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Neil McLachlan - A Spatial Theory of Rhythmic Resolution

LEONARDO MUSIC JOURNAL, Vol. 10, pp. 61–67, 2000


Cyclic arrays, such as clock faces, have advantages over linear arrays for conceptualizing repetitive rhythmic structures. The author maps rhythms from African and Indonesian musics into cyclic arrays and analyzes them using concepts from Gestalt psychology, mathematical group theory and psycho-acoustics. The perceptual structures thus revealed exist between the different musical parts played on various instruments and contradict the usual processes of auditory segregation according to the physical locations of instrumentation. This prompts a proposal for a theory of musical despatialization to explain the psychological efficacy of these rhythms.

Excerpt 1

J. Becker [10] has used a cyclic array to map a traditional Javanese gamelan piece, as the cyclic array more accurately represents the concepts of time described by the gamelan musicians themselves than do the linear models and notations used by Western musicologists.
Becker shows how repeating musical forms generate temporal symmetries and balance in accordance with Hindu and Buddhist beliefs. She also describes the similarity in form between traditional Javanese gamelan music when mapped in this way and Hindu mandalas, while pointing out that both evolved in ritual and meditative practices.

10. J. Becker, “Hindu-Buddhist Time in Javanese Gamelan Music,” in J.T. Fraser, ed., Study of Time IV (New York: Springer-Verlag, 1981).

Excerpt 2

At this point it is necessary to introduce the psycho-acoustic concept of auditory streams. A.S. Bregman [18] has described how the application of Gestalt and other perceptual principles may be employed to explain people’s natural ability to differentiate sound sources occurring in the world about them. The complex mixtures of spectral envelopes that are commonly sensed by the ear are grouped into auditory streams relating to individual sources through the analysis of similarities, continuities and discontinuities in spectral, temporal and binaural sensory data. These spatialized auditory streams generate an auditory scene or field, described by D. Ihde [19] as a specific form of opening into the physical world with its own center and horizon, independent but related to other sensory fields.

In the Kecak chant described above, two spatially and temporally distinct musical parts combine to form a structure (perceptual or auditory stream) that is powerfully cohesive due to its simplicity relative to its components. This perceptual grouping contradicts the auditory streams relating to individual sources that define the spatialized auditory field and so may lead the listener to experience the music as a despatialized entity.

This may then contribute to the experiences and emotions of removal from normal acoustic time and space reported by performers of or listeners to this music [20]. Other common musical devices, such as the simultaneous sounding of different instruments or musical harmony creating spectral fusion, may also create despatialized auditory streams and have similar effects [21].

18. A.S. Bregman, Auditory Scene Analysis: The Perceptual Organisation of Sound (Cambridge, MA: MIT Press, 1990).

19. D. Ihde, Listening and Voice, A Phenomenology of Sound (Athens, OH: Ohio Univ. Press,1976).

20. J. Becker, “Music and Trance,” Leonardo Music Journal 4 (1994) pp. 41–51.

21. N. McLachlan, “Consonance and Self,” Critical inQueeries 2, No. 1, 45–57 (1998).



Iteration of all possible groups in an 8-element array.


Reflection through points 1 and 5 of 3-element group.
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