NOLL, Thomas - Global Music Theory

8/14/2019 NOLL, Thomas - Global Music Theory

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Global Music Theory

Thomas Noll

Research Group KIT-MaMuTh for Mathematical Music Theory

Institute for Telecommunication Systems, Technical University of Berlin

Sekr. FR 6-10, Franklinstr. 28/29, D-10587 Berlin

[email protected], www.mamuth.de

AbstractThe article investigates aspects of globalitywith respect to music the-

ory and especially mathematical and computer-aided music theory. Thelocal/global dichotomy is applied (a) to the discipline such from a cul-tural semiotic point of view, (b) to the strategies of scientific knowledgemanagement dogmatics, modeling and hermeneutics, and (c) to the music-theoretical discourse subjects. A detailed discussion is dedicated to thestudy of Guerino Mazzolas proposal for a mathematical denotator systemfor music-theoretical objects which is finally applied to Daniel Harrisonsdual network of harmonic concepts.

1 Music Theory and Scientific Culture

The Klangart 99 Global Village - Global Brain - Global Music was de-voted to the global nature of musical culture and presented approaches tointerdisciplinary music research. Hence, on a metalevel, it is worth study-ing the global nature of music research. Interdiscplinary collaborationleads to a complex interplay of research interests. Although the dynamicsof research interests in a scientific community work on their own and arecertainly not fully controlled by institutions and individuals, it is nev-ertheless useful to reflect upon these processes. We will argue in favourof the idea that globality of music-theoretical knowledge in some respectmirrors the globality of its object domain. However, global knowledgeis not the same as knowledge about globality. In contemporary physicsone observes a strong desire for Grand Unified Theories turning the mainaccepted working theories1 of the principal interactions into one.

It is not clear wether it is legitimate and of great benefit to compare

physics with musicology. What are the accepted local working music theo-ries to be unified? Is there a desire to do so? These questions are not only

Financed by the Volkswagen-Stiftung1General Relativity is one local perspective among others. Nevertheless it is a theory about

global structures.

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directed towards epistemology, but to a high degree towards the mecha-nisms of scientific culture. Therefore we refer to a semiotically motivated

tripartition of culture and discuss it with respect to the local/global di-chotomy, because these are constituents that may be suitably associatedwith the Klangart topic. The basic semiotic anatomy and mechanismsof culture are based on the interaction of three domains, namely social,material and mental culture (cf. [13]).

1. Social culture is constituted by a community in which individualsand institutions occupy different positions in a structure of inter-dependences, that regulate their actual behavior through manifoldkinds of stimulation and restriction. For our considerations of ascientific community we are especially interested in the regulationof research and communication on the background of specializationand division of labour.

2. Material culture is constituted by all kinds of artifacts produced

and consumed by its members, like musical instruments, computers,etc. - including all kinds of sign vehicles, like scores and datafiles.We are especially interested in the conditions under which computerprograms and electronic musical corpora may contribute to a newexperimental paradigm.

3. Mental culture is constituted by knowledge domains, natural lan-guages, musical and other sign systems, theories, etc. We are espe-cially interested in the way theories and other knowledge domainsmay coexists and/or influence each other.

Music Theory has to be characterized as an open substructure of alarger surrounding culture having many operlaps with and ramificationsinto musical, scientific and technological domains. We may presuppose thepenetration of alien disciplines into music theory as something natural

with regard to the cultural mechanism. But we focuss our attention tophenomena of globality inside the music-theoretical subculture though itsunquestionable openness. 2

It is very popular to illustrate globalitywith a network-metaphor: Ev-erything can be linked to everything else. But there is a difference betweena mere reference from one object to another and two objects being gluedalong a shared substructure. The latter happens when geographers recon-struct the globe from an atlas of overlapping maps. The idea of gluinglocal maps is behind the mathematical concept of global structures. Theoverlap of the maps allows a controlled transition from one coordinatesystem to the other.

Mathematical models of global structure can be applied to the music-theoretical domain in two ways:

1. in order to conceive musical structures as global ones,

2With respect to ongoing discussions about globalization of knowledge through internettechnology one should not appraise globality naively. Global accessibility to informationwill perhaps support a more general process of knowledge globalization towards a new typeof encyclopedism. But such a process will heavily depend on further fundamental research.Mazzola (cf. [7], [8]) argues in favour of a programmatic role of music within such a movement.We nevertheless prefer to continue our attention to the needs of Music Theory.

