Thank you for your conclusions:
bubudi wrote:(...) about many older rhythms having what we would term today as odd time signatures. although there is plenty of proof for this in the indo-european traditions, i do not see much evidence in african music although there are two 18/16 rhythms (often referred to as 9/8) in the mande repertoire. the rest are 12/8, 6/8, 4/4 etc.
bubudi wrote:(...) the dogon rhythms are the only examples of 10/8 and 5/4. this music has not been formally studied so there are no references i can point you to other than two recordings that should still be available. if you do a search in google you will surely find them.
bubudi wrote:(...) 'absence of evidence is not evidence of absence'. therefore it is possible that either: a) 'odd' time signatures signifies a more ancient tradition only in euroindian traditions, or b) africa did have many more 'odd beat' rhythms through the ages, but due to loss of popularity and lack of any written record of rhythms, they were eventually lost forever.
e2c wrote:Still, I think that a lot of Indian and Indonesian music might be more polyrhythmic than we imagine... partly due to trade ties and other contacts with Arabs and East African peoples + people of African descent living in India and Pakistan.
I want to add to this my small analysis based on the paper linked below:
http://ehess.modelisationsavoirs.fr/ati ... ES2005.pdf
I have selected
only rhythm patterns somehow related to Africa, plus some basic concepts and author's conclusions. Citations from this scientific article are presented in form of
italic text.
During the past thirty years a number of researchers have approached the study of rhythmic timelines using generative methods, notably Kubik {77}, Locke {80}, Pressing {101}, Rahn {105}, {106}, Anku {4}, Toussaint {125}, {126}, {127}, {128}, and Agawu {1}.
Agawu {1} provides an in-depth analysis of these methods applied to African timelines.
(...) Euclidean rhythms are closely related to a family of rhythms known as aksak rhythms, which have been studied from the combinatorial point of view for some time now {23}, {33}, {7}.
Béla Bartók {11} and Constantin Brăiloiu {23}, respectively, have used the terms Bulgarian rhythm and aksak to refer to those meters which use units of durations 2 and 3, and no other durations. Furthermore, the rhythm or meter must contain at least one duration of length 2 and at least one duration of length 3. Arom {7} referes to these durations as binary cells and ternary cells, respectively.
Arom {7} has generated an inventory of all the theoretically possible aksak rhythms for values of n ranging from 5 to 29, as well as a list of those that are actually used in traditional world music. He has also proposed a classification of these rhythms into several classes, based on structural and numeric properties.
Three of his classes are considered here: authentic-aksaks, quasi-aksaks, and pseudo-aksaks.
- An aksak rhythm is authentic if n is a prime number.
- An aksak rhythm is quasi-aksak if n is an odd number that is not prime.
- An aksak rhythm is pseudo-aksak if n is an even number.
The Euclidean rhythms that are favoured in classical music and jazz are also Euclidean strings (the first group). Furthermore, this group is not popular in African music.
The following Euclidean rhythms are authentic aksak and Euclidean strings at the same time:
Code: Select allE(2,5)={x.x..}=(23) (classical music, jazz, Greece, Macedonia, [u]Namibia[/u], Persia, [u]Rwanda[/u]), (authentic aksak).
It is a rhythm found in Greece, Namibia, Rwanda and [u]Central Africa[/u] {7}.
It is the pattern of the [u]N-geru and Yalli rhythms[/u] used in heroic ballads by the [u]Tuareg nomadic people of the Sahara desert[/u] {135}.
When started on the second onset as in {x..x.} it is a rhythm found in [u]Central Africa[/u], Bulgaria, Turkey, Turkestan and Norway {7}.
E(3,7)={x.x.x..}=(223) (Bulgaria, Greece, [u]Sudan[/u], Turkestan), (authentic aksak).
The following Euclidean rhythms are quasi-aksak and Euclidean strings at the same time:
Code: Select allE(4,9)={x.x.x.x..}=(2223) It is the rhythmic ostinato of a lullaby discovered by Simha Arom in [u]south-western Zaı̈re[/u] {7}.
The Euclidean rhythms that are reverse Euclidean strings (the second group) appear to have a much wider appeal. Finding musicological explanations for the preferences apparent in these mathematical properties raizes interesting ethnomusicological questions.
The following Euclidean rhythms are pseudo-aksak and reverse Euclidean strings at the same time:
Code: Select allE(3,8)={x..x..x.}=(332) ([u]Central Africa[/u], Greece, India, [u]Latin America, West Africa, Sudan[/u]), (pseudo-aksak).
