Computational Models of Rhythm and Meter by Georg Boenn
Author:Georg Boenn
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham
We have seen that it is possible to use the Farey sequence as a form of representation of rhythms in Western notation. We will show in Chap. 9 that it can also be used in order to notate onsets recorded from music performances. The method uses Farey sequences to form grid points for rhythm quantisation; the outcome of the quantisation process is then a sub-set of some Farey Sequence . We showed already in Boenn (2007) that Farey sequences can represent rhythms of many diverse styles from different historical and cultural contexts, because of their scalability and self-similarity and because appropriate filters can be found. Results obtained by Desain and Honing (2003) and Papadelis and Papanikolaou (2004) suggest that human listeners categorise performed rhythms in such a way that classes of short rhythmic patterns are formed in the vicinity of relatively small integer ratios. Filtered Farey Sequences are capable of representing such categories.
Graham et al. (1994) have shown that the Stern-Brocot tree can serve to approximate decimal fractions by integer fractions, with arbitrary precision. See also (Martelli et al. 2005, p. 675) for a Python algorithm that approximates floating point numbers into Farey fractions. One of the main advantages of the Farey sequence is that it is scalable to different musical and cognitive timescales and does not rely on context-related manifestations of meter and bar. However, metrical hierarchies on any level can be modelled easily by using filtered Farey sequences.
Musically Relevant Structure of the Farey Sequence
Figure 7.3 shows a plot of that correlates the position of the ratios a / b in the interval with the unit fraction 1 / b built from their corresponding denominators. The unit fractions represent the metrical subdivision by b of the reference duration 1.
Fig. 7.3Correlation of and 1 / b in the interval
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