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CONCEPTUAL BLENDING IN MUSIC CADENCES: A FORMAL MODELAND SUBJECTIVE EVALUATION.
Asterios ZacharakisSchool of Music Studies,
Aristotle University ofThessaloniki, [email protected]
Maximos Kaliakatsos-PapakostasSchool of Music Studies,
Aristotle University ofThessaloniki, [email protected]
Emilios CambouropoulosSchool of Music Studies,
Aristotle University ofThessaloniki, [email protected]
Conceptual blending is a cognitive theory whereby ele-ments from diverse, but structurally-related, mental spacesare ‘blended’ giving rise to new conceptual spaces. Thisstudy 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 betweenharmonic cadences, which arguably constitute the most fun-damental harmonic concept. The system creates a varietyof blends combining elements of the penultimate chords oftwo input cadences and it further estimates the expected re-lationships between the produced blends. Then, a prelimi-nary subjective evaluation of the proposed blending systemis presented. A pairwise dissimilarity listening test wasconducted 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 twoconfigurations showed that the system is capable of mak-ing fair predictions of the perceived dissimilarities betweenthe blended cadences. This implies that this conceptualblending approach is able to create perceptually meaning-ful blends based on self-evaluation of its outcome.
Conceptual blending is a cognitive theory developed byFauconier and Turner  whereby elements from diverse,but structurally-related, mental spaces are ‘blended’ givingrise to new conceptual spaces that often possess new pow-erful interpretative properties allowing better understand-ing of known concepts or the emergence of novel conceptsaltogether. The general framework within which the cur-rent work is placed, comprises a formal model for concep-tual blending  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, MaximosKaliakatsos-Papakostas, Emilios Cambouropoulos. “Conceptual Blend-ing in Music Cadences: A formal model and subjective evaluation.”, 16thInternational Society for Music Information Retrieval Conference, 2015.
portant interdisciplinary research advances from cognitivescience, artificial intelligence, formal methods and com-putational creativity. To substantiate its potential, a proof-of-concept autonomous computational creative system thatperforms melodic harmonisation is being developed.
Musical meaning is to a large extent self-referential;themes, motives, rhythmic patterns, harmonic progressionsand so on emerge via self-reference rather than externalreference 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 conceptualblending (musical cross-domain blending is discussed byAntovic , Cook , Zbikowski ). Indeed, structuralconceptual blending is omnipresent in music making: fromindividual pieces harmoniously combining music charac-teristics of different pieces/styles, to entire musical styleshaving emerged as a result of blending between diversemusic idioms.
Suppose we have a basic tonal ontology where onlydiatonic notes are allowed and dissonances in chords aremostly forbidden (except possibly using minor 7th inter-vals as in the dominant seventh chord). We assume thatsome basic cadences have been established as salient har-monic functions around which the harmonic language ofthe 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 computationalsystem to enrich its learned tonal ontology by inventing‘new’ meaningful cadences based on blending betweenknown cadences?
Figure 1: Conceptual blending between the basic perfectand Phrygian cadences gives rise to the Tritone Substitu-tion progression/cadence. The Backdoor progression canalso be derived as a weaker blend since less attributes ofthe two input spaces are retained leading to a lower ratingby the system.