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2. in order to understand knowledge about musical structures as global.

Of course, without being forced by the music-theoretical content, onewould not leave a suitable coordinate system. Therefore we start with asimple musical example of a chord sequence resisting against interpreta-tion within a specific local coordinate system: the Euler Tone-Net.3 Thiscoordinate system is explicitly or implicitly favoured by many authors.It is spanned by fifths (horizontal direction) and major thirds (verticaldirection). Triads correspond to triangles.

Figure 1: A simple chord sequence with tied notes

F C G

A E

dd

d

dd

d

D A

F C

dd

d

dd

d

G D A

B

F

dd

dd

dd

C G D

E B

dd

d

dd

d

s

s

s

s

s s

s s

The four figures above represent four Euler Tone-Net Maps each con-sisting of two successive triads in the sequence. What counts here is nottheir succession as such, but the fact that tied notes occupy identical po-sitions. The little circular nodes within the triangles together with theconnecting edges represent prescricptions for triangles to be glued. Theresulting global object cannot be embedded into the Tone-Net.4.

This structure somehow reflects the global character of knowledgeabout harmony. We may conceive the above construction as a proto-col of a mental experiment of hermeneutic nature. Instead of directlyconcluding that the chord sequence is a global structure, we could inspectother theoretical viewpoints. We list some of them:

1. to neglect the independence of fifth- and third-kinship. This is whatmost scholars do in practice, although often not in their theoreticalreasoning (e.g. Heinrich Schenker and his school),

2. to neglect the homogeneity of the tone space, (e.g., not to considerthe triad of scale degree II as a proper one, like Moritz Hauptmann)

3Positions in the Euler Tone-Net correspond to o ctave classes in just tuning, cf. [6],[12].4In Mazzolas terminology this is an example for a non-interpretable global composition.

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3. to neglect that tied notes should occupy identical tone positions.This is what Martin Vogel suggests.

To detect the actual global nature of our present knowledge aboutharmony would need to make an inventory of those approaches havingworking parts, i.e., explanatory power for conrete phenoma of interest.Encouraging points of departure are those where music theorists recognizetheir situation as a crisis - either of music or of their theory. Presently, oneobserves a solidarity between local approaches and musical periods andstyles (e.g., Schenkerian analysis for Baroque music till early Romantics,Neo-Riemannian analysis for late Romantic Music). From the viewpoint ofhistorical relativism in the second half of the 20th century, there appearedto be no need to theoretically trace contiguity along the diachronic axisof music history.

We now leave the concrete example in order to comment upon somebasic research strategies. The internal dynamics of mental culture can

hardly be explained as an intended result of the collective behavior insocial culture. Especially an apperance of unauthorized aliens may passwithout any noticable effect, but it may lead to unpredicted turbulencesas well. Growth, maintenance and evaluation of knowledge within thementality of a scientific culture are internally caused by a fundamentaldrive towards local and global coherence. Processes of localization andglobalization occur in systematic interaction.

We suggest to distinguish the following three types of knowledge man-agement:

Dogmatics preserves local coherence within a domain of knowledgeby selective incorporation, i.e., through filtering.

Modeling obtains local coherence within a domain of knowledgethrough construction.

Hermeneutics collects and compares varying viewpoints on givenobjects of interest and hence forces globalization of knowledge.

If there is any kind of native interest in the object domain at all, it isthe hermeneutician, who is in control of it. It should be stressed that thetwo localization strategies involve a specific normative/creative behaviorintervening with such native interest. Nevertheless, one should not un-derestimate the role of model and dogma. While there is always somedanger for a modelist to confuse a model with reality, there is also a par-allel source of confusion for his critics, who are likely to take engagementfor a model already as confusion between model and reality. Somethingsimilar holds for the honest dogmatist and his critics. We come back tothis issue in the following section.

Whenever hermeneutic activity discovers a partially global coherence,this already implies knowledge about a global object and hence might

lead to further modeling activity. The systematic interaction betweenknowledge localization and globalization is of recursive nature.