E(5,12)={x..x.x..x.x.}=(32322) (Macedonia, [u]South Africa[/u]), (pseudo-aksak).
E(7,16)={x..x.x.x..x.x.x.}=(3223222) ([u]Brazilian[/u], Macedonian, [u]West African necklaces[/u]), (pseudo-aksak).
E(11,24)={x..x.x.x.x.x..x.x.x.x.x.}=(32222322222) ([u]Central African[/u] and Bulgarian necklaces), (pseudo-aksak).
It is a rhythm necklace of the [u]Aka Pygmies of Central Africa[/u] {6}. It is usually started on the seventh onset.
The following Euclidean rhythms are not classified as aksak, but are reverse Euclidean strings:
Code: Select allE(2,3)={xx.}=(12)
When started on the second onset as in {x . x}, it is a [u]hand-clapping pattern used by the Bantu people of Africa[/u] {66}, as well as the [u]first rhythm taught to beginners of Mandinka drumming[/u] {74}.
It is also found in [u]Cuba[/u], as for example, the [u]conga rhythm of the (6/8)-time Swing Tumbao[/u] {73}. It is common in [u]Latin American music[/u], as for example in the [u]Cueca[/u]{131}, and the [u]coros de clave[/u] {111}.
It is common in Arab music, as for example in the [u]Al Táer rhythm of Nubia[/u] {58}. It is a [u]Tuareg rhythm[/u] played on the [u]tende drums[/u] {135}.
The Drum Dance of the [u]Slavey Indians of Northern Canada[/u] {8}. The “ancestral” rhythm obtained from a phylogenetic analysis of Steve Reich’s [u]Clapping Music[/u] {36}.
When started on the silent pulse (anacrusis) as in {. x x}, it is used to complement [u]certain African rhythms[/u] {31}.
E(4,7)={x.x.x.x}=(2221)
When started on the third onset it is the Kalamátianos Greek dance rhythm {58}, as well as the [u]Shaigie rhythmic pattern of Nubia[/u] {58}.
When started on the fourth (last) onset it is the rhythmic pattern of the [u]Dar daasa al mutawasit[/u] of [u]Yemen[/u] {58}.
E(5,7)={x.xx.xx}=(21211) is the [u]Nawakhat[/u] pattern, another popular Arab rhythm {121}.
In [u]Nubia[/u] it is called the [u]Al Noht rhythm[/u] {58}.
E(5,9)={x.x.x.x.x}=(22221) When started on the second onset, it is a drum pattern used by the [u]Venda in South Africa[/u] {105}.
E(3,10)={x..x..x...}=(334), when started on the second onset, is the metric pattern of several [u]Tuareg rhythms[/u] played on tende drums {135}.
The Euclidean rhythms that are neither Euclidean strings nor reverse Euclidean strings (group three) fall into two categories: those consisting of interval lengths ‘1’ and ‘2’, and those consisting of interval lengths ‘2’ and ‘3’.
The latter group is used only in Bulgaria, and the former is used in Africa.
The following Euclidean rhythms are neither Euclidean nor reverse Euclidean strings:
Code: Select allE(5,8)={x.xx.xx.}=(21212) (Egypt, Korea, [u]Latin America, West Africa[/u]).
E(7,12)={x.xx.x.xx.x.}=(2122122) [u](West Africa), (Central African, Nigerian, Sierra Leone necklaces).[/u]
E(9,14)={x.xx.xx.xx.xx.}=(212121212) [u](Algerian necklace).[/u]
E(9,16)={x.xx.x.x.xx.x.x.}=(212221222) [u](West and Central African, and Brazilian necklaces).[/u]
E(13,24)={x.xx.x.x.x.x.xx.x.x.x.x.}=(2122222122222) [u](Central African necklace).[/u]
It is another rhythm necklace of the [u]Aka Pygmies of the upper Sangha[/u] {6}.
When started on the penultimate (12-th) onset it is the [u]Bobangi[/u] metal-blade pattern used by the [u]Aka Pygmies[/u].
As mentioned in the introduction, during the past thirty years a number of researchers have approached the study of rhythmic timelines using generative methods, notably Kubik {77}, Locke {80}, Pressing {101}, Rahn {105}, {106}, Anku {4}, Toussaint {125}, {126}, {127}, {128}, and Agawu{1}.
Agawu{1} provides an in-depth analysis of these methods applied to African timelines. On the other hand, the Euclidean algorithm exposed here is a mathematical model of rhythmic timeline generation that applies to music from all over the world.
A notable exception are the Indian talas, which tend to have longer timelines than other music (as many as 128 beats per cycle), and therefore use a greater variety of duration intervals, thus violating the maximal evenness of Euclidean rhythms.
