Page  396 ï~~A COMPUTER-BASED EDITOR FOR LERDAHL AND JACKENDOFF'S RHYTHMIC STRUCTURES Bruno Degazio The Artificial Evolution Studio ( 192 Spadina Ave " suite 512 Â~ Toronto " Ontario " Canada Â~ MST 2C2 In A Generative Theory of Tonal Music, Lerdahl and Jackendoff (hereafter L+J) discuss two forms of rhythmic structure which they call metrical structure and grouping structure. Together these constitute a basis for an analytical understanding of rhythmic structure in music. A software based editor has been written which allows the 414 interactive exploration of these two types of hierarchical a 4, a, ii,0,, 4,,,,,, a,, rhythmic structures. It bears some relationship to simple time 2 based editors found in commercial software, but represents an 2.. extension of such software to a much wider range of rhythmic "'"-' ', *oooo*oooooo features 4* [ ll *VbI t N i t 00 O IN. OUteIs.l tw T 60Ih[: Figure 1- regular metrical structure based on hierarchical duple divisions The Metrical Structure Editor allows the user to specify up to eight hierarchical levels, each with strong-weak beat I- [ l Ish cld toN1..... pattern up to sixteen elements long. The pattern can also be 6,, -', "9,6/,/ ",, /,/4 3A 4 14 15/,, 4 1,/6 broken arbitrarily and restarted at any point, a feature common 2..': to the higher (i.e. greater than measure length) levels of,.:........... metrical structure. The software strictly enforces L+J's 2 Metrical Well Formedness Rules 1 and 2:,,2.,t,r-.,* woo....*I _ Figure 2 - complex metres created by multiple layers of S structured as 2+3 1. Every attack point must be associated with a beat at the smallest metrical level present. 2. Every beat at a given level must also be a beat at all smaller levels Metrical Well Formedness Rules 3 and 4 are implemented in a freer form: 3. At each metrical level, strong beats are spaced either two or three beats apart. 4. The tactus and immediately larger metrical levels must consist of beats equally spaced. The relaxation of these latter two rules allows for compound metrical structures such as are found in Bulgarian folk music, and for free metrical structures such as..................... recitative and for specification of metrical ",. structures impossible to notate conventionally,, such as poly-compound meters where both the 0e tactus and the measure are comprised of a recurring. Â~-,",.,' pattern of strong/weak beats (e.g 3+2+3)...;: _ -::- ":: -: - The Metrical Structure Editor is shown in figures 1 and 2 abovc. Figurc 1 shows a regular metrical "! structure based on aduple division at all levels. I I...,,.,.,..,,, twolevlsempoya fvebea patensruurd,,,lL6 Â~r l) - z.,.om-,tÂ~.,,r...| Figure 2 shows a more complex structure where u,, o+-, o.4EOO.Ull-, z. Lii~mi teIO CratUIUIP I*to.m ~I fiF A internally as 2+3. The user can specify a strong- _____" __" _________"'____"_______ weakpaternforeac ofeigh leelsof etrcal Figure 3 - metrical structures displayed superimpose~d on a sequence of notes structure. Two of these levels have a special status: the measure and the tactus, designated with the letters 'in' and 't' along the left edge of the editing screen. The tactus is the metrical level used, if possible, for the denominator portion of the time signatures, and usually moves along at approximately the rate of the Degazio 396 ICMC Proceedings 1996

Page  397 ï~~human heartbeat (about 60 to 140 beats per minute). The measure is the level used for the numerator portion of the time signatures, which are shown along the top of the editing screen. These levels have a special meaning in other portions of the software as well, notably in the Selection Filter, where choices can be made based on an event's measure or lactus characteristics. Levels of metrical structure can be displayed in the graphic event editor as a set of 'barlines' of varying weight (figure 3). The Grouping Structure Editor allows for the specification of three levels of grouping structure, arranged hierarchically according to L+J's Grouping Well-Formedness Rules: 1. Any contiguous series of events, and only contiguous events, can constitute a group. 2. A piece constitutes a group. 3. A group may contain smaller groups. 4. If a group G 1 contains a group G2, it must contain all of G2. Grouping structure is displayed along the bottom of the graphic editor screen (figure 4), using lines with curved ends which resemble L+J's notation. Six levels of grouping structure are allowed, with the current levels designated by a small letter 'G' along the left edge of the screen. A series of notes can be designated as a group by selecting them with the mouse and choosing 'Group Notes' from the Edit Menu. Notes already grouped can be selected with the 'Select Group' command. L+J's Grouping Well-Formedness Rules are not enforced at the time of selection; instead a 20I I o w 74~44 ~ 4 ~ 1 4f~240 ~5.e~up JC~a. O~r~oa 04P20rAl IL. ~ ~ ( * 32TITO CNMLD separate pass can be made using the 'Check Figure 4 - graphic editor with grouping structure displayed along bottom Group Structure..." command. Two other commands allow the automatic creation of group structures. "Make Repeating Groups..." (figure 5) creates regularly - repeating groups at the current level of grouping structure. This _ creEtREGLR__PRTN__GROUPS #+" t may be used, for example, to Start at: 12 (note number) y End at: 28 (note number) group measures or beats, if I, Auto-grouping uses: these are suitably regular. A G Group euer: 24 (ticks) 25 0 pitch more complex command is Group level: (1-6, 0 means 0 'volume Start at: "Auto Group..." (figure 6), current level) o- 0 duration 12 ]which uses a measurement group this leuel... 0 0- articulation End at: similar to that discussed by, cance, 99 0 pitch class 28 Jamcs Tcnncy and Larry,__..UP cancel...,,, ] Polansksi to autom atically aUTO-GROUP PFigure 5 - "Create Repeating Groups...' dialog create groups based on a 'Figure 6 ""Auto Group...' dialog selection of intrinsic musical parameters. These include: pitch, velocity, duration, articulation (i.e. onset delay) and pitch class. Only pitch weighting is currently implemented. Acknowledgements This research was carried out with the financial assistance of the Canada Council, Media Arts Section. The Artificial Evolution Studio wishes to thank Karl Mohr for his help in the preparation of this paper. References [Degazio, Bruno 1988] Context Sensitive Editing in the MIDIFORTH Computer Music System. Proceedings of the International Computer Music Conference: Cologne, 1989. [Degazio. Bruno 1993] New Software Composition Tools. Fourth Biennial Arts and Technology Symposium: New London. Connecticut. 1993. [Lerdhal, F, Jackendoff. R 1983] A Generative Theory of Tonal Music. MIT Press: Cambridge. 1983. Tenney.J and PolanskiL] Hierarchical Temporal Gestalt Perception in Music: A "Metric Space" Model. York University, Toronto. Ontario, 1.978 ICMC Proceedings 1996 397 Degazio