EFFICIENT EXTRACTION OF CLOSED MOTIVIC PATTERNS IN MULTI-DIMENSIONAL SYMBOLIC REPRESENTATIONS OF MUSIC

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1 EFFICIENT EXTRACTION OF CLOSED MOTIVIC PATTERNS IN MULTI-DIMENSIONAL SYMBOLIC REPRESENTATIONS OF MUSIC Olvr Lrtllot Unvrsty o Jyväskylä Dprtmnt o Mus ABSTRACT In ts ppr, w prsnt n nt mol or sovrn rpt pttrns n symol rprsnttons o mus. Comntorl runny nrnt to t pttrn sovry prm s usully ltr trou lol sltv mnsms, s on pttrn rquny n lnt. Our ppro s s nst on t onpt o los pttrn, nsurn losslss ruton y ptvly sltn most sp srptons n t mult-mnsonl prmtr sp. A noton o yl pttrn s ntrou, llown t ltrn o notr orm o omntorl runny provok y sussv rpttons o pttrns. T us o yl pttrns mpls nssry ronolol snnn o t p, n t ton o mnsms ormlzn prtulr Gstlt prnpls. Ts stuy sows tror tt t utomt nlyss o mus nnot rly on smpl mtmtl or sttstl ppros, ut n rtr omplx n tl moln o t ontv systm ruln t lstnn prosss. T rsultn lortm s l to or or t rst tm ompt n rlvnt motv nlyss o smpl monos, n my tror ppl to utomt nxn o symol mus tss. Numrous tonl mnsms n to n orr to onsr ll spts o mus xprsson, nlun polypony n omplx motv trnsormtons.. INTRODUCTION Ts ppr s ous on utomt srpton o symol mus, n prsnts n nt lortm or sovrn rpt pttrns. Rpt pttrns r struturs sly prv y lstnrs, xprn or not, n rprsnt tror on o t most slnt rtrsts o musl works, somtms ll prlllsm [8]. In ts ppr, pttrns wll sr rom symol rprsnttons o mus rtr tn rom uo snls. A pttrn xtrton tsk on t symol lvl, ltou tortlly smplr, rmns xtrmly ult to rry out, n ts utomton s not n v up to now. In, omputr rsrs on ts sujt rly or rsults los to lstnrs or musolosts xpttons. Hn t pttrn sovry tsk s so omplx tt t nnot ulll rtly rom t uo snl, wtout pror trnsrpton rom t uo to t symol rprsnttons, n orr to rry out t nlyss on onptul lvl. W prvously sow tt t pttrn sovry tsk ls to prolm o omntorl runny, w ns to rully ontroll [6]. W propos tror urst s on mxmlly sp srpton o pttrn lsss, tt my n t rlt to t onpt o los pttrn []. W lso ntrou noton o mplton rlton twn mult-mnsonl pttrn srpton, tt my rlt to t suonpt-supronpt rlton n n t Forml Conpt Anlyss (FCA) tory [5], n tt oms rom t Gllos onnton twn pttrn srpton n pttrn lsss. W sow lswr t nssty to mol t pnomnon o pttrn ylty [7]. Ts wll n mor prsly n t son prt o ts ppr, w wll lso prsnt n xtnson o t suonpt-supronpt rlton to yl pttrns, nln smpl molln n nt ontrol o t omplx struturl onurtons oun n vry musl p, vn smpl ons.. CLOSED PATTERN DISCOVERY Ts ston prsnts t s prolm o pttrn sovry n ntrous t noton o los pttrn... Dntons Lt S =<... N > squn o lmnts o som st A. A susqun S,l o nx [, N] n o lnt l [, N + ] s t squn : S,l =< l >. () W sy tt susqun S,k s nlu n notr susqun S j,l, n w wrt S,k S j,l j n j+l +k. W my tn propos rst smpl nton o pttrn P P(S) o lnt l: P P(S) (, j) [, N], P = S,l = S j,l. () T support o pttrn P, not σ(p ), s t numr o ourrns o tt pttrn,.. σ(p ) = { [, N], S,l = P }. (3).. Runny Fltrn T tsk o sovrn rpt pttrns ls to omntorl prolms, tt v n onsr n prtulr

