Klaus-Peter Brenner, Laurent Bartholdi, Radhika Gupta

The ›12-step harmonic standard sequence‹ of the Shona mbira music of Zimbabwe: Computer animated visualization of its rotational symmetric structure on plane and torus, with synchronized music example.

Video clip (Göttingen, 2013) downloadable as MPEG-4 movie (60mb, H.264 in 1280x720 resolution), and visible on youtube.

Article downloadable as PDF.

  1. Subject, music example, and text by Klaus-Peter Brenner (University of Goettingen, Germany, Department of Musicology, Lecturer in Ethnomusicology and Curator of the Collection of Musical Instruments).
  2. Film outline and mathematical specifications by Laurent Bartholdi (University of Goettingen, Germany, Mathematical Institute, Professor of Mathematics and Curator of the Collection of Mathematical Models and Instruments).
  3. Programming by Radhika Gupta (University of Utah, Department of Mathematics) by means of the software POV-Ray.

Tags: Ethnomathematics, Ethnomusicology, African Music, Shona, Zimbabwe, Mbira, Torus, Symmetry.

Since the emergence of the Great Zimbabwe state and its impressive stone wall architecture in the 13th century, Bantu-speaking peoples of the Shona linguistic branch established a number of large and culturally influential kingdoms in the area between Zambezi and Limpopo (cf. Beach 1980; Mudenge 1988; Böhmer-Bauer 2000). Economically and technologically, these were based on a Later Iron Age type of agriculture and cattle breeding complex combined with surface gold mining and elephant hunting. Their control of substantial gold and ivory resources connected them to the intercontinental sea trade networks on the Indian Ocean. Ideologically, these kingdoms were based upon a system of politico-religious authority whose backbone was their ancestor cult with its hierarchy of ancestral spirits, spirit media and possession rituals. While, due to the Zulu expansion as well as Portuguese and British colonialism, the last remnants of those kingdoms declined during the 19th century, Shona religion has, to a certain extent, survived this demise and thereby resisted Christian missionary pressure.

Long since, most likely for centuries, and down to the present day, lamellophone music has been an integral part of its ritual practice (cf. Gelfand 1959, 1962; Tracey/Zanzinger 1975b-e; Berliner 1978; Ranger 1982). It was this cultural context that fostered the emergence, and fuelled the co-evolutionary expansion, of both a large family of mbira type lamellophones (cf. Tracey 1972, 1974; Kubik 1998, 2002a, 2002b) and the highly sophisticated polyphonic musical style associated with them. The hallmark of the grammar underlying this musical style is a complex, yet coherent, system of distinctively structured harmonic sequences, primarily consisting of circular patterns of fifth dyads (and their respective octave equivalents) progressing in leaps of thirds and fourths within the framework of a hexa- or heptatonic scale. These patterns constitute a deep structural level of Shona mbira music, and it is this kind of harmonic patterning to which the latter owes much of its uniqueness (cf. Tracey 1961, 1970, 1989; Tracey/Zanzinger 1975a; Kaemmer 1975; Kubik 1988; Brenner 1997, 2004b, 2013 i.p.; Grupe 1998, 2004; Berliner/Magaya 2013 i.p.).

Most surprisingly, an explorative analysis of the harmonic sequences of mbira music brought to light a coherent complex of geometrical properties, more specifically: of perfect rotational as well as partial translational symmetries, inherent to, and mutually permeating in, these patterns (Brenner 1997: 66-135; cf. Brenner 2004a). The evidence of such a body of implicit geometrical knowledge, embedded in the deep structures of an orally transmitted repertoire of music, was a most exciting ethnomathematical finding, not least because ethnomathematics used to be predominantly concerned with visual manifestations of culturally embedded mathematical thinking such as tangible artifacts, graphic traditions, games and the like (cf. Ascher 1991; Gerdes 1999; Kubik 1987a, 1987b; Washburn/Crowe 1992; Zaslavsky 1990).

The actual video clip highlights the basic phenotype of the so-called ›12-step harmonic standard sequence‹ and demonstrates the most striking of its geometric properties, namely its two-fold rotational symmetry – in musicological terminology: its identity with its own retrograde inversion. This sequence consists of the fifth dyads on the heptatonic scale degrees 1 3 5 1 3 6 1 4 6 2 4 6, i.e. the dyads 1-5, 3-7, 5-2, 1-5, 3-7, 6-3, 1-5, 4-1, 6-3, 2-6, 4-1, 6-3. Moreover, due to its logic of internal repetition versus offsetting of elements it shows an inherent segmentation in four groups of three dyads each: 1 3 5, 1 3 6, 1 4 6, 2 4 6.

