2 { fis8[ r fis] r[ a r] }
  }
>>
</score> [[File:Ghanaian gyil.mid|frameless]]

====European music====
In compound time ({{Time signature|6|8}} or {{Time signature|6|4}}), where a regular pattern of two beats to a measure is established at the start of a phrase, this changes to a pattern of three beats at the end of the phrase.
[[File:Archaic hemiola.mid|thumb]]
[[File:Archaic hemiola.png|thumb|center|500px|Archaic hemiola]]
The minuet from [[Johann Sebastian Bach|J. S. Bach]]'s keyboard [[Partitas for keyboard (Bach)|Partita No. 5]] in G major articulates groups of 2 times 3 quavers that are really in {{Time signature|6|8}} time, despite the {{Time signature|3|4}} metre stated in the initial time-signature.<ref>Alison Latham (ed.), "Cross-rhythm", ''[[The Oxford Companion to Music]]'' (Oxford and New York: Oxford University Press, 2002).</ref> The latter time is restored only at the cadences (bars 4 and 11–12):
[[File:Bach Minuet from Partita 5 in G bars 1-12.wav|thumb|Bach: Minuet from Partita 5 in G bars 1–12]]
[[File:Bach Minuet from Partita 5 in G bars 1-12.png|thumb|center|500px|Bach: Minuet from Partita 5 in G bars 1–12]]

Later in the same piece, Bach creates a conflict between the two metres ({{Time signature|6|8}} against {{Time signature|3|4}}):
[[File:Bach Minuet from Partita 5 in G bars 37-52.wav|thumb|Bach Minuet from Partita 5 in G bars 37–52]]
[[File:Bach Minuet from Partita 5 in G bars 37-52 0002.png|thumb|center|500px|Bach: Minuet from Partita 5 in G, bars 37–52]]

Hemiola is found in many Renaissance pieces in triple rhythm. One composer who exploited this characteristic was the 16th-century French composer [[Claude Le Jeune]], a leading exponent of [[musique mesurée|musique mesurée à l'antique]]. One of his best-known chansons is "Revoici venir du printemps", where the alternation of compound-duple and simple-triple metres with a common counting unit for the beat subdivisions can be clearly heard:
[[File:Claude LeJeune, Revoici venir du printemps.wav|thumb|Claude LeJeune, Revoici venir du printemps]]
[[File:Le Jeune Reveci bars 1-4 upper vocal line.png|thumb|center|500px|Claude LeJeune, "Revoici venir du printemps", bars 1–4 of the upper vocal line. {{YouTube|D8hzcpkoh24|Listen|link=no}}]]

The hemiola was commonly used in [[baroque music]], particularly in [[dance]]s, such as the [[courante]] and [[minuet]]. Other composers who have used the device extensively include [[Arcangelo Corelli|Corelli]], [[George Frideric Handel|Handel]], [[Carl Maria von Weber|Weber]] and [[Ludwig van Beethoven|Beethoven]]. A spectacular example from Beethoven comes in the scherzo from his [[String Quartet No. 6 (Beethoven)|String Quartet No. 6]]. As [[Philip Radcliffe]] puts it, "The constant cross-rhythms shifting between {{Time signature|3|4}} and {{Time signature|6|8}}, more common at certain earlier and later periods, were far from usual in 1800, and here they are made to sound especially eccentric owing to frequent sforzandi on the last quaver of the bar... it looks ahead to later works and must have sounded very disconcerting to contemporary audiences."<ref>[[Philip Radcliffe]], ''Beethoven's String Quartets'' (London: Hutchinson, 1965): 41.</ref>
[[File:Beethoven Scherzo from Op 18 No 6 Quartet version for audio.wav|thumb|Beethoven Scherzo from 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 {{YouTube|8theuK4BgNs|Listen|link=no}}]]
Later in the nineteenth century, [[Tchaikovsky]] frequently used hemiolas in his waltzes, as did [[Richard Strauss]] in the waltzes from ''[[Der Rosenkavalier]]'', and the third movement of [[Robert Schumann]]'s [[Piano Concerto (Schumann)|Piano Concerto]] is noted for the ambiguity of its rhythm. [[John Daverio]] says that the movement's "fanciful hemiolas... serve to legitimize the dance-like material as a vehicle for symphonic elaboration."<ref>[[John Daverio]], ''Robert Schumann: Herald of a New Poetic Age'' (Oxford and New York: Oxford University Press, 1997): 314. {{ISBN| 978-0-19-509180-9}}.</ref>
[[File:Schumann Piano Concerto Finale bars 120-127.wav|thumb|Schumann Piano Concerto Finale bars 120–127]]
[[File:Schumann Piano Concerto Finale bars 120-128.png|thumb|center|600px|Schumann Piano Concerto Finale bars 120–127]]

[[Johannes Brahms]] was particularly famous for exploiting the hemiola's potential for large-scale thematic development. Writing about the rhythm and meter of Brahms's [[Symphony No. 3 (Brahms)|Symphony No. 3]], Frisch says "Perhaps in no other first movement by Brahms does the development of these elements play so critical a role. The first movement of the third is cast in {{Time signature|6|4}} meter that is also open, through internal recasting as {{Time signature|3|2}} (a so-called hemiola). Metrical ambiguity arises in the very first appearance of the motto [opening theme]."<ref>{{cite book |first=Walter |last=Frisch |title=Brahms: The Four Symphonies |location=New Haven and London |publisher=[[Yale University Press]] |date=2003 |page=95 |isbn=978-0-300-09965-2}}</ref>
[[File:Brahms Symphony No. 3, opening bars.wav|thumb|Brahms Symphony No. 3, opening bars]]
[[File:Brahms 3 opening.png|thumb|center|500px|Brahms, Symphony No. 3, opening bars]]

At the beginning of the second movement, {{Lang|fr|Assez vif – très rythmé}}, of his [[String Quartet (Ravel)|String Quartet]] (1903), [[Maurice Ravel|Ravel]] "uses the ''[[pizzicato]]'' as a vehicle for rhythmic interplay between {{Time signature|6|8}} and {{Time signature|3|4}}."<ref>[[Roger Nichols (musical scholar)|Roger Nichols]], ''Ravel'' (London: Dent, 1977): 24.</ref>
[[File:Ravel Quartet, second movement.wav|thumb|Ravel Quartet, second movement]]
[[File:Second movement of Ravel Quartet.png|thumb|center|400px|Second movement of Ravel Quartet]]