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Hemiola

Inmusic,hemiola(alsohemiolia) is theratio3:2. The equivalent Latin term issesquialtera.Inrhythm,hemiolarefers to threebeatsof equal value in the time normally occupied by two beats. In pitch,hemiolarefers to the interval of aperfect fifth.

Etymology

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The wordhemiolacomes 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."[1]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 amonochordin this ratio produces theintervalof aperfect fifth.Beginning in the 15th century, both words were also used to describe rhythmic relationships, specifically the substitution (usually through the use ofcoloration—red notes in place of black ones, or black in place of "white", hollow noteheads) of threeimperfectnotes (divided into two parts) for two perfect ones (divided into three parts) intempus perfectumor inprolatio maior.[2][3]

Rhythm

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Inrhythm,hemiolarefers to threebeatsof equal value in the time normally occupied by two beats.[4]

Vertical hemiola: sesquialtera

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The Oxford Dictionary of Musicillustrates hemiola with a superimposition of three notes in the time of two and vice versa.[5]

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… issesquialtera."[6]The New Harvard Dictionary of Musicstates that in some contexts, a sesquialtera is equivalent to a hemiola.[7]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,[8][3]althoughDavid 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.[9]

Sub-Saharan African music

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A repeating vertical hemiola is known aspolyrhythm,or more specifically,cross-rhythm.The most basicrhythmic cellof 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."[10]Agawustates: "[The] resultant [3:2] rhythm holds the key to understanding... there is no independence here, because 2 and 3 belong to a singleGestalt."[11]

Ghanaian gyil

In the following example, a Ghanaiangyilplays a hemiola as the basis of anostinatomelody. The left hand (lower notes) sounds the two main beats, while the right hand (upper notes) sounds the three cross-beats.[12]

European music

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In compound time (6
8or6
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.

Archaic hemiola

The minuet fromJ. S. Bach's keyboardPartita No. 5in G major articulates groups of 2 times 3 quavers that are really in6
8time, despite the3
4metre stated in the initial time-signature.[13]The latter time is restored only at the cadences (bars 4 and 11–12):

Bach: Minuet from Partita 5 in G bars 1–12
Bach: Minuet from Partita 5 in G bars 1–12

Later in the same piece, Bach creates a conflict between the two metres (6
8against3
4):

Bach Minuet from Partita 5 in G bars 37–52
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 composerClaude Le Jeune,a leading exponent ofmusique 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:

Claude LeJeune, Revoici venir du printemps
Claude LeJeune, "Revoici venir du printemps", bars 1–4 of the upper vocal line.Listenon YouTube

The hemiola was commonly used inbaroque music,particularly indances,such as thecouranteandminuet.Other composers who have used the device extensively includeCorelli,Handel,WeberandBeethoven.A spectacular example from Beethoven comes in the scherzo from hisString Quartet No. 6.AsPhilip Radcliffeputs it, "The constant cross-rhythms shifting between3
4and6
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. "[14]

Beethoven Scherzo from Op 18 No 6
Beethoven Scherzo from Op. 18, No. 6, violin and cello onlyListenon YouTube

Later in the nineteenth century,Tchaikovskyfrequently used hemiolas in his waltzes, as didRichard Straussin the waltzes fromDer Rosenkavalier,and the third movement ofRobert Schumann'sPiano Concertois noted for the ambiguity of its rhythm.John Daveriosays that the movement's "fanciful hemiolas... serve to legitimize the dance-like material as a vehicle for symphonic elaboration."[15]

Schumann Piano Concerto Finale bars 120–127
Schumann Piano Concerto Finale bars 120–127

Johannes Brahmswas particularly famous for exploiting the hemiola's potential for large-scale thematic development. Writing about the rhythm and meter of Brahms'sSymphony 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 in6
4meter that is also open, through internal recasting as3
2(a so-called hemiola). Metrical ambiguity arises in the very first appearance of the motto [opening theme]. "[16]

Brahms Symphony No. 3, opening bars
Brahms, Symphony No. 3, opening bars

At the beginning of the second movement,Assez vif – très rythmé,of hisString Quartet(1903),Ravel"uses thepizzicatoas a vehicle for rhythmic interplay between6
8and3
4."[17]

Ravel Quartet, second movement
Second movement of Ravel Quartet

Horizontal hemiola

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Peter Manuel, in the context of an analysis of theflamencosoleásong form, refers to the following figure as ahorizontal hemiolaor "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 a6
8bar followed by one in3
4,for a 3 + 3 + 2 + 2 + 2 pattern.[18]

