Page  207 ï~~A Model for Musical Rhythm.Jc/f Biilmre s < edu> T h1 NI1' 1i ed i a Laboratory Massaclhusetts Illstitut.' of l'.'echnology.ltl)li(lge, NIA 021:19 ABSTRACT At least four elements characterize mtusica.! rlitytl llll 1) Ietric content, a quantized attribute; 2) ametric phrases, those unrelated to any tactus; 3) tempo variatiol, change in the number of beats per second; anLd I) event shifting, time deviations from a steady beat. This deconstruction provides a means to represent timing features front nsical styles. Both tempo variation and event shift information can operate at (ifferent. lei,s of tie I n llhl(' Iierarcly folllld inI mus)Iic TIis representation will facilitate the analysis and prodI( l't oll of illsic;dul sl e tI ne ex t.that. it is rhytl iicat ly expressed. 1. Introduction This paper introduces a mo(lel fot re esent itng the expressive tinting in insica.l rhythm. The motivation for the model is to facilitate tie accura.te pitedlictioli atid p1(ro1lctioll of hulnai-like rhythmic phrases by computer. In the lllodel, rhythmic style is a. cotiSpiCtots l)a.ra.nteter titat is, analyses of rhythmic plhrases from differing nltusica.l styles should prollice different ittoclel para.tmeters. 2. Rhythm At least four elenits ciaracterize nitsica il iyt. lii " n etric contentit.,, a.ntetric characteristics, tempo variation, and eweitt shifting. Each of these elettieit~s ('alb) modleled separately. The metric content, of musical rtythl hni is the l)ercel)t.ual relation of successive rhythmic events to an evenly-spaced title grid. The tidte gri(l (loterliies the "bea.t; within which all musical events are heard. Our perceptiot of rltythtn, sontetities relerre(l to as.,bjec-I-'iv,( /ruhyhihnuzzuion(IFraisse, 1983), is a psyclological linkitg of se(ltential event stinlii lot the effect to occtr, the evett inter-onset times must be bounded in tine, rantging fromr about 120 tttsec to about. 2 sec. 'raditiotla.l Western musical notation is one example of a mo(lel which represents the Itettic" ollt.t. Other exall)les include the output of a music quantizer and representations utsed by cotililt tI(I-nlisic se(linetcet-s. Ametric models of rhythm replresent those lhrases t lat (10 oot Ihave an associated beat. A musician produces these p) without t si ug a collstatit beat as a reference for not.e ilacement. There are two musical styles where these )11rases o(cur: I ) (' ert.i" ic has no Plerceptible beat and rhythmic phrases are completely unanchored front ally fixed t.illt, grid. I l)ue t.o thle tiature of thlese ph~rases, no beat-based representation is suitable. 2) \V\Ileti the(re is a clearly (Illls( beat., sontlet.itltes a performner will only coincide with that beat otic:e every )ltrase (e.g., t1lie perl'orlller will 'oiicidle 'vitli a. i)ea t every eight bars, but in the interim, the perform~er"S notes occurt ittleeli (letltly ). Inl this c'as.e, th le tttodlel Ilust accoutr for the very low frequency beats def'ined by di~e occasiotta.l ('oitlc'i(lellce(. I loivever, we will tiot consider ametric rhythmic models in this paper since mnany tvyl)es of 11111Sic" ('ati be(,l)tesel.e(l usin~g th~e thrtee othier illodels. Tempo variatioti models of rhyt~li lave( rece(,ll lvbeet gi v'et son ic attention. Modlels in th~is category are usually functions tha.t nap) titite (I t r'atiotts io (lelotie(l ittie (lt rations or tltat. tuapI a bea.t tiutiber (essentially a tite posit ion ) tIo a b~eat (t;lit atot.I \' a pp~rop nrat ely v'a ryi g tli~e tent 1)0 of a. musical lpiece 207

