Editing Hemiola

{{Short description|Musical ratio of 3:2}}
In [[music]], '''hemiola''' (also '''hemiolia''') is the [[ratio]] 3:2. The equivalent Latin term is '''[[Superparticular number|sesquialtera]]'''. In [[rhythm]], ''hemiola'' refers to three [[beat (music)|beats]] of equal value in the time normally occupied by two beats. In pitch, ''hemiola'' refers to the interval of a [[perfect fifth]].

==Etymology==
The word ''hemiola'' comes from the Greek adjective ἡμιόλιος, ''hemiolios'', meaning "containing one and a half," "half as much again," "in the ratio of one and a half to one (3:2), as in musical sounds."<ref>[[Henry George Liddell]] and [[Robert Scott (philologist)|Robert Scott]], ''A Greek-English Lexicon'', 9th edition (Oxford: Clarendon Press, 1940).</ref> The words "hemiola" and "sesquialtera" both signify the ratio 3:2, and in music were first used to describe relations of pitch. Dividing the string of a [[monochord]] in this ratio produces the [[interval (music)|interval]] of a [[perfect fifth]]. Beginning in the 15th century, both words were also used to describe rhythmic relationships, specifically the substitution (usually through the use of [[Color (medieval music)#Coloration (mensural notation device)|coloration]]—red notes in place of black ones, or black in place of "white", hollow noteheads) of three [[Mensural notation#Proportions and colorations|imperfect]] notes (divided into two parts) for two perfect ones (divided into three parts) in ''tempus perfectum'' or in ''prolatio maior''.<ref name=Randel>[[Don Michael Randel]], "Hemiola, hemiolia", ''[[Harvard Dictionary of Music]]'', fourth edition. Cambridge: The Belknap Press of Harvard University Press, 2003, {{ISBN|978-0-674-01163-2}}</ref><ref name=Rushton>[[Julian Rushton]], "Hemiola [hemiolia]", ''[[The New Grove Dictionary of Music and Musicians]]'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (musicologist)|John Tyrrell]]. London: Macmillan, 2001.</ref>

==Rhythm==
In [[rhythm]], ''hemiola'' refers to three [[beat (music)|beats]] of equal value in the time normally occupied by two beats.{{sfn|Randel|1986|p=376}}

===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]]

===Horizontal hemiola===
Peter Manuel, in the context of an analysis of the [[flamenco]] [[soleá]] song form, refers to the following figure as a ''horizontal hemiola'' or "sesquialtera" (which mistranslates as: "six that alters"). It is "a cliché of various Spanish and Latin American musics ... well established in Spain since the sixteenth century", a twelve-beat scheme with internal accents, consisting of a {{Time signature|6|8}} bar followed by one in {{Time signature|3|4}}, for a 3&nbsp;+ 3&nbsp;+ 2&nbsp;+ 2&nbsp;+ 2 pattern.<ref>Peter Manuel, "Flamenco in Focus: An Analysis of a Performance of Soleares", in ''Analytical Studies in World Music'', edited by [[Michael Tenzer]], 92–119 (Oxford and New York: Oxford University Press, 2006): 102.</ref>

[[File:Horizontal hemiola.png|thumb|center|400px|Horizontal hemiola[[File:Horizontal hemiola.mid]]]]

This figure is a common African [[bell pattern]], used by the [[Hausa people]] of [[Nigeria]], in [[Haitian Vodou drumming]], Cuban [[Palo (religion)|palo]], and many other drumming systems. The horizontal hemiola suggests [[metric modulation]] ({{Time signature|6|8}} changing to {{Time signature|3|4}}). This interpretational switch has been exploited, for example, by [[Leonard Bernstein]], in the song [[America (West Side Story song)|"America"]] from ''[[West Side Story]]'', as can be heard in the prominent motif (suggesting a duple beat scheme, followed by a triple beat scheme):
[[File:Alternating time signatures2.gif|thumb|center|400px|Horizontal hemiola in [[Leonard Bernstein|Bernstein's]], [[America (West Side Story song)|"America"]] from ''[[West Side Story]]''[[File:Alternating time signatures2.mid]]]]

==Pitch==

===The perfect fifth===
''Hemiola'' can be used to describe the ratio of the lengths of two strings as three-to-two (3:2), that together sound a [[perfect fifth]].<ref name="Randel" /> The early Pythagoreans, such as [[Hippasus]] and [[Philolaus]], used this term in a [[music theory|music-theoretic]] context to mean a [[perfect fifth]].{{sfn|Barker|1989|pp=31, 37–38}}

[[File:Just perfect fifth on C.mid|thumb|Just perfect fifth on C]]
The [[just intonation|justly tuned]] [[interval ratio|pitch ratio]] of a perfect fifth means that the upper note makes three vibrations in the same amount of time that the lower note makes two. In the [[Cent (music)|cent]] system of pitch measurement, the 3:2 ratio corresponds to approximately 702 cents, or 2% of a semitone wider than seven semitones. The just perfect fifth can be heard when a [[violin]] is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune. Just perfect fifths are the basis of [[Pythagorean tuning]], and are employed together with other just intervals in [[just intonation]]. The 3:2 just perfect fifth arises in the justly tuned C [[major scale]] between C and G.<ref>[[Oscar Paul]], ''[https://books.google.com/books?id=4WEJAQAAMAAJ&q=musical+interval+%22pythagorean+major+third%22 A Manual of Harmony for Use in Music-Schools and Seminaries and for Self-Instruction]'', trans. [[Theodore Baker]] (New York: G. Schirmer, 1885), p. 165</ref>

===Other intervals===
Later Greek authors such as [[Aristoxenus]] and [[Ptolemy]] use the word to describe smaller intervals as well, such as the hemiolic [[Genus (music)#Chromatic|chromatic]] ''[[pyknon]]'', which is one-and-a-half times the size of the [[semitone]] comprising the [[Genus (music)#Enharmonic|enharmonic]] ''pyknon''.{{sfn|Barker1989|pp=164–165, 303}}

==See also==
*[[Syncopation]]

==References==
{{reflist}}

'''Sources'''
* {{cite book|last=Barker|first=Andrew|author-link=Andrew Barker (classicist)|title=Greek Musical Writings: [vol. 2] Harmonic and Acoustic Theory|location=Cambridge|publisher=Cambridge University Press|year=1989}}
* {{cite book|editor-last=Randel|editor-first=Don Michael|editor-link=Don Michael Randel|title=[[The New Harvard Dictionary of Music]]|location=Cambridge, Massachusetts|publisher=Belknap Press of Harvard University Press|year=1986}}{{Full citation needed|date=April 2014|reason=It is customary when citing dictionaries to specify the article title, rather than a page number.}}

== Further reading ==
*Brandel, Rose (1959). ''The African Hemiola Style'', ''[[Ethnomusicology (journal)|Ethnomusicology]]'', 3(3):106–117, correction, 4(1):iv.
*[[Ottó Károlyi|Károlyi, Ottó]] (1998). ''Traditional African & Oriental Music'', Penguin Books. {{ISBN|0-14-023107-2}}.

{{Rhythm and meter}}

[[Category:Ratios]]
[[Category:Musical techniques]]
[[Category:Musical terminology]]
[[Category:Rhythm and meter]]