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

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

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

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

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Tn-sets of cardinality 3 Tn-setsemitones 3-1012 3-2A013 3-2B023 3-3A014 3-3B034 3-4A015 3-4B045 3-5A016 3-5B056 3-6024 3-7A025 3-7B035 3-8A026 3-8B046 3-9027 3-10036 3-11A037 3-11B047 3-12048

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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 spacingof each tone doubling surface parameters duration loudnessof each tone timbre

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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)

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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

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Root-support intervals Root- support interval P1, P8 P5, P12 M3, M10 m7, m14 M2, M9 074102 weight105321 Estimation of root-support weights Music-theoretic intuition predictions of model intuitively correct? Comparison of predictions with data Krumhansl & Kessler (1982), Parncutt (1993)

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Harmonic series template

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Octave-generalised template

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Circular representation of template

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Matrix multiplication model notes x template = saliences notes 1 0 0 0 1 0 0 1 0 0 0 0 saliences 18 0 3 10 6 2 10 3 7 1 0 template

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Major triad 047 notes pitches

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Minor triad 037 notes pitches

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Diminished triad 036 notes pitches

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Augmented triad 048 notes pitches

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Experimental data Diamonds: mean ratings Squares: predictions

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Krumhansls key profiles

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Tonal profiles Probability that a tone perceived as the tonic Algorithm: Krumhansls 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

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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

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The major and minor triads pitch class semitones01234567891011 letter name CDEFGAB major triad 3-11B (047) harmonic profile 34066191141961320 tonal profile 220135171002241349 minor triad 3-11A (037) harmonic profile 29242501501915426 tonal profile 1471012811714108118

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Tn-sets of cardinality 3 a h : harmonic ambiguity a t : tonal ambiguity r: correlation between harmonic and tonal profiles Tn-setsemi- tones ahah atat r 3-10122.293.260.72 3-2A0132.293.110.75 3-2B0232.293.110.75 3-3A0142.203.130.75 3-3B0342.203.130.75 3-4A0152.052.970.83 3-4B0452.052.970.83 3-5A0162.053.080.73 3-5B0562.053.080.73 3-60242.123.110.75 3-7A0252.052.950.85 3-7B0351.932.950.85 3-8A0262.053.140.59 3-8B0462.203.140.59 3-90271.982.820.90 3-100362.513.110.62 3-11A0372.052.950.84 3-11B0471.872.950.84 3-120482.203.150.74

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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