Page  00000001 ETNA - a New Advanced Graphical Tree Representation of Music Wolfgang Chico-TSpfer Martinstr.72 D-64285 Darmstadt e-mail: Abstract This paper presents ETNA, a new advanced graphical tree representation of music seen from a composer's point of view of music as a process of elaboration. It emphasizes and encloses a proper formal foundation and a one-to-one relation to a practical linear symbolic notation. ETNA is inspired by the Generative Theory of Tonal Music (GTTM), but abstract enough to match other transformational music theories. First applied in AVA, a semi-automated two-phase composition system, ETNA demonstrated a high degree of practicability and straight-forward implementability. ETNA is predestined to be reused beyond AVA, for instance in computer-assisted music analysis and/or synthesis. 1 Why yet another representation? ETNA1 delivers a representation with features that I found lacking in other reduction/elaborationorientated music representations: 1. Represent "music as a result of elaboration": The representation should show the music it represents from a composer's view. It should clearly display the various parts as they are elaborated into more refined sub parts which in turn are elaborated further. Such a "constructive" view seems best suited to express the organic character of a composition, making its offals available for inspection, analysis or/and further (re-)composition. 2. Provide a solid formal foundation: The representation should support scientific methodology. A formalization is needed to obtain precision, clarity and unambiguity. 3. Include a linear notation for the graphical representation: The representation should be easily transferrable into a notation (and vice-versa) that allows algorithmic processing which is otherwise difficult if only a graphical representation is available. 4. Stress the visualization of elaboration/reduction and of different elaboration levels: The representation should clearly denote the pairs of music connected by a relation of elaboration/reduction i.e. it should always be clearly visible which part is an elaboration of another part. Moreover, its topology should show all elements ordered according to their degree of elaboration. 5. Allow superposition: The representation should harmonize with a superposed representation to point out certain elaborations while keeping representation clarity. This is achieved by any tie-like notation (e.g. the ties in CMN2) such as the representation of so-called groundlines [Chico-TSpfer; 1998, Chico-TSpfer; 2001]. They illustrate how visually clear superposing can be as a consequence of carefully designing a representation to accomodate a superposition. For space reasons, this is not further detailed in this paper. Consequently, the approach is to properly define a number of required terms which is done mathematically (point 1). This results in the formulation of a notation as required in point 2. Finally, it is transformed into a graphical counterpart that complies with points 3-5. 1Elaboration-orientated Tree Notation for AVA 2Conventional Music Notation

Page  00000002 2 What needs to be defined? First of all we need some basic conventions (for a complete discussion see [Hoos; 1994]): * let TONE be the set of tone description terms i.e. of all notes and pauses3 * let a tone t e TONE be a 4-tuple t = (t ~,6tI, t, t) where 7t is the pitch, 6t the duration, Lt the intensity and yt the applied instrument; they may be extracted by pitch(t): = 7t, dur(t) 6t, intens(t):= -t, instr(t):= -yt The above definitions are needed to define what pitch events and sequences are: Definition 2.1 (Pitch Event and Sequence) A pitch event is a term P E POLY. The set POLY is recursively defined as follows: 1. Vp eTONE: pE POLY 2. P1,p2 e POLY > poly(pl,p2) E POLY A sequence is a term S E SEQ. SEQ is recursively defined: 1. VpE POLY: pe SEQ 2. P1,P2 e SEQ > seq(p1,p2) e SEQ The function poly yields a "vertical tone layer" i.e. music exlusively made up of tones that start to sound at the same time; seq yields a sequence i.e. music exclusively made up of tones or vertical tone layers that sound one after the other. Both functions may be formally defined [Hoos; 