9th EUROPEAN MUSIC ANALYSIS CONFERENCE — EUROMAC 9
Erica Bisesi
Royal Institute of Technology – Stockholm, Sweden & Comenius University – Bratislava, Slovakia
[email protected] , [email protected]
Measuring and Modelling Perceived Distance Among Collections in Post-Tonal Music:
Music theory Meets Music Psychology
ABSTRACT
Can we perceive similarity among chords or keys, and
which computational models best account for it? How distance
profiles are related to other structural aspects (e.g. grouping,
melody, rhythm) in different pieces of music?
Background
Models for distance between pitch-class sets include psychoacoustical models between chord–roots (Parncutt 1988;
1993), chord Tonnetz (Cohn 1998; Tymoczko 2011), interval
proximity (Rahn 1979–1980; Morris 1980; Lewin 1987), and
voice leading (Straus 2003; 2005; 2014). Krumhansl (1998)
tested and supported three different neo-Riemannian models of
triadic distance against psychological data. Roger and
Callender (2006) explored correlations between perceived
distance between triads and factors like tuning environment,
direction of motion, and relationship of moving voices. Milne
and Holland (2016) provided a psychoacoustical explanation
for perceived triadic distance based on a spectral pitch-class
representation.
Aims and repertoire studied
Our repertoire includes three post-tonal pieces belonging to
different stylistic conventions: Webern Canon for Voice and
Clarinet Op. 16 No. 2, Bartok Mikrokosmos No. 84 (Merriment), and Scriabin Poem Op. 69 No. 1.
In this paper, we discuss preliminary results for the first of
these pieces, the Webern Canon for Voice and Clarinet Op. 16
No. 2. This piece belongs to a series of vocal works composed
by Anton Webern in the 1910s and 1920s, and exemplifies how
his seemingly austere, constructivist methods can be combined
with expressive achievements. It ‘also emphasize(s) Webern’s
penchant for canonical structures — indeed, the entire collection is rife with echoed contours of lines in counterpoint and
complementary motivic shapes as melodies unfolding in mirror
images, like the angular peaks of a mountain range reflected in
inversion on the surface of a lake’ (Jeremy Grinshaw).
We compare five contrasting models of the distance between
collections in different post-tonal pieces and test them against
perceptual data. The models include an adaptation of the
Parncutt (1988) psychoacoustical model of distance between
chord-roots, three models of distance based on interval similarity (ASYM, MEMBn, REL: Morris 1980; Rahn 1979–1980;
Lewin 1987), and atonal voice-leading (uniformity, balance,
smoothness: Straus 2003).
Methods
1) Music Analysis. In a first stage, the profiles of distance
between sets are computed by means of algorithms (Ariza 2002;
Bisesi et al. in preparation). To this purpose, the piece has been
preliminarily segmented into chunks, each chunk containing
one pair of sets to be compared. Given the specific symmetry in
the music structure, i.e. the form of canon, we segmented the
piece at the level of single bars and compared, for each bar, the
upper with the lower voice. By ignoring the first and the last
bars (which contains only one pair), we extracted in total 11
pairs. Example 1 provides an example of segmentation for the
first three bars, where sets have been named according to the
Set Theory. The complete list of 11 pairs, as resulting from this
preliminary analysis, is listed in Table 1.
0236
0156
I2
0236 I2
I2
0156
0126
Ex. 1. Webern Canon for Voice and Clarinet Op. 16 No. 2, bars 1-3:
segmentation and set analysis.
Table 1. Pairwise comparison between sets in the Webern Canon
for Voice and Clarinet Op. 16 No. 2.
Sets
Pair
Bar
1
2
3
4
5
6
7
8
9
10
11
2
3
4
5
6
7
8
9
10
11
12
Forte number
4-12A / 4-8
4-8 / 4-5B
4-5A / 4-z15B
4-z15A / 4-13B
4-13A / 3-2A
3-2B / 4-4B
4-4A / 4-7
4-7 / 4-4B
4-4A / 4-4B
4-4A / 4-11B
4-11A / 3-3B
Set Classes
0236 / 0156
0156 / 0126
0126 / 0146
0146 / 0136
0136 / 013
013 / 0125
0125 / 0145
0145 / 0125
0125 / 0125
0125 / 0135
0135 / 014
Figure 1 reports the values of the distance between each
pair of sets listed in Table 1, as calculated for each of the
models considered in this study. The algorithms being used are:
9th EUROPEAN MUSIC ANALYSIS CONFERENCE — EUROMAC 9
• Parncutt (1988): Pearson’s correlation coefficients between the pitch-salience profiles of Set X and Y in each
pair XY, as calculated over the 12 pitch classes.