A rhythm is said to be maximally even if its representation on the circle of time maximizes the sum of its pairwise inter-onset straight line distances {34}, {35}. For example, of the 35 Sulaadi talas only 4 are maximally even, and of the 108 Astottara Sata talas only 9 are maximally even {94}. It has been shown by Demaine et al., {45} that a rhythm is maximally even if and only if it is Euclidean, or a rotation of a Euclidean rhythm. Euclidean (maximally even) rhythms have many interesting mathematical and musical properties. For example, the complement of a Euclidean rhythm is also Euclidean {124}.
[/i]
Selected bibliography more or less -related to African rhythms:
Code: Select all{1} Kofi Agawu. Structural analysis or cultural analysis? Competing perspectives on the “standard pattern” of West African rhythm. Journal of the American Musicological Society, 59(1):1–46, 2006.
{2} Samuel Akpabot. Theories on African music. African Arts, 6(1):59–62, Autumn 1972.
{4} Willie Anku. Circles and time: A theory of structural organization of rhythm in African music. Music Theory Online, 6(1), January 2000.
{6} Simha Arom. African Polyphony and Polyrhythm. Cambridge University Press, Cambridge, England, 1991.
{8} Michael I. Asch. Social context and the musical analysis of Slavey drum dance songs. Ethnomusicology, 19(2):245–257, May 1975.
{12} Gregory Barz. Music in East Africa. Oxford University Press, Oxford, England, 2004.
{14} Gerard Behague. Bossa and Bossas: recent changes in Brazilian urban popular music. Ethnomusicology, 17(2):209–233, 1973.
{15} Gerard Behague. Improvisation in Latin American musics. Music Educators Journal, 66(5):118–125, January 1980.
{19} John Blacking. Tonal organization in the music of two Venda initiation schools. Ethnomusicology, 14(1):1–56, 1970.
{20} Rose Brandel. The African hemiola style. Ethnomusicology, 3(3):106–117, September 1959.
{22} Roy Brewer. The use of Habanera rhythm in rockabilly music. American Music, 17:300–317, Autumn 1999.
{30} John Miller Chernoff. African Rhythm and African Sensibility. The University of Chicago Press,
Chicago, 1979.
{31} John Miller Chernoff. The rhythmic medium in African music. New Literary History, 22(4):1093–1102, Autumn 1991.
{49} A. Dworsky and B. Sansby. A Rhythmic Vocabulary. Dancing Hands Music, Minnetonka, 1999.
{50} Issam El-Mallah and Kai Fikentscher. Some observations on the naming of musical instruments and on the rhythm in Oman. Yearbook for Traditional Music, 22:123–126, 1990.
{53} Bob Evans. Authentic Conga Rhythms. Belwin Mills Publishing Corporation, Miami, 1966.
{54} Mary Farquharson. Africa in America. Discos Corazon, Mexico, 1992. {CD}.
{60} Lennart Hallstrom. African Drum Rhythms for Djembes, Bass Drums and Bells. Lennart Hallstrom,
Stockholm, 2000.
{62} Royal Hartigan, Abraham Adzenyah, and Freeman Donkor. West African Rhythms for Drum Set.
Manhattan Music, Inc., 1995.
{63} Christopher F. Hasty. Meter as Rhythm. Oxford University Press, Oxford, England, 1997.
{66} A. M. Jones. African rhythm. Africa: Journal of the International African Institute, 24(1):26–47,
January 1954.
{67} A. M. Jones. Studies in African Music. Oxford University Press, Amen House, London, 1959.
{68} S. A. Floyd Jr. The sources and rescources of classic ragtime music. Black Music Research Journal, 4:22–59, 1984.
{69} S. A. Floyd Jr. Black music in the circum-Caribbean. American Music, 17(1):1–38, 1999.
{70} R. Kauffman. African rhythm: A reassessment. Ethnomusicology, 24(3):393–415, Sept. 1980.
{73} T˝om Kl˝ower. The Joy of Drumming: Drums and Percussion Instruments from Around the World. Binkey Kok Publications, Diever, Holland, 1997.
{74} Roderic Knight. Mandinka drumming. African Arts, 7(4):24–35, Summer 1974.
{77} Gerhard Kubik. Oral notation of some West and Central African time-line patterns. Review of
Ethnology, 3(22):169–176, 1972.
{78} Gerhard Kubik. Central Africa: An introduction. In Ruth M. Stone, editor, The Garland Handbook of African Music, pages 260–290. Garland Publishing, Inc., New York and London, 2000.