1994]. Further conventions and short definition extensions are convenient to work comfortably with a pitch event p = poly(tl, poly(t2,.., poly(t,_i1, t)..)) and a sequence s = seq(pi, seq(p2,.., seq(p,_1, p)..)): * we need to work with the duration of p and s: dur(p):= max{dur(ti),.., dur(tn)} and dur(s) dur(pi) + dvr(p2) +.. + d2r(pn) * we need to refer to the pitches, intensities and instruments of p and s: H(p):= {pitch(ti),pitch(t2),.., pitch(tn)}, II(s):= (p) U II(p2)... U II(p) I(p):= {irntens(ti), intens(t2),.., interns(t,)}, I(s):= I(pi) U I(p2)... U I(p) F(p):= {instr(ti), irnstr(t2),.., instr(tn)}, F(s):= F(p) U F(p2)... U F(p) Now we can properly define a reduction: Definition 2.2 (Reduction) Let P:- {pi,..,pn C POLY be the set of pitch events of a music piece m4.A reduction is a term R E RED; RED is recursively defined: 1. VpEP:pE RED 2. x e RED > red(x) E RED; xi,x2 e RED > seq(xi, x2) E RED Note that RED is very flexible as it is practically independent of any particular music theory: One of the few assumptions is the implicit notion that a reduction is a reduced version of a sequence or a pitch event x (the elaboration). Musically speaking, a drawback may seem that a number of terms in RED do not really represent reductions. However, RED is the first non-trivial term set to specify closer what a reduction is i.e. we have a term set that clearly describes the structure of a reduction. Only one term in RED is expected to be a musical reduction according to a rsp. music theory. Observe also that RED includes sequences which may seem strange since seq has an extending effect. There are two reasons for this: First, there are cases where no true reduction seems possible or advisable. Typically, this may occur when cadences are reduced, see cadential retention [Jackendoff, Lerdahl; 1983, pp.155-158]. Secondly, seq is needed when an elaboration is generated from a sequence as a whole. To keep above mentioned flexibility we define red partially by deferring theory-sensitive and context-sensitive criteria to a function /m: 2POLY, POLY that decides which reduction to prefer (the context m is the music piece see Def.2.2): 3a tone may be a pause, just like a number may be zero 4note: m is not defined since only its elements need to be defined 2

Page  00000003 x: x ePOLY red() m(O): x = seq(xi,x2),xi,x2 e SEQ A:= {pp Ep POLY A nI(p) C fI(x) A I(p) C I(x) A r(p) C F(x) A n(p) n I(xi) 0 A dur(p) E {dur(x), dur(xi)}} Note that a definition of /)m could be an outcome of a full formalization of the GTTM or another music theory; this is beyond the scope of this paper. Yet we have clearly defined a set of reductions 0x for a sequence x. It is a result of observations that characterize our reduction: * red yields a pitch event: the reduction always yields a pitch event as a result i.e. it is always truly reductive because it does not extend anything, which is conceptually clean * red is based on the importance of pitch events: red(seq(x, x2)) implies x, to be more important than x2 which allows to determine pivotal elements of the reduction in xl, thus at least one tone pitch must come from xl; red works according to the notion that the musical surface contains elaborated versions of elements that make up the reductions * red either keeps the same duration or that of the important part: Normally, the duration remains unchanged; however, because there are exceptions such as upbeats which durationally disappear in a reduction, it may also be that the rsp. reduction has only a duration of dur(xi) (preferring the duration of x, to that of the less important x2) * semper idem sed non eodem modo: all tone pitches, instruments and intensities come from x, the reduction does not add anything new; it selects elements from x and transforms a given elaboration to its former state We are ready now to develop our linear notation in two steps: 1. simplify Def.2.2: instead of 2.2.2 we could write x1, x2 RED seq(xx,x2), red(seq(xi,x2)) E RED. Because for x e POLY we have red(x) = x, this second version is equivalent (trivial proof) 2. derive an infix notated equivalent: we replace above second version, this time by its infix5 version x1,x2 e RED = xl - x2, xl - x2 (trivial equivalence proof) Still we need an adequate linear notation for pitch events themselves. Let us use a GuIDO6-like notation i.e. simplified so-called complex segments [Chico-T6pfer; 1998] since they are straight-forward and clear enough. For instance, {f/4 a c} stands for the F major chord with a quarter duration. Indications of intensity and instrumentation are omitted for simplicity. 