Horizontal hemiola

This figure is a common Africanbell pattern,used by theHausa peopleofNigeria,inHaitian Vodou drumming,Cubanpalo,and many other drumming systems. The horizontal hemiola suggestsmetric modulation(6
8changing to3
4). This interpretational switch has been exploited, for example, byLeonard Bernstein,in the song"America"fromWest Side Story,as can be heard in the prominent motif (suggesting a duple beat scheme, followed by a triple beat scheme):

Horizontal hemiola inBernstein's,"America"fromWest Side Story

Pitch

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The perfect fifth

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Hemiolacan be used to describe the ratio of the lengths of two strings as three-to-two (3:2), that together sound aperfect fifth.[2]The early Pythagoreans, such asHippasusandPhilolaus,used this term in amusic-theoreticcontext to mean aperfect fifth.[19]

Just perfect fifth on C

Thejustly tunedpitch ratioof a perfect fifth means that the upper note makes three vibrations in the same amount of time that the lower note makes two. In thecentsystem 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 aviolinis 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 ofPythagorean tuning,and are employed together with other just intervals injust intonation.The 3:2 just perfect fifth arises in the justly tuned Cmajor scalebetween C and G.[20]

Other intervals

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Later Greek authors such asAristoxenusandPtolemyuse the word to describe smaller intervals as well, such as the hemiolicchromaticpyknon,which is one-and-a-half times the size of thesemitonecomprising theenharmonicpyknon.[21]

See also

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References

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  1. ^Henry George LiddellandRobert Scott,A Greek-English Lexicon,9th edition (Oxford: Clarendon Press, 1940).
  2. ^abDon Michael Randel,"Hemiola, hemiolia",Harvard Dictionary of Music,fourth edition. Cambridge: The Belknap Press of Harvard University Press, 2003,ISBN978-0-674-01163-2
  3. ^abJulian Rushton,"Hemiola [hemiolia]",The New Grove Dictionary of Music and Musicians,second edition, edited byStanley SadieandJohn Tyrrell.London: Macmillan, 2001.
  4. ^Randel 1986,p. 376.
  5. ^Michael Kennedy,"Hemiola, Hemiolia",The Oxford Dictionary of Music(Oxford and New York: Oxford University Press, 2002).
  6. ^Paul Cooper,Perspectives in Music Theory; An Historical-Analytical Approach(New York: Dodd, Mead, 1973): 36.
  7. ^Randel 1986,p. 744.
  8. ^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 byGeorge 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
  9. ^David Hiley,E. Thomas Stanford, andPaul R. Laird,"Sesquialtera",The New Grove Dictionary of Music and Musicians,2nd edition, 29 vols., edited byStanley SadieandJohn Tyrrell(London: Macmillan, 2001): 23:157–159.
  10. ^Eugene Domenic Novotney (1998)."The 3:2 Relationship as the Foundation of Timelines in West African Musics(thesis). Urbana, Illinois: University of Illinois. p. 201. Archived fromthe originalon 2016-08-20.(blurb)
  11. ^V. Kofi Agawu,Representing African Music: Postcolonial Notes, Queries, Positions(New York: Routledge, 2003): 92.ISBN0-415-94390-6.
  12. ^David Peñalosa,The Clave Matrix; Afro-Cuban Rhythm: Its Principles and African Origins(Redway, California: Bembe Inc., 2009): 22.ISBN1-886502-80-3.
  13. ^Alison Latham (ed.), "Cross-rhythm",The Oxford Companion to Music(Oxford and New York: Oxford University Press, 2002).
  14. ^Philip Radcliffe,Beethoven's String Quartets(London: Hutchinson, 1965): 41.
  15. ^John Daverio,Robert Schumann: Herald of a New Poetic Age(Oxford and New York: Oxford University Press, 1997): 314.ISBN978-0-19-509180-9.
  16. ^Frisch, Walter (2003).Brahms: The Four Symphonies.New Haven and London:Yale University Press.p. 95.ISBN978-0-300-09965-2.
  17. ^Roger Nichols,Ravel(London: Dent, 1977): 24.
  18. ^Peter Manuel, "Flamenco in Focus: An Analysis of a Performance of Soleares", inAnalytical Studies in World Music,edited byMichael Tenzer,92–119 (Oxford and New York: Oxford University Press, 2006): 102.
  19. ^Barker 1989,pp. 31, 37–38.
  20. ^Oscar Paul,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
  21. ^Barker1989,pp. 164–165, 303.

Sources

Further reading

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