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CONCEPTUAL BLENDING IN MUSIC CADENCES: A FORMAL MODEL
AND SUBJECTIVE EVALUATION.
Asterios Zacharakis
School of Music Studies,
Aristotle University of
Thessaloniki, Greece
aszachar@mus.auth.gr
Maximos Kaliakatsos-Papakostas
School of Music Studies,
Aristotle University of
Thessaloniki, Greece
maxk@mus.auth.gr
Emilios Cambouropoulos
School of Music Studies,
Aristotle University of
Thessaloniki, Greece
emilios@mus.auth.gr
ABSTRACT
Conceptual blending is a cognitive theory whereby ele-
ments from diverse, but structurally-related, mental spaces
are ‘blended’ giving rise to new conceptual spaces. This
study focuses on structural blending utilising an algorith-
mic formalisation for conceptual blending applied to har-
monic concepts. More specifically, it investigates the abil-
ity of the system to produce meaningful blends between
harmonic cadences, which arguably constitute the most fun-
damental harmonic concept. The system creates a variety
of blends combining elements of the penultimate chords of
two input cadences and it further estimates the expected re-
lationships between the produced blends. Then, a prelimi-
nary subjective evaluation of the proposed blending system
is presented. A pairwise dissimilarity listening test was
conducted using original and blended cadences as stimuli.
Subsequent multidimensional scaling analysis produced spa-
tial configurations for both behavioural data and dissimi-
larity estimations by the algorithm. Comparison of the two
configurations showed that the system is capable of mak-
ing fair predictions of the perceived dissimilarities between
the blended cadences. This implies that this conceptual
blending approach is able to create perceptually meaning-
ful blends based on self-evaluation of its outcome.
1. INTRODUCTION
Conceptual blending is a cognitive theory developed by
Fauconier and Turner [8] whereby elements from diverse,
but structurally-related, mental spaces are ‘blended’ giving
rise to new conceptual spaces that often possess new pow-
erful interpretative properties allowing better understand-
ing of known concepts or the emergence of novel concepts
altogether. The general framework within which the cur-
rent work is placed, comprises a formal model for concep-
tual blending [7] based on Goguen’s initial ideas of a Uni-
fied Concept Theory [9, 18]. This model incorporates im-
c
Asterios Zacharakis, Maximos Kaliakatsos-Papakostas,
Emilios Cambouropoulos.
Licensed under a Creative Commons Attribution 4.0 International Li-
cense (CC BY 4.0). Attribution: Asterios Zacharakis, Maximos
Kaliakatsos-Papakostas, Emilios Cambouropoulos. “Conceptual Blend-
ing in Music Cadences: A formal model and subjective evaluation.”, 16th
International Society for Music Information Retrieval Conference, 2015.
portant interdisciplinary research advances from cognitive
science, artificial intelligence, formal methods and com-
putational creativity. To substantiate its potential, a proof-
of-concept autonomous computational creative system that
performs melodic harmonisation is being developed.
Musical meaning is to a large extent self-referential;
themes, motives, rhythmic patterns, harmonic progressions
and so on emerge via self-reference rather than external
reference to non-musical concepts. Since musical mean-
ing largely relies on structure and since conceptual blend-
ing involves mapping between different conceptual struc-
tures, music seems to be an ideal domain for conceptual
blending (musical cross-domain blending is discussed by
Antovi´
c [1], Cook [6], Zbikowski [24]). Indeed, structural
conceptual blending is omnipresent in music making: from
individual pieces harmoniously combining music charac-
teristics of different pieces/styles, to entire musical styles
having emerged as a result of blending between diverse
music idioms.
Suppose we have a basic tonal ontology where only
diatonic notes are allowed and dissonances in chords are
mostly forbidden (except possibly using minor 7th inter-
vals as in the dominant seventh chord). We assume that
some basic cadences have been established as salient har-
monic functions around which the harmonic language of
the idiom(s) has been developed, such as the authentic/per-
fect cadence, the half cadence, the plagal cadence and,
even, older 15th century modal cadences such as the Phry-
gian cadence (Figure 1). The main question to be ad-
dressed is the following: Is it possible for a computational
system to enrich its learned tonal ontology by inventing
‘new’ meaningful cadences based on blending between
known cadences?