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2 Mathematical and Computer-Aided

Music TheoryOur discussion of computer-aided experiments in Mathematical MusicTheory is especially motivated by the concept of the RUBATO softwarefor musical analysis and performance.5 For both disciplines it is charac-teristic that the involved researchers spend a lot of time and energy forlarge portions of scientific work that are not directly motivated by musictheoretical interests. However, there is a complex interaction between thedirect and indirect research interests, that can hardly be classified intoproper research on the one side and service on the other. Another do-main makes this evident: the influence of Psychology and Sociology onSystematic Musicology in the last decades goes far beyond mere service,because there is a significant influence of these disciplines on the dynamicsof musicological interest.

First we give a very short characterization of Mathematical Music The-ory as a subdiscipline of General Music Theory. We especially refer to theZurich School of Mathematical Music Theory, which has been initiated,developed and programmatically inspired by Guerino Mazzola.

There are two complementary research interests within General MusicTheory, namely

1. Analysis, i.e., understanding of concrete ideosyncratic musical struc-tures

2. Theory, i.e., understanding of general principles and rules behindmusical structures

The dymanics of interest in General Music Theory is characterized by apermanent change of focus between analytical and theoretical approaches.

Something similar can also be observed within Mathematical Music

Theory. On the one hand, there are approaches providing methods in-tended to represent concrete musical structures in terms of concrete math-ematical objects: Denotators. These denotators are then further investi-gated by suitable mathematical methods in order to obtain insights intothe concrete musical structures. On the other hand, there are approachesaiming at solving a specific problem within music theory by explanatorypower of a suitable mathematical model. Hence there are four channelsof transfer to be considered:

Musical Structures

General Music Theory

T

c' E

' E

Denotators

Mathematical Music Theory

T

c

5This software has been developed by Guerino Mazzola and Oliver Zahorka (Universityof Zurich) for NEXTSTEP and ported to Mac OSX by Jorg Garbers (Techical University ofBerlin). Its further development is subject of an OpenSource-Project.

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The vertical arrows denote shifts of interest between ideosyncraticstructures and theoretical problems. The horizontal arrows denote math-

ematical modeling (left to right) and music-theoretical interpretation ofmathematical facts (right to left). Mazzolas earlier investigations (e.g.,on modulation and counterpoint) started from theoretical problems andare hence located on the top side of the square. The analytical appli-cability of these approaches is limited to very special musical structures.As a consequence, Mazzolas later investigations within the context ofthe RUBATO-project aimed at reaching the bottom side of the squareas well. The analytical RUBETTEs are software tools to be used withinthe RUBATO frame application. They provide methods for the analysisof scores (given in MIDI-format) through the transformation of empiri-cally given sets of tones (in the onset-pitch space) into highly structuredmathematical objects and finally encoding them in analytical weigths.These transformations are motivated by general ideas on paradigmaticsand syntagmatics, but they are not comparable to grammars. There is no

normative distinction between well-formed and ill-formed musical struc-tures. The RUBETTEs work on any set of tones. As a consequence, thereis a division of labour involved between

1. RUBETTE-authors who offer analytical transformations of musicalinput data into mathematical structures and analytical weights,

2. and RUBETTE-users who interpret these structures and weights inthe context of other analytical methods and/or by means of experi-mental performance.

In the case of the existing RUBETTES6 it is interesting to characterizethis division of labour from a scientific point of view. To call RUBETTE-authorship a mere service would certainly miss the point. The transfor-mations from musical input data into the analytical data include (in the

given cases) a lot of theoretical ideas - including music-theoretical andsemiotical ones. On the other hand, these approaches start from the leftbottom corner of the square and end up at the top right one. Thereis not a strong music-theoretical hypothesis behind each RUBETTE tobe falsified. The user is invited to find out what is interesting in givenideosycratic structures under the specific perspective - provided by theRUBETTE. The creative scientific work as well as the responsibility arehence distributed among both: authors and users.

We recall the remark about the engagement of a modelist in his modelresulting in a shifting focus of scientific interest. Once a mathematicalmodel of a specific music-theoretical situation is established, it asks forseparate attention. What distinguishes a model from a mere description isits metaphorical mechanism. The more knowledge about the model can betransferred into the music-theoretical domain, the richer its explanatory

power. Hence, gaining knowledge about the model is a necessary pre-requisite for the metaphorical transfer. It is essential that the researchermoves with the focus of his interest from the object domain to the modeland back.