Figure 1 presents a conceptual blending example wherethe perfect and the 15th century modal Phrygian cadencesare used as input spaces. These have been chosen in thisexample as they are both final cadences to the tonic andat the same time, they are very different (i.e. the Phry-gian mode does not have an upward leading note to thetonic but rather a downward ‘leading note’ from the IIbto the I). Initially, these cadences are formally describedas simply pitch classes with reference to a tonal centre (Ctonality was adopted in this case). Assuming that the fi-nal chord is always a common tonic chord, blending takesplace by combining pitches of the penultimate chords be-tween different cadences. Rather than mere combinationof pitches, other characteristic attributes of a cadence arealso 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 relativestrength 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 tocomply with a certain chord type such as the major or mi-nor chord (in this instance the characteristic major chordwith 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 twocadence types create a large number of possible combina-tions. We start with combinations (at least 3 notes) of thehighest priority/salience notes (notes connected with boldlines in Figure 1). Many of these combinations are nottriadic chords and may be filtered out using a set of con-straints (in this instance our constraint is to have a chordthat is a standard characteristic tonal chord such as a dom-inant seventh), while also a note completion step mightbe 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 highlyrated one based on priorities values is the tritone substi-tution progression (IIb7-I) of jazz harmony. This simpleblending mechanism ‘invents’ a chord progression that em-bodies characteristics of the Phrygian cadence (root/bassdownward motion by semitone) and the dominant seventhchord (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 BackdoorProgression also appears in the potential blends albeit withmuch lower priority (i.e. a much weaker blend) – see Fig-ure 1. A number of other applications and uses of harmonicblending  and, more specifically, chord blending arereported in .
Following the above, a challenging question that needsto be addressed concerns the evaluation of the outcomeproduced by such a creative system. The mere definitionof creativity is problematic and not commonly accepted asmany authors approach it from different perspectives (e.g.,[3,5,17,21], for a comprehensive discussion see ). Ap-
plications of computational creativity to music pose theextra issue of aesthetic quality judgement since creativ-ity may not always be accompanied by aesthetic value andvice versa. In terms of assessing a creative system, thetwo usual approaches are to either directly evaluate the fi-nal product or to evaluate the productive mechanism .The present work is concluded by an empirical experimentthat attempts to address the former by shedding some lighton how the system’s output is perceived leaving -for themoment- the issue of aesthetic value intact.
To this end, the output of the cadence blending systemdescribed in the following section is used to set up a pre-liminary subjective evaluation of the conceptual blendingalgorithm 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 cadencesand it further estimates the expected relationships betweenthe produced blends. A number of blends between the per-fect and the Phrygian cadence were produced in order totest 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 andconducted. Subsequent multidimensional scaling (MDS)analysis was utilised to obtain geometric configurations forboth behaviourally acquired pairwise distances and dissim-ilarity estimations by the algorithm. Comparison of thetwo configurations showed that the system can model theperceptual space quite accurately.
2. FORMAL CONCEPTUAL BLENDING MODEL
This section begins with a description of the conceptualblending mechanism utilised by the system for cadenceconstruction. It then proceeds with a consideration of anaive 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) twochords that conclude a phrase, section or piece of mu-sic . In our case we have examined the simplest caseof 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 penultimatechords of the cadences. The penultimate chords shouldtherefore 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 consideredin combination with ‘key-related’ characteristics (i.e. theirrelations to the final chord). For instance, a ‘chord-type’and distinctive characteristic of the penultimate chord inthe 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
142 Proceedings of the 16th ISMIR Conference, Malaga, Spain, October 26-30, 2015
it includes the leading tone to the tonic (expressed as thepitch class 11 relative to the local key) and b) that its rootmoves 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 allcharacteristics (‘chord-type’ or ‘key-related’) are equallysalient.
The blending framework employed in this paper forproducing novel cadences through concept blending hasbeen presented in . This framework follows Goguen’sproposal to model conceptual spaces as algebraic spec-ifications, while the utilised specifications defined in avariant of Common Algebraic Specification Language(CASL)  are extended with priority values associatedto axioms. These specifications incorporate symbols asbasic building blocks, over which more refined specifica-tions are constructed, beginning from the sort ‘Note’ thatis utilised to built the sort ‘Chord’. The sort Chord rep-resents the penultimate chord of the cadence which is infact the notion of the cadence as previously described. ANote can receive values between 0 and 11, indicating the12 pitch classes. In addition, a ‘+’ operator is consideredfor arithmetics of addition in a modulo 12. For example,7 + 9 = 4 denotes that a sixth plus a fifth is a major third.
A Chord 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 betweenthe scale’s root note and the notes comprising the chord,e.g., a major chord with minor seventh and with chord rooton the fifth degree of the major scale (i.e. pitch class 7) hasthe following key-related notes: [7, 11, 2, 5].