Figure 1: Conceptual blending between the basic perfect
and Phrygian cadences gives rise to the Tritone Substitu-
tion progression/cadence. The Backdoor progression can
also be derived as a weaker blend since less attributes of
the two input spaces are retained leading to a lower rating
by the system.
Figure 1 presents a conceptual blending example where
the perfect and the 15th century modal Phrygian cadences
are used as input spaces. These have been chosen in this
example as they are both final cadences to the tonic and
at the same time, they are very different (i.e. the Phry-
gian mode does not have an upward leading note to the
tonic but rather a downward ‘leading note’ from the IIb
to the I). Initially, these cadences are formally described
as simply pitch classes with reference to a tonal centre (C
tonality was adopted in this case). Assuming that the fi-
nal chord is always a common tonic chord, blending takes
place by combining pitches of the penultimate chords be-
tween different cadences. Rather than mere combination
of pitches, other characteristic attributes of a cadence are
also taken into account. Weights/priorities that reflect rel-
ative prominence (e.g., root, upwards or downwards lead-
ing note, dissonant note that requires resolution – see Fig-
ure 1 where lines of variable thickness illustrate relative
strength of voice-leading connections in cadences) are as-
signed to each chord note according to a human expert.
The ‘blended’ penultimate chord is also constrained to
comply with a certain chord type such as the major or mi-
nor chord (in this instance the characteristic major chord
with minor seventh was preferred).
Let us examine the particular blending example be-
tween the perfect and the Phrygian cadences more closely.
Notes from the two input penultimate chords of the two
cadence types create a large number of possible combina-
tions. We start with combinations (at least 3 notes) of the
highest priority/salience notes (notes connected with bold
lines in Figure 1). Many of these combinations are not
triadic chords and may be filtered out using a set of con-
straints (in this instance our constraint is to have a chord
that is a standard characteristic tonal chord such as a dom-
inant seventh), while also a note completion step might
be required (adding notes to incomplete blending results)
if the examined combination incorporates too few notes;
more details regarding constraints will be given in sub-
section 2.1. Among the accepted blends, the most highly
rated one based on priorities values is the tritone substi-
tution progression (IIb7-I) of jazz harmony. This simple
blending mechanism ‘invents’ a chord progression that em-
bodies characteristics of the Phrygian cadence (root/bass
downward motion by semitone) and the dominant seventh
chord (resolution of tritone). Thus, it creates a new har-
monic ‘concept’ that was actually introduced in jazz, cen-
turies later than the original input cadences. The Backdoor
Progression also appears in the potential blends albeit with
much lower priority (i.e. a much weaker blend) – see Fig-
ure 1. A number of other applications and uses of harmonic
blending [11] and, more specifically, chord blending are
reported in [7].
Following the above, a challenging question that needs
to be addressed concerns the evaluation of the outcome
produced by such a creative system. The mere definition
of creativity is problematic and not commonly accepted as
many authors approach it from different perspectives (e.g.,
[3,5, 17,21], for a comprehensive discussion see [10]). Ap-
plications of computational creativity to music pose the
extra issue of aesthetic quality judgement since creativ-
ity may not always be accompanied by aesthetic value and
vice versa. In terms of assessing a creative system, the
two usual approaches are to either directly evaluate the fi-
nal product or to evaluate the productive mechanism [16].
The present work is concluded by an empirical experiment
that attempts to address the former by shedding some light
on how the system’s output is perceived leaving -for the
moment- the issue of aesthetic value intact.
To this end, the output of the cadence blending system
described in the following section is used to set up a pre-
liminary subjective evaluation of the conceptual blending
algorithm applied to cadence invention. As stated previ-
ously, the computational system is capable of creating a va-
riety of blends combining elements of two input cadences
and it further estimates the expected relationships between
the produced blends. A number of blends between the per-
fect and the Phrygian cadence were produced in order to
test the ability of the cadence blending system to accu-
rately predict their perceived relations (i.e. the function-
ality of the blends) using an ‘objective’ distance metric
(see subsection 2.2). To achieve this, a pairwise dissim-
ilarity listening test for the nine cadences (two original,
four blends and three miscellaneous) was designed and
conducted. Subsequent multidimensional scaling (MDS)
analysis was utilised to obtain geometric configurations for
both behaviourally acquired pairwise distances and dissim-
ilarity estimations by the algorithm. Comparison of the
two configurations showed that the system can model the
perceptual space quite accurately.