• Morris (1980): a similarity index SYM (X,Y), comparing the different elements of vectors X = [x1, x2, …, xN]
and Y = [y1, y2, … ,yN].
• Rahn (1979-1980): embedding ATMEMBn (S, X, Y), as
Trial No. 9 was used as a benchmark. In total, each participant
rated 33 stimuli.
Results
Measured and predicted similarities between sets are shown
respectively in Figure 1 and 2.
defined by:
12
ATMEMB( A , B) =
å MEMB (X , A, B)
n
2
n=2
#A
+2
#B
-(# A + #B+ 2)
,
where MEMBn (S, X, Y) is a count of all subsets S of a
specified size n embedded mutually in Sets X and Y, and
the symbol # stands for cardinality.
• Lewin (1987): a measure of the extent of relatedness
based on a probabilistic approach, REL (X,Y).
• Straus (2003): the minimum distance, defined as
dmin = min (U min , Bmin ),
where Umin is the minimum uniformity (measured by the
minimum extent to which the voices move by the same
or nearly-the-same interval), and Bmin is the minimum
balance (the minimum extent to which the voices can be
understood to flip symmetrically around some common
axis) in the voice/leading space mapping Set X onto Set
Y.
As far as the Parncutt model is concerned, the algorithm has
been implemented in a customized Matlab code. For the other
models, calculations have been performed by means of Athena
CL-1.4.9.
2) Music Perception. In a second stage, set distances have
been innovatively explored from the viewpoint of music perception. The main purpose of this pilot study was to both investigate the best arrangement for the trials, and to test the
experimental procedure. In this regard, since the beginning we
were faced with two main question marks. First, what are the
best realizations for presenting the stimuli — i.e. should they
be chords or arpeggios, and, in the latter case, in which pitch
order? Second, are the stimuli to be presented with exactly the
same pitches as in the notation, with the advantage of being
respectful of the musical score but the disadvantage of introducing a high degree of inhomogeneity in the stimuli in terms
of the pitch classes that are involved? To answer the first
question, each stimulus was presented to the participants in
three different realizations: chords, ascending arpeggios, and
notated arpeggios (with subsequent pitches following the same
order as in the score). Half participants received the three
blocks in this order: ascending arpeggios, chords, notated arpeggios; the other half of the participants received the notated
arpeggios first, then the chords, and finally the ascending arpeggios. Inside each block, the stimuli were randomized. To
answer the second question, we chose to preserve the notated
pitch position, in order to discriminate whether absolute pitch
position might affect ratings for set similarity. Besides this, all
the pairs were normalized to have the first note as a C4 (pc = 0).
For each pair of sets listed in Table 1, we asked 15 musicians
(age = 46.2 ± 31.26%, 9 males and 6 females) to rate similarity
between the two sets in each pair on a rating scale from 1 to 3.
Fig. 1. Predicted similarities between sets in Webern Canon Op.
16 No. 2, according to the five models considered in this study.
Fig. 2. Mean ratings for similarity between sets in Webern Canon
Op. 16 No. 2. The three blocks correspond respectively to ascending arpeggios, chords, and notated arpeggios. The vertical
bars denote a 0.95 confidence interval.
1) Comparison Between Models. Although featuring different scaling, the five models considered in this study are
fairly consistent in terms of tendencies and overall predictions
for macroformal structure. In particular, we observe that all of
the models predict a high level segmentation of the piece into
two main blocks — respectively from bar 2 to bars 6-7 and
from bars 6-7 to bar 12. The two blocks exhibit a similar profile
— a monotonic increasing / decreasing function with peaks
respectively at bar 5 (sets 4-z15A and 4-13B) and bar 10 (set
4-4A and 4-4B). This segmentation might be regarded as an
evidence of internal symmetry. (Regarding the meaning of
peaks in the context of each model, see the reference literature).
Interestingly, this internal symmetry is confirmed by perceptual data, though peak positions do match in only one case: the
set pair 4-z15A / 4-13B of bar 5 in block 1. Other two pairs,
4-4A / 4-4B in bar 10 and block 3, and 4-5A / 4-z15B in bar 4
and blocks 2 and 3 feature consistency between model predictions and data at a lower level. The reason because the sets in
9th EUROPEAN MUSIC ANALYSIS CONFERENCE — EUROMAC 9
pair 4-4A / 4-7 of bar 7 were rated as more similar as predicted
might be identified with higher physical closeness between the
two sets in this trial.
We observe that the main difference between the Parncutt
model and the other four models is that the former focuses on
absolute pitches, while the latter are applied to interval vectors.
