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Version: 1.3
31. August 2001
Modeling Global Musical Structures
Thomas Noll
Abstract
The 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 Coherence
Collective Musical Performance is a considerably diffcult research sub-
ject and the considerations of this article may only partially contribute to
some aspects of its investigation. This rst paragraph is of preliminary
nature and intents to motivate the remainder of the article for the work-
shop subject.

What enables a group of musicians to perform a complex collective action,
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 have been developed in the context of speech act theory (cf. [11]). Musical intentions deal with an entire musical world, and the explanation of intentional coherence should be hidden in music itself, i.e. in the structure of musical sign complexes.
 

2 Musical Syntactics
It 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 [12]). 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-
fronted with fundamental problems within syntactics. There is no reason to assume a stable interplay of a grammar with a dictionary, like in natural languages. According to the Saussurean dichotomy langue / parole music is essentially parole (c.f. [4]: 10). For late romantic composers it was a challenge, to produce new and new sequences of harmonies. Max Reger said in his classrooms: "Any chord can follow another chord". In an ahistoric provocative statement one could even generalize: "Any musical sign can be in a musically meaningful relation to another one". But what is the subject of musical syntactics then? Is it just combinatorics?

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 syntactics consists in detecting possible motivations between possible musical signs, and, furthermore, in the study of the interplay of several motivations at once. Jean Molino and, following him, Jean-Jacques Nattiez postulated a neutral level of musical analysis, in addition to and as a basis for the poietic or aesthetic ones. (cf. [7]). Guerino Mazzola's investigations in Mathematical Music Theory can be considered as an attempt to develop a working theory of musical analysis on the neutral level (cf. [3], [4], [5]). We refer to Roland Posners threefold subdivision of syntactics. Syntactics1 investigates formal aspects of signs, Syntactics2 investigates relations between signs and Syntactics3 investigates principles, according to which sign complexes are built (cf. [10]).
 

2.1 Syntactics1: Ambient Spaces
Whatever 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. [5]) 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.
In other words, each form not only serves the bookkeeping of a certain type of denotators, but it o ers an inherent motivation principle for the establishment of mutual relations between these denotators.
 
2.2 Syntactics2: Paradigmatics and Syntagmatics
Paradigmatics 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.
  • Class Membership: Signs are classi ed according to an equivalence relation. If the signs in question are described as objects of a matematical category, one has the isomorphy classes of this category (chord classes, motiv classes,..) at ones disposal. In the latter case, however, the existence of an isomorphism already provides a modality to actively reach other signs of the same class, which is an instance of the 3rd type of paradigmatic relations: transformation. Anothertype of sign classes are bres of functions (de ned on a repertoire of signs) and as a poor special case one obtains extensions of predicates as bres of their characteristic functions.
  • Neighbourhood and Distance: - Signs are described in terms of topological or metric spaces. In Mathematical Music Theory, there two approaches dealing with motivic topologies and harmonic topology. In both cases, musical similarity is modeled in terms of neighbourhoods (c.f. [9], [5]).
  • Transformation: - Signs are connected through a suitable transformation. Appart from geometric transformations there is the large domain of rewriting and transformational grammers, where such paradigmatic relations are studied.
Syntagmatic relations are studied in various theories of musical form. Many approaches are concerned with segmental analysis, which is based on the study of sequences of disjoint timespans. A much more tolerant concept of musical syntagm is mathematically modeled in terms of a carrier set of a musical parameter space together with a covering family of local patches. To study such a syntagm means to study the mutual intersection pattern of all these patches.1 The main task of musical analysis with regard to Syntactics2 is the study of the interplay of syntagmatic and paradigmatic relations. The projection of the paradigmatic axis to the syntagmatic one is what Roman Jacobson calls the poetic function. Paradigmatic relations of signs are highlighted through suitable arrangement of their mutual positions in a syntagm. Analysis at the neutral level studies the variety of instances of poetic function and aims at relating them to one another.

1 Mazzola speaks of an atlas of local charts or local compositions and defnes global compositions even in a situation where the carrier set is not necessarily embedded into the ambient space.
 

2.3 Syntactics3: Whole versus Global
Syntactics3 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 being global. This term was introduced into geometry in order to refer to objects that are achieved by "gluing together" local ones. According to Mazzola's defnition of a global composition, a musical piece is being modeled by a patchwork of local compositions.
 

3 Analytical Strategies
Musical 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 variety 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 Presuppositions
Whenever 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 
(cf. [2]).
 
3.2 Paradigmatic Presuppositions
As 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 Analysis
The 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
the regulative role of the metric hierarchies, inner metric analysis makes a paradigmatic presupposition: The onsets of the piece are a patchwork of local meters, i.e. artithmetic sequences of onsets. Instead of the outer syntagmatic role of a single onset within the hierarchy, one studies its inner metric context, i.e., the set of all local meters, passing through it. The resulting syntagms are often very complex. Through quantifcation one obtains inner metric weights, that suit for analysis as well as for experimental performance. Anja Fleischer introduced the notion of metrical coherence in order to characterize regularities in the inner metric weight that correspond to the outer metrical structure. She was able to show that many phenomena discussed in studies of metric chow traces in the inner metric weights (see also her contribution to these proceedings).
 
3.4 Melodic and Rhythmic Patch Analysis
In 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. [8]) proposes several quantitative measures that he uses for explorative studies on choral melodies and children songs. The experimental
work with larger musical pieces is a topic of recent research.
 
4 Conclusion
Are 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. 
 
References
[1] Fleischer, A., Mazzola, G., Noll, Th: Computergest? utzte
Musiktheorie, Musiktheorie, 4(2000), 314-325 .

[2] Lerdahl F., Jackendoff R. (1983): A Generative Theory of
Tonal Music, MIT Press, Cambridge.

[3] Mazzola, G. (1985): Gruppen und Kategorien in der Musik, Hel-
dermann, Berlin.

[4] Mazzola, G. (1990): Geometrie der T?one, Birkh? auser, Basel.

[5] Mazzola, G. (2001): The Topos of Music, Birkh? auser, Basel (to
appear).

[6] Monelle, R.: Lingusitics and Semiotics of Music, Harwood, Chur
1992.

[7] Nattiez, J.J.: Fondements d'une Semiologie de la Musique, Paris
1975.

[8] Nestke, A.: Paradigmatic Motivic Analysis, Electronic Bulletin of
the Mexican Mathematical Society, Mexico City 2001 (to appear).

[9] Noll, Th. (2001): Geometry of Chords, Electronic Bulletin of the
Mexican Mathematical Society, Mexico City (to appear).

[10] Posner, R.: Syntactics, Encyclopedic Dictionary of Semiotics:
Mouton de Gruyter, Berlin and New York (2) 1042-1061.

[11] Searle J. R. (1983). Intentionality: An Essay in the Philosophy of
Mind. Cambridge University Press, New York.

[12] Tarasti, E. (1994): A Theory of Musical Semiotics, Indiana Uni-
versity Press, Bloomington.