2 n t Frqunt Itmst Mnn (FIM) ommunty. For pttrn P P(S), ll ts l susquns Q r lso pttrns o S : (P P(S) n Q P ) Q P(S). (4) All ts supttrns r tror xpltly sovr y ny s pttrn sovry lortm []. On ommon wy to solv ts prolm onssts n ousn on mxml pttrns P o t squn S, not P M(S), tt r pttrns o S not nlu nto ny otr pttrn o S : { P P(S) P M(S) (5) Q P(S), P Q. Ts urst nls snnt ruton o t numr o sovr pttrn, ut ls lso to loss o normton. In, not ll t supttrns my mmtly ronstrut known t mxml pttrns. For nstn, t ry supttrn n ur s runnt s t n smply rtrv s sux o t lk pttrn. Howvr, n ur, t sm ry supttrn s not runnt ny mor, us ts support (4) s lrr tn t support o t lk pttrn (). Fur. T ry supttrn s not los pttrn: t s smpl sux o t lk pttrn. Fur. T ry supttrn s now los: ts support s r tn t support o t lk pttrn. A pttrn P wll ll los, not P C(S), n only tr xsts no propr suprpttrn Q o sm support : P C(S) P P(S) Q P(S), { P Q σ(p ) = σ(q). T st o los pttrns v ompt n losslss srpton o squn..3. An Inrmntl n Cronolol Appro Our ppro to sovrn los pttrns n musl squns s nrmntl n ronolol. Ts wll just n ston 4, sown t nssty to mol sussv rpttons o sm pttrn s trvrsl trou (6) on snl yl pttrn, n n to onsr ronolol ppro o mus. A tl n llustrt srpton o t lortm s n prsnt n [6]..3.. Inrmntl ppro T sussv prxs o pttrn r sovr prorssvly: ty r onsr s sussv ntrmry stts o pttrn n (PC) wos nl stt rprsnts t wol pttrn. At stp o t prorssv onstruton, ll t ourrns o t lst sovr prx r onsr. Intl ontnutons orm nw xtnsons o t prx, rprsnt s lrn o t urrnt stt. Sn stt n pt svrl lrn, t st o ll pttrns orm tr, ll pttrn tr (PT). T xplt rprsntton o prxs s ntrmry stts o pttrn ns nus smply lnr omplxty: pttrn o lnt l s rprsnt y l stts (w s snntly lowr tn t l non-los supttrns). Smlrly, pttrn ourrn s lso rprsnt s n o stts ll pttrn ourrn n (POC) turn t sussv prxs too. E stt o POC s rlt to ts orrsponn PC. As pttrn ourrn n lso pt svrl rnt possl ontnutons, t st o ll pttrn ourrns tt r ntt y on not orms tr, ll pttrn ourrn tr (POT). T root o POT s ssot to t root o t PT (no n ur 4), w rprsnts t smpl onpt o not, n s tror ll not pttrn. Sn ll nots n potntlly ntt POT, ty r ll ourrns o t not pttrn. T nluson rlton twn pttrns my ompos s prout o two su-rltons: prx n sux rltons. Any supttrn wll tn onsr s prx o sux o pttrn. T losur o pttrn P my tn ssss ollown ts two rltons.. Pttrn P s prx-los tr os not xst pttrn Q o w P s prx o sm support.. T rmnn, n most omplx, prt o t stuy onrns pttrns n sux rltons. Pttrn P s sux-los tr os not xst ny pttrn Q o w P s sux o sm support. Consr pttrn P, w s n sovr s n xtnson o ts prx P. T st o pttrns Q o w P s sux s smply onstrut rom t st o pttrns Q o w P s sux..3.. Cronolol ppro T pttrn sovry pross s ronolol: t mn routn o t lortm onssts n snl trvrsl o t squn S, rom t rst lmnt to t lst lmnt N. E nw lmnt nus n upt o t wol pttrn tr. For ts purpos, s-tls mmors ll t possl ontnutons o P, tt s, ll t possl lmnts pprn just tr o ts ourrn. I prvous