2. FORMAL CONCEPTUAL BLENDING MODEL
This section begins with a description of the conceptual
blending mechanism utilised by the system for cadence
construction. It then proceeds with a consideration of a
naive distance metric for pairs of cadences based on repre-
sentation of cadences according to the system.
2.1 Cadence generation through chord blending
A cadence is described as a progression of (at least) two
chords that conclude a phrase, section or piece of mu-
sic [2]. In our case we have examined the simplest case
of two chords, a penultimate and a final chord. If the final
– destination – chord is considered fixed, then blending be-
tween two cadences can occur by blending the penultimate
chords of the cadences. The penultimate chords should
therefore be described in a way that reflects the ‘functional’
role of their constitutive components. To this end, ‘chord-
type’ properties of the penultimate chords (i.e. characteris-
tics of type such as major, minor etc.) should be considered
in combination with ‘key-related’ characteristics (i.e. their
relations to the final chord). For instance, a ‘chord-type’
and distinctive characteristic of the penultimate chord in
the perfect cadence (V7) is the fact that it includes a tri-
tone (between the third and the minor seventh), while two
‘key-related’ important characteristics are a) the fact that
it includes the leading tone to the tonic (expressed as the
pitch class 11 relative to the local key) and b) that its root
moves by a perfect fifth to the tonic. Additionally, the spec-
ification of cadences (penultimate chords) should incorpo-
rate priority values, taking into account the fact that not all
characteristics (‘chord-type’ or ‘key-related’) are equally
salient.
The blending framework employed in this paper for
producing novel cadences through concept blending has
been presented in [7]. This framework follows Goguen’s
proposal to model conceptual spaces as algebraic spec-
ifications, while the utilised specifications defined in a
variant of Common Algebraic Specification Language
(CASL) [14] are extended with priority values associated
to axioms. These specifications incorporate symbols as
basic building blocks, over which more refined specifica-
tions are constructed, beginning from the sort ‘Note’ that
is utilised to built the sort ‘Chord’. The sort Chord rep-
resents the penultimate chord of the cadence which is in
fact the notion of the cadence as previously described. A
Note can receive values between 0 and 11, indicating the
12 pitch classes. In addition, a ‘+’ operator is considered
for arithmetics of addition in a modulo 12. For example,
7 + 9 = 4 denotes that a sixth plus a fifth is a major third.
AChord specification incorporates two kinds of attribu-
tes that relate to the aforementioned ‘chord-type’ and ‘key-
related’ attributes, respectively ‘chordNote’ and ‘keyNote’.
The ‘chordNote’ property indicates semitone distances be-
tween the chord’s root and the notes comprising the chord,
e.g., a major chord with minor seventh has the follow-
ing relative notes: [0,4,7,10]. On the other hand, the
‘keyNote’ property indicates semitone distance between
the scale’s root note and the notes comprising the chord,
e.g., a major chord with minor seventh and with chord root
on the fifth degree of the major scale (i.e. pitch class 7) has
the following key-related notes: [7,11,2,5].
The salient characteristics of penultimate chords, and
in extension of cadences, are defined for the two input
spaces by employing human knowledge 1. The salience
of a penultimate chord property is input to the system as a
priority value which is then directly linked to this property.
The output of conceptual blending, i.e. a conceptual blend,
should incorporate the most salient features of the two in-
put spaces – reflected by higher priority values. Additional
constraints that concern further knowledge about chords
are imposed. For the system employed in this paper, pre-
sented in more detail in [7], the additional constraints con-
cern the facts that a chord should not have a major and a
minor third (‘chordNote 3 and 4) at the same time, it should
not have a minor second (‘chordNote’ 1) and it should not
have both a perfect and a diminished fifth (‘chordNote’ 6
and 7) at the same time. When a new blendoid 2emerges,
these constraints are enforced in the form of a consistency
1In this study, for convenience, they are determined manually by a
music expert.
