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Unifying Conceptual Spaces: Concept Formation in Musical Creative Systems

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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

  1. 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.

  2. 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.

  3. 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 \({\fancyscript{C}}^{s}\).

  4. While “up” and “down” are standard terminology, sequences of them are not.

  5. 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.

  6. 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.

  7. 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.

  8. MDS was carried out using the cmdscale function from the R statistical package (R Development Core Team 2010).

References

  • Aisbett, J., & Gibbon, G. (2001). A general formulation of conceptual spaces as a meso level representation. Artificial Intelligence, 133(1–2), 189–232.

    Article  MATH  MathSciNet  Google Scholar 

  • Baars, B. J. (1988). A cognitive theory of consciousness. Cambridge: Cambridge University Press.

    Google Scholar 

  • Benjamin, W. E. (1984). A theory of musical meter. Music Perception, 1(4), 355–413.

    Google Scholar 

  • Boden, M. A. (1998). Creativity and artificial intelligence. Artificial Intelligence Journal, 103, 347–356.

    Article  MATH  MathSciNet  Google Scholar 

  • Boden, M. A. (2004). The creative mind: Myths and mechanisms, (2nd ed.). London, UK: Routledge.

    Google Scholar 

  • Bown, O., & Wiggins, G. A. (2009). From maladaptation to competition to cooperation in the evolution of musical behaviour. Musicæ Scientiæ (special issue on evolution of music).

  • Brachman, R. J., & Levesque, H. J., (Eds.). (1985). Readings in knowledge representation. Los Altos: Morgan Kaufmann.

    MATH  Google Scholar 

  • Caclin, A., Brattico, E., Tervaniemi, M., Näätänen, R., Morlet, D., Giard, M.-H., et al. (2006). Separate neural processing of timbre dimensions in auditory sensory memory. Journal Cognitive Neuroscience, 18(12), 1959–1972.

    Article  Google Scholar 

  • Carota, F., Posada, A., Harquel, S., Delpuech, C., Bertrand, O., & Sirigu, A. (2009). Neural dynamics of the intention to speak. Cerebral Cortex, 20(8), 1891–1897.

    Article  Google Scholar 

  • Chew, E. (2000). Towards a mathematical model of tonality. PhD thesis, Department of Operations Research. Massachusetts: MIT, Cambridge.

  • Clarke, E. F. (1999). Rhythm and timing in music. In D. Deutsch (Eds.), Psychology of music (2nd ed. pp. 473–500). San Diego, CA: University of California.

    Chapter  Google Scholar 

  • Cooper, G. W., & Meyer, L. B. (1960). The rhythmic structure of music. Chicago, IL: University of Chicago Press.

    Google Scholar 

  • Dowling, W. J. (1978). Scale and contour: Two components of a theory of memory for melodies. Psychological Review, 85(4), 341–354.

    Article  Google Scholar 

  • Fraisse, P. (1978). Time and rhythm perception. In E. Carterette, & M. Friedmans (Eds.), Handbook of perception (Vol. 2, pp. 203–254). New York, NY: Academic Press

    Google Scholar 

  • Gabrielsson, A. (1993). The complexities of rhythm. In T. J. Tighe, & W. J. Dowling (Eds.), Psychology and music: The understanding of melody and rhythm (pp. 93–120). Hillside, NJ: Lawrence Erlbaum Associates, Inc.

    Google Scholar 

  • Gärdenfors, P. (2000). Conceptual spaces: The geometry of thought. Cambridge, MA: MIT Press.

    Google Scholar 

  • Gärdenfors, P. (2007). Representing actions and functional properties in conceptual spaces. In T. Ziemke, J. Zlatev, & R. M. Frank (Eds.), Body, language and mind (Vol. 1, pp. 167–195). Berlin, Germany: Mouton de Gruyter.

    Google Scholar 

  • Guilford, J. (1967). The nature of human intelligence. New York: McGraw-Hill.

