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Melcomplexity Escom 20090729

Talk held at the ESCOM 2009 in Jyäskylä

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Melcomplexity Escom 20090729

  1. 1. Melodic Complexity Klaus Frieler Universität Hamburg Musikwissenschaftliches Institut ESCOM 2009, Jyväskylä 12.8.2009
  2. 2. <ul><li>Melodic Complexity </li></ul><ul><li>Perceived complexity is a complex process generated within a signal/receiver system </li></ul><ul><li>Hypothesis: Perceived complexity is a function of (objective) signal complexity </li></ul>
  3. 3. <ul><li>Melodic Complexity </li></ul><ul><li>Idea : Test various algorithmical melodic complexity measures in psychological experiments </li></ul><ul><li>If there are significant correlations, build a model </li></ul>
  4. 4. <ul><li>Melodic Complexity </li></ul><ul><li>Complexity algorithms for n-gram sequences: </li></ul><ul><li>Entropies </li></ul><ul><li>Zipf complexity </li></ul><ul><li>N-gram redundancy </li></ul>
  5. 5. <ul><li>Algorithm Construction </li></ul><ul><li>Given: Melody as onset/pitch sequences with metrical annotations </li></ul><ul><li>Apply basic transformations </li></ul><ul><li>Here: pitches, intervals, durations, metrical circle map (cf. later) </li></ul>
  6. 6. <ul><li>Algorithm Construction </li></ul><ul><li>Main transformation: n-gram sequences, i.e. sequences of subsequences of length n </li></ul><ul><li>Calculate histograms of n-grams </li></ul><ul><li>Here: n = 1, 2, 3, 4, variable </li></ul>
  7. 7. <ul><li>Entropies </li></ul><ul><li>Entropies of n-gram distribution </li></ul><ul><li>Norm by max. entropy </li></ul>
  8. 8. <ul><li>Zipf complexity </li></ul><ul><li>Zipf‘s law: The ordered sequence of term frequencies obeys a power law (k = rank): </li></ul><ul><li>h(k) ~ k -s </li></ul><ul><li>log h(k) ~ - s log k </li></ul>
  9. 9. <ul><li>Zipf complexity </li></ul>Source: Wikipedia
  10. 10. <ul><li>Zipf complexity </li></ul><ul><li>Ordered n-gram frequencies </li></ul><ul><li>Regression on log-log data with slope s </li></ul><ul><li>Define c := 2 s as Zipf complexity </li></ul><ul><li>s = 0  c = 1, s = -1  c = 0.5, </li></ul><ul><li>s = -   c = 0 </li></ul>
  11. 11. <ul><li>N-gram redundancy </li></ul><ul><li>Number of distinct elements in a sequence is a simple measure of redundancy. </li></ul><ul><li>The more distinct elements the more „complex“ </li></ul>
  12. 12. <ul><li>N-gram redundancy </li></ul><ul><li>Let |n(s)| be the count of distinct n-grams in a sequence s of length N . Then </li></ul>
  13. 13. <ul><li>N-gram redundancy </li></ul><ul><li>Extensions: Weighted sum of n-gram redundancies up to a fixed or variable n max </li></ul>
  14. 14. Metrical Circle Map
  15. 15. Ex.: „Mandy“ by Barry Manilow
  16. 16. <ul><li>Experiments </li></ul><ul><ul><li>Two listening experiments with a total of 47 subjects </li></ul></ul><ul><ul><li>Stimuli: 12 folk songs and 3 jazz saxophon chorusses, 9 melodies identical in both experiments </li></ul></ul><ul><ul><li>Task: Judgement of melodic complexity on a scale from 1-7 </li></ul></ul>
  17. 17. <ul><li>Results </li></ul><ul><li>Normal distributed, reliable judgements </li></ul><ul><li> Pooling of data from both experiments and </li></ul><ul><li> Using subject means for further comparisions </li></ul>
  18. 18. <ul><li>Results </li></ul><ul><li>42 complexity measures: </li></ul><ul><ul><li>Note count </li></ul></ul><ul><ul><li>Metrical Markov entropies (0th, 1th order) </li></ul></ul><ul><ul><li>Zipf complexities (int, pitch, dur) </li></ul></ul><ul><ul><li>N-gram redundancies (int, pitch, dur) </li></ul></ul><ul><ul><li>Entropies (int, pitch, dur) </li></ul></ul>
  19. 19. <ul><li>Correlations </li></ul>Note count r = ,869**
  20. 20. <ul><li>Results </li></ul><ul><li>Note count explains judgement nearly perfect! </li></ul><ul><li>Calculate partial correlations for other measures  Only metrical entropies left </li></ul>
  21. 21. <ul><li>Correlations </li></ul>0 th order metrical entropy r = ,934** r‘= ,837**
  22. 22. <ul><li>Correlations </li></ul>1 st order metrical entropy r = ,944** r‘= ,867**
  23. 23. <ul><li>Correlations </li></ul>Pitch entropy r = ,132 r‘= ,092
  24. 24. <ul><li>Linear Regression </li></ul><ul><li>Stepwise regression of variables with highest correlation </li></ul><ul><li>Corrected R 2 = .929 </li></ul><ul><li>zsubjmean = .345 * znotecount </li></ul><ul><li> + .677 * zmeter1ent </li></ul>
  25. 25. <ul><li>Conclusion </li></ul><ul><li>Good agreement with measured complexity </li></ul><ul><li>1 st order Metrical Markov entropy shows highest correlation </li></ul><ul><li>But: Note count explains most of all correlations </li></ul><ul><li> Rather simple complexity ?! </li></ul>
  26. 26. <ul><li>Conclusion </li></ul><ul><li>No partial correlation with any pitch/interval based measure could be found </li></ul><ul><li>Meter is the most important dimension… </li></ul>
  27. 27. <ul><li>Outlook </li></ul><ul><li>We plan experiments with note counts kept constant </li></ul><ul><li>Pretests show, that metrical entropies might be suited to predict „hit-potential“ of pop melodies </li></ul>
  28. 28. Thank you!
  29. 29. Metrical Intervals
  30. 30. Metrical Markov chain 0 th order
  31. 31. Metrical Markov chains 1 st order