{80} David Locke. Principles of offbeat timing and cross-rhythm in Southern Ewe dance drumming.
Ethnomusicology, 26:217–246, 1982.
{81} David Locke and Godwin Agbeli. Drum language in Adzogbo. The Black Perspective in Music,
9(1):25–50, Spring 1981.
{82} Wendell Logan. The ostinato idea in Black improvised music: A preliminary investigation. The Black Perspective in Music, 12(2):193–215, Autumn 1984.
{87} Peter Manuel. The anticipated bass in Cuban popular music. Latin American Music Review, 6(2):249–261, Autumn-Winter 1985.
{90} Matthew Montfort. Ancient Traditions–Future Possibilities: Rhythmic Training Through the Traditions of Africa, Bali and India. Panoramic Press, Mill Valley, 1985.
{91} Robin Moore and Elizabeth Sayre. An Afro-Cuban Bat´a piece for Obatal´a, king of the white cloth. In Michael Tenzer, editor, Analytical Studies in World Music, pages 120–160. Oxford University Press, New York, 2006.
{93} Larry Morris. Rhythm Catalog. The Internet, http://www.drums.org/djembefaq, 2001.
{97} John P. Murphy. Music in Brazil. Oxford University Press, New York and Oxford, 2006.
{98} J. H. Nketia. Drumming in Akan Communities of Ghana. Thomas Nelson and Sons Ltd., Edinburgh, Scotland, 1963.
{99} J. H. Kwabena Nketia. African Music in Ghana. Northwestern Univ. Press, Britain, 1963.
{101} Jeff Pressing. Cognitive isomorphisms between pitch and rhythm in world musics: West Africa, the Balkans and Western tonality. Studies in Music, 17:38–61, 1983.
{103} Putumayo. Congo to Cuba. Publisher, Address, 2002.
{104} Jose Luis Quintana and Chuck Silverman. Changuito: A Master’s Approach to Timbales. Belwin-Mills Publishing Corp., Miami, 1998.
{105} Jay Rahn. Asymmetrical ostinatos in sub-saharan music: time, pitch, and cycles reconsidered. In Theory Only, 9(7):23–37, 1987.
{106} Jay Rahn. Turning the analysis around: African-derived rhythms and Europe-derived music theory. Black Music Research Journal, 16(1):71–89, 1996.
{110} John Donald Robb. Rhythmic patterns of the Santo Domingo corn dance. Ethnomusicology,
8(2):154–160, May 1964.
{111} Olavo Al´en Rodr´ıguez. Instrumentos de la Musica Folok´lrico-Popular de Cuba. Centro de Investigaci´on y Desarrollo de la Musica Cubana, Havana, Cuba, 1997.
{112} Rene V. Rosalia. Migrated Rhythm: The Tamb´u of Curac¸ao. CaribSeek, 2002.
{121} James A. Standifer. The Tuareg: their music and dances. The Black Perspective in Music, 16(1):45–62, Spring 1988.
{123} Ruth M. Stone. Music in West Africa. Oxford University Press, Oxford, England, 2005.
{125} Godfried T. Toussaint. A mathematical analysis of African, Brazilian, and Cuban clave rhythms. In Proceedings of BRIDGES: Mathematical Connections in Art, Music and Science, pages 157–168, Towson University, Towson, MD, July 27-29 2002.
{126} Godfried T. Toussaint. Classification and phylogenetic analysis of African ternary rhythm timelines. In Proceedings of BRIDGES: Mathematical Connections in Art, Music and Science, pages 25–36, Granada, Spain, July 23-27 2003.
{127} Godfried T. Toussaint. A mathematical measure of preference in African rhythm. In Abstracts of Papers Presented to the American Mathematical Society, volume 25, page 248, Phoenix, Arizona, January 7-10 2004. American Mathematical Society.
{130} Ed Uribe. The Essence of Brazilian Persussion and Drum Set. CCP/Belwin Inc., Miami, Florida,
1993.
{132} John Varney. Colombian Bambuco: The Evolution of a National Music Style. Grifith University,
South Brisbane, Australia, 1999. Ph.D. Thesis.
{135} Caroline Card Wendt. Tuareg music. In Ruth M. Stone, editor, The Garland Handbook of African Music, pages 206–227. Garland Publishing, Inc., New York and London, 2000.
{137} Luis Felipe Ram´on y Rivera. Rhythmic and melodic elements in Negro music of Venezuela. Yearbook of the International Folk Music Council, 14:56–60, 1962.
