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)
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
Matrix multiplication model notes x template = saliences
notes 1 0 0 0 1 0 0 1 0 0 0 0
saliences18033
1062
103710
template
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