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Tonal implications of harmonic and melodic Tn-sets

Tonal implications of harmonic and melodic Tn-sets. Richard Parncutt University of Graz, Austria Presented at Mathematics and Computation in Music (MCM2007) Berlin, Germany, 18-20 May, 2007. “Atonal” music is not atonal!. Every… interval sonority melodic fragment

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Tonal implications of harmonic and melodic Tn-sets

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  1. Tonal implications of harmonic and melodic Tn-sets Richard Parncutt University of Graz, Austria Presented at Mathematics and Computation in Music (MCM2007) Berlin, Germany, 18-20 May, 2007

  2. “Atonal” music is not atonal! Every… • interval • sonority • melodic fragment …has tonal implications. Exceptions: • null set (cardinality = 0) • chromatic aggregate (cardinality = 12)

  3. Finding “atonal” pc-sets • Build your own • avoid octaves and fifth/fourths • favor tritones and semitones • listening (trial and error) • Borrow from the literature

  4. Aim of this study Systematic search for pc-sets with specified • cardinality • strength of tonal implication

  5. Tn-sets of cardinality 3

  6. What influences tonal implications? Intervals of a Tn-set • pc-set • inversion, if not symmetrical • e.g. minor (037, 3-11A) vs major (047, 3-11B) Realisation • voicing • register • spacing …of each tone • doubling • surface parameters • duration • loudness …of each tone • timbre

  7. Perceptual profile of a Tn-set perceptual salience of each chromatic scale degree Two kinds: • harmonic profile of a simultaneity • model: pitch of complex tones (Terhardt) • tonal profile when realisation not specified • model: major, minor key profiles (Krumhansl)

  8. Harmonic profile • probability that each pitch perceived as root Parncutt (1988) chord-root model, based on • virtual pitch algorithm (Terhardt et al., 1982) • chord-root model (Terhardt, 1982) “Root is a virtual pitch”

  9. Root-support intervals Estimation of root-support weights • Music-theoretic intuition • predictions of model intuitively correct? • Comparison of predictions with data • Krumhansl & Kessler (1982), Parncutt (1993)

  10. Harmonic series template

  11. Octave-generalised template

  12. Circular representation of template

  13. Matrix multiplication modelnotes x template = saliences notes 1 0 0 0 1 0 0 1 0 0 0 0 template saliences 18 0 3 3 10 6 2 10 3 7 1 0

  14. Major triad 047notes pitches

  15. Minor triad 037notes pitches

  16. Diminished triad 036notes pitches

  17. Augmented triad 048notes pitches

  18. Experimental data Diamonds: mean ratings Squares: predictions

  19. Krumhansl’s key profiles

  20. Tonal profiles Probability that a tone perceived as the tonic Algorithm: • Krumhansl’s key profiles: 24 stability values • subtract 2.23 from all  minimum stability = 0 • estimate probability that Tn-set is in each key (just add stability values of tones in that key) • tonal profile = weighted sum of 24 key profiles

  21. Ambiguity of a tone profile • flear peak: low ambiguity • flat: high ambiguity Algorithm: • add 12 values • divide by maximum • take square root cf. number of tones heard in a simultaneity

  22. The major and minor triads

  23. Tn-sets of cardinality 3ah: harmonic ambiguityat: tonal ambiguity r: correlation between harmonic and tonal profiles

  24. Musical prevalence of a Tn-set Depends on: • ambiguity • roughness (semitones, tritones…) • whether subset of a prevalent sets of greater cardinality • e.g. 036 is part of 0368

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