2The term blendoid refers to a possible result of blending, which, how-
ever, is not necessarily consistent or optimal. Additional criteria either
validate or discard the consistency of a blendoid as well as evaluate it as
optimal (based on ‘blending optimality principles’ or on domain-specific
characteristics inherited to the blendoid).
check on the chord specification. Thereby, inconsistent
blends are discarded.
The input cadences that have been selected to demon-
strate blending of harmonic concepts were the perfect and
the Phrygian, with their attributes and priorities depicted
in Table 1. For both cadences, the highest priorities are
assigned in such a way that the most musically salient as-
pects of the penultimate chords are highlighted. For the
perfect cadence, the most highlighted features include the
leading note (keyNote: 11) to the tonic and the fact that its
type includes a tritone (chordNote: 4 and chordNote: 10).
For the Phrygian cadence, the musically salient feature is
the descending leading note (keyNote: 1) to the tonic.
perfect Phrygian
attribute priority attribute priority
keyNote: 7 p: 2 keyNote: 10 p: 1
keyNote: 11 p: 3 keyNote: 1 p: 3
keyNote: 2 p: 1 keyNote: 5 p: 2
keyNote: 5 p: 2
chordNote: 0 p: 1 chordNote: 0 p: 1
chordNote: 4 p: 3 chordNote: 3 p: 1
chordNote: 7 p: 1 chordNote: 7 p: 1
chordNote: 10 p: 3
Table 1: Attributes and priorities (higher values indicate
higher priority) considered in the blending system for the
input penultimate chords in the perfect and Phrygian ca-
dences. Common attributes of both cadences (the generic
space [7]) appear in boxes.
tritone backdoor
attribute priority attribute priority
keyNote: 1 p: 3 keyNote: 10 p: 1
keyNote: 11 p: 3 keyNote:2 p: 1
keyNote: 5 p: 2 keyNote: 5 p: 2
keyNote:8 p:1 keyNote:8 p:1
chordNote: 0 p: 1 chordNote: 0 p: 1
chordNote: 4 p: 3 chordNote: 4 p: 3
chordNote: 7 p: 1 chordNote: 7 p: 1
chordNote: 10 p: 3 chordNote: 10 p: 3
Table 2: Attributes and priorities (higher values indicate
higher priority) in the tritone substitution and backdoor ca-
dences that result as blends from the perfect and Phrygian
cadences. The completion step adds the keyNote:8.
The computational chord blending framework com-
bines the salience of chord features and core ideas of the
notion of Amalgams [15], resulting in a process that itera-
tively produces blendoids with descending salience in their
characteristics. However, the produced blendoids poten-
tially require completion, i.e. additional reasoning mech-
anisms that fill-in incomplete properties. Let us consider
the example of the tritone substitution cadence blend to
elucidate the completion step, as demonstrated in Table 2.
The tritone substitution cadence is acquired by preserv-
ing the most salient keyNote attributes (with priority 3)
from both input spaces: [1,5,11], and all the chordNote
attributes of the perfect cadence: [0,4,7,10]. However,
the utilisation of the pitch classes [1,5,11] does not sat-
isfy the requirements for a full dominant seventh chord of
type [0,4,7,10]. The completions step for the pilot study
presented in [7] is performed manually, although it is pos-
sible to develop an automatic completion algorithm based
on the chord root provided by the utilisation of the Gen-
eral Chord Type (GCT) [4] algorithm. For instance, in
the tritone substitution, pitch class 1 is assigned as a root
note, a fact that leads to the completion of the pitch class
(keyNote:) 8 as a perfect fifth (to match the chordNote:
7). The backdoor cadence preserves the keyNote attributes
spaces: [2,5,10], which are not the ones with the highest
priorities, and again all the chordNote attributes of the per-
fect cadence: [0,4,7,10]. Similarly, the completion step
assigns the pitch class 10 as a root note, while the require-
ment for a minor seventh (chordNote: 10) leads to im-
porting the pitch class (keyNote:) 8 into the blend. Since
no background knowledge about the role of the attribute
keyNote: 8 is given, the ‘default’ priority 1 is inserted,
which will also be the case for all the examples in this pa-
per: if attributes emerge through completion that have not
been modelled in the input spaces, the default priority 1 is
assigned.
