Subsumption of Vertical Viewpoint Patterns

Transcription

1 Susumption of Vertil Viewpoint Ptterns Mthieu Bergeron 1 nd Drrell Conklin 2 1 CIRMMT, MGill University, Montrel, Cnd mthieu.ergeron@mil.mgill. 2 Deprtment of Computer Siene nd AI Universidd del Pís Vso, Sn Sestián, Spin IKERBASQUE, Bsque Foundtion for Siene, Bilo onklin@ikersque.org Astrt. This pper formlizes the vertil viewpoint pttern lnguge for polyphoni pttern representtion. The semntis of ptterns is given in terms of trnsltion to reltionl network form. The lnguge supports pttern susumption, n essentil inferene for pttern mining, development, nd refinement. Though omputed in wy entirely different to reltionl network mthing, this pper proves tht susumption inferenes re sound nd omplete with respet to the underlying reltionl lnguge. Keywords: Pttern susumption, Polyphoni ptterns, Musi representtion, Sore sliing, Computtionl musiology. 1 Motivtion Pttern reognition nd disovery re importnt topis in omputtionl musiology. Ptterns in musi n e used to desrie repetition within piees nd lso reurrene within orpus of piees. Ptterns my refer to, for exmple, melodi lines, hord sequenes, or to more omplex polyphoni strutures where temporl reltions exist etween overlpping events. Considering polyphony, nturl nd powerful representtion of polyphoni ptterns is grph strutures, where the nodes represent notes, or more generlly fetures of notes, nd edges represent their temporl reltions. Grph representtions require the omputtion of sugrph isomorphism to determine pttern to instne reltions, nd this hs onstrined their use in omputtionl musiology. The im of this pper is to elorte new formlism lled VVP (vertil viewpoint ptterns) for the representtion of polyphoni ptterns. The representtion omines the viewpoint formlism [1] with the ide of segmenting the polyphoni texture into slies [2]. As slie n ontin the ontinution of musil event, vriety of temporl reltions re supported. While this representtion does not hve the full power of grphs, it is suitly expressive for mny types of musil ptterns, speifilly ptterns where notes re temporlly ompt nd our in fixed numer of voies. The method ssumes tht ptterns hve the sme numer of voies thn the musil soure from whih instnes re drwn. This n require the extrtion C. Agon et l. (Eds.): MCM 2011, LNAI 6726, pp. 1 12, Springer-Verlg Berlin Heidelerg 2011

2 2 M. Bergeron nd D. Conklin of fixed voie textures from tul musil dt (e.g. [3] onsidered voie pirs extrted out of the four-voie texture of Bh horle hrmoniztions). A similr tehnique is lso required when using Humdrum, well-known toolkit for pttern mthing in symoli musi dt [4,5]. Although Humdrum supports polyphony, it n e diffiult to use for even simple ptterns [6]. For exmple, Figure 1 shows some Humdrum dt (fter preproessing to otin expliit event ontinutions nd extr spines with musil fetures) nd Humdrum pttern tht ptures the ide of ontrpuntl suspension. The regulr expressions in eh line of Humdrum pttern do two things : i) mth the eginning of events or the ontinution of events (this is the purpose of rkets in the pttern of Fig. 1, rket mthes the ontinution of n event nd the sene of rket mthes the eginning of n event); nd ii) mth fetures of those events y mthing orresponding vlues in dditionl olumns (this is the purpose of the tokens ending the lines of the pttern, dissonne feture is mthed on the first line nd, on the seond line, fetures enoding the melodi ontour of step down nd onsonne re mthed). ss tenor ont qul D B +s ons BB (B). ons FF# (B). diss (FF#) A# -s ons [-g A-G]+ [- # n]* [ˆ)]* [(] [-g A-G]+ [- # n]*.* diss$ [-g A-G]+ [- # n]* [)] [ˆ(]* [-g A-G]+ [- # n]* -s ons$ Fig. 1. A musil frgment in Humdrum (top) nd suspension pttern (ottom). The musil dt is preproessed to otin expliit event ontinutions, denoted y rkets, nd to dd extr spines ontining musil fetures: ont for melodi ontour nd qul for hrmoni intervl. In ddition to the required expertise in regulr expressions, more fundmentl diffiulties with developing Humdrum ptterns rise due to their opity nd lk of denottionl semntis. It is in generl not possile for omputtionl musiologist to inspet pttern nd from tht dedue wht frgments will e mthed. Lking denottionl semntis, it my e intrtly diffiult to develop nd refine ptterns. For exmple, simply dding nother melodi feture to pttern requires oth the existene of spine ontining tht feture nd, to referene the vlues of tht feture, the modifition of every line of the pttern in wy tht respets the spine ordering. This pper formlly defines the semntis of VVP showing how it expresses reltions etween temporlly relted events in ompt regions of time. The method offers ler syntx for ptterns tht re very similr to the ptterns tht Humdrum n express. In ddition, it is possile to formlly define the notion of pttern susumption in VVP, inditing when the piees mthed y one pttern re lwys superset of the piees mthed y nother. With forml

