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

 A divisive (or, alternately, 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  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 Ahmed Adnan Saygun . 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: 

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 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

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, which means "limping" (; ). 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 .

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  indicates each bar is eight quavers long, and has four beats, 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, who was influenced by similar rhythms in Bulgarian folk music. Stravinsky's Octet for Wind Instruments "ends with a jazzy 3+3+2 = 8 swung coda" . 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 metres. In jazz, Dave Brubeck's song "Blue Rondo à la Turk" features bars of nine quavers grouped into patterns of  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" . "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" .

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.

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. 

"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 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 . In Middle Eastern 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 . 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.