Additive rhythm and divisive rhythm

Inmusic,the termsadditiveanddivisiveare used to distinguish two types of bothrhythmandmeter:

• Adivisive(or, alternately,multiplicative)rhythmis arhythmin which a larger period of time is divided into smallerrhythmic unitsor, conversely, some integer unit is regularly multiplied into larger, equal units.
• This can be contrasted withadditive rhythm,in which larger periods of time are constructed byconcatenating(joining end to end) a series of units into larger units of unequal length, such as a⁵
  ₈meter produced by the regular alternation of²
  ₈and³
  ₈.

When applied to meters, the termsperfectandimperfectare sometimes used as the equivalents ofdivisiveandadditive,respectively.

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 while⁵
₈is additive.

The termsadditiveanddivisiveoriginate with Curt Sachs's bookRhythm and Tempo(1953),while the termaksak rhythmwas introduced for the former concept at about the same time byConstantin Brăiloiu,in agreement with the Turkish musicologistAhmet 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 theNew 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.

Winold recommends that, "metric structureis best described through detailed analysis ofpulse groupingson various levels rather than through attempts to represent the organization with a single term ".

Sub-Saharan African musicand mostEuropean (Western) musicis divisive, whileIndianand otherAsianmusics may be considered as primarily additive. However, many pieces of music cannot be clearly labeled divisive or additive.

Divisive rhythm

For example:⁴
₄consists 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 in⁴
₄time

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.This type of rhythm is also referred to in musicological literature by the Turkish wordaksak,which means "limping".In the special case oftime signaturesin which the upper numeral is not divisible by two or three without a fraction, the result may alternatively be calledirregular,imperfect,orunevenmeter, and the groupings into twos and threes are sometimes calledlong beatsandshort beats.

Additive rhythm in³⁺³⁺²
₈time

Additive rhythm³⁺³⁺²
₈time. 1 whole note = 8 eighth notes = 3 + 3 + 2.

The termadditive rhythmis also often used to refer to what are also incorrectly calledasymmetric rhythmsand evenirregular rhythms– that is,meterswhich have a regular pattern of beats of uneven length. For example, thetime signature⁴
₄indicates eachbaris eightquaverslong, and has fourbeats,each acrotchet(that is, two quavers) long. The asymmetric time signature³⁺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, byBéla Bartók,who was influenced by similar rhythms inBulgarian Folk Music.The third movement of Bartók'sString Quartet No. 5,a scherzo markedalla bulgaresefeatures a "⁹
₈rhythm (4+2+3) ".Stravinsky'sOctetfor Wind Instruments "ends with a jazzy 3+3+2 = 8 swung coda".Stravinsky himself found a kinship with additive rhythms in music of therenaissanceandbaroqueperiods. For example, he marvelled at theLaudate PuerifromMonteverdi'sVespers 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

"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 ofPhilip Glass,and otherminimalists,most noticeably the "one-two-one-two-three" chorus parts inEinstein 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 of2+2+2+3at the start.George Harrison's song "Here Comes the Sun"onThe Beatles' albumAbbey Roadfeatures a rhythm "which switches between¹¹
₈,⁴
₄and⁷
₈on 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 ofStravinskythan of the Beatles ".

Olivier Messiaenmade 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" fromThe Quartet for the End of Timeis a bracing example. A gentler exploration of additive patterns can be found in "Le Regard de la Vierge" from the same composer's piano cycleVingt regards sur l'enfant-Jésus.

György Ligeti'sÉtudeNo. 13, "L'escalier du diable" features patterns involving quavers grouped in twos and threes. The rhythm at the start of the study follows the pattern2+2+3,then2+2+2+3.According to the composer's note, the¹²
₈time signature "serves only as a guideline, the actual meter consists of 36 quavers (three 'bars'), divided asymmetrically".

Sub-Saharan African rhythm

Main articles:Sub-Saharan African music traditionsandRhythm in Sub-Saharan Africa

A divisive form ofcross-rhythmis 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".

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.

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 ".

Tresillo: divisive and additive interpretations

Main article:Tresillo (rhythm)

In divisive form, the strokes oftresillocontradict the beats. In additive form, the strokes of tresilloarethe 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 fromMoroccoin North Africa toIndonesiain South Asia. Use of the pattern inMoroccan musiccan be traced back to slaves brought north across the Sahara Desert from present-dayMali.This pattern may have migrated east from North Africa to Asia through the spread ofIslam.In Middle Eastern and Asian music, the figure is generated through additive rhythm.

Divisive structure

The most basic duple-pulse figure found in theMusic of Africaand music of theAfrican diasporais a figure the Cubans calltresillo,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.

See also

• Counting (music)

References

Sources

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