Additive rhythm and divisive rhythm
Encyclopedia
In music
, additive and divisive are terms used to distinguish two types of both rhythm
and meter
.
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).
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.
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" (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, metre
s 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 beat
s, 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.
"the entire African rhythmic structure . . . is divisive in nature" (Novotney 1998, 147).
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.
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.
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, additive and divisive are terms used to distinguish two types of both rhythm
Rhythm
Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...
and meter
Meter (music)
Meter or metre is a term that music has inherited from the rhythmic element of poetry where it means the number of lines in a verse, the number of syllables in each line and the arrangement of those syllables as long or short, accented or unaccented...
.
A divisive (or, more commonly, multiplicative) rhythm is a rhythm
Rhythm
Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...
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
Concatenation
In computer programming, string concatenation is the operation of joining two character strings end-to-end. For example, the strings "snow" and "ball" may be concatenated to give "snowball"...
(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).
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
Music of India
The music of India includes multiple varieties of folk, popular, pop, classical music and R&B. India's classical music tradition, including Carnatic and Hindustani music, has a history spanning millennia and developed over several eras. It remains fundamental to the lives of Indians today as...
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 signatureTime signature
The time signature is a notational convention used in Western musical notation to specify how many beats are in each measure and which note value constitutes one beat....
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" (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, metre
Metre (music)
Meter or metre is a term that music has inherited from the rhythmic element of poetry where it means the number of lines in a verse, the number of syllables in each line and the arrangement of those syllables as long or short, accented or unaccented...
s which have a regular pattern of beats of uneven length. For example, the time signature
Time signature
The time signature is a notational convention used in Western musical notation to specify how many beats are in each measure and which note value constitutes one beat....
4/4 indicates each bar
Bar (music)
In musical notation, a bar is a segment of time defined by a given number of beats of a given duration. Typically, a piece consists of several bars of the same length, and in modern musical notation the number of beats in each bar is specified at the beginning of the score by the top number of a...
is eight quavers long, and has four beat
Beat (music)
The beat is the basic unit of time in music, the pulse of the mensural level . In popular use, the beat can refer to a variety of related concepts including: tempo, meter, rhythm and groove...
s, 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
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...
, who was influenced by similar rhythms in Bulgarian folk music, and in some music of Philip Glass
Philip Glass
Philip Glass is an American composer. He is considered to be one of the most influential composers of the late 20th century and is widely acknowledged as a composer who has brought art music to the public .His music is often described as minimalist, along with...
, and other minimalists
Minimalism
Minimalism describes movements in various forms of art and design, especially visual art and music, where the work is set out to expose the essence, essentials or identity of a subject through eliminating all non-essential forms, features or concepts...
, most noticeably the "one-two-one-two-three" chorus parts in Einstein on the Beach
Einstein on the Beach
Einstein on the Beach is an opera that premiered on July 25, 1976 at the Avignon Festival in France, scored and written by Philip Glass and designed and directed by theatrical producer Robert Wilson. It also contains writings by Christopher Knowles, Samuel M. Johnson and Lucinda Childs...
. 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).
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 MoroccoMorocco
Morocco , officially the Kingdom of Morocco , is a country located in North Africa. It has a population of more than 32 million and an area of 710,850 km², and also primarily administers the disputed region of the Western Sahara...
in North Africa to Indonesia
Indonesia
Indonesia , officially the Republic of Indonesia , is a country in Southeast Asia and Oceania. Indonesia is an archipelago comprising approximately 13,000 islands. It has 33 provinces with over 238 million people, and is the world's fourth most populous country. Indonesia is a republic, with an...
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
Mali
Mali , officially the Republic of Mali , is a landlocked country in Western Africa. Mali borders Algeria on the north, Niger on the east, Burkina Faso and the Côte d'Ivoire on the south, Guinea on the south-west, and Senegal and Mauritania on the west. Its size is just over 1,240,000 km² with...
. This pattern may have migrated east from North Africa to Asia through the spread of Islam
Islam
Islam . The most common are and . : Arabic pronunciation varies regionally. The first vowel ranges from ~~. The second vowel ranges from ~~~...
(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 AfricaMusic of Africa
Africa is a vast continent and its regions and nations have distinct musical traditions. The music of North Africa for the most part has a different history from sub-Saharan African music traditions....
and music of the African diaspora
African diaspora
The African diaspora was the movement of Africans and their descendants to places throughout the world—predominantly to the Americas also to Europe, the Middle East and other places around the globe...
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)