Additive rhythm and divisive rhythm

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In music, the terms additive and divisive are used to distinguish two types of both rhythm and meter:

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When applied to meters, the terms perfect and imperfect are sometimes used as the equivalents of divisive and additive, respectively . [2]

Additive and divisive meters. Additive and divisive meters.png
Additive and divisive meters.

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, 4
8
(or, more commonly, 2
4
) is divisive while 5
8
is additive.

The terms additive and divisive originate with Curt Sachs's book Rhythm and Tempo (1953), [3] 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. [4] 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, Justin London states that:

[i]n discussions of rhythmic notation, practice or style, few terms are as confusing or used as confusedly as 'additive' and 'divisive'. … These confusions stem from two misapprehensions. The first is a failure to distinguish between systems of notation (which may have both additive and divisive aspects) and the music notated under such a system. The second involves a failure to understand the divisive and additive aspects of meter itself. [1]

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". [5]

Sub-Saharan African music and most European (Western) music is divisive, while Indian 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: 4
4
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 &).

Divisive rhythm in 4
4
time
Divisive Rhythm. 1 whole note = 2 half notes = 4 quarter notes = 8 eighth notes = 16 sixteenth notes Divisive rhythm in 4-4 time.png
Divisive Rhythm. 1 whole note = 2 half notes = 4 quarter notes = 8 eighth notes = 16 sixteenth notes

Additive rhythm features nonidentical or irregular durational groups following one another at two levels, within the bar and between bars or groups of bars. [6] This type of rhythm is also referred to in musicological literature by the Turkish word aksak , which means "limping". [7] [4] In the special case of time signatures 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. [8]

Additive rhythm in 3+3+2
8
time
Additive rhythm
8 time. 1 whole note = 8 eighth notes = 3 + 3 + 2. Additive rhythm 3+3+2 over 8.png
Additive rhythm
8
time. 1 whole note = 8 eighth notes = 3 + 3 + 2.

The term additive rhythm is also often used to refer to what are also incorrectly called asymmetric rhythms and even irregular rhythms[ citation needed ] – that is, meters which have a regular pattern of beats of uneven length. For example, the time signature 4
4
indicates each bar is eight quavers long, and has four beats, each a crotchet (that is, two quavers) long. The asymmetric time signature 3+3+2
8
, 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, who was influenced by similar rhythms in Bulgarian Folk Music. The third movement of Bartók's String Quartet No. 5, a scherzo marked alla bulgarese features a "9
8
rhythm (4+2+3)". [9] Stravinsky's Octet for Wind Instruments "ends with a jazzy 3+3+2 = 8 swung coda". [10] Stravinsky himself found a kinship with additive rhythms in music of the renaissance and baroque periods. For example, he marvelled at the Laudate Pueri from Monteverdi'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:

Monteverdi opening of Laudate Pueri Monteverdi opening of Laudate Pueri.png
Monteverdi opening of Laudate Pueri

"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". [11] [12]

Additive patterns also occur in some music of Philip Glass, and other minimalists, most noticeably the "one-two-one-two-three" chorus parts in Einstein on the Beach . They may also occur in passing in pieces which are on the whole in conventional meters. In jazz, Dave Brubeck's song "Blue Rondo à la Turk" features bars of nine quavers grouped into patterns of 2+2+2+3 at the start. George Harrison's song "Here Comes the Sun" on The Beatles' album Abbey Road features a rhythm "which switches between 11
8
, 4
4
and 7
8
on the bridge". [13] "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 than of the Beatles". [14]

Olivier Messiaen 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's Étude 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 2+2+3, then 2+2+2+3. According to the composer's note, the 12
8
time signature "serves only as a guideline, the actual meter consists of 36 quavers (three 'bars'), divided assymetrically". [15]

Sub-Saharan African rhythm

A divisive form of cross-rhythm 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. [16]

"the entire African rhythmic structure... is divisive in nature". [17]

Do African musicians think additively? The evidence so far is that they do not. Writing in 1972 about the Yoruba version of the standard pattern, Kubik stated. 'There is no evidence that the musicians themselves think it as additive.' I have argued elsewhere that additive thinking is foreign to many African musicians' ways of proceeding. ... Then, too, there appears to be no trace of an additive conception in the discourses of musicians, whether directly or indirectly. … It would seem, then, that whereas structural analysis (based on European metalanguage) endorses an additive conception of the standard pattern, cultural analysis (originating in African musicians' thinking) denies it, ... no dancer thinks in cycles of 12 when interpreting the standard pattern. The evidence of the rate at which the dance feet move is that 4, not 12, is the reckoning that most closely approximates the regulative beat. ... what can be said for sure is that the cycle of four beats is felt and thus relied upon. This is cultural knowledge that players and especially dancers possess; without such knowledge, it is difficult to perform accurately. [18]

The African rhythmic structure which generates the standard pattern is a divisive structure and not an additive one … the standard pattern represents a series of attack points that outline the onbeat three-against-two / offbeat three-against-two sequence, not a series of durational values". [19]

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 in North Africa to Indonesia in South Asia. Use of the pattern in Moroccan music can be traced back to slaves brought north across the Sahara Desert from present-day Mali. This pattern may have migrated east from North Africa to Asia through the spread of Islam. [20] In Middle Eastern and Asian music, the figure is generated through additive rhythm.

