Editing Additive rhythm and divisive rhythm

{{Use dmy dates|date=January 2023}}
{{Use shortened footnotes|date=February 2021}}
In [[music]], the terms '''''additive''''' and '''''divisive''''' are used to distinguish two types of both [[rhythm]] and [[meter (music)|meter]]:

* A '''divisive''' (or, alternately, '''multiplicative''') '''rhythm''' is a [[rhythm]] in which a larger period of time is divided into smaller [[rhythmic unit]]s 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 [[concatenation|concatenating]] (joining end to end) a series of units into larger units of unequal length, such as a {{music|time|5|8}} meter produced by the regular alternation of {{music|time|2|8}} and {{music|time|3|8}}.{{sfn|London|2001|loc=§ I.8}}

When applied to meters, the terms ''perfect'' and ''imperfect'' are sometimes used as the equivalents of ''divisive'' and ''additive'', respectively .{{sfn|Read|1969|p=150}}[[Image:Additive and divisive meters.png|thumb|250px|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, {{music|time|4|8}} (or, more commonly, {{music|time|2|4}}) is divisive while {{music|time|5|8}} is additive.

The terms ''additive'' and ''divisive'' originate with Curt Sachs's book ''Rhythm and Tempo'' (1953),{{sfn|Sachs|1953}} while the term ''[[Aksak|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]].{{sfn|Fracile|2003|p=198}} 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:

{{blockquote|[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.{{sfn|London|2001|loc=§ I.8}}}}

Winold recommends that, "[[metric structure]] is best described through detailed analysis of [[pulse group]]ings on various levels rather than through attempts to represent the organization with a single term".{{sfn|Winold|1975|p=217}}

[[Sub-Saharan African music traditions|Sub-Saharan African music]] and most [[European music|European (Western) music]] is divisive, while [[Music of India|Indian]] and other [[Music of Asia|Asian]] musics may be considered as primarily additive. However, many pieces of music cannot be clearly labeled divisive or additive.

==Divisive rhythm==

For example: {{music|time|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&nbsp;&&nbsp;2&nbsp;&&nbsp;3&nbsp;&&nbsp;4&nbsp;&).
[[File:Divisive rhythm in 4-4 time.wav|thumb|Divisive rhythm in {{music|time|4|4}} time]]
[[File:Divisive rhythm in 4-4 time.png|thumb|center|300px|Divisive Rhythm. 1&nbsp;whole note = 2&nbsp;half notes = 4&nbsp;quarter notes = 8&nbsp;eighth notes = 16&nbsp;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.{{sfn|Agawu|2003|p=86}} This type of rhythm is also referred to in musicological literature by the Turkish word ''[[aksak]]'', which means "limping".{{sfn|Brăiloiu|1951}}{{sfn|Fracile|2003|p=198}} In the special case of [[time signature]]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''.{{sfn|Beck & Reiser|1998|pp=181–182}}
[[File:Additive rhythm in 8-8 time.wav|thumb|Additive rhythm in {{music|time|3+3+2|8}} time]]
[[File:Additive rhythm 3+3+2 over 8.png|thumb|center|300px|Additive rhythm {{music|time|3+3+2|8}} time. 1 whole note = 8 eighth notes =&nbsp;3&nbsp;+&nbsp;3&nbsp;+&nbsp;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|date=April 2010}} – that is, [[metre (music)|meter]]s which have a regular pattern of beats of uneven length. For example, the [[time signature]] {{music|time|4|4}} indicates each [[bar (music)|bar]] is eight [[quaver]]s long, and has four [[beat (music)|beat]]s, each a [[crotchet]] (that is, two quavers) long. The asymmetric time signature {{music|time|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 (Bartók)|String Quartet No. 5]], a scherzo marked ''alla bulgarese'' features a "{{music|time|9|8}} rhythm (4+2+3)".{{sfn|Walsh|1982|p=66}} [[Stravinsky]]'s [[Octet (Stravinsky)|''Octet'']] for Wind Instruments "ends with a jazzy 3+3+2 = 8 swung coda".{{sfn|Walsh|1988|p=127}} 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: [[File:Monteverdi opening of Laudate Pueri.png|thumb|center|500px|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".<ref>Stravinsky, I. (1972, p120) Themes and Conclusions. London, Faber.</ref><ref>{{Cite web|url=https://www.youtube.com/watch?v=ylEbnaiGBz4|title = Monteverdi: Vespro della Beata Vergine, SV 206 – IV. Laudate, pueri Dominum a 8 (Live)|via = [[YouTube]]}}</ref>

