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music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...
, the terms ''additive'' and ''divisive'' are used to distinguish two types of both
rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular re ...
and
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
: * A divisive (or, alternately, multiplicative) rhythm is a
rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular re ...
in which a larger period of time is divided into smaller rhythmic units or, conversely, some integer unit is regularly multiplied into larger, equal units. * This can be contrasted with additive rhythm, in which larger periods of time are constructed by
concatenating In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenati ...
(joining end to end) a series of units into larger units of unequal length, such as a meter produced by the regular alternation of and . When applied to meters, the terms ''perfect'' and ''imperfect'' are sometimes used as the equivalents of ''divisive'' and ''additive'', respectively . For example, 4 may be evenly divided by 2 or reached by adding 2 + 2. In contrast, 5 is only evenly divisible by 5 and 1 and may be reached by adding 2 or 3. Thus, (or, more commonly, ) is divisive while is additive. The terms ''additive'' and ''divisive'' originate with Curt Sachs's book ''Rhythm and Tempo'' (1953), while the term '' aksak rhythm'' was introduced for the former concept at about the same time by Constantin Brăiloiu, in agreement with the Turkish musicologist
Ahmet Adnan Saygun Ahmet Adnan Saygun (; 7 September 1907 – 6 January 1991) was a Turkish composer, musicologist and writer on music. One of a group of composers known as the Turkish Five who pioneered western classical music in Turkey, his works show a mast ...
. The relationship between additive and divisive rhythms is complex, and the terms are often used in imprecise ways. In his article on rhythm in the second edition of the ''
New Grove Dictionary of Music and Musicians ''The New Grove Dictionary of Music and Musicians'' is an encyclopedic dictionary of music and musicians. Along with the German-language ''Die Musik in Geschichte und Gegenwart'', it is one of the largest reference works on the history and theo ...
,'' Justin London states that: Winold recommends that, " metric structure is best described through detailed analysis of pulse groupings on various levels rather than through attempts to represent the organization with a single term". Sub-Saharan African music and most European (Western) music is divisive, while
Indian Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...
and other Asian musics may be considered as primarily additive. However, many pieces of music cannot be clearly labeled divisive or additive.


