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Modeling Global Musical StructuresThomas Noll
AbstractThe article compares different strategies in the investigation of musical syntactic structure and argues in favour of an experimental comparison on a neutral level of analysis. Special attention is paid to syntagmatic and paradigmatic presuppositions. The rst paragraph intents to motivate the remainder of the article for the workshop subject.
1 Intentional CoherenceCollective Musical Performance is a considerably diffcult research sub-
What enables a group of musicians to perform a complex collective
like a concert? Such a collective action is composed of many autonomous
local actions of the individual agents. These local actions, consist of individual intentions, doings and experiences.
Local action control is highly based on the evaluation of individual experience with regard to the individual intentions. The individual doings are manipulated by intentions as well. What distinguishes the participation in a global action from an isolated individual action is the scope of intentions and experiences. While every agent's doings are still performed by himself, his intentions and experiences are directed towards a larger range. Each individual success, hence largely depends on the doings of other agents as well. In the extreme case of a pure-listener-agent satisfaction depends on the interplay of intentions and experiences solely.
What constitutes a global action? On the one hand, it would be very naive to postulate a coincidence of all individual intentions and experiences. But, on the other hand, what does it mean that local actions contribute to a global action? How to evaluate the success of a global action?
Our reflections suggest the following characterization: While each agent controls his doings on the basis of his individual intentions and experiences, at the same time he contributes to the satisfaction of other agent's actions, although the conditions for these satisfactions are only given in terms of the individual intentions and experiences of the others. In order to refer to a patchwork of individual intentions, that allow for simultaneous satisfaction through individual local doings, we suggest the term intentional coherence. The establishment of intentional coherence in music can hardly be achieved through metamusical decisions and argumentative discourses. They are collective discoveries made through experimental practice und are typically conveyed to others by imitation. Interestingly, individual musicians in "illiterate" music cultures sometimes can not articulate their own "part" in the absence of the other co-musicians. But also in western music culture, where inspired composers create complicated musical scores, intentional coherence has to be achieved in each single performance through sensible collective interpretation.
These considerations ask for a theory of musical intentionality as a
prerequisit of a theory of musical action. It is debatable, whether one
can take ones bearings directly from theories of intentionality that
been developed in the context of speech act theory (cf. ). Musical
intentions deal with an entire musical world, and the explanation of
coherence should be hidden in music itself, i.e. in the structure of musical
2 Musical SyntacticsIt is an open question, whether music can be properly grasped by a suit able semiotic approach. At least there is no straightforward way to apply semiotic concepts to the study of musical structures as sign complexes. However, the question is open in two directions in so far as both disciplines - Music Theory and Semiotics - will perhaps further develop and may inspire each other from time to time. At rst sight, the main problem with music semiotics seems to be semantics: What is music about? Within the mainstream music semiotics there are several interesting attempts towards extramusical semantics (e.g. see ). Nevertheless, we will leave them aside here. Instead, we are concerned with the study of pure musical syntactics. What legitimates the sign-concept is the possibility of intramusical signi cation through Jacobsons poetic function (see below).But without access to (extramusical) signi eds one is automatically con-
Our syntactic investigations are based on an additional principle, that can be stated as follows: Syntactic relations between musical signs are typically motivated. This principle does by no means exclude extramusical semantics, but it does exclude it as a general necessary prerequisite for syntactic investigations. The absence of motivation is arbitrariness. In natural language, many words and relations between words carry partially "frozen" motivations, that are studied in etymology. But, typically, it is not necessary to activate these motivations in order to understand the words. Instead, natural language expressions are loaded with motivation on the semantic level.
According to above motivation principle, the main task of musical
consists in detecting possible motivations between possible musical
and, furthermore, in the study of the interplay of several motivations
at once. Jean Molino and, following him, Jean-Jacques Nattiez
a neutral level of musical analysis, in addition to and as a
for the poietic or aesthetic ones. (cf. ). Guerino Mazzola's
in Mathematical Music Theory can be considered as an attempt to
develop a working theory of musical analysis on the neutral level (cf.
, , ). We refer to Roland Posners threefold subdivision of
Syntactics1 investigates formal aspects of signs, Syntactics2
investigates relations between signs and Syntactics3
principles, according to which sign complexes are built (cf. ).
2.1 Syntactics1: Ambient SpacesWhatever one might consider a musical sign, it not interesting as an isolated entity, but as an element of a sign systems, where it participates in various relations to other signs. According to the motivation principle we are looking for systems of musical signs with a "natural" and rich intensional structure. Geometric spaces are of this kind. Geometric objects are studied on the basis of a inherent repertoire of geometric transformations. Basic musical parameters like those of pitch or onset can be modeled using geometric spaces. The transposition of a chord is a typical instance of a geometrically motivated relation. Mazzola's general system of musical denotators and forms (cf. ) follows the same idea in a more general context: musical denotators could be considered as mathematical models of musical signs. They are "points" within suitable forms, (i.e. generalized ambient spaces) and local compositions are sets of such denotators and inherit their mutual relations from natural transformations between their ambient spaces.