2.2 Model-based distance metric
A naive method to compute the distances between pairs of
cadences is by comparing their common features with the
set of all their distinct features. In our case, since the final
chord is always the same minor tonic, the comparison boils
down to the features of their penultimate chords. Thereby,
the more features these chords have in common, the more
similar the cadences should be. For two cadences, Ciand
Cjtwo sets are considered: the intersection,∩(Ci, Cj),
and the union,∪(Ci, Cj)of their penultimate chord fea-
tures. The intersection is the set of their common features
and the union is the sum of all the features appearing in
both cadences without repetitions. For instance, for the ca-
dences indexed 1 and 3 in Table 3:
∩(C1, C3) = [[5,11],[0,4,7,10]],
∪(C1, C3) = [[1,2,5,7,8,11],[0,4,7,10]].
The considered distance based on the intersection and
union of the features of penultimate chords is computed by
dividing the number of elements in the intersection with
the number of elements in the union. If N(X)is the num-
ber of elements in a set X, then the distance between two
cadences is computed as
d(Ci, Cj) = N(∩(Ci, Cj))
N(∪(Ci, Cj)).
In the aforementioned example, d(C1, C3)=6/10.
3. EMPIRICAL EVALUATION
In order to investigate the functionality of the blended ca-
dences (i.e. the perceived relationships between them) we
conducted a pairwise dissimilarity rating listening experi-
ment using as stimuli the nine selected cadences described
below. This approach is widely adopted in psychoacoustics
because it enables the construction of perceptual spaces
by employing multidimensional scaling (MDS) analysis on
the obtained dissimilarity matrices.
3.1 Stimuli
The stimulus set consisted of the two input cadences (per-
fect and Phrygian), four blends of the input spaces and
three miscellaneous cadences (Figure 2). More specifi-
cally, seven selected blends were as follows: blend 3 was
the tritone substitution progression, blends 4 and 5 were
the backdoor progression (the latter without seventh), ca-
dence 6 was a plagal cadence (it was input manually as a
cadence instance that was not a blend and was rather dif-
ferent to the two input cadences), cadence 7 contained a
minor dominant penultimate chord, cadence 8 was essen-
tially a French-sixth chord-type (similar in principle to the
tritone substitution) and cadence 9 was a manually con-
structed non-blend chromatic chord. Note that all the ca-
dences were assumed to be in C minor and each cadence
was preceeded by the notes C and F to reinforce perception
of tonal context – the only chord that changed in each stim-
ulus was the penultimate chord. Table 3 illustrates the ca-
dences used in the subjective experiment with the keyNote
and chordNote features grouped in two arrays. Therefore,
since the system is able to produce blended cadences ac-
cording to these features (keyNote and chordNote), the sim-
ilarity between two cadences in terms of the system’s mod-
elling should depend merely on them.
Figure 2: Score annotation of the two input cadences (1-
2), 4 blends of the input spaces (3-6) and 3 miscellaneous
cadences (7-9).
3.2 Participants
Fifteen listeners (aged 19-48, mean age: 26.5, 8 female)
participated in the listening test. All reported normal hear-
ing and long term music practice (years on average: 18.7,
range: 6 to 43). Participants were students in the Depart-
ment of Music Studies of the Aristotle University of Thes-
saloniki. All participants were naive about the purpose of
the test.
input blends miscellaneous
index 1 2 3 4 5 6 7 8 9
keyNote [7,11,2,5] [10,1,5] [1,5,8,11] [10,2,5,8] [10,2,5] [2,5,9,0] [7,10,2] [1,5,7,11] [3,7,10,1]
chordNote [0,4,7,10] [0,3,7] [0,4,7,10] [0,4,7,10] [0,4,7] [0,3,7,10] [0,3,7] [0,4,6,10] [0,4,7,10]
Table 3: The penultimate cadence chords for the experiments along with their features and their respective indexes.