Divisive rhythm

For example: consists of one measure (whole note: 1) divided into a stronger first beat and slightly less strong second beat (half notes: 1, 3), which are in turn divided, by two weaker beats (quarter notes: 1, 2, 3, 4), and again divided into still weaker beats (eighth notes: 1 & 2 & 3 & 4 &). Additive rhythm features nonidentical or irregular durational groups following one another at two levels, within the bar and between bars or groups of bars. This type of rhythm is also referred to in musicological literature by the Turkish word ''
aksak In Ottoman musical theory, ''aksak'' is a rhythmic system in which pieces or sequences, executed in a fast tempo, are based on the uninterrupted reiteration of a matrix, which results from the juxtaposition of rhythmic cells based on the alternat ...
'', which means "limping". In the special case of
time signature The time signature (also known as meter signature, metre signature, or measure signature) is a notational convention used in Western musical notation to specify how many beats (pulses) are contained in each measure (bar), and which note va ...
s in which the upper numeral is not divisible by two or three without a fraction, the result may alternatively be called ''irregular'', ''imperfect'', or ''uneven'' meter, and the groupings into twos and threes are sometimes called ''long beats'' and ''short beats''. The term ''additive rhythm'' is also often used to refer to what are also incorrectly called ''asymmetric rhythms'' and even ''irregular rhythms'' – that is,
meter The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
s which have a regular pattern of beats of uneven length. For example, the
time signature The time signature (also known as meter signature, metre signature, or measure signature) is a notational convention used in Western musical notation to specify how many beats (pulses) are contained in each measure (bar), and which note va ...
indicates each
bar Bar or BAR may refer to: Food and drink * Bar (establishment), selling alcoholic beverages * Candy bar * Chocolate bar Science and technology * Bar (river morphology), a deposit of sediment * Bar (tropical cyclone), a layer of cloud * Bar ( ...
is eight
quaver 180px, Figure 1. An eighth note with stem extending up, an eighth note with stem extending down, and an eighth rest. 180px, Figure 2. Four eighth notes beamed together. An eighth note (American) or a quaver ( British) is a musical note pla ...
s long, and has four
beat Beat, beats or beating may refer to: Common uses * Patrol, or beat, a group of personnel assigned to monitor a specific area ** Beat (police), the territory that a police officer patrols ** Gay beat, an area frequented by gay men * Battery (c ...
s, each a crotchet (that is, two quavers) long. The asymmetric time signature , on the other hand, while also having eight quavers in a bar, divides them into three beats, the first three quavers long, the second three quavers long, and the last just two quavers long. These kinds of rhythms are used, for example, by
Béla Bartók Béla Viktor János Bartók (; ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt are regarded as Hu ...
, who was influenced by similar rhythms in
Bulgarian folk music The music of Bulgaria refers to all forms of music associated with the country of Bulgaria, including classical, folk, popular music, and other forms. Classical music, opera, and ballet are represented by composers Emanuil Manolov, Pancho Vlad ...
. The third movement of Bartók's String Quartet No. 5, a scherzo marked ''alla bulgarese'' features a " rhythm (4+2+3)".
Stravinsky Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential 20th-century clas ...
's ''Octet'' for Wind Instruments "ends with a jazzy 3+3+2 = 8 swung coda". Stravinsky himself found a kinship with additive rhythms in music of the
renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
and
baroque The Baroque (, ; ) is a style of architecture, music, dance, painting, sculpture, poetry, and other arts that flourished in Europe from the early 17th century until the 1750s. In the territories of the Spanish and Portuguese empires including ...
periods. For example, he marvelled at the ''Laudate Pueri'' from
Monteverdi Claudio Giovanni Antonio Monteverdi (baptized 15 May 1567 – 29 November 1643) was an Italian composer, choirmaster and string player. A composer of both secular and sacred music, and a pioneer in the development of opera, he is consider ...
’s Vespers of 1610, where the music follows the natural accentuation of the Latin words to create metrical groupings of twos, threes and fours at the very start: “I know of no music before or since…. which so felicitously exploits accentual and metrical variation and irregularity, and no more subtle rhythmic construction of any kind than that which is set in motion at the beginning of the ‘Laudate Pueri,’ if, that is, the music is sung according to the verbal accents instead of… the editor’s bar-lines.” Additive patterns also occur in some music of
Philip Glass Philip Glass (born January 31, 1937) is an American composer and pianist. He is widely regarded as one of the most influential composers of the late 20th century. Glass's work has been associated with minimalism, being built up from repetitive ...
, and other minimalists, most noticeably the "one-two-one-two-three" chorus parts in ''
Einstein on the Beach ''Einstein on the Beach'' is an opera in four acts composed by Philip Glass and directed by theatrical producer Robert Wilson, who also collaborated with Glass on the work's libretto. The opera eschews traditional narrative in favor of a formali ...
''. They may also occur in passing in pieces which are on the whole in conventional meters. In jazz,
Dave Brubeck David Warren Brubeck (; December 6, 1920 – December 5, 2012) was an American jazz pianist and composer. Often regarded as a foremost exponent of cool jazz, Brubeck's work is characterized by unusual time signatures and superimposing contrasti ...
's song "
Blue Rondo à la Turk "Blue Rondo à la Turk" is a jazz standard composition by Dave Brubeck. It appeared on the album ''Time Out'' in 1959. It is written in time, with one side theme in and the choice of rhythm was inspired by the Turkish aksak time signatures. It ...
" features bars of nine quavers grouped into patterns of at the start.
George Harrison George Harrison (25 February 1943 – 29 November 2001) was an English musician and singer-songwriter who achieved international fame as the lead guitarist of the Beatles. Sometimes called "the quiet Beatle", Harrison embraced Indian c ...
's song "
Here Comes the Sun "Here Comes the Sun" is a song by the English rock band the Beatles from their 1969 album ''Abbey Road''. It was written by George Harrison and is one of his best-known compositions. Harrison wrote the song in early 1969 at the country house o ...
" on the
Beatles The Beatles were an English rock band, formed in Liverpool in 1960, that comprised John Lennon, Paul McCartney, George Harrison and Ringo Starr. They are regarded as the most influential band of all time and were integral to the developm ...
' album ''
Abbey Road ''Abbey Road'' is the eleventh studio album by the English rock band the Beatles. It is the last album the group started recording, although '' Let It Be'' was the last album completed before the band's break-up in April 1970. It was mostly ...
'' features a rhythm "which switches between , and on the bridge". "The special effect of running even eighth notes accented as if triplets against the grain of the underlying backbeat is carried to a point more reminiscent of
Stravinsky Igor Fyodorovich Stravinsky (6 April 1971) was a Russian composer, pianist and conductor, later of French (from 1934) and American (from 1945) citizenship. He is widely considered one of the most important and influential 20th-century clas ...
than of the Beatles".
Olivier Messiaen Olivier Eugène Prosper Charles Messiaen (, ; ; 10 December 1908 – 27 April 1992) was a French composer, organist, and ornithology, ornithologist who was one of the major composers of the 20th-century classical music, 20th century. His m ...
made extensive use of additive rhythmic patterns, much of it stemming from his close study of the rhythms of Indian music. His "Danse de la fureur, pour les sept trompettes" from ''The Quartet for the End of Time'' is a bracing example. A gentler exploration of additive patterns can be found in "Le Regard de la Vierge" from the same composer's piano cycle '' Vingt Regards sur l'enfant-Jésus''.
György Ligeti György Sándor Ligeti (; ; 28 May 1923 – 12 June 2006) was a Hungarian-Austrian composer of contemporary classical music. He has been described as "one of the most important avant-garde composers in the latter half of the twentieth century ...
's
Étude An étude (; ) or study is an instrumental musical composition, usually short, designed to provide practice material for perfecting a particular musical skill. The tradition of writing études emerged in the early 19th century with the rapidl ...
No. 13, "L'escalier du diable" features patterns involving quavers grouped in twos and threes. The rhythm at the start of the study follows the pattern , then . According to the composer's note, the time signature "serves only as a guideline, the actual meter consists of 36 quavers (three 'bars'), divided assymetrically".