    Google Scholar 

  • Haase, K. (1995). Too many ideas, just one word: a review of Margaret Boden’s The creative mind: Myths and mechanisms. Artificial Intelligence Journal, 79, 69–82.

    Article  Google Scholar 

  • Hasty, C. F. (1997). Metre as rhythm. Oxford: Oxford University Press.

    Google Scholar 

  • Jain, A., Nandakumar, K., & Ross, A. (2005). Score normalization in multimodal biometric systems. Pattern Recognition, 38(12), 2270–2285.

    Article  Google Scholar 

  • Koestler, A. (1964). The act of creation. London: Hutchinson & Co.

    Google Scholar 

  • Lemström, K., & Wiggins, G. A. (2009). Formalizing invariances for content-based music retrieval. In G. Tzanetakis & K. Hirata (Eds.), Proceedings of ISMIR 2009.

  • Lerdahl, F., & Jackendoff, R. (1983). A generative theory of tonal music. Cambridge, MA: MIT Press.

    Google Scholar 

  • London, J. (2004). Hearing in time: Psychological aspects of music metre. Oxford, UK: Oxford University Press.

    Google Scholar 

  • Lustig, R. (1995). Margaret Boden, the creative mind: Myths and mechanisms. Artificial Intelligence Journal, 79, 83–96.

    Article  Google Scholar 

  • MacKay, D. J. C. (1998). Introduction to Monte Carlo methods. In M. I. Jordan (Ed.), Learning in graphical models, NATO science series (pp. 175–204). Dordrecht, The Netherlands: Kluwer.

  • Müllensiefen, D. (2009). Fantastic: Feature analysis technology accessing statistics (in a corpus). Technical report, Goldsmiths, London, UK: University of London.

  • Parncutt, R. (1994). A perceptual model of pulse salience and metrical accent in musical rhythms. Music Perception, 11, 409–464.

    Google Scholar 

  • Parsons, D. (1975). The directory of tunes and musical themes. Cambridge, UK: S. Brown.

    Google Scholar 

  • Patel, A. D. (2008). Music, language, and the brain. Oxford: Oxford University Press.

    Google Scholar 

  • Pearce, M. T., & Wiggins, G. A. (2006). Expectation in melody: The influence of context and learning. Music Perception, 23(5), 377–405.

    Article  Google Scholar 

  • Perkins, D. (1995). An unfair review of Margaret Boden’s The Creative Mind from the perspective of creative systems. Artificial Intelligence Journal, 79, 97–109.

    Article  Google Scholar 

  • Povel, D.-J. (1984). A theoretical framework for rhythm perception. Psychological Research, 45(4), 315–337.

    Article  Google Scholar 

  • R Development Core Team. (2010). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. ISBN 3-900051-07-0.

  • Ram, A., Wills, L., Domeshek, E., Nersessian, N., & Kolodner, J. (1995). Understanding the creative mind: A review of Margaret Boden’s Creative Mind. Artificial Intelligence Journal, 79, 111–128.

    Article  Google Scholar 

  • Raubal, M. (2004). Formalizing conceptual spaces. In A. Varzi & L. Vieu (Eds.), Formal ontology in information systems, proceedings of the third international conference (FOIS 2004), Vol. 114 of frontiers in artificial intelligence and applications (pp. 153–164). Amsterdam, NL: IOS Press.

  • Raubal, M. (2008a). Cognitive modeling with conceptual spaces. In M. Ragni, H. Schultheis, & T. Barkowsky (Eds.), Workshop on cognitive models of human spatial reasoning (pp. 7–11). Freiburg, Germany.

  • Raubal, M. (2008b). Representing concepts in time. In C. Freksa, N. S. Newcombe, Peter Gärdenfors, & S. Wölfl (Eds.), Spatial cognition VI. Learning, reasoning, & talking about space. Lecture Notes in Computer Science (Vol. 5248, pp. 328–343). Berlin: Springer.

  • Rickard, J. T. (2006). A concept geometry for conceptual spaces. Fuzzy Optimization and Decision Making, 5(4), 311–329.