2.2 Syntactics2: Paradigmatics and SyntagmaticsParadigmatics are concerned with the investigation of relations between signs, disregarding the contexts where these signs occur, while Syntagmatics are concerned with the investigation of the relative locations of signs as parts of sign complexes, disregarding the other relationships between them. There are at least three basic types of paradigmatic relations that play a fundamental role in music.
1 Mazzola speaks of an atlas of local charts or local
compositions and defnes global compositions even in a
where the carrier set is not necessarily embedded into the ambient
2.3 Syntactics3: Whole versus GlobalSyntactics3 deals with the investigation of principles according to which complex signs or sign complexes are built. The confrontation of these two terms is not just a play on words. It refers to di erent views on the nature of musical pieces.
The term "complex sign" puts the main emphasis on the aspect of wholeness. The whole composition is the starting point. According to the principles of Schenkerian analysis, a musical composition is to be understood much like an organism who unfolds from archetypical background structures along a repertoire of transformations into the foreground structure of the given score.
The term "sign complex" puts the main emphasis on the aspect of
global. This term was introduced into geometry in order to refer to
that are achieved by "gluing together" local ones. According to
defnition of a global composition, a musical piece is
modeled by a patchwork of local compositions.
3 Analytical StrategiesMusical analysis and especially the search for occurences of poetic function is essentially of hermeneutic nature. Several local perspectices, like metric, rhythmic, motivic, contrapuntal, harmonic and others, are taken in order to discover various kinds of correspondence between them. Often, there is even no a-priori access to single local perspectives without simultaneous access to others. The experienced hermeneutician discovers such dependencies and his strategy might be characterized as a search for analytical coherence.
However, it is a theoretically hard problem to control the rich
of possible analyses on the neutral level. There are two types of more
restrictive analyses, which are less open towards the expected results:
3.1 Syntagmatic PresuppositionsWhenever an analysis is based on a prede ned domain of possible syntagmatic structures, one may speak of syntagmatic presuppositions. Such presuppositions impose practical restrictions on the domain of possible signs to be considered. As a typical approach of this kind, we mention Lerdahl's and Jackendo 's "Generative Theory of Tonal Music" (GTTM), where syntagmatic presuppositions are presented through well-formedness rules. The analytical task can be viewed as an optimization of paradigmatic structure among the well-formed syntagms, where the analyst is guided through preference rules
3.2 Paradigmatic PresuppositionsAs an analytical "counterpoint" to syntagmatic presuppositions, we discuss examples of paradigmatic presuppositions. In these approaches the analyst is, in rst instance, interested in a restricted repertoire of signs and their paradigmatic relations. The analytical task consists in the investigation of resulting syntagms. In the following two paragraphs we sketch two approaches that have been developed in the context of Mathematical Music Theory.
3.3 Inner Metric AnalysisThe term inner metric analysis is opposed to outer metric analysis. The latter is a typical approach with syntagmatic presuppositions. It studies the notes in a piece in terms of metrical hierarchies (e.g., in GTTM). We call it outer analysis, because the actual onsets of notes are embedded into an outer regular structure of musical time. Inner metric analysis, instead, looks for metric regularities inside the given piece. It looks upon a composition as if it were written for an ensemble of metronomes. They vary in period and phase and are switched on and of, whenever they can click along the onsets of the notes of the piece. This analysis is especially interesting if one does not take into account time signature and barlines. While outer analysis makes syntagmatic presuppositions about
3.4 Melodic and Rhythmic Patch AnalysisIn metric analysis, a very narrow set of paradigms is considered: Local meters can be classi ed according to their period, phase, and length, and all local meters within the piece are taken into account. Melodic (and rhythmic) analysis is concerned with much wider repertoire of possible signs. The motivating presupposition of melodic patch analysis is paradigmatic repetition. What makes a set of notes interesting is the fact that it reoccurs as the image under a suitable symmetry (translation in time, transposition, augmention in time, inversion, or a composition of those). Therefore, these signs are called symmetry patches. This analysis can be applied to any melody in order to study its inner symmetry patches. It can also be applied for the comparison of two or more melodies. After detection of all symmetry patches with respect to a paradigmatic symmetry group, the analysis focusses on the combinatorial structure of these patches. The syntagms are described in terms of simplicial complexes and can be studied with methods of combinatorial topology. Andreas Nestke (cf. ) proposes several quantitative measures that he uses for explorative studies on choral melodies and children songs. The experimental
4 ConclusionAre there convincing arguments in favour of a higher priority of the syntagmatic axis over the paradigmatic in the realization of a poetic function? This question might be answered on the basis of an experimental hermeneutic approach comparing several syntagmatic and paradigmatic presuppositions on the neutral level. One way to make this comparison of restrictive analytical perspectives explicit is the study of quantitative weights measuring analytical coherence, like in the studies of metric.
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