Page  208 ï~~-- that is, by varying the ut iiiben of grid ilarks that, pass byI ill a. given a.niount of tinle, or, by varying the beat duration - we create what is )en('eived to contain more expression. Functions describing tempo variation have been discovered thfat corresp~ond closely to real musical )erfoilinainces. Some examples are time maps (Ja.ffe, 198,5), time deforinationis (Anderson Y lKuivila, 1989), selt.ic curves (Clines, 1977), and force model constructs (Feldiman, Epsteinii, k iclards, res). One important difference between Western classical musiC and ethnic music (such a.s African) or modern music (such as Africa.n-American jazz) is that expressive timing in the former can often be described using only tempo variation functiols. To describe exJ)ressive timing in the latter (and to better describe timing in classical music) requires the developument of new l.iili ig representations called event shift models. Event shift models of musical rhytlili are, like tell)po variation models, functions of a time position (or metric position); but, in addition, hley' are fu nctioiis of insla taileours tem1)O. Iliese functions, when given a metric position and an insta.lntaleous teiil)O, provide a duration. That (I is used to shift, ahead or behind in time, a (musical event occurring at that iiietr iic position. Tli us, such a fuction can be used to model Africani and jazz (or swiig) mllusic \vlere )erformers deviate in tine fron a rela.tively uniform beat, and canIn odel (for exa.inple) pialo n tiisi in tle Iolllanic style where ii(liv (Iiia.l voices are slifted. It is commonly believed that iiiusic with.i allreciabl)e (leviations froi i a stable bea.t (event slhifting music), such a.s jazz, l a.s an iin)phcit fixed ti inn( grid B i thow is a fixed 1tne gri d implicitly defined, especially in music whlere al lper forens (levia.e froi ite gri. Il.IIJazz. people ale saidt to play "tbeliind tle beat," "right on the beat," or "in frond (Or oii torl) of tie e)eat" whreiiot,e ('veilt.s 0CCi reslectively la.ter than, right o1, or eailier tliani lite gril n arks. li (;-ses \Vie<ie 1 lie mn ajority of perfor ners play on the beat, the time grid is exp)licitly (lefi ned -- althbought e itlusk Ii igllt. be (onsilered "lifeless". Hiowever, in cases where the majority of perfolnres play b)ot hi bell iid and 11 fron of the beat, how (10 we know where the fixed time grid lies? If all l)erforners in ail e1se imile play behind the bea.t, wi do we not perceive the time grid a.s being shifted forward in tirne, and tulls perceive all performers a.s playing on the beat? Several reasons are possible: 1) Performers do not deviate by the sa.nie a.niount,. In ensembles, certain inst ruments, whose role is to define the back-beat, typically play more on the beat, than others instruments; for example, the in a. jazz enseml)le and a. sulpport (frut in an Affricani enselml)le tend to play more on the beat. 2)The amount a performer plays behind or ahead of lle bea.l is i ine-varying and can e rel)resented as a. function that maps a. metric position and a. t1eml) to a trinle (-ra.t.ioii that is used to adjust a, note event. For example, at a section beginning, a perfoi-li eilllighl play rigll 0) the bieat. andd then deviate appreciably at the section Imiddle. Event shift. funct.iois (-all easily model both of tlese situations, thus demonstrating the sufficiency of representing event shifting lii sic with models that. are based on a. fixed time grid. 3. Hierarchical Rhythmic Deconstruction Music, and its rhythm, can be hierarchically (lecolist.ructel based onthe metric content. One simple example of a metric hierarchy can be onta i ed froii t!le:32 bar jazz A A B3A form: the form repeats indefinitely. Each repetition is divided into foumr sections. eam-Ih section into eight. measures, each measure into four beats, and eacl beat into t.mi-ee l)ilses. 'l'liei-e ai-e many fom-ms tlhat can be disassembled in this way; the point is that each nay be (lecolsi mted according 1.o its oxvm imlicit hiemarchical structure. The arra.ngemeiit of tlme h~ierarch~y ima.v' chiamige ovei- t.iime. F'-or exam)))le, each beat might for a time be divided into four-, rather- th~an 1mmce, l)lhses t.o 1)1-0(1 mce a "'straigh~t cights" feel rather than a "swing" feel. Also, a jazz tune might alternativehy m~ove fi-omn a sti-uctured jazz form (e.g., AABA) to free form. During free form, the hie-am-chiy still coitrai ns the imieasu ie level on down, but, thicie is no~ need for higher levels. A time-varying hieram-chical strtuctume thmis vaiies (Ilie 1.0 1)01.1 ch aiges in thle musiciam) and changes in the lAnd perhaps even inmsical initent - emnoliomial 11101, el C. 208

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