3 ourrn o pttrn P s n ontnut y n lmnt ntl to, tn n xtnson P o pttrn P my sovr ( not lry) n rprsnt s on o ts l. T sux-los onton soul owvr pply: tr soul not xst pttrn Q o w P s sux o sm support. Durn t ronolol nlyss, pttrn P tt ws rst onsr s non-los my om los on sovrn nw ourrn tt s not n ourrn o t suprpttrn Q (n xmpl wll sown n prrp 3.3). Suprpttrns Q n sly rtrv, s xtnsons o t orrsponn suprpttrns Q o P y t sm lmnt. Not ll t suprpttrns Q n to onsr, ut only tos ontnn n ourrn onlu y prvous lmnt. 3. MULTI-DIMENSIONAL CLOSED PATTERNS In prvous suston, pttrn ws sr n squns o lmnts S =<... N >, A. Mus, on t ontrry, s xprss n mult-mnsonl prmtr sp. 3.. Mult-Prmtr Dsrpton o Mus T mol prsnt n ts ppr only nlyss mono squns,.. sussons o nots wtout suprposton. W n tror rprsnt ny mono squn s prvously : S =< n n... n N >, n N, (7) wr N s t prmtr sp o nots. In our mol, ts not sp s smply ru to : wr N = t ro ryt (8) t, or ton pt sp, rprsnts pts s postons n t mplt tonl sl. Dton trnspostons wll tt n ts sp. ro, or romt pt sp, rprsnts pts s postons on t pno kyor. Follown t MIDI stnr, wt ml C s ssot t vlu 60. ryt, or mtrl sp, rprsnts tmporl postons n trm o t stn rom t nnn o t musl squn. T rytm unt o t mtrl sp s vn y t tm sntur. T pttrn sovry tsk nnot rtly ppl on ts not squn S, us sussv not s rlt to stnt mtrl poston n s tror stnt. Evn t mtrl sp s sr, ntr rytm pttrns nor trnspos mlo pttrns my sovr. Most umputtonl ppros, nlun ours, mol tror musl squns s susson o ntrvls twn sussv nots : S = ( n n n n 3... n N n N ). (9) An ntrvl n n + N s vtor twn two ponts o t not sp N n = (t =, ro =, ryt = t ) n + = (t = +, ro = +, ryt = t + ) n n tror sr y t tr oornts : n n + = t( n n + ) ro( n n + ) ryt( = + +. n n + ) t + t () t: ro: ryt: pttrn ourrns: pttrn : t: ryt:.5.5 Fur 3. Mult-mnsonl srpton o musl squn, w ontns two ourrns o pttrn. 3.. Forml Contxt Rprsntton o Pttrns (0) W wll rprsnt musl pttrns wt t lp o onptul rmwork tt ns ojts ssot wt rnt kns o ttruts [5]. Ts ttruts onsst not only o t rnt musl mnsons, ut lso o t rnt supttrns n suprpttrns. Follown t nrmntl n ronolol ppro xpln n prrp.3, w n rstrt our stuy o t nluson rltons twn pttrns to sux rltons. In ts rspt t ojts o t pttrn srptons r t sussv nots o t musl squn ormn t st N (S). E not n N (S) rlts to sp tmporl ontxt, n y t prt o t musl squn onlu y ts not n. E not n s sr rstly y t rnt musl rtrsts o t prn ntrvl: n n. D 0, s (n ) : s( n n ) =, s {t, ro, ryt}, s. () E not n s lso sr y t musl rtrsts o t prvous ntrvls: D j, s (n ) = s( n j n j ) =, s {t, ro, ryt}, s. (3) Tn t pttrn srpton o t squn S my xprss s orml ontxt (N (S), D, I) wr : t st o ojts s N (S): t st o nots n S,