On the left-hand side of the video screen this structure is visualized on the plane as a constellation of white noteheads within a grid of 12 cyclically repeating time units (represented as vertical lines along the x-axis) and 7 cyclically repeating heptatonic scale degrees or pitch classes (represented as horizontal lines along the y-axis). Additionally, the noteheads are connected by white auxiliary lines in such a way that both the inherent segmentation and the rotational symmetry become prominent in an eye-catching way. The – shiftable – black frame marks the object of the cyclical repetition in both dimensions of the plane. Black bullets mark the center points of the perfect rotational symmetry.

The two-fold circularity of a plane pattern can be non-redundantly represented on the surface of a ring-shaped three-dimensional object: a torus. The result of having transferred the plane pattern described above onto a torus (however, with the white auxiliary lines omitted) is shown on the right-hand side of the video screen. In the original publication (Brenner 1997: 119-120) a series of photographs of a tangible model made of styrofoam had been used to visualize this. The idea of this torus type of music notation was inspired, firstly, by David Rycroft’s circular notation of Nguni vocal polyphony (Rycroft 1967), secondly, by Geza Révész’s distinction between ›tone chroma‹ and ›tone height‹ as described in his psychoacoustic two-component theory of pitch perception (Révész 1912, 1926) and incorporated in modern pitch class theory (cf. Shepard 1964, 1982) and, thirdly, by John Blacking’s grammatical concept of ›harmonic equivalence‹ (Blacking 1967: 168; cf. Kaemmer 1975: 91; Berliner 1978: 98).

Both of these visual representations as well as the operations applied to them are shown strictly in parallel throughout the video clip.

In the first part of the film a yellow cursor moves along the time axis of the structure, and the visualization is synchronized with the respective music example (Brenner 1997: CD I: Track 16) that was played on the instrument shown in picture 1. It should be noted here that the music example is an extremely condensed beginners’ version of an actual mbira piece called »Kariga Mombe«, in fact the – typically much more elaborate – surface structure is reduced here to a sounding abstraction of the harmonic deep structure.

In the second part of the film the two-fold rotational symmetry of the ›12-step harmonic standard sequence‹ is demonstrated by an actual 180°-rotation of both the plane and the torus representation. The four different center points of the rotational symmetry shown in the plane representation correspond to those four points on the torus where the rotational axis pierces through its surface.

The film lends itself to be looped: due to the symmetry under discussion it loops back into its beginning.

* * *

The idea for the realization of this video clip emerged from a collaboration between the Collection of Musical Instruments and the Collection of Mathematical Models and Instruments at the University of Göttingen during the exhibition »Objects of Knowledge« which was presented at the Pauliner Church Göttingen on occasion of the 275th anniversary of the University of Göttingen in 2012. The computer animation was programmed by Laurent Bartholdi and Radhika Gupta using the software POV-Ray, in 2012. Two years before Radhika Gupta, then a 4th year Engineering Physics undergraduate student at the Indian Institute of Technology Bombay, had already constructed computer animations centered on the torus, as a part of her summer internship under Laurent Bartholdi’s supervision; they were published on her project website (cf. Gupta 2010; Bartolomaeus 2012). The present video clip was collaboratively realized as an outgrowth of that project towards musicology.


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Picture 1: Mbira dzaVadzimu, by Rinos Mukuwurirwa Simboti, Harare/Mufakose, Zimbabwe, ca. 1980.
University of Göttingen, Collection of Musical Instruments, Inventory No. 1303. Photo: Stephan Eckardt.

Picture 2: Mbira dzaVadzimu, played by Cephas B. C. Machaka, during a kurova guva ceremony
held in Munaku village, Communal Land Mondoro, Zimbabwe, 1993. Photo: Klaus-Peter Brenner.

Picture 3: The mbira dzaVadzimu players Alois and Sydney Musarurwa Nyandoro, supported by a
hosho (rattle) player and Cephas B. C. Machaka on a ngoma (drum) during a bira ceremony held in
Dzama village, Communal Land Mondoro, Zimbabwe, 1993. Photo: Klaus-Peter Brenner.