Additional ones which can be not related to Africa, but still would be very interesting:
Code: Select all{7} Simha Arom. L’aksak: Principes et typologie. Cahiers de Musiques Traditionnelles, 17:12–48, 2004.
{9} Marcia Ascher. Mathematics Elsewhere: An Exploration of Ideas Across Cultures. Princeton University Press, Princeton and Oxford, 2002.
{10} Anthony Ashton. Harmonograph–A Visual Guide to the Mathematics of Music. Walker and Company, New York, 2003.
{23} Constantin Br˘ailoiu. Le rythme aksak. Revue de Musicologie, 33:71–108, 1951.
{25} Viggo Brun. Euclidean algorithms and musical theory. Enseignement Math´ematique, 10:125–137, 1964.
{27} Marc Chemillier. Ethnomusicology, ethnomathematics. The logic underlying orally transmitted artistic practices. In G. Assayag, H. G. Feichtinger, and J. F. Rodrigues, editors, Mathematics and Music, pages 161–183. Springer-Verlag, 2002.
{32} Martin Clayton. Time in Indian Music. Oxford University Press, Inc., New York, 2000.
{33} J´erˆome Cler. Pour une th´eorie de l’aksak. Revue de Musicologie, 80:181–210, 1994.
{34} J. Clough and J. Douthett. Maximally even sets. Journal of Music Theory, 35:93–173, 1991.
{39} Aaron Copland. Jazz structure and influence. Modern Music, 4(2):9–14, 1927.
{56} Jos´e Manuel Gamboa. Cante por Cante: Discolibro Didactico de Flamenco. New Atlantis Music,Alia Discos, Madrid, 2002.
{57} David Goldsworthy. Cyclic properties of Indonesian music. Journal of Musicological Research,
24:309–333, 2005.
{58} Kobi Hagoel. The Art of Middle Eastern Rhythm. OR-TAV Music Publications, Kfar Sava, Israel,
2003.
{59} Man-Young Hahn. The four musical types of Buddhist chant in Korea. Yearbook for Traditional
Music, 15, East Asian Musics:45–58, 1983.
{84} Justin London. Hearing in Time: Psychological Aspects of Musical Meter. Oxford University Press,Inc., New York, 2004.
{85} M. Lothaire. Algebraic Combinatorics on Words. Cambridge University Press, Cambridge, England,2002.
{88} Thomas J. Mathiesen. Rhythm and meter in ancient Greek music. Music Theory Spectrum, 7:159–180, Spring 1985.
{89} Alan P. Merriam. Flathead indian instruments and their music. The Musical Quarterly, 37:368–375, July 1951.
{92} Sidney Moore. Thai songs in 7/4 meter. Ethnomusicology, 13(2):309–312, May 1969.
{94} Robert Morris. Sets, scales, and rhythmic cycles: A classification of talas in Indian music. In 21st
National Convention of the Society of Music Theory, Chapel Hill, North Carolina, December 1998.
{95} Robert Morris. Architectonic composition in South Indian classical music. In Michael Tenzer, editor, Analytical Studies in World Music, pages 303–331. Oxford University Press, New York, 2006.
{108} Timothy Rice. Aspects of Bulgarian musical thought. Yearbook of the International Folk Music
Council, 12:43–66, 1980.
{109} Timothy Rice. Music in Bulgaria. Oxford University Press, Oxford, England, 2004.
{113} Curt Sachs. Rhythm and Tempo: A Study in Music History. W. W. Norton, New York, 1953.
{116} Patricia K. Shehan. Teaching music through Balkan folk dance. Music Educators Journal, 71(3):47–51, November 1984.
{119} Alice Singer. The metrical structure of Macedonian dance. Ethnomusicology, 18(3):379–404,
September 1974.
{128} Godfried T. Toussaint. The Euclidean algorithm generates traditional musical rhythms. In Proc. of BRIDGES: Mathematical Connections in Art, Music and Science, pages 47–56, Banff, Canada, July 31 - August 3 2005.
{129} Leo Treitler. Regarding meter and rhythm in the ‘ars antiqua’. The Musical Quarterly, 65(4):524–558, October 1979.
{134} Marnix St. J. Wells. Rhythm and phrasing in Chinese tune-title lyrics; old eight-beat and its 3-2-3 meter. Asian Music, 23(1):119–183, (Autumn, 1991 - Winter, 1992).
{136} O. Wright. The Modal System of Arab and Persian Music AD 1250-1300. Oxford University Press, Oxford, England, 1978.
[/size]
Further readings, this ones are related to rhythm & tonality in human languages:
tone
http://wals.info/chapter/13
rhythm
http://wals.info/chapter/17