3 Examples of linear and graphical ETNA representations How are graphical ETNA representations built and related to linear ones? The example7 (((c2 - c) - (g - g)) - ((a - a) <-- (g - g))) - (((f - f) <-- (e e)) - ((d (tr <-- d2/8.)) <-- (e2/16 <-- c2/2))) shows that we need to apply the reduction arrow -> in its counter direction to keep the information related to the time when a pitch event occurs. This also allows to read the term from left to right like CMN. Equivalence with Def.2.2 is kept since we can trivially extend RED to generate x2 - xl with x2 <- x1:= x1 -- x2; now we can define a graphical representation that complies with 1.3-1.5: A reduction xl -- x2 is made into a graphical view by bending the arrow twice so that it resembles a squared bracket whose ends are directly over x1, x2; the same is done for a sequence xl - x2 (without an arrow peak). Graphical ETNA representations are principally built by nesting such "squared bracket arrows" one over another w.r.t. Def.2.2 (e.g. Fig.2). So all linear ETNA representations can be expressed as graphical ones. Conversely, the latter can always be transformed into their linear versions if they are built as described. 5note that x2 -t X2 stands for the infix version of red(seq(xi, 2)) 6see [Hoos et al.; 1998] 7note that tr stands for Triller i.e. the tone is elaborated by a specific ornamentation 3

Page  00000004 X1 X2 Figure 1: A reduction x1 - +X2. The arrow peak always points to the elaboration With ETNA we can also establish a well-ordered structure orientated along piece-specific duration levels of the rsp. reductions. They can be derived from the piece's group structure8. Note that this is not necessary. But it achieves clarity by displaying reductions that do not have piece-specific durations on the same level as the next higher duration level. E.g. if reduction e/4 -> e/8. has a duration of 1/4 + 1/8 + 1/16, then it may be displayed on level 1/2. {f#0/4 el a} {g#0/8 el b} {a0/4 e c} {d0/8 fl b d}{e0/4 e1 a c#2} {e-1/8 dl e g b} {a-1/4 c#1 e a} Figure 2: A graphical ETNA of K.331 mm 7-8 where cadential retention yields {e1/8 dl eg b}- {a-1/4 cil ea} 4 Conclusion and Further Remarks ETNA, a powerful superposable tree notation to represent musical transformation, is formally founded and based on a well-defined linear notation. The ETNA Builder [Chico-T6pfer; 2001] makes its visualization easy and supports their export to other formats. ETNA is part of the AVA Project which aims at delivering an open component-orientated music system for various musician user groups. The AVA web site is planned to go online in the 3rd quarter of 2001. References [Chico-T6pfer; 1998] Wolfgang Chico-T6pfer. AVA: An Experimental, Grammar/Case-based Composition System to Variate Music Automatically Through the Generation of Scheme Series; ICMC 1998 Proceedings, pp.98-101. [Chico-T6pfer; 2001] Wolfgang Chico-T6pfer. ETNA Builder - Interactively Building Advanced Graphical Tree Representations of Music; SBCM 2001 Proceedings. VIII Brazilian Symposium on Computer Music. [Hoos; 1994] Holger H. Hoos. Strukturorientierte Beschreibung und Erzeugung von Musik. Student research project, June 1994. Darmstadt University of Technology. [Hoos et al.; 1998] Holger H. Hoos et al. The GUIDO Notation Format - a Novel Approach for Adequately Representing Score-Level Music; ICMC 1998 Proceedings, pp.451-454. [Jackendoff, Lerdahl; 1983] Ray Jackendoff, Fred Lerdahl. A Generative Theory of Tonal Music. MIT Press Series on Cognitive Theory and Mental Representation. J.Bresnan, L.Gleitman, S.J.Keyser editors. MIT Press, 1983. 8see [Jackendoff, Lerdahl; 1983]; note that a graphical ETNA fulfills all the properties demanded there by definition (if time-dependency is respected as described). Of course, all pitch events of the piece must be considered. 4