    Article  MATH  MathSciNet  Google Scholar 

  • Rickard, J. T., Aisbett, J., & Gibbon, G. (2007a). Knowledge representation and reasoning in conceptual spaces. In Foundations of Computational Intelligence, 2007. FOCI 2007. IEEE Symposium on (pp. 583–590).

  • Rickard, J. T., Aisbett, J., & Gibbon, G. (2007b). Reformulation of the theory of conceptual spaces. Information Sciences, 177(21), 4539–4565.

    Article  MATH  MathSciNet  Google Scholar 

  • Roberson, D., Davidoff, J., Davies, I. R. L., & Shapiro, L. R. (2006). Colour categories and category acquisition in Himba and English. In N. Pitchford & C. Bingham (Eds.), Progress in colour studies (pp. 159–172). Amsterdam: John Benjamins.

    Google Scholar 

  • Schank, R., & Foster, D. (1995). The engineering of creativity: A review of Boden’s The Creative Mind. Artificial Intelligence Journal, 79, 129–143.

    Article  Google Scholar 

  • Schwering, A., & Raubal, M. (2005a). Measuring semantic similarity between geospatial conceptual regions. In GeoSpatial semantics—First international conference, GeoS 2005, Vol. 3799 of LNCS (pp. 90–106). Berlin, Germany: Springer.

  • Schwering, A., & Raubal, M. (2005b). Spatial relations for semantic similarity measurement. In Perspectives in conceptual modeling, Vol. 3770 of LNCS (pp. 259–269). Berlin, Germany: Springer.

  • Shepard, R. N. (1982). Structural representations of musical pitch. In D. Deutsch (Ed.), Psychology of music (pp. 343–390). New York: Academic Press.

    Google Scholar 

  • Shepard, R. N. (1987). Toward a universal law of generalization for psychological science. Science, 237(4820), 1317–1323.

    Article  MathSciNet  Google Scholar 

  • Steedman, M. J. (1977). The perception of musical rhythm and metre. Perception, 6(5), 555–569.

    Article  Google Scholar 

  • Thornton, C. (2007). How thinking inside the box can become thinking outside the box. In A. Cardoso & G. Wiggins (Eds.), Proceedings of the 4th international joint workshop on computational creativity.

  • Thornton, C. (2009). Personal communication.

  • Turner, S. R. (1995). Margaret Boden, The Creative Mind. Artificial Intelligence Journal, 79, 145–159.

    Article  Google Scholar 

  • Wallas, G. (1926). The art of thought. New York: Harcourt Brace.

    Google Scholar 

  • Wiggins, G. A. (2006a). A preliminary framework for description, analysis and comparison of creative systems. Journal of Knowledge Based Systems, 19(7), 449–458.

    Article  Google Scholar 

  • Wiggins, G. A. (2006b). Searching for computational creativity. New Generation Computing, 24(3), 209–222.

    Article  MATH  Google Scholar 

  • Wiggins, G. A., Harris, M., & Smaill, A. (1989). Representing music for analysis and composition. In M. Balaban, K. Ebcioglu, O. Laske, C. Lischka, & L. Soriso (Eds.), Proceedings of the second workshop on AI and music (pp. 63–71). Menlo Park, CA: AAAI.

  • Wiggins, G. A., Pearce, M. T., & Müllensiefenllensiefen, D. (2009). Computational modelling of music cognition and musical creativity. In R. Dean (Ed.), Oxford handbook of computer music and digital sound culture. Oxford: Oxford University Press.

    Google Scholar 

  • Zajonc, R. B. (1968). Attitudinal effects of mere exposure. Journal of Personality and Social Psychology, Monograph Supplement, 9, 1–27.

    Article  Google Scholar 

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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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  1. Department of Computing & Centre for Cognition, Computation and Culture, Goldsmiths, University of London, London, UK

    Jamie Forth, Geraint A. Wiggins & Alex McLean

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  1. Jamie Forth
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Correspondence to Jamie Forth.

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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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