3.3 Procedure
In the pairwise dissimilarity listening test, participants
were asked to compare all the pairs among the nine sound
stimulus set using the free magnitude estimation method
[23]. Therefore, they rated the perceptual distances of
forty-five pairs (same pairs included) by freely typing in
a number of their choice to represent dissimilarity of each
pair (i.e., an unbounded scale) with 0 indicating a same
pair.
Listeners became familiar with the different cadences
during an initial presentation of the stimulus set in random
order. This was followed by a brief training stage where
listeners rated four selected pairs of stimuli. For the main
part of the experiment participants were allowed to listen to
each pair of sounds as many times as needed prior to sub-
mitting their rating. The pairs were presented in random
order and listeners were advised to retain a consistent rat-
ing strategy throughout the experiment. In total, the listen-
ing test sessions, including instructions and breaks, lasted
around twenty minutes for most of the participants.
4. EXPERIMENTAL RESULTS
The proposed formal conceptual blending framework en-
ables the generation of multiple cadences with different
values of ‘importance’, as reflected by the priorities of the
attributes preserved into the penultimate chords of the re-
sulting cadences. For the purpose of this study, the system-
wise ‘objective’ distance metric between cadences (see
subsection 2.2) is merely based on the common features
of the penultimate chords, not taking priority values into
account. The aim of this study is to examine whether
the pairwise distances between several cadences, as ex-
pressed by this ‘objective distance’ is aligned with the cog-
nitive/perceptual distances that musically trained partici-
pants assign.
A non-metric, weighted individual differences scaling
(INDSCAL) MDS analysis as offered by the SPSS PROX-
SCAL (proximity scaling) algorithm [13] was applied to
the dissimilarity matrices. INDSCAL computes weights
that represent the importance attributed to each perceptual
dimension by each participant and then uses these weights
to reconstruct a common perceptual space. Additionally,
the ‘ordinal’ option applies a rank ordering transforma-
tion to the raw dissimilarities within each participant’s re-
sponses. The non-metric approach was adopted since it has
been proven robust to the presence of monotonic transfor-
mations or random error in the data [19, 22].
A two-dimensional solution of the behavioural data
with the following goodness of fit measures: Stress-I: .228
1st dimension
-1 0 1
2nd dimension
-1.5
-1
-0.5
0
0.5
1
1.5
2
1
2
3
4
5
6
7
8
9
(a)
1st dimension
-0.5 0 0.5 1
2nd dimension
-1
-0.5
0
0.5
1
1
23
4
5
6
7
8
9
(b)
Figure 3: The perceptual (a) and the algorithmic (b) spa-
tial configurations for the nine selected cadences. The ca-
dences are labelled according to the indexes of table 3.
and Dispersion Accounted for (DAF): .947 3was favoured.
Considering the number of objects in combination with the
number of dimensions, the achieved Stress-I value does not
imply an adequate fit between the MDS model produced
disparities and the actual distances reported by the partic-
ipants. This fact can be attributed to the high level of un-
certainty present in the subjective responses. However, the
satisfactory interpretability of the two dimensional config-
uration (as will be shown below) supports the acceptance
of this solution.
The dissimilarity matrix that was produced by the dis-
tance metrics of the cadence-blending-system was also
analysed through non-metric MDS. The two-dimensional
3Stress-I is a measure of missfit where a lower value indicates a better
fit (with a minimum of zero) and DAF is a measure of fit where a higher
value indicates a better fit (with a maximum of one).
solution featured both acceptable Stress-I (.123) and DAF
(.985). The configurations of both spaces are shown in Fig-
ure 3.
Visual inspection of the perceptual space reveals that
prior expectations regarding cadence positioning are gen-
erally fulfilled. The perfect (no.1) and the Phrygian (no.2)
input cadences are positioned far away from each other on
the 1st dimension. This dimension could be interpreted as
‘modal vs tonal’ since negative values coincide with ab-
sence of the leading note [11] while positive values sig-
nify presence of the leading note. Cadences no.4 (back-
door with seventh) and 5 (backdoor without seventh) are
naturally closely related. The clustering of no.4 and no.5
with the Phrygian could be explained by their shared notes
[5,10] and also by the absence of the leading note [11]
that moves them away from the perfect cadence territory.