3 Susumption of Vertil Viewpoint Ptterns 3 susumption reltion the omputtionl musiologist my develop ptterns y progressive pttern speiliztion nd generliztion, ringing in fewer or more mthing dtse piees s desired. Furthermore, with pttern susumption it is possile to define dt mining lgorithms tht serh refinement grphs of the pttern spe [7]. This pper is orgnized s follows. Setion 2 defines R, reltionl pttern lnguge we replite nd extend from [8]. In tht lnguge, temporl reltions etween musil events re expliit nd n e visulized s network; nd the notion of pttern susumption is strightforwrd nd intuitive. Setion 3 developssemntisforvvp in terms of R, tht is, in terms of the reltions VVP pttern expresses etween musil events. Setion 4 defines pttern susumption for VVP nd shows tht it is sound nd omplete with respet to susumption in R. This is the key ontriution of this pper s it estlishes tht the denottionl semntis we give to VVP orretly ptures the mening of ptterns s they re eing speilized or instntited. This extends the work presented in [8], where the instntition of ptterns ws not onsidered. Finlly, Setion 5 further disusses the pplitions nd limittions of the urrent work. 2 Reltionl Ptterns A reltionl pttern expresses temporl reltions over event vriles. The temporl reltions we onsider here re: i) m(, ) ( meets : finishes when egins), ii) st(, ) ( nd strt together), nd iii) sw(, ) ( strts while is sounding). Formlly, the reltions re defined in Fig. 2 (see [8] for omprison etween these nd the well-known temporl reltions etween intervls [9]). This pper extends the reltionl pttern lnguge of [8] y inluding two new types of tomi formuls: i) toms of the form voie(ε, γ), whih indite tht some event ε elongs to voie γ; nd ii) toms of the form τ(ε,,ε,μ), whih either reord some si musil informtion out n event, for exmple pith(, A4#), or some musil reltion ouring etween mny events, for exmple intervl(,, M3). This yields the lnguge R, defined s follows: m(, ) when: offset(, μ), onset(, μ) st(, ) when: onset(, μ), onset(, μ) [!h] sw(, ) when: onset(, μ 1), onset(, μ 2), offset(, μ 3), μ 1 >μ 2, μ 1 <μ 3 Fig. 2. Reltionl definition of the three temporl reltions onsidered in this rtile

