Additive rhythm and divisive rhythm
A divisive (or, more commonly, multiplicative) rhythm is a rhythm 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 (joining end to end) a series of units into larger units of unequal length, such as a 5/8 meter produced by the regular alternation of 2/8 and 3/8 (London 2001, §I.8). When applied to meters, the terms "perfect" and "imperfect" are sometimes used as the equivalents of "divisive" and "additive", respectively (Read 1964,[page needed]).
For example, 4 may be evenly divided by 2 (4/2 = 2) or reached through repeatedly adding 2 (2 + 2 = 4), while 5 is only evenly divisible by 5 and 1 (5/2 = 2.5; 5/3 = 1.66) and may be reached by repeatedly adding 1 or 5 (2 + 2 = 4, 4 + 2 = 6; 3 + 3 = 6); thus 4/8 is divisive while 5/8 is additive.
The terms additive and divisive originate with Curt Sachs's book Rhythm and Tempo (1953) (Agawu 2003, 86), while the term akshak rhythm was introduced for the former concept at about the same time by Constantin Brăiloiu (1951), in agreement with the Turkish musicologist Ahmed Adnan Saygun (Fracile 2003, 198). The relationship between additive and divisive rhythms is complex, and the terms are often used in imprecise ways. Justin London states in his article on rhythm in the second edition of the New Grove Dictionary of Music and Musicians that "In 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 metre itself" (London 2001, §I.8). 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" (Winold 1975, 217)
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 consist of one measure (whole note: 1) divided into a stronger first beat and slightly less strong second beat (half notes: 12), which are in turn divided, by two weaker beats (quarter notes: 1234), and again divided into still weaker beats (eighth notes: 1 & 2 & 3 & 4 &).
Additive rhythm 
Additive, as opposed to divisive, rhythm, features nonidentical or irregular durational groups following one another at two levels, within the bar and between bars or groups of bars (Agawu 2003, 86). This type of rhythm is also referred to in musicological literature by the Turkish word aksak, which means "limping" (Braïloiu 1951; Fracile 2003, 198). 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" (Beck and Reiser 1998, 181–82).
The term additive rhythm is also often used to refer to what are also incorrectly called asymmetric rhythms and even irregular rhythms – that is, metres 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, and 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 metres.
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. (Ladzekpo 1995)
"the entire African rhythmic structure . . . is divisive in nature" (Novotney 1998, 147).
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 (Agawu 2003, 94).
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". (Novotney 1998, 158)
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 (Peñalosa 2009, 236). In Middle East and Asian music the figure is generated through additive 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.
African-based music has a divisive rhythm structure (Novotney 1998, 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 ÷ 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. (Agawu 2003, 87)
See also 
- Agawu, Victor Kofi (2003). Representing African Music: Postcolonial Notes, Queries, Positions. New York: Routledge. ISBN 0-415-94390-6.
- 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).
- Brăiloiu, Constantin. 1951. "Le rythme Aksak" Revue de Musicologie 33, nos. 99 and 100 (December): 71–108.
- 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.
- Ladzekpo, C. K. (1995). "The Myth of Cross-Rhythm", Foundation Course in African Dance-Drumming (webpage, accessed 24 April 2010).
- London, Justin (2001). "Rhythm". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
- Novotney, Eugene D. (1998). The Three Against Two Relationship as the Foundation of Timelines in West African Musics. Urbana, IL: University of Illinois.
- Peñalosa, David (2009). The Clave Matrix; Afro-Cuban Rhythm: Its Principles and African Origins. Redway, CA: Bembe Inc. ISBN 1-886502-80-3.
- Read, Gardner. 1964. Music Notation: A Manual of Modern Practice. Boston: Allyn and Bacon, Inc.
- Sachs, Curt (1953). Rhythm and Tempo: A Study in Music History. New York: Norton. Reprinted 1988, New York: Columbia University Press. ISBN 0-231-06910-3 (cloth); ISBN 0-231-06911-1 (pbk).
- Winold, Allen (1975). "Rhythm in Twentieth-Century Music". In Aspects of Twentieth-Century Music, edited by Gary Wittlich,[page needed]. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0-13-049346-5.