Also, the close positioning of cadences no.3 (tritone sub-
stitution) and no.8 is explained by the fact that the former
is a German-sixth-type while the latter is a French-sixth-
type both sharing three basic notes [1,5,11]. These two
cadences are additionally positioned more closely to the
perfect cadence (no.1) than to the Phrygian showing that
although the tritone substitution is created by incorporat-
ing the most salient attributes of the two input cadences
(see subsection 2.1), it is not perceived as being equidistant
between them. This can be explained by the fact that both
no.3 and no.8 take the leading note [11] and the seventh
[5] (that needs to be resolved) from the perfect cadence but
only take note [1] (base of the Phrygian) from the Phry-
gian. Cadence no.6 -the plagal- is positioned in the middle
between the perfect and Phrygian along dimension 1 but is
expectedly an outlier along dimension 2.
The comparison between the perceptual and algorith-
mic configurations was performed using Tucker’s congru-
ence coefficient [20]. As a guideline, for the congruence
coefficient, values larger than .92 are considered good/fair,
and values larger than .95 practically show equality be-
tween configurations [12]. In our case, the congruence co-
efficient between the perceptual and the algorithmic space
was computed to be .944 indicating that the system can
make a very good estimation of the relationships between
cadences.
5. CONCLUSIONS
According to the theory of conceptual blending developed
by Fauconier and Turner, novel conceptual spaces can be
created by blending elements from diverse input concep-
tual spaces. Based on this theory and its category-theoretic
interpretation proposed by Goguen this study presented
initial developments of a system for blending between har-
monic structures, using cadence blending as a proof of con-
cept. To this end, two input spaces with simple formalisa-
tions of the perfect and the Phrygian cadences were used
to produce several blended cadences.
The two input spaces along with the produced blends,
and other cadences, were subjected to a pairwise dissim-
ilarity rating listening test and subsequent MDS analysis
in order to evaluate the output produced by the cadence
blending system. The basic aim of the study was to ex-
amine whether perceptual distances between pairs of ca-
dences, as rated by the participants, were actually reflected
by an objective distance metric that related to the formal-
isation of cadences in the blending system. Indeed, the
comparative results showed that the system is capable of
making fair predictions of the perceived dissimilarities be-
tween the blended cadences. Given the uncertainty intro-
duced by both the demanding nature of the behavioural
task and the MDS analyses for the two sets of data, this
result is deemed rather satisfactory and leads to the follow-
ing implications:
1. The presented cadence description framework is
meaningful. Although the representation of knowl-
edge in cadences is very elementary (just describing
the penultimate chords with their absolute and rela-
tive notes), the derived results align with human per-
ception/cognition.
2. The utilised blending methodology produces consis-
tent results in the sense that resulting blends do in-
deed match the perceptual/cognitive attributes of the
input spaces.
The utilisation of more sophisticated system-oriented
metrics is expected to increase the accuracy of the self-
evaluation process within the system so as to produce
meaningful results for a wider combination of input ca-
dences (also ending in different final chords) or even for
more complex harmonic structures. As an obvious next
step, the parameters of the system distance metric can be
refined to optimise the fit between the algorithm’s predic-
tion and the actual perception of cadence dissimilarities.
Cadence blending is a proof-of-concept example of the
computational framework for conceptual blending that is
being developed in the context of the COINVENT project
[18]. Overall, the results of the subjective experiment,
even with this elementary representation of cadences, in-
dicate the effectiveness of this framework towards creating
meaningful output. The long term objective is the appli-
cation of the computational blending approach for devel-
oping melodic harmonisation methodologies that facilitate
structural blending between harmonies of diverse music id-
ioms. This will require the development of ontologies ca-
pable of describing significantly more complex harmonic
concepts compared to a simple harmonic cadence. At the
same time, the employed subjective evaluation will need
to be enriched by more elaborate experiments that will not
only be able to assess the aesthetic value and functional-
ity of the blends but to also address the challenge of rating
longer stimuli.
6. ACKNOWLEDGEMENTS
This work is funded by the COINVENT project. The
project COINVENT acknowledges the financial support of
the Future and Emerging Technologies (FET) programme
within the Seventh Framework Programme for Research
of the European Commission, under FET-Open grant num-
ber: 611553.
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