Sub-Saharan African rhythm

A divisive form of
cross-rhythm In music, a cross-beat or cross-rhythm is a specific form of polyrhythm. The term ''cross rhythm '' was introduced in 1934 by the musicologist Arthur Morris Jones (1889–1980). It refers to when the rhythmic conflict found in polyrhythms is th ...
is the basis for most Sub-Saharan African music traditions. Rhythmic patterns are generated by simultaneously dividing a span of musical time by a triple-beat scheme and a duple-beat scheme.
In the development of cross rhythm, there are some selected rhythmic materials or beat schemes that are customarily used. These beat schemes, in their generic forms, are simple divisions of the same musical period in equal units, producing varying rhythmic densities or motions. At the center of a core of rhythmic traditions within which the composer conveys his ideas is the technique of cross-rhythm. The technique of cross-rhythm is a simultaneous use of contrasting rhythmic patterns within the same scheme of accents or meter... By the very nature of the desired resultant rhythm, the main beat scheme cannot be separated from the secondary beat scheme. It is the interplay of the two elements that produces the cross-rhythmic texture.
"the entire African rhythmic structure... is divisive in nature".


Tresillo: divisive and additive interpretations

In divisive form, the strokes of tresillo contradict the beats. In additive form, the strokes of tresillo ''are'' the beats. From a metrical perspective then, the two ways of perceiving tresillo constitute two different rhythms. On the other hand, from the perspective of simply the pattern of attack-points, tresillo is a shared element of traditional folk music from the northwest tip of Africa to southeast tip of Asia.