6The same holds for RUBETTES currently being developed by the KIT-MaMuTh-groupat the Technical University of Berlin

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The situation becomes more difficult when the division of labour be-comes institutionalized. The investigation of a mathematical model can

be done more effectively by a mathematician, who himself might not beable to judge its applicability in music theory. The RUBETTE-conceptprovides a suitable means to support the interdisciplinary communication.As an example, we mention Anja Fleischers investigations into metricalcoherence, that are empirically based on the work with the MetroRU-BETTE (cf. [2], [1]). In a first step, Mazzola defined a model of metricalregularity with mathematical intuition and tested it on a small corpusof examples. However, the musictheoretical meaning of these structureswas still unclear, when an algorithm was implemented. Later on, AnjaFleischer (and other RUBATO-experimenters as well) had a closer look onthe resulting inner metrical weights of many musical pieces and comparedthem to other analytical structures, especially to the outer bar structure.Various types of correspondence between the two turned out to provide in-sight into the general phenomenon of metricity and meanwhile suggested

a refinement of the tools for inner metrical analysis. In this particularcase the division of labour and the communication through the mediat-ing software worked quite well and lead to an effect of interdisciplinarysynergy. Furthermore, the concept of metrical coherence exemplifies thehermeneutic strategy mentioned above: to detect global coherence. Inthis specific case the coherence occurs between outer metrical bar struc-ture - an accepted standard in music theory, and inner metrical structure- a mathematical analysis of the pure onset structure that deliberatelyneglects any information concerning bar lines and time signature.

In this discussion we already reached a meeting point of Mathematicaland Computer-aided Music Theory. The role of the RUBETTE-author issomehow located between mere service and working (Mathematical) Mu-sic Theory. He has an initial idea and provides tools intended to betterunderstand his idea. He creates these tools not just in private, but invites

other researchers as well to develop Music theory in collaboration. Theinterface to performance experiments provides another channel of com-munication. Users may include their aesthical judgements about artificialperformances produced by the PerformanceRUBETTE on the basis ofanalytical weights in order to decide about their scientific interest in aspecific mathematical model. 7

Now recall the two fundamental directions of interest in General Mu-sic Theory: interest toward ideosyncratic structures and interest towardsystematization. Analysis of a given piece is a hermeneutic activity. Theideosyncratic structure of a piece is typically reflected in specific corre-spondences of local analyses. In other words: The analyst collects globalknowledge about the piece with the intention to construct a global objectfrom it. The existing analytical RUBETTEs are specialized tools for localanalysis8. The RUBATO concept includes the idea of a communicative

platform for this hermeneutic activity. As a consequence, another research

7As a side-effect users may even consume analyses for performance reasons without in-specting them.

8Due to the mentioned recursivity of the local/global dichotomy local viewpoints may yieldglobal objects in their own right

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interest enters the field: Software Integration Techniques.A single computer-aided music-theoretical experiment essentially con-

sists in

1. a problem or a question that motivates the experiment,

2. a program whose behaviour can help to better understand or evensolve the problem or to answer the question,

3. musical data that are used as input data for the program.

While the original motivation - to find out something about the prob-lem - may be a special task from a larger music-theoretical context, theexperiment itself splits into subtasks that are not directly connected withthat context. If a suitable program and/or corresponding data are notavailable to the experimenter, he (or she) has to prepare the experimentfirst by writing a program and/or encoding data. An individual music-theorist may do so whenever an experiment comes to his mind. He will

consider these subtasks as necessary steps within the organisation of hiswork. He will reuse programs or data whenever possible, but ideally hewould not change his motivating question just because of problems with aprogram or with lacking data. The motivation for the experiment governsall subsequent intentions. The experiment is sucessfull, if an answer orsome new insight into the problem has been gained. From this local per-spective, there is no reason to invest time and energy into the managmentof further single experiments of the same kind, unless they are necessaryin the same concrete music-theoretical context in which the experimenteris involved.