The salient characteristics of penultimate chords, andin extension of cadences, are defined for the two inputspaces by employing human knowledge 1 . The salienceof a penultimate chord property is input to the system as apriority 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. Additionalconstraints that concern further knowledge about chordsare imposed. For the system employed in this paper, pre-sented in more detail in , the additional constraints con-cern the facts that a chord should not have a major and aminor third (‘chordNote 3 and 4) at the same time, it shouldnot have a minor second (‘chordNote’ 1) and it should nothave both a perfect and a diminished fifth (‘chordNote’ 6and 7) at the same time. When a new blendoid 2 emerges,these constraints are enforced in the form of a consistency
1 In this study, for convenience, they are determined manually by amusic expert.
2 The term blendoid refers to a possible result of blending, which, how-ever, is not necessarily consistent or optimal. Additional criteria eithervalidate or discard the consistency of a blendoid as well as evaluate it asoptimal (based on ‘blending optimality principles’ or on domain-specificcharacteristics inherited to the blendoid).
check on the chord specification. Thereby, inconsistentblends are discarded.
The input cadences that have been selected to demon-strate blending of harmonic concepts were the perfect andthe Phrygian, with their attributes and priorities depictedin Table 1. For both cadences, the highest priorities areassigned in such a way that the most musically salient as-pects of the penultimate chords are highlighted. For theperfect cadence, the most highlighted features include theleading note (keyNote: 11) to the tonic and the fact that itstype includes a tritone (chordNote: 4 and chordNote: 10).For the Phrygian cadence, the musically salient feature isthe descending leading note (keyNote: 1) to the tonic.
perfect Phrygianattribute priority attribute priority
keyNote: 7 p: 2 keyNote: 10 p: 1keyNote: 11 p: 3 keyNote: 1 p: 3keyNote: 2 p: 1 keyNote: 5 p: 2
keyNote: 5 p: 2chordNote: 0 p: 1 chordNote: 0 p: 1chordNote: 4 p: 3 chordNote: 3 p: 1
chordNote: 7 p: 1 chordNote: 7 p: 1chordNote: 10 p: 3
Table 1: Attributes and priorities (higher values indicatehigher priority) considered in the blending system for theinput penultimate chords in the perfect and Phrygian ca-dences. Common attributes of both cadences (the genericspace ) appear in boxes.
tritone backdoorattribute priority attribute priority
keyNote: 1 p: 3 keyNote: 10 p: 1keyNote: 11 p: 3 keyNote: 2 p: 1keyNote: 5 p: 2 keyNote: 5 p: 2keyNote: 8 p: 1 keyNote: 8 p: 1
chordNote: 0 p: 1 chordNote: 0 p: 1chordNote: 4 p: 3 chordNote: 4 p: 3
chordNote: 7 p: 1 chordNote: 7 p: 1chordNote: 10 p: 3 chordNote: 10 p: 3
Table 2: Attributes and priorities (higher values indicatehigher priority) in the tritone substitution and backdoor ca-dences that result as blends from the perfect and Phrygiancadences. The completion step adds the keyNote: 8.
The computational chord blending framework com-bines the salience of chord features and core ideas of thenotion of Amalgams , resulting in a process that itera-tively produces blendoids with descending salience in theircharacteristics. However, the produced blendoids poten-tially require completion, i.e. additional reasoning mech-anisms that fill-in incomplete properties. Let us considerthe example of the tritone substitution cadence blend toelucidate 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 chordNoteattributes of the perfect cadence: [0, 4, 7, 10]. However,
Proceedings of the 16th ISMIR Conference, Malaga, Spain, October 26-30, 2015 143
the utilisation of the pitch classes [1, 5, 11] does not sat-isfy the requirements for a full dominant seventh chord oftype [0, 4, 7, 10]. The completions step for the pilot studypresented in  is performed manually, although it is pos-sible to develop an automatic completion algorithm basedon the chord root provided by the utilisation of the Gen-eral Chord Type (GCT)  algorithm. For instance, inthe tritone substitution, pitch class 1 is assigned as a rootnote, 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 attributesspaces: [2, 5, 10], which are not the ones with the highestpriorities, and again all the chordNote attributes of the per-fect cadence: [0, 4, 7, 10]. Similarly, the completion stepassigns 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. Sinceno background knowledge about the role of the attributekeyNote: 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 notbeen modelled in the input spaces, the default priority 1 isassigned.