4 4 M. Bergeron nd D. Conklin Definition 1. r R::= ω,,ω with ω ::= m(ε, ε) st(ε, ε) sw(ε, ε) voie(ε, γ) τ(ε,,ε,μ) Reltionl networks re used to visulize the temporl reltions of pttern in R. Figure 3 exemplifies reltionl network (left) nd reltionl pttern (middle; the pttern is one of the mny disovered y the musi mining pproh developed in [3]). Musil informtion is given, for exmple, y the tom voie(, lto)whih speifies tht event is in the lto voie; or the tom ont(, -s), whih speifies tht event is rehed y downwrd melodi step. An event hs the musil feture ont(ε, μ) whih desries melodi ontour nd tkes the possile vlues lep up: +l; lepdown:-l; step up: +s; stepdown:-s; repet: r. The other musil feture used in this pper is qul(ε, ε, μ), the hrmoni intervl (non-ompound ditoni intervl etween lower nd higher pithes in the slie), tking the vlues ons (intervls P1, m3, M3, P5, m6, M6) nd diss (ll other intervls). Susumption in R n e defined s speil se of θ-susumption [10]: pttern r 1 susumes pttern r 2 iff there exists mpping θ of the event vriles of r 1 to the event vriles of r 2 suh tht r 1 θ is suset of r 2.For exmple, the following ptterns re in susumption reltion, denoted y R : Exmple 1. st(, d), voie(, lto) R with θ = nd dθ = st(, ), voie(, lto), voie(, tenor), qul(,, ons) R Instnes st m sw m(, ), st(, ), sw(, ), voie(, lto), voie(, lto), voie(, tenor), qul(,, diss) qul(,, ons) Fig. 3. Ptterns in R nd instnes from Bh horle hrmoniztions

5 3 Vertil Viewpoints Susumption of Vertil Viewpoint Ptterns 5 The vertil viewpoint method [3] represents polyphoni musi s slies. The ide is to segment the musil soure into series of time-spns, or slies. A new slie is reted whenever new event ppers. Events tht re still sounding t the onset time of tht new event hve ontinution reorded. This results in representtion of polyphoni musi tht is thoroughly sequentil, s the definition of VVP mkes ler. Definition 2. v VVP ::= [α,α,,α] with α ::= {δ,δ,,δ} δ ::= τ[γ] [γ] :μ Aording to Definition 2, VVP pttern v is sequene of feture sets. Eh feture set α enodes slie. A feture τ[γ] [γ] :μ inludes feture nme τ, voie lels γ nd feture vlue μ. For exmple, onsider Fig. 4. Note tht the sequene is displyed without the rkets nd tht omms re omitted within the sets (heneforth, we disply VVP ptterns tht wy to simplify presenttion). The pttern ontins two slies. In eh slie, feture of the form strts[γ] : μ indites whether, for some voie γ, the slie onerns the eginning of new event (strts[γ] : t) or the ontinution of some previously existing event (strts[γ] : f). To relte slies nd events, whih is neessry to give reltionl semntis to VVP, we lel these fetures with event vriles (not prt of the syntx of VVP). In Figure 4, the first slie onerns the ontinution of some event nd the eginning of new event, while the seond slie lso onerns the ontinution of nd the eginning of new event (this time lelled ). This yields the reltionl network shown in the figure. In ddition, the pttern speifies musil fetures, suh s melodi ontours in voie 1 (e.g. feture ont[1] :-l) nd onsonnt intervls etween voie 1 nd voie 0 (e.g. feture qul[1][0] :ons). In generl, the lelling of events in VVP pttern is done s follows. For the first feture set of the pttern, every voie is lelled with fresh event vrile. Then for the following feture sets, every voie exhiiting strts[γ i ]:t feture is lelled with fresh event vrile. Any other voie is lelled with its orreponding event vrile from the previous set. Figure 5 nd the exmple elow illustrte the lelling proess. VVP strts[0] : f strts[0] : f strts[1] : t strts[1] : t qul[1][0] : ons qul[1][0] : ons ont[1] : -l ont[1] : +l sw sw m 4 3 Instnes Fig. 4. A pttern in VVP nd the orresponding reltionl network. Instnes re tken from Bh horle hrmoniztions.

