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Music, emotion and Zipf’s law

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  1. Quantitative linguistics
    1. Frequency distributions
    2. Entropy
  2. Previous research
    1. Balancing effort
    2. Data types
    3. Corpus size
  3. Music and Zipf 's law
    1. Music and communication
    2. Music and emotion
    3. Research question
    4. Experimental setup
  4. Results
    1. The rank-frequency distribution of notes and intervals
    2. Single music compositions
  5. Discussion and conclusion
  6. Bibliography

The hypothesis of Zipf concerning a universal Principle of Least Effort, manifesting itself in Zipf 's law and modeled by Ferrer i Cancho and Sol'e in a signal-object reference matrix, gave rise to the idea that maybe the elements
in music that elicit our emotional responses can be identified. The unde- niable relation between music and emotion was the reason to consider a possible signal-emotion reference system analogous to the signal-object ref- erence in natural human language. Following Zipf 's line of reasoning, music
as an exponent of human behaviour is subject to the Principle of Least Effort and is consequently structured in such a way that the distribution pattern
of the signals that ?communicate the musical message? follows a power law.
In this thesis the possibility to deploy the characteristics of the Zipf curve to gain more insight into the relation between music and emotion was investi- gated. From the elements that, viewed in the framework of a signal-emotion reference could qualify as the signals that elicit emotion, two were investi- gated, viz. notes and intervals. Experiments were performed on 18 single classical music pieces from 18 different composers and on a larger classical music corpus. Results indicate that intervals have a distribution pattern that comes closest to a Zipf curve, but neither data type exhibits a gen- uine Zipf distribution. Further research will be needed to decide whether the method that was applied could be a useful tool in the search for the elements that elicit emotion.

[...] Extension of Zipf 's law to words and phrases. In Proceedings of the 19th International Conference on Computational Linguistics, volume pages 315?320. Ha, L., Sicilia-Garcia, E., Ming, J., and Smith, F. J. (2003). Extension of Zipf 's law to word and character n-grams for English and Chinese. In Journal of Computational Linguistics and Chinese Language Processing (CLCLP), volume pages pp. 77?102. Ha, L., Sicilia-Garcia, E., and Smith, F. (2004). Zipf and type-token rules for the English, Irish and Latin languages. [...]

[...] These results support the idea that Zipf 's law can be explained as the result of a balance between the minimized effort of both speaker and hearer Data types Zipf 's law has been tested over many data types, some of which were derived from different kinds of human and non-human communication. All elements that make up human language and are countable could serve as a subject of investigation. Zipf 's research was targeted at characteristics like, for example, word frequency, word length, word intervals, morphemes and phonemes and comprised languages like English, German, Hebrew and Chinese (Zipf, 1949). [...]

[...] Based on the hypothesis that the rank-frequency curves of the emotion evoking elements in music should have the shape of a Zipf curve, the curves of notes and intervals were compared to the Zipf curve. Although the curves of intervals approximated the Zipf curve better than the curves of notes, neither data type exhibited a genuine Zipf distribution. In this exploratory study, the number of works that have been investigated was rather small, which re- duces the reliability and complicates the interpretation of the results. [...]

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