2.2 Model-based distance metric
A naive method to compute the distances between pairs ofcadences is by comparing their common features with theset of all their distinct features. In our case, since the finalchord is always the same minor tonic, the comparison boilsdown to the features of their penultimate chords. Thereby,the more features these chords have in common, the moresimilar the cadences should be. For two cadences, Ci andCj two 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 featuresand the union is the sum of all the features appearing inboth 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 andunion of the features of penultimate chords is computed bydividing the number of elements in the intersection withthe number of elements in the union. If N(X) is the num-ber of elements in a set X , then the distance between twocadences is computed as
d(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 describedbelow. This approach is widely adopted in psychoacousticsbecause it enables the construction of perceptual spacesby employing multidimensional scaling (MDS) analysis onthe obtained dissimilarity matrices.
The stimulus set consisted of the two input cadences (per-fect and Phrygian), four blends of the input spaces andthree miscellaneous cadences (Figure 2). More specifi-cally, seven selected blends were as follows: blend 3 wasthe tritone substitution progression, blends 4 and 5 werethe backdoor progression (the latter without seventh), ca-dence 6 was a plagal cadence (it was input manually as acadence instance that was not a blend and was rather dif-ferent to the two input cadences), cadence 7 contained aminor dominant penultimate chord, cadence 8 was essen-tially a French-sixth chord-type (similar in principle to thetritone 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 cadencewas preceeded by the notes C and F to reinforce perceptionof 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 keyNoteand 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 miscellaneouscadences (7-9).
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 ofthe test.
144 Proceedings of the 16th ISMIR Conference, Malaga, Spain, October 26-30, 2015
input blends miscellaneousindex 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.
In the pairwise dissimilarity listening test, participantswere asked to compare all the pairs among the nine soundstimulus set using the free magnitude estimation method. Therefore, they rated the perceptual distances offorty-five pairs (same pairs included) by freely typing ina number of their choice to represent dissimilarity of eachpair (i.e., an unbounded scale) with 0 indicating a samepair.
Listeners became familiar with the different cadencesduring an initial presentation of the stimulus set in randomorder. This was followed by a brief training stage wherelisteners rated four selected pairs of stimuli. For the mainpart of the experiment participants were allowed to listen toeach pair of sounds as many times as needed prior to sub-mitting their rating. The pairs were presented in randomorder 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, lastedaround twenty minutes for most of the participants.
4. EXPERIMENTAL RESULTS
The proposed formal conceptual blending framework en-ables the generation of multiple cadences with differentvalues of ‘importance’, as reflected by the priorities of theattributes preserved into the penultimate chords of the re-sulting cadences. For the purpose of this study, the system-wise ‘objective’ distance metric between cadences (seesubsection 2.2) is merely based on the common featuresof the penultimate chords, not taking priority values intoaccount. The aim of this study is to examine whetherthe 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  was applied tothe dissimilarity matrices. INDSCAL computes weightsthat represent the importance attributed to each perceptualdimension by each participant and then uses these weightsto 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 hasbeen proven robust to the presence of monotonic transfor-mations or random error in the data [19, 22].
A two-dimensional solution of the behavioural datawith the following goodness of fit measures: Stress-I: .228
1st dimension-1 0 1
1st dimension-0.5 0 0.5 1
(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 3 was favoured.Considering the number of objects in combination with thenumber of dimensions, the achieved Stress-I value does notimply an adequate fit between the MDS model produceddisparities 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, thesatisfactory interpretability of the two dimensional config-uration (as will be shown below) supports the acceptanceof this solution.
