Transcript 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,

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
semitones
3-1
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
– doubling
…of each tone
• surface parameters
– duration
– loudness
– timbre
…of each tone
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
Rootsupport
interval
weight
P1,
P8…
P5,
P12…
0
10
7
5
M3,
m7,
M10… m14…
4
3
M2,
M9…
10
2
2
1
Estimation 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
poids (1/n)
1
0
40
36
32
28
24
20
16
12
8
4
0
interval (semitones)
weight
Octave-generalised template
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
interval class (semitones)
9
10 11
Circular representation of template
0
11
1
10
2
9
3
8
4
7
5
6
Matrix multiplication model
notes x template = saliences
1
0
0
0
1
notes
0
0
template
1
0
0
0
0
saliences
18
0
3
3
10
6
2
10
3
7
1
0
Major triad 047
notes
pitches
0
11
0
1
10
11
2
9
10
3
8
4
7
5
6
1
2
9
3
8
4
7
5
6
Minor triad 037
notes
pitches
0
11
0
11
1
10
10
2
9
3
8
4
7
5
6
1
2
9
3
8
4
7
5
6
Diminished triad 036
notes
pitches
0
11
0
1
10
11
2
9
10
3
8
4
7
5
6
1
2
9
3
8
4
7
5
6
Augmented triad 048
notes
pitches
0
0
11
1
11
2
10
10
3
9
4
8
5
7
6
1
2
9
3
8
4
7
5
6
Experimental data
Diamonds:
mean ratings
Squares:
predictions
Krumhansl’s key profiles
Ratings for C Major
7
6
5
4
3
2
1
0
C
C#
D
D#
E
F
F#
G
G#
A
A#
B
G
G#
A
A#
B
Tone
Ratings for C Minor
7
6
5
4
3
2
1
0
C
C#
D
D#
E
F
F#
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:
• flat:
low ambiguity
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
semitones
0
letter
name
C
major
triad
3-11B
(047)
harmonic
profile
34
0
6
tonal
profile
22
0
minor
triad
3-11A
(037)
harmonic
profile
29
tonal
profile
14
pitch
class
1
2
3
4
5
E
F
6
19
11
4
19
6
13
2
0
13
5
17
10
0
22
4
13
4
9
2
4
25
0
15
0
19
15
4
2
6
7
10
12
8
11
7
14
10
8
11
8
D
6
7
8
G
9
10
A
11
B
Tn-sets of
cardinality 3
ah: harmonic
ambiguity
at: tonal
ambiguity
r: correlation
between
harmonic and
tonal profiles
Tn-set
semitones
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