Additive rhythm and divisive rhythm

Divisive structure

The most basic duple-pulse figure found in the Music of Africa and music of the African diaspora 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.

Additive rhythm and divisive rhythm

African-based music has a divisive rhythm structure. [21] 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 [main beats], 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. [22]

See also

Related Research Articles

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In music, syncopation is a variety of rhythms played together to make a piece of music, making part or all of a tune or piece of music off-beat. More simply, syncopation is "a disturbance or interruption of the regular flow of rhythm": a "placement of rhythmic stresses or accents where they wouldn't normally occur". It is the correlation of at least two sets of time intervals.

The time 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 value is equivalent to a beat.

<span class="mw-page-title-main">Metre (music)</span> Aspect of music

In music, metre or meter refers to regularly recurring patterns and accents such as bars and beats. Unlike rhythm, metric onsets are not necessarily sounded, but are nevertheless implied by the performer and expected by the listener.

<span class="mw-page-title-main">Polyrhythm</span> Simultaneous use of two or more conflicting rhythms

Polyrhythm is the simultaneous use of two or more rhythms that are not readily perceived as deriving from one another, or as simple manifestations of the same meter. The rhythmic layers may be the basis of an entire piece of music (cross-rhythm), or a momentary section. Polyrhythms can be distinguished from irrational rhythms, which can occur within the context of a single part; polyrhythms require at least two rhythms to be played concurrently, one of which is typically an irrational rhythm. Concurrently in this context means within the same rhythmic cycle. The underlying pulse, whether explicit or implicit can be considered one of the concurrent rhythms. For example, the son clave is poly-rhythmic because its 3 section suggests a different meter from the pulse of the entire pattern.

In music, hemiola is the ratio 3:2. The equivalent Latin term is sesquialtera. In rhythm, hemiola refers to three beats of equal value in the time normally occupied by two beats. In pitch, hemiola refers to the interval of a perfect fifth.

In music, duration is an amount of time or how long or short a note, phrase, section, or composition lasts. "Duration is the length of time a pitch, or tone, is sounded." A note may last less than a second, while a symphony may last more than an hour. One of the fundamental features of rhythm, or encompassing rhythm, duration is also central to meter and musical form. Release plays an important part in determining the timbre of a musical instrument and is affected by articulation.

In music, an ostinato is a motif or phrase that persistently repeats in the same musical voice, frequently in the same pitch. Well-known ostinato-based pieces include classical compositions such as Ravel's Boléro and the Carol of the Bells, and popular songs such as Donna Summer and Giorgio Moroder's "I Feel Love" (1977), Henry Mancini's theme from Peter Gunn (1959), The Who's "Baba O'Riley" (1971), and The Verve's "Bitter Sweet Symphony" (1997).

<span class="mw-page-title-main">Clave (rhythm)</span> Rhythmic pattern in Cuban music

The clave is a rhythmic pattern used as a tool for temporal organization in Cuban music. In Spanish, clave literally means key, clef, code, or keystone. It is present in a variety of genres such as Abakuá music, rumba, conga, son, mambo, salsa, songo, timba and Afro-Cuban jazz. The five-stroke clave pattern represents the structural core of many Cuban rhythms.

<span class="mw-page-title-main">Bar (music)</span> Segment of time corresponding to a specific number of beats

In musical notation, a bar is a segment of time corresponding to a specific number of beats in which each beat is represented by a particular note value and the boundaries of the bar are indicated by vertical bar lines. Dividing music into bars provides regular reference points to pinpoint locations within a musical composition. It also makes written music easier to follow, since each bar of staff symbols can be read and played as a batch.

Ewe music is the music of the Ewe people of Togo, Ghana, and Benin, West Africa. Instrumentation is primarily percussive and rhythmically the music features great metrical complexity. Its highest form is in dance music including a drum orchestra, but there are also work, play, and other songs. Ewe music is featured in A. M. Jones's Studies in African Music.