Additive structure

"Tresillo" is also found within a wide geographic belt stretching from
Morocco Morocco (),, ) officially the Kingdom of Morocco, is the westernmost country in the Maghreb region of North Africa. It overlooks the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria to A ...
in North Africa to
Indonesia Indonesia, officially the Republic of Indonesia, is a country in Southeast Asia and Oceania between the Indian and Pacific oceans. It consists of over 17,000 islands, including Sumatra, Java, Sulawesi, and parts of Borneo and New Gui ...
in South Asia. Use of the pattern in
Moroccan music Moroccan music varies greatly between geographic regions and social groups. It is influenced by musical styles including Arab, Berber, Andalusi, Mediterranean, Saharan, West African, and others. Musical styles vary by geography. Andalusi music ...
can be traced back to slaves brought north across the Sahara Desert from present-day
Mali Mali (; ), officially the Republic of Mali,, , ff, 𞤈𞤫𞤲𞥆𞤣𞤢𞥄𞤲𞤣𞤭 𞤃𞤢𞥄𞤤𞤭, Renndaandi Maali, italics=no, ar, جمهورية مالي, Jumhūriyyāt Mālī is a landlocked country in West Africa. Ma ...
. This pattern may have migrated east from North Africa to Asia through the spread of
Islam Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God (or '' Allah'') as it was revealed to Muhammad, the ...
. In Middle Eastern and Asian music, the figure is generated through additive rhythm. : \new RhythmicStaff


Divisive structure

The most basic duple-pulse figure found in the
Music of Africa Given the vastness of the African continent, its music is diverse, with regions and nations having many distinct musical traditions. African music includes the genres amapiano, Jùjú, Fuji, Afrobeat, Highlife, Makossa, Kizomba, and othe ...
and music of the
African diaspora The African diaspora is the worldwide collection of communities descended from native Africans or people from Africa, predominantly in the Americas. The term most commonly refers to the descendants of the West and Central Africans who were ...
is a figure the Cubans call ''tresillo'', a Spanish word meaning 'triplet' (three equal beats in the same time as two main beats). However, in the vernacular of Cuban popular music, the term refers to the figure shown below. : \new RhythmicStaff African-based music has a divisive rhythm structure. Tresillo is generated through cross-rhythm: 8 pulses ÷ 3 = 2 cross-beats (consisting of three pulses each), with a remainder of a partial cross-beat (spanning two pulses). In other words, 8 ÷ 3 = 2,  r2. Tresillo is a cross-rhythmic fragment. Because of its irregular pattern of attack-points, "tresillo" in African and African-based musics has been mistaken for a form of additive rhythm.
Although the difference between the two ways of notating this rhythm may seem small, they stem from fundamentally different conceptions. Those who wish to convey a sense of the rhythm's background
ain beats Ain (, ; frp, En) is a department in the Auvergne-Rhône-Alpes region in Eastern France. Named after the Ain river, it is bordered by the Saône and Rhône rivers. Ain is located on the country's eastern edge, on the Swiss border, where it ...
and who understand the surface morphology in relation to a regular subsurface articulation, will prefer the divisive format. Those who imagine the addition of three, then three, then two sixteenth notes will treat the well-formedness of 3 + 3 + 2 as fortuitous, a product of grouping rather than of metrical structure. They will be tempted to deny that African music has a bona fide metrical structure because of its frequent departures from normative grouping structure.


See also

*
Counting (music) In music, counting is a system of regularly occurring sounds that serve to assist with the performance or audition of music by allowing the easy identification of the beat. Commonly, this involves verbally counting the beats in each measure as ...


References

Sources * * * * * * * * * * * * * Reprinted 1988, New York: Columbia University Press. (cloth); (pbk). * * * {{DEFAULTSORT:Additive Rhythm And Divisive Rhythm Rhythm and meter fr:Division du temps (solfège) pl:Rytmika zmienna pl:Rytmika okresowa sv:Asymmetrisk rytm