The dissimilarity matrix that was produced by the dis-tance metrics of the cadence-blending-system was alsoanalysed through non-metric MDS. The two-dimensional
3 Stress-I is a measure of missfit where a lower value indicates a betterfit (with a minimum of zero) and DAF is a measure of fit where a highervalue indicates a better fit (with a maximum of one).
Proceedings of the 16th ISMIR Conference, Malaga, Spain, October 26-30, 2015 145
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 thatprior 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 onthe 1st dimension. This dimension could be interpreted as‘modal vs tonal’ since negative values coincide with ab-sence of the leading note  while positive values sig-nify presence of the leading note. Cadences no.4 (back-door with seventh) and 5 (backdoor without seventh) arenaturally closely related. The clustering of no.4 and no.5with the Phrygian could be explained by their shared notes[5, 10] and also by the absence of the leading note 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 formeris a German-sixth-type while the latter is a French-sixth-type both sharing three basic notes [1, 5, 11]. These twocadences are additionally positioned more closely to theperfect cadence (no.1) than to the Phrygian showing thatalthough 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 equidistantbetween them. This can be explained by the fact that bothno.3 and no.8 take the leading note  and the seventh (that needs to be resolved) from the perfect cadence butonly take note  (base of the Phrygian) from the Phry-gian. Cadence no.6 -the plagal- is positioned in the middlebetween the perfect and Phrygian along dimension 1 but isexpectedly an outlier along dimension 2.
The comparison between the perceptual and algorith-mic configurations was performed using Tucker’s congru-ence coefficient . As a guideline, for the congruencecoefficient, values larger than .92 are considered good/fair,and values larger than .95 practically show equality be-tween configurations . In our case, the congruence co-efficient between the perceptual and the algorithmic spacewas computed to be .944 indicating that the system canmake a very good estimation of the relationships betweencadences.
According to the theory of conceptual blending developedby Fauconier and Turner, novel conceptual spaces can becreated by blending elements from diverse input concep-tual spaces. Based on this theory and its category-theoreticinterpretation proposed by Goguen this study presentedinitial 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 usedto 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 analysisin 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 reflectedby an objective distance metric that related to the formal-isation of cadences in the blending system. Indeed, thecomparative results showed that the system is capable ofmaking fair predictions of the perceived dissimilarities be-tween the blended cadences. Given the uncertainty intro-duced by both the demanding nature of the behaviouraltask and the MDS analyses for the two sets of data, thisresult is deemed rather satisfactory and leads to the follow-ing implications:
1. The presented cadence description framework ismeaningful. Although the representation of knowl-edge in cadences is very elementary (just describingthe 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 theinput spaces.
The utilisation of more sophisticated system-orientedmetrics is expected to increase the accuracy of the self-evaluation process within the system so as to producemeaningful results for a wider combination of input ca-dences (also ending in different final chords) or even formore complex harmonic structures. As an obvious nextstep, the parameters of the system distance metric can berefined 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 thecomputational framework for conceptual blending that isbeing developed in the context of the COINVENT project. Overall, the results of the subjective experiment,even with this elementary representation of cadences, in-dicate the effectiveness of this framework towards creatingmeaningful output. The long term objective is the appli-cation of the computational blending approach for devel-oping melodic harmonisation methodologies that facilitatestructural blending between harmonies of diverse music id-ioms. This will require the development of ontologies ca-pable of describing significantly more complex harmonicconcepts compared to a simple harmonic cadence. At thesame time, the employed subjective evaluation will needto be enriched by more elaborate experiments that will notonly be able to assess the aesthetic value and functional-ity of the blends but to also address the challenge of ratinglonger stimuli.
This work is funded by the COINVENT project. Theproject COINVENT acknowledges the financial support ofthe Future and Emerging Technologies (FET) programmewithin the Seventh Framework Programme for Researchof the European Commission, under FET-Open grant num-ber: 611553.
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