6 6 M. Bergeron nd D. Conklin slie A slie B slie C C5 B4 A4 G4 F4 E4 D4 C4 B3 A3 G3 d f g d f g F3 E3 D3 C3 B2 A2 G2 Fig. 5. Sliing of the polyphoni texture nd lelling of events Exmple 2. slie A slie B slie C strts[0] :t d strts[0] :t f strts[0] :t strts[1] :t strts[1] :f g strts[1] :t strts[2] :f strts[2] :f strts[2] :f The semntis of VVP ptterns is defined in terms of trnsltion to R. Wedo this vi the five rules presented in Fig. 6. The first three rules onern temporl reltions nd ll refer to the strt of new event (indited y feture of the form strts[γ] :t). The first rule onerns the se where two events strt in the sme slie nd st(, ) temporl reltion is reted. The seond rule speifies when to rete sw(, ) reltion. The event vrile orresponds to n event tht is strting in the urrent slie, while vrile orresponds to n event tht hs strted in the pst nd is eing prolonged into the urrent slie. The third rule ssigns m(, ) temporl reltion whenever n event strts. The event hs to e in the sme voie s the event (Fig. 6 nd heneforth, we indite this y horizontlly ligning the lels). Finlly, rules four nd five simply trnslte the informtion out voies nd musil fetures into their reltionl form. In rule 5, for every voie present in the musil feture τ[γ i ] : μ, the tom τ(,,μ) must refer to the event vrile tht lels tht voie.

7 Susumption of Vertil Viewpoint Ptterns 7 1) 2) 3) VVP R strts[γ i] : t st(, ) strts[γ j] : t strts[γ i] : t sw(, ) strts[γ j] : f strts[γ i] : t m(, ) VVP R 4) strts[γ i] : t voie(, γi) strts[γ] : t 5) τ(,,μ) τ[γ] : μ Fig. 6. Interprettion of VVP in terms of R Exmple 3. The following VVP pttern is trnslted to reltionl pttern y pplying the rules of Fig. 6: strts[0] : t strts[1] : t qul[0][1] : ons { strts[0] : f strts[1] : t } rule 1) st(, ) rule 2) sw(, ) rule 3) m(, ) rule 4) voie(, 0), voie(, 1), voie(, 1) rule 5) qul(,, ons) Note tht the semntis presented here onerns restrited form of VVP,where for exmple the strts[γ] :μ fetures re lwys present to expliitly indite the temporl onfigurtion of slie. The restrition is defined y the three well-formedness rules expressed elow. Definition 3. A well-formed VVP pttern ontins only feture sets of the following form nd stisfying the following rules: strts[γ 0] : μ strts[γ 1] : μ strts[γ n] : μ τ[γ i] : μ Every feture set refers to the sme voies nd for every voie it ontins feture of the form strts[γ] :μ There is t lest one feture strts[γ] :t per set Fetures of the form τ[γ i ] : μ only pper in set where the feture strts[γ i ]:t ppers