4 t st o ttruts s D: t st o lmntry musl srptons n y qutons n 3, n I s t nry rlton twn N (S) n D, ll nn, n y: (n, ) I (n ) s tru. (4) W tn n t rv srpton C o st o nots C N (S) s t ommon srpton o ll ts nots: C = { D n A, (n, ) I }. (5) T nots n C r tror ourrns o sm pttrn, w s mxmlly sr y C. Is ully n t rv lss D o srpton D D s t ollown lss : D = { n N (S) D, (n, ) I }. (6) T pttrn sovry tsk onssts n nn xustv lss D srn sm srpton D. T troul s, lots o rnt srptons D my l to sm lsss D Forml Conpt Rprsntton o Pttrns T rvtors oprtons n y quton 5 n 6 stls Gllos onnton twn t powr st ltts on N (S) n D. T Gllos onnton ls to ul somorpsm twn two losur systms, wos lmnts, ll orml onpt o (S(S), D, I) orrspons xtly to t los pttrns P = (C, D), vryn: C N (S), D D, C = D, n D = C. (7) For los pttrn P = (C, D), C s ll t xtnt o D n D t ntnt o C. W my smply ll C n D rsptvly t lss n t srpton o P. Hn, or st o pttrns P = (D, D ) o sm lss D = C, t los pttrn P = (C, D) s sr usn t rv oprtor C n n quton 5: t ontns ll t lmntry srptons ommon to ll nots o t lss C. In otr wors, los pttrns r sr s prsly s possl. Howvr, s lstnrs tn to prv only rptton o onnx susquns, w soul slt only t srptons o t lonst st o ontuous ntrvls (D j D j... D 0 ) ln to t ontxt not n sr olr srptons D j+k tr s no srpton D j+ ssot to stp j +. W v propos urtr ontnuty onstrnt, tt sms to orrspon mor ply to lstnrs prpton, sttn tt ontuous ntrvls soul sr y sm mnsons. [, j], s {t, ro, ryt}, (, ) s, D j, s n Dj, s (8) Clos pttrns, s orml onpts, r nturlly orr y t suonpt-supronpt rlton n y (C, D ) < (C, D ) C C ( D D ). (9) (C, D ) my tror onsr s mor sp tn (C, D ). T nrmntl n ronolol pttrn sovry mtooloy prsnt n ston.3 my nrlz usn ts mult-prmtr nton o los pttrn. For nstn, pttrn (mor smply not ), n ur 4, turs mlo n rytml srptons: ( t = 0 ryt =.5 t = 0 ryt =.5 t = ryt =.5 ryt = 4 ) wrs pttrn (or ) only turs ts rytm prt: (ryt =.5 ryt =.5 ryt =.5 ryt = 4). Hn pttrn s mor sp tn pttrn. Wn only t two rst ourrns r nlyz, s ot pttrns v sm support, only t mor sp pttrn soul xpltly rprsnt. But t lss sp pttrn wll rprsnt on t lst ourrn s sovr, sn t s not n ourrn o t mor sp pttrn. t: ryt: mor sp tn Fur 4. T rytm pttrn s lss sp tn t mloo-rytm pttrn Optml sor srpton In orr to ru t sp omplxty o t pttrn rprsntton, n lso to smply s mu s possl t pttrn srpton o t musl sor, to los pttrn P = (C, D) wll ssot sp lss SC(P ) w onssts o t st o ourrns tt r not nlu nto lsss o mor sp pttrns: SC(P ) = C\ C. (0) (C,D )<(C,D) Rvrsly, t nrl pttrn lsss my rtrv trou n unon o ts sp lss n t unon o t sp lsss o ll t mor sp pttrns : C = SC(P ) SC(P ). () P <P Durn t ronolol nlyss o t musl sor, only t sp lsss r onstrut. But tm sp pttrn ourrn s sovr, ll t lss sp pttrns n to rll y t lortm, us tr xtnsons my l to t sovry o nw sp