The growing practice of making computer-aided experiments, the ex-istence of already written programs and encoded data provides two otherdirections of possible scientific activity:

1. reuse of programs in similar experiments with varying data,2. reuse of data in other experiments with varying questions.

But the hermeneutic interest of an analyst has its own dynamics -depending on findings in a specific situation. Thus one has typicallya tension between intended experminents and immediate practicability.The challenge of the RUBATO concept consists in its support capabilityto flexible analytic experiments including the possibility of a division oflabour among several researchers. See also Jorg Garbers contribution tothis volume ([3]). With regard to mathematical modelling this is related toanother field that attracts scientific interest, namely the ongoing processof systematization within Mathematical Music Theory. The developmentof software integration techniques includes two roles of mathematical mod-els, namely data models and models for music theoretical objects, which

come in close interaction, but must not be confused.

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3 Metalanguage

Already in his first book Mazzola (cf. [5]) suggested a theoretical frame-work comprizing all of his concrete music-theoretical models. In his mostrecent book (cf. [9]) this aspect plays a central role and led to an extendedmeta theory. Hence the attempts of localizing specific mathematical mod-els within the framework of a scientific metalanguage is another aspect ofa shifting scientific interest. One should mention that Mazzolas metalan-guage goes beyond a mere descriptive framework. It includes classificationmethods, that apply to a large class of possible models and it supportsthe construction of complex models from given simpler ones.

Music-theoretical knowledge is based on denotation and predication.Denotators identify discourse subjects, while predicates load them withmeaning in various ways. We give a short portrait9 of the Denotatorsystem. Its design has been motivated by two main ingrediences as shownbelow:

Musical parametersand their transformations

Complex objectsand universal constructions

Denotators

ddd

The upper left corner represents denotative aspects in the systematicinvestigation of generalized musical parameter spaces as proposed in [5]and [6]. The upper right side labels denotative aspects of general conceptanalysis and construction which gained Mazzolas interest in connectionwith data modeling for RUBATO. It seems useful not to directly enterMazzolas system. For the most part of this section we restrict our con-

siderations to a simplified dollhouse- ontology corresponding only to theupper right corner. In order to avoid confusion we mark the dollhouse-Denotators with the prefix , with the idea in mind that these are builtupon simple acts of pointing at (arbitrary) objects. At the end of this por-tait we comment on the refinements which come into play in Mazzolasontology together with the integration of the upper left corner.

The accessability of Denotators is controlled though Forms. A Formbasically consists of an AmbientSet having Denotators as its elements.Extensionally, the AmbientSetserves as a Denotator-container. In addi-tion, a Form includes the definition of its construction relative to otherForms. This attributes a formal intension to the Denotators in addi-tion to the mere elementship.

We now adopt Mazzolas definition to our narrowed pointing-ontology.A Denotatoris constituted by its name, its Form and its Coordinates.

This can be written as

Name : Form(Coordinates)

9The reader should consider our remarks as a vademecum rather than a proper introductionand is refered to Mazzolas exposition in [9], Chapter 6.

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The coordinates identify the Denotator as an element of its Ambi-entSet. How is this related to the corresponding Form F?. The Ambi-

entSetAS(F) is the core of the Form. It is embedded into the FrameSetFS(F) by an injective set-map which is called the Identifier:

Identifier : AS(F) FS(F)

In many practical cases, AmbientSetand FrameSetcoincide, and the Iden-tifier is the Identity map. The FrameSet is defined through one of fiveconstruction types that apply to a Coordinator, i.e., to the source struc-ture to be used in the construction. The definition of a Form looks likethis:

Name Type(Coordinator)Identifier

The five construction types are labeled as follows:

Simple Syn Power Limit Colimit

It is useful to discuss these construction types and the role of theinvolved coordinators in some detail and to illustrate them by musicalexamples. We skip the Syn-type, which stands for Synonymy.10

The coordinator of a Simple Form is an arbitrary one-elementedset. Pointing at some isolated object is the basic activity that motivatessimple Denotation. In the pointing ontology simple is equivalent tosingle. Hence, the Coordinator of a Simple Form is the pointer itself.The FrameSetF S(S) of any simple Form S is always the same, namely = {}, while its AmbientSet AS(S) can be any one-elemented set. Thedefinition a SingleTone-Form might look like this:

SingleTone Simple(){}

Power,

Limitand

Colimitare known as universal constructions in cat-egory theory. For readers not familiar with that we recall the three strate-

gies of knowledge management mentioned in the first section: hermeneu-tics, dogma and model. A closer inspection of the ways how to manipulatedenotators following these strategies gives a suitable heuristics for theseconstructions.

