Abstract
We examine Gärdenfors’ theory of conceptual spaces, a geometrical form of knowledge representation (Conceptual spaces: The geometry of thought, MIT Press, Cambridge, 2000), in the context of the general Creative Systems Framework introduced by Wiggins (J Knowl Based Syst 19(7):449–458, 2006a; New Generation Comput 24(3):209–222, 2006b). Gärdenfors’ theory offers a way of bridging the traditional divide between symbolic and sub-symbolic representations, as well as the gap between representational formalism and meaning as perceived by human minds. We discuss how both these qualities may be advantageous from the point of view of artificial creative systems. We take music as our example domain, and discuss how a range of musical qualities may be instantiated as conceptual spaces, and present a detailed conceptual space formalisation of musical metre.
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Notes
For example, Boden’s combinatorial creativity seems to coincide precisely with Koestler’s bisociation of matrices. On the other hand, while Guilford’s descriptions of divergent and convergent thinking may evidently be applicable within a framework such as that which Boden provides, they are necessarily vague descriptions of the high-level behaviour of a very complex system—sufficiently high-level not to be helpful in actually defining it.
Photoshop users evidently do this all the time: there is a tool for sampling the colour of a region, which allows the user to refer to “the colour of that region” without having a name for it. “The colour of that region” is, however, as much a symbolic concept as taupe, turquoise or turnip.
It is clear that geometrical transformations can change the nature of the space itself, perhaps rendering it non-Euclidean or changing the magnitude of its dimensions, but this, in our understanding, is not the same kind of transformation as Boden’s transformational creativity, which involves a change in the content of \({\fancyscript{C}}^{s}\).
While “up” and “down” are standard terminology, sequences of them are not.
For a tutorial on the difference between major and minor tonality, listen to “Ev’ry Time We Say Goodbye” by Cole Porter: the perceptual experience of tonal function is available to every listener, even if he or she has not studied music theory, nor become consciously aware of the differences in sound so produced.
In Gärdenfors’ terms, this is a natural property, too, but since we are considering it, here, in its own right, and not as part of a more complicated concept, of something which might have red as a property, we think of it as a concept.
We adorn functions specialised to a dimension and/or domain with a subscript identifying that dimension. Since these dimensions are subsets of infinite sets, these operations may not be everywhere defined.
MDS was carried out using the cmdscale function from the R statistical package (R Development Core Team 2010).
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Acknowledgments
We are grateful to our colleagues in the Intelligent Sound and Music Systems group at Goldsmiths, and the computational creativity community at large, for many years of richly fruitful discussion. We also thank several anonymous reviewers for constructive comments on earlier drafts. The work reported here was supported by an Arts and Humanities Research Council Doctoral Studentship awarded to the first author and an Engineering and Physical Sciences Research Council DTA Doctoral Studentship awarded to the third author.
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Forth, J., Wiggins, G.A. & McLean, A. Unifying Conceptual Spaces: Concept Formation in Musical Creative Systems. Minds & Machines 20, 503–532 (2010). https://doi.org/10.1007/s11023-010-9207-x
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DOI: https://doi.org/10.1007/s11023-010-9207-x