<span class="mw-page-title-main">Period (music)</span> Musical unit of two interdependent phrases

In music, the term period refers to certain types of recurrence in small-scale formal structure. In twentieth-century music scholarship, the term is usually used as defined by the Oxford Companion to Music: "a period consists of two phrases, antecedent and consequent, each of which begins with the same basic motif." Earlier usage varied somewhat, but usually referred to similar notions of symmetry, recurrence, and closure. The concept of a musical period originates in comparisons between music structure and rhetoric at least as early as the 16th century.

Afro-Cuban jazz is the earliest form of Latin jazz. It mixes Afro-Cuban clave-based rhythms with jazz harmonies and techniques of improvisation. Afro-Cuban music has deep roots in African ritual and rhythm. Afro-Cuban jazz emerged in the early 1940s with the Cuban musicians Mario Bauzá and Frank Grillo "Machito" in the band Machito and his Afro-Cubans in New York City. In 1947, the collaborations of bebop trumpeter Dizzy Gillespie and percussionist Chano Pozo brought Afro-Cuban rhythms and instruments, such as the tumbadora and the bongo, into the East Coast jazz scene. Early combinations of jazz with Cuban music, such as "Manteca" and "Mangó Mangüé", were commonly referred to as "Cubop" for Cuban bebop.

Ewe drumming refers to the drumming ensembles of the Ewe people of Ghana, Togo, and Benin. The Ewe are known for their experience in drumming throughout West Africa. The sophisticated cross rhythms and polyrhythms in Ewe drumming are similar to those in Afro-Caribbean music and late jazz. The original purpose of Ewe drumming were sung or performed by warriors. Now the songs and performed to celebrate or for recreational use. For example, Agbadza was originally used as a warrior dance but is now used to celebrate events.

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 alternation of binary and ternary quantities, as in 2+3, 2+2+3, 2+3+3, etc. The name literally means "limping", "stumbling", or "slumping", and has been borrowed by Western ethnomusicologists to refer generally to irregular, or additive meters.

<span class="mw-page-title-main">Bell pattern</span> Rhythmic pattern of striking a hand-held bell or other instrument

A bell pattern is a rhythmic pattern of striking a hand-held bell or other instrument of the idiophone family, to make it emit a sound at desired intervals. It is often a key pattern, in most cases it is a metal bell, such as an agogô, gankoqui, or cowbell, or a hollowed piece of wood, or wooden claves. In band music, bell patterns are also played on the metal shell of the timbales, and drum kit cymbals.

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 the basis of an entire musical piece.

<span class="mw-page-title-main">Rhythm in Sub-Saharan Africa</span> Periodicity in music developed by sub-Saharan African peoples

Sub-Saharan African music is characterised by a "strong rhythmic interest" that exhibits common characteristics in all regions of this vast territory, so that Arthur Morris Jones (1889–1980) has described the many local approaches as constituting one main system. C. K. Ladzekpo also affirms the profound homogeneity of approach. West African rhythmic techniques carried over the Atlantic were fundamental ingredients in various musical styles of the Americas: samba, forró, maracatu and coco in Brazil, Afro-Cuban music and Afro-American musical genres such as blues, jazz, rhythm & blues, funk, soul, reggae, hip hop, and rock and roll were thereby of immense importance in 20th century popular music. The drum is renowned throughout Africa.

Tresillo is a rhythmic pattern used in Latin American music. It is a more basic form of the rhythmic figure known as the habanera.

William Oscar "Willie" Anku was a Ghanaian music theorist, ethnomusicologist, composer, and performer. His work combined Western set theory with computer programming and experience in working with performers of various West African musical traditions to create a comprehensive theory of African rhythm. He was "unique among Africa-based music theorists in attracting the attention of the US-based Society for Music Theory," being invited to give plenary lectures and receiving tributes from prominent US-based theorists.

References

  1. 1 2 London 2001, § I.8.
  2. Read 1969, p. 150.
  3. Sachs 1953.
  4. 1 2 Fracile 2003, p. 198.
  5. Winold 1975, p. 217.
  6. Agawu 2003, p. 86.
  7. Brăiloiu 1951.
  8. Beck & Reiser 1998, pp. 181–182.
  9. Walsh 1982, p. 66.
  10. Walsh 1988, p. 127.
  11. Stravinsky, I. (1972, p120) Themes and Conclusions. London, Faber.
  12. "Monteverdi: Vespro della Beata Vergine, SV 206 – IV. Laudate, pueri Dominum a 8 (Live)" via YouTube.
  13. Margotin & Guesdon 2013, p. 576.
  14. Pollack n.d.
  15. Ligeti 1994, p. [ page needed ].
  16. Ladzekpo 1995.
  17. Novotney 1998, p. 147.
  18. Agawu 2003, p. 94.
  19. Novotney 1998, p. 158.
  20. Peñalosa 2009, p. 236.
  21. Novotney 1998, p. 100.
  22. Agawu 2003, p. 87.

Sources