8 8 M. Bergeron nd D. Conklin The purpose of the first rule is to ensure tht ptterns n e interpreted s ompt in time. The seond rule is neessry s the opposite yields set where no new event is strting, whih does not hve ler interprettion in terms of events. The third is neessry s τ[γ i ] : μ is interpreted s ssigning feture to the urrent event in voie γ i. To give ler interprettion to VVP in terms of R, we must void ses where feture would e ssigned to the ontinution of n event (s this is impossile to represent in R). Notie how well-formed VVP pttern oinides with the typil use of regulr expressions in Humdrum pttern s desried in Set. 1: slie ontins informtion out the strt or ontinutions of events, nd out musil fetures. 4 Susumption for Vertil Viewpoints In this setion, the reltion of susumption for VVP is defined. This reltion is entirely different thn the θ-susumption of R defined in Set. 2 nd it diretly orresponds to how pttern mthing is implemented in VVP. We show tht susumption in VVP is sound nd omplete with respet to susumption in R, tking into ount the VVP to R trnsltion defined in the previous Set. 3. Susumption of vertil viewpoints is defined s follows. Definition 4. Given two VVP ptterns v 1 nd v 2,wesythtv 1 susumes v 2, written v 1 VVP v 2, iff there is mpping from the sets of v 1 to the sets of v 2 suh tht onseutive sets in v 1 re mpped to onseutive sets in v 2 ; nd tht tht every set in v 1 is mpped to superset in v 2. Informlly, we sy tht v 1 n e ligned with v 2 suh tht ligned sets re in superset reltion. Figure 7 illustrtes the notion of susumption in VVP. The figure shows suessive refinements of suspension pttern, strting from two-event pttern tht does not ontin ny musil reltions nd ulminting with three-event pttern tht expresses the pproprite temporl reltions, dissonne nd onsonne, nd resolution y melodi step down. Clim 1. Susumption for VVP is sound nd omplete with respet to susumption in R. Tht is, given v 1 nd v 2, nd their orresponding reltionl ptterns R(v 1 )ndr(v 2 ), we hve v 1 VVP v 2 iff R(v 1 ) R R(v 2 ). Cse v 1 VVP v 2. Suppose tht it is not true tht R(v 1 ) R R(v 2 ), then for ll vrile sustitution θ, itmustethesethtr(v 1 )θ ontins t lest one tom tht R(v 2 ) does not ontin. Then there must e one set in v 1 tht ontins feture tht its orresponding set in v 2 does not ontin. This ontrdits v 1 VVP v 2, so lerly we must hve R(v 1 ) R R(v 2 ).

9 Susumption of Vertil Viewpoint Ptterns 9 VVP { } strts[0] : f strts[1] : t VVP strts[0] : f strts[1] : t qul[1][0] : diss sw Instnes VVP strts[0] : f { } strts[0] : t strts[1] : t strts[1] : f qul[1][0] : diss VVP strts[0] : t strts[0] : f strts[1] : f strts[1] : t qul[0][1] : ons qul[1][0] : diss ont[0] : -s R m sw sw Fig. 7. Susumption in VVP nd the susumption of orresponding reltionl networks. Instnes re seleted from Bh horle hrmoniztions. Cse R(v 1 ) VVP R(v 2 ). The proof rests on three oservtions: i) given the lelling of events in v 1 nd v 2, the sustitution θ implied y R(v 1 ) R R(v 2 ) indues mpping from the feture sets of v 1 to the feture sets of v 2 ; ii) ny two onseutive sets in v 1 re mpped to onseutive sets in v 2 ; nd iii) ny set in v 1 is neessrily mpped to superset in v 2. From these onditions, it follows tht v 1 VVP v 2. First oservtion. Consider ny set in v 1. By definition, there is t lest one strts[γ i ]:t feture, nd it is ssigned fresh lel : v 1 : v 2 : strts[γ i] : t strts[γ j] : t strts[γ i] : t strts[γ j] : t θ = θ =

