Tonal implications of harmonic and melodic Tn-sets Richard Parncutt University of Graz, Austria Presented at Mathematics and Computation in Music (MCM2007)Mathematics.

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

…has tonal implications.

Exceptions: • null set (cardinality = 0)• chromatic aggregate (cardinality = 12)

Finding “atonal” pc-sets

• Build your own– avoid octaves and fifth/fourths– favor tritones and semitones– listening (trial and error)

• Borrow from the literature

Aim of this study

Systematic search for pc-sets with specified– cardinality – strength of tonal implication

Tn-sets of cardinality 3

Tn-set semitones3-1 012

3-2A 013

3-2B 023

3-3A 014

3-3B 034

3-4A 015

3-4B 045

3-5A 016

3-5B 056

3-6 024

3-7A 025

3-7B 035

3-8A 026

3-8B 046

3-9 027

3-10 036

3-11A 037

3-11B 047

3-12 048

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

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)

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”

Root-support intervals

Root-support interval

P1, P8…

P5, P12…

M3, M10…

m7, m14…

M2, M9…

0 7 4 10 2

weight 10 5 3 2 1Estimation of root-support weights • Music-theoretic intuition

– predictions of model intuitively correct?

• Comparison of predictions with data – Krumhansl & Kessler (1982), Parncutt (1993)

Harmonic series template

0

1

0 4 8 12

16

20

24

28

32

36

40

interval (semitones)

po

ids

(1

/n)

Octave-generalised template

0

24

68

10

0 1 2 3 4 5 6 7 8 9 10 11

interval class (semitones)

we

igh

t

Circular representation of template

0

1

2

3

4

5

6

7

8

9

10

11

Matrix multiplication model notes x template = saliences

notes 1 0 0 0 1 0 0 1 0 0 0 0

saliences18033

1062

103710

template

Major triad 047

notes pitches

0

1

2

3

4

5

6

7

8

9

10

11

0

1

2

3

4

5

6

7

8

9

10

11

Minor triad 037

notes pitches0

1

2

3

4

5

6

7

8

9

10

11

0

1

2

3

4

5

6

7

8

9

10

11

Diminished triad 036

notes pitches0

1

2

3

4

5

6

7

8

9

10

11

0

1

2

3

4

5

6

7

8

9

10

11

Augmented triad 048

notes pitches0

1

2

3

4

5

6

7

8

9

10

11

0

1

2

3

4

5

6

7

8

9

10

11

Experimental data

Diamonds: mean ratings

Squares: predictions

Krumhansl’s key profiles

0

1

2

3

4

5

6

7

C C# D D# E F F# G G# A A# B

Tone

Rat

ing

s fo

r C

Maj

or

0

1

2

3

4

5

6

7

C C# D D# E F F# G G# A A# B

Tone

Ra

tin

gs

fo

r C

Min

or

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

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

The major and minor triads

pitch class

semitones 0 1 2 3 4 5 6 7 8 9 10 11

letter name

C D E F G A B

major triad3-11B (047)

harmonic profile

34 0 6 6 19 11 4 19 6 13 2 0

tonal profile

22 0 13 5 17 10 0 22 4 13 4 9

minor triad 3-11A (037)

harmonic profile

29 2 4 25 0 15 0 19 15 4 2 6

tonal profile

14 7 10 12 8 11 7 14 10 8 11 8

Tn-sets of cardinality 3

ah: harmonic ambiguity

at: tonal ambiguity

r: correlation between harmonic and tonal profiles

Tn-set semi-tones

ah at r

3-1 012 2.29 3.26 0.72

3-2A 013 2.29 3.11 0.75

3-2B 023 2.29 3.11 0.75

3-3A 014 2.20 3.13 0.75

3-3B 034 2.20 3.13 0.75

3-4A 015 2.05 2.97 0.83

3-4B 045 2.05 2.97 0.83

3-5A 016 2.05 3.08 0.73

3-5B 056 2.05 3.08 0.73

3-6 024 2.12 3.11 0.75

3-7A 025 2.05 2.95 0.85

3-7B 035 1.93 2.95 0.85

3-8A 026 2.05 3.14 0.59

3-8B 046 2.20 3.14 0.59

3-9 027 1.98 2.82 0.90

3-10 036 2.51 3.11 0.62

3-11A 037 2.05 2.95 0.84

3-11B 047 1.87 2.95 0.84

3-12 048 2.20 3.15 0.74

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