Additive patterns also occur in some music of [[Philip Glass]], and other [[minimalism|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 {{serif|'''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 {{music|time|11|8}}, {{music|time|4|4}} and {{music|time|7|8}} on the bridge".{{sfn|Margotin & Guesdon|2013|p=576}} "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".{{sfn|Pollack|n.d.}}

[[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 [[Quatuor pour la fin du temps|''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 [[Études (Ligeti)|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 {{serif|'''2+2+3'''}}, then {{serif|'''2+2+2+3'''}}. According to the composer's note, the {{music|time|12|8}} time signature "serves only as a guideline, the actual meter consists of 36 quavers (three 'bars'), divided asymmetrically".{{sfn|Ligeti|1994|p={{page needed|date=February 2021}}}}

==Sub-Saharan African rhythm==
{{Main|Sub-Saharan African music traditions|Rhythm in Sub-Saharan Africa}}

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.

<blockquote>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.{{sfn|Ladzekpo|1995}}</blockquote>

"the entire African rhythmic structure... is divisive in nature".{{sfn|Novotney|1998|p=147}}

{{blockquote|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.{{sfn|Agawu|2003|p=94}}}}

{{blockquote|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".{{sfn|Novotney|1998|p=158}}}}

==Tresillo: divisive and additive interpretations==
{{Main|Tresillo (rhythm)}}
In divisive form, the strokes of [[Tresillo (rhythm)|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]].{{sfn|Peñalosa|2009|p=236}} In Middle Eastern and Asian music, the figure is generated through additive rhythm.

:<score>
\new RhythmicStaff {
   \clef percussion
   \time 8/16
   \autoBeamOff
   \repeat volta 2 { c8. c c8 }
}
</score>
[[File:Tresillo divisive.mid]]

===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.

:<score sound="1">
\new RhythmicStaff {
   \clef percussion
   \time 2/4 \set Staff.midiInstrument = #"woodblock"
   \repeat volta 2 { c8. c16 r8 c }
}
</score>

African-based music has a divisive rhythm structure.{{sfn|Novotney|1998|p=100}} 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&nbsp;÷&nbsp;3&nbsp;=&nbsp;2,&nbsp;[[Remainder|r]]2. 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.

<blockquote>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&nbsp;+&nbsp;3&nbsp;+&nbsp;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.{{sfn|Agawu|2003|p=87}}</blockquote>

==See also==
*[[Counting (music)]]