10 10 M. Bergeron nd D. Conklin The lel is is mpped to some y the sustitution θ implied y R(v 1 ) R R(v 2 ). As every strts[γ i ]:t feture in v 2 is lso ssigned fresh lel, the unmiguously refers to extly one set in v 2. Tht is, there is single set in v 2 where the lel is ssoited with strts[γ i ]:t feture. Note tht euse the feture strts[γ i ]:t implies n tom voie(,i)inr(v 1 )θ, it follows tht R(v 2 ) must lso ontin tht tom nd tht the strts[γ i ]:t fetures in v 1 nd v 2 re in the sme voie. In the se where seond fresh lel exists (e.g. in the exmple ove), it must lso mp to the sme set in v 2.Otherwise,wewouldhvest(, )in R(v 1 )θ, ut R(v 2 ) would not ontin st(, )sthestrts[γ] :t fetures ssoited with nd would pper in different sets, whih would ontrdit R(v 1 ) R R(v 2 ). This holds for ny other fresh lel nd ensures tht we hve proper mpping: ny set in v 1 is oupled with extly one set in v 2. Seond oservtion. Consider two onseutive sets in v 1 nd their orresponding sets in v 2 : { } { } strts[γ i] : t v 1 : v 2 : { } strts[γ i] : t { } strts[γ i] : f { } strts[γ i] : t θ = θ = Clerly, R(v 1 )θ nd R(v 2 ) oth ontin m(, ) reltion. This is only possile if v 2 ontins the pproprite suessive sets. Now suppose s ove tht the suessive sets in v 2 re not onseutive: { } { } v 1 : v 2 : { } d/ {strts[γ j] : t strts[γ i] : t strts[γ j] : μ } { } strts[γ i] : t d θ = θ = θ = dθ = d Consider one intervening feture set in v 2 (shown ove with lels, ). Tht setmusthvetlestonestrts[γ j ]:t feturethtwesupposehereinvoiej. Both vlues of strts[γ j ]:μ in the seond set of v 1 led to ontrdition. If strts[γ j ]:t, thenr(v 1 )θ ontins m(,d )reltionthtr(v 2 ) lerly does not ontin. If strts[γ j ]:f, then the lel is prolonged in v 1 nd R(v 1 )θ ontins sw(, )reltionthtr(v 2 ) nnot ontin s the equivlent lel is not prolonged y virtue the strts[γ j ]:t feture in the intervening set, whih retes the fresh lel. A similr resoning pplies for more thn one intervening set nd hene the oservtion holds. Third oservtion. The mpping indued y R(v 1 ) R R(v 2 ) neessrily mps every set in v 1 to superset in v 2. Suppose this is not true. Then some set in v 1 ontins feture tht its orresponding set in v 2 does not. If tht feture is

11 Susumption of Vertil Viewpoint Ptterns 11 of the form strts[γ i ]:μ, then lerly R(v 1 )θ ontins some temporl reltion tht R(v 2 ) does not ontin. If tht feture is of the form τ[γ i ] : μ, then y definition the set in v 1 lso ontins strts[γ i ]:t feture nd y rule 5 of Fig. 6, R(v 1 )θ willontinntomoftheformτ(ε,,ε,μ)thtr(v 2 ) will not ontin. In oth ses, this ontrdits R(v 1 ) R R(v 2 ) nd we must hve tht the sets of v 1 re ligned with supersets in v 2. By the three oservtions ove, we must hve tht v 1 VVP v 2. The result is signifint s it estlishes tht the semntis given in Set. 3 is fully onsistent with the implementtion of pttern mthing in VVP,whihis diret pplition of susumption. The steps tht were tken in order to estlish the result (e.g. the well-formedness onstrints of Set. 3) revel the pities nd limittions of VVP ptterns. As pointed out in Set. 1, the nlysis presented here provides insight, y nlogy, for understnding Humdrum ptterns nd, in generl, polyphoni ptterns sed on the ide of vertil slies. 5 Disussion This pper hs formlized the vertil viewpoint lnguge VVP for expressing polyphoni ptterns. This lnguge inspired y the Humdrum pttern representtion nd the eventul gol of this work is to fully inorporte VVP within Humdrum serh tool. A semntis for the pttern lnguge ws provided y developing trnsltion of VVP ptterns to more powerful reltionl network representtion R, itself suset of predite logi. The VVP lnguge is restrited to ptterns tht re ompt in time, in prtiulr ptterns where suessive events re neessrily ontiguous. Pttern lnguges suh s Humdrum nd SPP re more generl s they llow onfigurtions tht express gps etween suessive events [8]. The VVP lnguge supports the importnt inferene of susumption, nd this is done in wy entirely different to the underlying R representtion, whih requires the omputtion of mpping etween the vriles in two reltionl networks. In VVP susumption is omputed in mnner similr to string mthing, with the exeption tht hrters re now sets nd tht insted of testing for equlity, we test for the suset reltion. In the future, we intend to detil how effiient string mthing lgorithms n e dpted to ompute susumption in VVP. Susumption is n essentil notion of ny pttern representtion, s it llows the development of ptterns y progressive speiliztion nd generliztion therey mthing fewer or more instnes. Susumption is lso essentil within ny dt mining pplition tht employs the pttern representtion s it permits refinement serh of susumption txonomy of ptterns [1]. A min result of this pper is tht susumption VVP is sound nd omplete with respet to susumption in R. In the future we intend to explore how susumption txonomies in VVP n e used to index lrge sore dtses in Humdrum formt for rpid retrievl to musiologil queries, in wy similr to the use of onepts in Desription Logis [11]. Finlly, we lso intend to nlyze other