5 pttrns. For nstn, n ur 5, roups n 3 r ourrns o pttrn, n roups 3 n 4 r ourrns o pttrn. Sn pttrn s mor sp, t lss sp pttrn os not n to ssot wt roup 4 (tt s wy t s rprsnt n ry). Howvr n orr to tt roups n 5 s ourrns o pttrn l, t s nssry to mpltly onsr roup 4 s n ourrn o pttrn. Hn, vn pttrn, sn lss sp tn, ws not xpltly ssot wt roup 4, t to onsr mpltly n orr to onstrut pttrn l. Implt normton s ronsttut trou trvrsl o t pttrn ntwork lon t suonpt-supronpt rltons..5.5 t: ryt: j - k l j k k j (t,ryt) +, k n rl z sp to n t n o j 0, rl k 0, +, 0, -3, 0, n z to n Fur 6. Prorssv sovry o t pttrn rpttons on t sor n t rsultn pttrn tr (low t sor). T nrlzton o pttrn (r 6) nto pttrn j (r 7) ls to t mplt nrlzton o pttrn nto pttrn k, tn n tror mmtly nt n r 8. 4 j k l l Fur 5. Group 4 n smply onsr s ourrn o pttrn. Howvr, n orr to tt roup 5 s ourrn o pttrn l, t s nssry to mpltly nr roup 4 s ourrn o pttrn too Gnrlzton o pttrns Nw pttrns n sovr s smpl nrlztons o lry known pttrns. In r 7 o ur 6, t two rst nots orm n ourrn o pttrn sr y t = + n ryt =. T tr not nnot owvr ulll t known xtnson o pttrn nto pttrn wt srpton t = 0 n ryt =, us t mlo srpton t = 0 os not t r. Howvr, s t rytm srpton ryt = ts, nw xtnson j s sovr s nrlzton o pttrn. T lss sp pttrns, ltou usully not xpltly rprsnt n t nlyss, soul upt nssry. In prtulr, wn nrlzton o pttrn s sovr, t nrlzton o ll ts mor nrl pttrns soul lso onsr. For nstn, s s n nrlz nto j, t soul lso nrr tt s nrlz nto k n t sm wy. In ts wy, t nlyss o t nxt r (8) onssts smply n ronzn ts nrl pttrn k lry known. 4. CYCLIC PATTERNS In ts ston, w prsnt notr mportnt tor o runny tt, ontrry to los pttrns, s not n stu n urrnt nrl lortm rsrs. 4.. Runny Du to Sussv Rpttons Comntory xploson n us y sussv rpttons o sm pttrn (or nstn t pttrn n ur 7, o srpton (ryt = ryt = )). As ourrn s ollow y t nnn o nw ourrn, pttrn n xtn (ln to pttrn o srpton (ryt = ryt = ryt = ) y nw ntrvl wos srpton (ryt = ) s ntl to t srpton o t rst ntrvl o t sm pttrn (.., pttrn ). Ts xtnson n prolon rursvly (nto,, t.), ln to omntorl xploson o pttrns tt r not prv u to tr omplx ntrtwnn. ryt: Fur 7. Multpl sussv rpttons o pttrn orm omplx ntrtwnn o non-prv struturs. 4.. Cyl Pttrns Our rprsntton (ur 7) sows tt t lst stt o ourrn o pttrn s synronz to t rst stt o t ollown ourrn. Lstnrs sm to tn to uson ts two stts, n to prv loop rom t lst stt () to t rst stt () (ur 8). T ntl yl pttrn ls tror to yl pttrn tt osllts twn two stts 0 0, orrsponn to rytm vlus n. In, wn lstnn to t rmnr o t rytm squn, w tully prv ts prorssv osllton twn ts two stts 0 n 0. Hn ts yl-s moln xplns