We also observe that consistency between all of the models is
higher when the Parncutt model is applied to absolute correlations between pitch-salience profiles, i.e. when only the magnitude of correlation is accounted for, regardless of its direction.
Reasoning in terms of similarity between sets, this suggests
that — according with the models — the more relevant information is carried by the interval class content of comparing
sets, rather than by the physical absolute pitch positions in each
set of a pair. Finally, we notice that the Rahn and the Lewin
models are very similar to each other, while the Morris model
looks closer to the Straus (2003) minimum distance profile.
2) Data Analysis: Main Effects. We performed 2-way repeated-measures ANOVA on factors Block and Stimulus,
where blocks correspond to the three different realizations in
which the stimuli were presented to the participants. Results
are reported in Table 2. Only the factor Stimulus is significant
at the 95% confidence level (p < 0.001), indicating that participants did distinguish between the trials — i.e. distances
between different sets are significantly perceivable — but not
across the blocks. Mean ratings for similarity between sets are
reported in Figure 2. Note the higher similarity between average ratings in blocks 1 and 2 (ascending arpeggios vs. chords; r
= .85) and 2 and 3 (chords vs. notated arpeggios; r = .79), than
in blocks 1 and 3 (ascending vs. notated arpeggios; r = .67).
Table 2. Two-way repeated-measures ANOVA for the pilot study
discussed in the text. 2P is the estimation of the effect size.
Effect
Block
Stimulus
Block x Stimulus
F
.54
9.18
1.35
P
.54
<.001***
.15
2P
.04
.40
.09
3) Interrater Reliability. The upper part of Table 3 shows
results of analysis of interrater reliability as an estimation of
internal consistency associated to the scores. Both single
blocks and different combinations of blocks are considered.
Note that reliability is pretty high, with the exception of block 3
(notated arpeggios). In the lower part of the table, we present
average and maximum Pearson’s correlations between all pairs
of raters and across all the musical examples. Both ratings
belonging to single blocks and average ratings between two or
more block are considered.
Table 3. Upper panel: Interrater reliability on the measure of
similarity between set pairs, inside single and combined blocks.
AA stands for ‘ascending arpeggios’, NA for ‘notated arpeggios’,
and C for ‘chords’. Lower panel: Average and maximum
Pearson’s correlation r between all pairs of raters and across all
of the music examples. Both ratings belonging to single blocks and
inter-block-averaged ratings are considered.
AA
NA
C
AA-NA
AA-C
Cronbach’s Alpha
.81
.53
.88
.82
.92
.87
.92
NA-C
AA-NA-C
Pairwise Correlation r
average
.24
.07
.33
.22
.38
.26
.20
AA
NA
C
<AA, NA>
<AA, C>
<NA, C >
<AA, NA, C>
maximum
.80
.79
.87
.86
.84
.84
.90
Correlations between participants are lower for notated arpeggios or blocks including notated arpeggios. This indicates that
perception of similarity between sets is affected by a systematic effect of pitch order, which in turn depends on physical
absolute pitch positions — in opposition to what asserted
above about the models. If correlations are, on average, not
very high, then this is because some pairs of raters are negatively correlated. However, in a considerable number of cases,
raters do correlate quite highly to each others.
4) Preliminary Comparison Between Models and Ratings.
At the stage of a preliminary investigation, any quantitative
reasoning about agreement/disagreement between models and
ratings is premature, as we are still exploring which are the best
realizations for the trials in terms of chords and/or arpeggios,
and in terms of pitch classes. For this reason, all the results
reported since now have to be considered as preliminary, as a
different realization in the trials might lead to different conclusions. In order to quantify the effect of pitch transposition
on set similarity, we measured the statistical significance of the
difference between measured and predicted values for similarity in the benchmark Trial n. 9 (4-4A / 4-4B), according to the
three summarizing models of Figure 1. According to all of the
models, this trial exhibits the highest expected similarity. Results for the combined block set are shown in Table 4. As we
can see, for all of the three models we should reject the null
hypothesis that data do correspond to model predictions. We
performed the t test on also separate blocks, and found that the
only block where there is a significant difference between the
two samples at the level of p < .001 is block 1 (ascending arpeggios) (t = 1.3). This confirms that the absolute pitch positions do affect ratings for similarity between set pairs significantly more than the structure of interval vectors, as well as that
there is a systematic effect of pitch order, when sets are arranged in arpeggios.
Table 4. Statistical ssignificance of the difference between
measured and predicted values for set similarity in Trial n. 9. f
are the degrees of freedom, and p is an estimator of the
significance of the results.