12 12 M. Bergeron nd D. Conklin representtion of polyphoni ptterns [12,13,14,15] y similrly expressing them in terms of reltionl lnguge. As ws the se for VVP, we ntiipte tht it will e only possile for susets of the lnguges, nd tht our efforts to find suh susets will help to shed light on the strengths nd weknesses of existing polyphoni pttern lnguges. Referenes 1. Conklin, D., Bergeron, M.: Representtion nd disovery of feture set ptterns in musi. Computer Musi Journl 32(1), (2008) 2. Mxwell, J.: An Artifil Intelligene Aproh to Computer-Implemented Anlysis of Hrmony in Tonl Musi. PhD thesis, Indin University (1984) 3. Conklin, D., Bergeron, M.: Disovery of ontrpuntl ptterns. In: ISMIR 2010: 11th Interntionl Soiety for Musi Informtion Retrievl Conferene, Utreht, The Netherlnds, pp (2010) 4. Huron, D.: Musi informtion proessing using the Humdrum T oolkit: Conepts, exmples, nd lessons. Comput. Musi J. 26, (2002) 5. Wild, J.: A review of the Humdrum Toolkit: Unix tools for musil reserh, reted y Dvid Huron. Musi Theory Online 2 (1996) 6. Jn, S.: Meme hunting with the Humdrum toolkit: Priniples, prolems, nd prospets. Computer Musi Journl 28(4), (2004) 7. De Redt, L.: A perspetive on indutive dtses. ACM SIGKDD Explortions Newsletter 4(2), (2002) 8. Bergeron, M., Conklin, D.: Temporl ptterns in polyphony. In: Chew, E., Childs, A., Chun, C.H. (eds.) MCM Communitions in Computer nd Informtion Siene, vol. 38, pp Springer, Heidelerg (2009) 9. Allen, J.: Mintining knowledge out temporl intervls. Communitions of the ACM 26(11), (1983) 10. Nienhuys-Cheng, S.-H., de Wolf, R.: Foundtions of Indutive Logi Progrmming. LNCS, vol Springer, Heidelerg (1997) 11. Bder, F.: Desription logis. In: Tessris, S., Frnoni, E., Eiter, T., Gutierrez, C., Hndshuh, S., Rousset, M.-C., Shmidt, R.A. (eds.) Resoning We. LNCS, vol. 5689, pp Springer, Heidelerg (2009) 12. Bergeron, M., Conklin, D.: Strutured polyphoni ptterns. In: Ninth Interntionl Conferene on Musi Informtion Retrievl, Phildelphi, USA, pp (2008) 13. Meredith, D., Lemström, K., Wiggins, G.A.: Algorithms for disovering repeted ptterns in multidimensionl representtions of polyphoni musi. Journl of New Musi Reserh 31(4), (2002) 14. Meudi, B., Sint-Jmes, E.: Automti extrtion of pproximte repetitions in polyphoni midi files sed on pereptive riteri. In: Wiil, U.K. (ed.) CMMR LNCS, vol. 2771, pp Springer, Heidelerg (2004) 15. Ukkonen, E., Lemström, K., Mäkinen, V.: Geometri lgorithms for trnsposition invrint ontent sed musi retrievl. In: Interntionl Conferene on Musi Informtion Retrievl (ISMIR), Bltimore, Mrylnd, USA, pp (2003)