6 ommon lstnn strty, n rsolvs t prolm o omntorl runny. ryt: Fur 8. T sussv rpttons o pttrn l to ts ylty, n to n osllton twn stts. Ts yl PC s onsr s ontnuton o t ornl yl PC (ur 8). In, t rst rptton o t rytm pro s not prv s pro s su ut rtr s smpl pttrn: ts sussv nots r smply lnk to t prorssv stts, n o t yl PC. On t ontrry, t ollown nots xtns t POC, w nnot tror ssot to t yl PC nymor, n r tror lnk to t sussv stts o t PC ( n ). T wol pro squn onsttuts tn snl POC rprsntn t trvrsl o t yl PC ollow y t yl PC. It n rmrk lso tt, y proprty o t yl PC, no smntton s xpltly rprsnt twn sussv rpttons. T pro squn n ur 8 n prv or nstn s orm y sussv rpttons o rytm pro ompos o susson o quvr n rot, or n t ontrry susson o rot n quvr. In, t lstnr my nln to smnt t ny ps o t yl PC (or not to smnt t ll). Ts tonl onpt mmtly solvs t runny prolm sr n t nnn o ts ston. In, t unqu POC tt s prorssvly xtn s mor sp tn ts suxs, w nnot tror xtn ny mor. Ts pnomnon o sussv rptton, ltou vry rqunt n musl xprsson, s n rrly stu []. T omntorl xploson nrt y t pnomnon s ru y sltn, on t nlyss s on, t pttrns turn mnml tmporl ovrlppn twn ourrns. As t slton s nrr lolly, rlvnt pttrns my sr. Bss omntorl runny rmn prolmt sn t ltrn s on tr t tul nlyss ps. Our onsrn o lol onurtons nls mor prs ltrn Gnrl n Sp Cyls T spty rlton n n prvous ston ns to ppl to yl pttrns too: yl pttrn C woul onsr s mor sp tn notr yl pttrn D wn t squn o srpton o pttrn D s nlu n t squn o srpton o pttrn C. Hr too, ts onpt o spty plys pvotl rol n mus prpton n nls soun lortm prossn o mus. (t,ryt) 0, 0, 0, +, 0, +, +, +, +, Fur 9. Mor tl nlyss o t prv yl onurtons. In Fur 9, t svn rst nots o t yl osllt roun t yl PC sr y: nrlzton nrlzton ryt = ryt = t = 0. Tn pprs mor sp yl sr y ryt = t = + ryt = t = 0 n s nrlz tr our nots to yl tt os not tur t unson ntrvl ny mor: ryt = t = + ryt =. Morovr, ollown t rul o nrlzton o nrlz pttrns xpln n prrp 3.5, t mor nrl yl too ns to nrlz nto yl wr t unson ntrvl s n sr: ryt = ryt =. Ts rnt yls r tully prv y t lstnr. Morovr, t ntrton o ts pnomnon nto t mol lps nsurn t rlvn o t rsults n von numrous unwnt omntorl runns T Fur/Groun Rul Anotr kn o runny pprs wn ourrns o pttrn, su s t mloo-rytm pttrn n ur 0, sr y t = n ryt = r suprpos to yl pttrn ( ), su tt t pttrn () s mor sp tn t yl pro, w s smply rytm: ryt =. In ts s, t ntrvls tt ollow ts ourrns r ntl, sn ty r rlt to t sm stt ( ) o t yl pttrn. Lolly t pttrn oul xtn y t sussv xtnsons o spton

7 t yl pttrns (ln to pttrns,, t.). Ts pnomnon, w my rquntly ppr n musl p, woul l to notr omntorl prolrton o runnt struturs not orrtly ontroll y rlvnt mnsms. On t ontrry, ollown t Gstlt Fur/Groun rul, lstnrs tn to prv t pttrn s sp ur tt mrs ov t pro kroun. Follown ts rul, t ur nnot xtn (nto ) y srpton tt n smply nt to kroun xtnson. (t,ryt) Trl -, - - Fur. Automt nlyss o rst tm o Mozrt Sont n A K Gsslrl Fur 0. Pttrn s sp ur, ov kroun nrt y t yl pttrn RESULTS Ts mol ws rst vlop s lrry o OpnMus []. A nw vrson wll nlu n t nxt vrson.0 o MIDItoolox [4], Mtl toolox