==References==
{{Reflist}}
'''Sources'''
{{div col|colwidth=45em}}
* {{cite book|last=Agawu|first=Victor Kofi|year=2003|title=Representing African Music: Postcolonial Notes, Queries, Positions|location=New York|publisher=Routledge|isbn=0-415-94390-6}}
* {{wikicite|ref={{harvid|Beck & Reiser|1998}}|reference=Beck, Jill, and Joseph Reiser (1998). ''Moving Notation: A Handbook of Musical Rhythm and Elementary Labanotation for the Dancer''. Performing Arts Studies 6. Amsterdam: Harwood Academic Publishers. {{ISBN|90-5702-178-1}} (cloth); {{ISBN|90-5702-179-X}} (pbk).}}
* {{wikicite|ref={{harvid|Brăiloiu|1951}}|reference=Brăiloiu, Constantin. 1951. "Le rythme Aksak" ''Revue de Musicologie'' 33, nos. 99 and 100 (December): 71–108.}}
* {{wikicite|ref={{harvid|Fracile|2003}}|reference=Fracile, Nice (2003). "The 'Aksak' Rhythm, a Distinctive Feature of the Balkan Folklore". ''Studia Musicologica Academiae Scientiarum Hungaricae'' 44, nos. 1 and 2:197–210.}}
* {{wikicite|ref={{harvid|Ladzekpo|1995}}|reference=Ladzekpo, C. K. (1995). [https://web.archive.org/web/20121123000213/https://home.comcast.net/~dzinyaladzekpo/Myth.html "The Myth of Cross-Rhythm"], ''Foundation Course in African Dance-Drumming'' (webpage, accessed 24 April 2010).}}
* {{cite book|last=Ligeti|first=György|date=1994|title=Études pour Piano|volume=Book 2|location=Mainz|publisher=Schott}} {{YouTube|Q6LmG9myHxA|Ligeti: Étude No. 13}}
* {{wikicite|ref={{harvid|London|2001}}|reference=London, Justin (2001). "Rhythm". ''[[The New Grove Dictionary of Music and Musicians]]'', second edition, edited by [[Stanley Sadie]] and [[John Tyrrell (musicologist)|John Tyrrell]]. London: Macmillan Publishers.}}
* {{wikicite|ref={{harvid|Margotin & Guesdon|2013}}|reference=Margotin, Philippe, and Jean-Michel Guesdon (2013). ''All the Songs: The Story Behind Every Beatles Release''. New York: Black Dog and Leventhal. {{ISBN|9781579129521}}.}}
* {{cite book|last=Novotney|first=Eugene D.|year=1998|title=The Three Against Two Relationship as the Foundation of Timelines in West African Musics|location=Urbana, Illinois|publisher=University of Illinois}}
* {{wikicite|ref={{harvid|Peñalosa|2009}}|reference=Peñalosa, David (2009). ''The Clave Matrix; Afro-Cuban Rhythm: Its Principles and African Origins''. Redway, California: Bembe Inc. {{ISBN|1-886502-80-3}}.}}
* {{wikicite|ref={{harvid|Pollack|n.d.}}|reference=Pollack, Alan (n.d.). "[http://www.icce.rug.nl/~soundscapes/DATABASES/AWP/hcts.shtml Notes on 'Here Comes the Sun']" (retrieved 14 February 2012).}}
* {{wikicite|ref={{harvid|Read|1969}}|reference=[[Gardner Read|Read, Gardner]] (1969). ''Music Notation: A Manual of Modern Practice'', second edition. Boston: Allyn and Bacon, Inc.}}
* {{cite book |last=Sachs |first=Curt |date=1953 |title=Rhythm and Tempo: A Study in Music History |location=New York City |publisher=W. W. Norton}} Reprinted 1988, New York: Columbia University Press. {{ISBN|0-231-06910-3}} (cloth); {{ISBN|0-231-06911-1}} (pbk).
* {{cite book|last=Walsh|first=Stephen|author-link=Stephen Walsh (writer)|year=1982|title=Bartók Chamber Music|series=BBC Music Guides|location=London|publisher=BBC|isbn=978-0563124658}}
* {{wikicite|ref={{harvid|Walsh|1988}}|reference=[[Stephen Walsh (writer)|Walsh, Stephen]] (1988). ''The Music of Stravinsky''. London: Routledge.}}
* {{wikicite|ref={{harvid|Winold|1975}}|reference=Winold, Allen (1975). "Rhythm in Twentieth-Century Music". In ''Aspects of Twentieth-Century Music'', edited by Gary Wittlich, Englewood Cliffs, New Jersey: Prentice-Hall. {{ISBN|0-13-049346-5}}. pp. 208-269.}}
{{div col end}}

{{Rhythm and meter}}

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[[Category:Rhythm and meter]]

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