Student’s t Test
Model
ABS (correlation pcSAL)
av (SYM, ATMEMB, REL)
minDIST
T
6.43
12.20
17.40
f
44
44
44
All Blocks
tcr (95%)
2.015
2.015
2.015
p
<.001***
<.001***
<.001***
9th EUROPEAN MUSIC ANALYSIS CONFERENCE — EUROMAC 9
Implications
The main results of this pilot study can be summarized as
follows: (i) computational analyses by means of five different
models of similarity between sets consistently predict a symmetrical structure in the macroform of the Webern Canon Op.
16 No. 2, which is associated to relatedness between pairs of
sets belonging to each bar; main peaks corresponding to pairs
sets 4-z15A / 4-13B and (unsurprisingly) 4-4A / 4-4B; (ii) this
internal symmetry is confirmed by perceptual ratings for similarity, though peak positions do not totally match between
models and data; (iii) mismatch between predicted and observed peak positions is likely a consequence of inhomogeneity
between trials in terms of inversion, register and pitch commonality; (iv) the pitch-time structure of trials (i.e., chords,
ascending arpeggios, or notated arpeggios) does not contribute
significantly to the overall variance.
On the base of these preliminary results, we planned to
perform a more rigorous study, differing from the pilot test in
two main aspects: first, in order to minimize the systematic
effects related to absolute pitch position, both sets in each pair
will be normalized to C4; second, although without any pretence of generalization, this approach will be applied on a
larger sample of pairs — in the specific case the ones outlined
by set analysis of Bartok Mikrokosmos No. 84 (Merriment) and
Scriabin Poem Op. 69 No. 1. A comparison between model
predictions and measurements by means of multiple regression
analysis will conclude the study. In a parallel study involving
the same pieces, we are looking at the correlations between
distance profiles and other structural aspects, in particular
melody and rhythm. Results of both studies will be presented at
the conference.
Our approach links together music theory/analysis and
perception using methods of statistical and computer sciences.
We are challenging these four disciplines to work more closely
together and take each other’s ideas and methods more seriously.
Keywords
music modelling, music perception, post-tonal music,
structure, distance
REFERENCES
Ariza, Christopher, 2002. AthenaCL. http://www.athenacl. org.
Bisesi, Erica, Friberg, Anders, and Parncutt, Richard, in preparation.
‘A Computational Model of Accent Salience in Tonal Music’.
Cohn, Richard, 1998. ‘Introduction to Neo-Riemannian Theory: A
Survey and a Historical Perspective’, Journal of Music Theory
42/2: 167–180.
Grinshaw,
Jeremy
http://www.allmusic.com/composition/canons-5-on-latin-texts-fo
r-voice-clarinet-bass-clarinet-op-16-mc0002356179
Krumhansl, Carol L., 1998. ‘Perceived Triad Distance: Evidence
Supporting the Psychological Reality of Neo-Riemannian
Transformations, Journal of Music Theory, 42/2: 265–281.
Lewin, David, 1988. Generalizaed Musical Intervals and Transformations. Oxford and New York: Oxford University Press.
Milne, Andrew J., and Holland, Simon, 2016. ‘Empirically Testing
Tonnetz, Voice-Leading, and Spectral Models of Perceived
Harmonic Distance’, Journal of Mathematics and Music, 10/1:
59–85.
Morris, Robert, 1987. ‘A Similarity Index for Pitch-Class Sets’,
Perspective of New Music 18/1–2: 445–460.
Parncutt, Richard, 1988. ‘Revision of Terhardt’s Psychoacoustical
Model of the Root(s) of a Musical Chord’, Music Perception 6:
65–94.
Rahn, John, 1979–1980. ‘Relating Sets’, Journal of Music Theory
49/1: 45–108.
Rogers, Nancy, and Callender, Clifton, 2006. ‘Judgments of Distance
Between Trichords’, in Mario Baroni, Anna Rita Addessi,
Roberto Caterina, and Marco Costa (eds.), Proceedings of the 9th
International Conference on Music Perception and Cognition.
Bologna, Italy: Bononia University Press, 1686–1691.
Straus, N. Joseph, 2003. ‘Uniformity, Balance, and Smoothness in
Atonal Voice Leading’, Music Theory Spectrum 25/2: 305–352.
Straus, N. Joseph, 2005. ‘Voice Leading in Set-Class Space’, Journal
of Music Theory, 49/1: 45–108.
Straus, N. Joseph, 2014. ‘Total Voice Leading’, Music Theory Online,
20/2.
Tymoczko, Dmitri, 2011. A Geometry of Music. Harmony and
Counterpoint in the Extended Common Practice. New York:
Oxford University Press.