t to mus nlyss. T mol n nlyz mono musl ps (.., onsttut y srs o non-suprpos nots) n lt t sovr pttrns on sor. Fur prsnts t rsultn nlyss o mvl son ll Gsslrl tt t lnust Nols Ruwt [], n on o t rst n most mous ttmpt to mol motv nlyss, propos s rst pplton o s mto. Our mol s t rst omputtonl systm l to or rlvnt n ompt nlyss o ts p. T p onsr r s owvr slt smplton o t tul p prsnt y Ruwt, w nlus svrl lol motv vrtons tt our mol nnot nl or t momnt. Morovr, u to t prsn o sux o pttrn B just or ts two sussv wol ourrns, our mol uns ts tr ourrns s yl pttrn n nnot smnt proprly t t n o ln o t sor. T xpt smntton ns t ntrton o nw onpts su s rrl smntton. 5.. Exmpls Tnks to t omplx moln o lstnn strts, t utomt nlyss systm s l to or lr pttrn srpton o smpl musl ps, orrsponn mostly to tully prv struturs. A B A 5... Bnnn o Mozrt Sont n A K 33 For nstn, t nlyss o t rst tm o Mozrt Sont n A K 33 (ur ) sows t s pttrn (), rpt n tonlly trnspos, n t 4-msur mn prs () rpt tw s ntnt n onsqunt. Howvr, slt rytm trnsormton t t n o t rst ourrn, w our systm nnot strt or t momnt, os not llow sovry o t wol prs (0 ). Is lso sown t sussv rptton o smpl rytm turn n t-not n ourtnot, ln to yl pttrn (). T ruton sows t nnn o nw ps o t yl, ut os not nssrly orrspon to prv smnttons, s xpln n prvous ston. A tl srpton o t yl pttrn sovr y t systm s n prsnt n Fur 9. Intrstnly nou, t sm lortm s l to sovr 4-msurs lon prss s wll s smpl sussons o snn () n rsn ( n ) onjunt lns. B B B Fur. Anlyss o Gsslrl (sltly smpl) []. Ntr ts lvls o prson or o prptv rlvn v n v or. Otrs pttrn sovry systms woul n nlu numrous st o runnt pttrns su s suxs or runnt xtnsons. Ts sows t nssty o mnsms o ptv runny ltrn su s tos propos n ts ppr. Howvr t nlyss rmn snntly rstrt, s numrous spts o musl xprsson v not n

8 tkn nto ount yt. 5.. Alortm Complxty T lortm omplxty my xprss lon two mnsons. Frst, trou t omplxty o sovr struturs: prolrton o runnt pttrns, or nstn, woul l to omntorl xploson, sn nw strutur ns propr prosss ssssn tr ntrrltonsps wt t otr struturs, n nrrn tr possl xtnsons. Hn mxmlly ompt srpton nsurs n t sm tm t lrty n rlvn o t rsults n t lmtton o omntorl xploson. Complxty my onsr lso wt rspt to t tnl mplmntton o t moln. Our propos lortms rmn yt n rt vrson n r or t tm n only prtlly optmz. Yt t moln s n onv wt t onstnt ojtv to mnmz omputtonl osts. Hn t ntton o smlr srptons s s on s tls, s xpln n prrp.3., w nsur optml tm omplxty. 6. CURRENT RESEARCHES Tnks to ts prptv mmry, t mol ors promsn rsults. Yt vors n to ontroll, n lr sop o musl xprsson su s polypony s not n tkn nto ount yt. 6.. Aton o Smntton Prnpls T struturs urrntly oun r s solly on pttrn rpttons. Soul notr mnsm oun on t mrn o nots los n tm or pt omn, n, rvrsly, on smntton twn stnt nots, ollown Gstlt ruls o proxmty n smlrty [8] []. Altou ts rul plys snnt rol n t prpton o lr-sl musl struturs, tr s no ommon rmnt on ts pplton to tl strutur, us t ly pns on t sujtv o o musl prmtrs us or t smnttons [3]. W propos to nvstt t ompttv/ollortv ntrrltons twn t two ruls o pttrn sovry n Gstlt smntton. For nstn, pttrn rptton my msk u to n mportnt tmporl p wtn on o t ourrns. 6.. Dtton o Musl Trnsormtons Our mol s l to tt not only xt rpttons o pttrns, ut lso prtl rpttons lon prtulr musl mnsons, ln to t sovry o ntrstn struturs. Yt otr spts o musl trnsormtons soul onsr too, su s t lol nsrton or lton o nots. Solutons v n propos [0], s on ynm prormmn n t stn, llown optml lnmnt twn smlr nots o ourrn. W xpt our mtooloy to or nw solutons to ts prolms, un on omplx moln o tl ontv strts From Monoy to Polypony Our ppro s lmt to t tton o rpt mono pttrns (.. squns o sussv nots) wtn mono musl p. Mus, n nrl, s polypon: t n ontn smultnous nots ormn ors, n prtulr, n smultnous mono lns ormn rnt vos. Rsrs v n rr out n ts omn [9], ous on t sovry o rpt xt pttrns lon rnt pr-sp mnsons. W r urrntly vlopn ruls o utomt sovry o mlo lns ns polypon sts o nots (or strm srton), s on ontv ursts. Our stuy wll tn ous on t ntrtons twn pttrn sovry n strm srton. W wll tn xtn t sop y onsrn pttrn o ors, w wll n ormlzn o nrl onpt o ntrvl twn sussv ors Appltons to Musl Dtss T utomt sovry o rpt pttrns n rtly ppl to utomt nxn o musl ontnt n symol mus tss. Ts ppro my nrlz ltr to uo tss, on roust n nrl tools or utomt trnsrpton o musl soun nto symol sors wll vll. A nw kn o smlrty stn twn musl ps my n, s on ts pttrn srptons, orn nw wys o rowsn ns mus ts usn pttrn-s smlrty stn. 7. REFERENCES [] G. Assy, C. Ru, M. Lurson, C. Aon, n O. Dlru. Computr ssst omposton t rm: From ptwork to opnmus. Computr Mus Journl, 3(3):59 7, 999. [] E. Cmouropoulos. Towrs Gnrl Computtonl Tory o Musl Strutur. PD tss, Unvrsty o Enur, 998. [3] I. Dlè. Groupn ontons n lstnn to mus: An ppro to lrl n jknos roupn prrn ruls. Mus Prpton, 4(4):35 350, 987. [4] T. Erol n P. Tovnn. Mr n mtl: T m toolox. In Prons o t 004 Intrntonl Conrn on Mus Inormton Rtrvl, 004. [5] B. Gntr n R. Wll. Forml Conpt Anlyss: Mtmtl Fountons. Sprnr- Vrl, 999.

9 [6] O. Lrtllot. A mult-prmtr n runny-ltrn ppro to pttrn ntton. In Prons o t 004 Intrntonl Conrn on Mus Inormton Rtrvl, 004. [7] O. Lrtllot. An ptv mult-prmtr n runny-ltrn ppro or motv pttrn sovry. In Prons o t 004 Intrntonl Conrn on Soun n Mus Computn, 004. [8] F. Lrl n R. Jkno. A Gnrtv Tory o Tonl Mus. T M.I.T. Prss, 983. [9] D. Mrt, K. Lmström, n G.A. Wns. Alortms or sovrn rpt pttrns n multmnsonl rprsnttons o polypon mus. Journl o Nw Mus Rsr, 3(4):3 345, 00. [0] P.-Y. Rolln. Dsovrn pttrns n musl squns. Journl o Nw Mus Rsr, 8: , 999. [] N. Ruwt. Mtos o nlyss n musoloy. Mus Anlyss, 6(-):4 39, 987. [] M. Zk. Ent lortms or mnn los tmsts n tr ltt strutur. IEEE Trnstons on Knowl n Dt Ennrn, 7(4):46 478, 005.