We start with those principles of Denotator construction that arenecessary for hermeneutic activity. A fundamental modality to go be-yond mere denotation of single objects is called Coproduct. It is thefree or unlimited special case of the more general Colimit type. Thehermeneutician makes use of it, when refers to Denotators of two ormore Forms as if they were of the same Form. One may even buildCoproduct of multiple copies of a given Form. This is a suitable way todefine a TwelveTone-Form:

TwelveTone Colimit(SingleTone,...,SingleTone)Identity 12 times

10The coordinator of Syn-construction is a Form, whose ambient space is the frame spaceof the constructed Form. Hence Synonymy allows to rename a Form and to narrow itsAmbientSet.

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The FrameSetF S(TwelveTone) = {0, . . . , 11} of this Form consitsof 12 copies of the element of AS(SingleTone) = {}, indexed by the

positions of the 12 identical Cofactors in the Coordinator of this Form.The structure of such Coordinators is explained below.

Another typical activity of a hermeneutician is to find out correspon-dences. Hence he has to be able to freely combine Denotators of variousForms. The corresponding Form-construction is called Product. TheFrameSet of such a Product-Form is the cartesian product of the un-derlying AmbientSets. The Product is a free or unlimited case of theLimit-type. Intervals can b e described as ordered pairs of TwelveTone-Denotators:

TwelveTonePair Limit(TwelveTone,TwelveTone)Identity

As third basic operation he needs the possibility to collectany Denotators

of interest. The construction ofPower-type Forms allows the denotationof Denotator collections. This construction avoids confusion betweenForms and Denotators. The coordinator of a Form G of type Powercan be any another Form F. Its FrameSet is supposed to be the powerset of the AmbientSetofF: FS(G) = 2AS(F). One may recursively collectDenotators in higher order Power-type Forms, e.g.:

TwelveToneChord Power(TwelveTone)Identity

TwelveToneChordSet Power(TwelveToneChord)Identity

...

Now we continue in our heuristics. The idealized hermeneutician isspecialized to find interesting objects and correspondences between themand hence - in a purely descriptive mood - he is not interested in puttinglimitations on the Denotators themselves. He minimizes his denotativeefforts to the necessary operations and is mainly involved in acts of pred-ication. For the dogmatist it is typical that he transfers systematicallyoccuring dependencies from the concrete to the formal level. Instead of ob-serving dependencies between Denotators - as the hermeneutician does- he formulates dependencies between Forms. Dependencies betweenForms are suitably described in terms ofForm-Diagrams. Such a dia-gram consists of a directed graph , whose nodes are loaded with Formsand whose arrows are loaded with set-maps beween the AmbientSets ofthese Forms. The set-maps are the carriers of formal dependencies. Thefigure below shows a typical abstract graph consisting of three nodes

and five arrows:

TT'&% s s sE E'

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In order to complete this graph into a Form-Diagram D, considerthree Forms F1, F2 and F3. Let A1 = AS(F1), A2 = AS(F2) and

A3 = AS(F3) be the corresponding AmbientSets. Consider further fiveset-maps:

f, g : A1 A1, h : A1 A2, i : A2 A3, j : A3 A2.

The figure below shows the resulting Form-Diagram D:

TT'&%

F1 F2 F3E E'

fg

h i

j

In order to understand the construction of a Limit-type FormL from thisdiagram we first recall specific hermeneutic activity that likely motivates

such a diagram. In order to describe dependencies between Denotators,that are formally expressed in the arrows of the diagram D, a hermeneu-tician would base his observations on the free product of F1, F2 and F3,i.e., on a diagram D0 having three nodes loaded with F1, F2 and F3, butnot having arrows.11

F123 Limit(F1, F2, F3) = Limit(D0)Identity

In his discourse the hermeneutician refers to specific denotators

d : F123(f1, f2, f3)

which attracted his interest because of an observation, that is expressedin the following predicate P:

P(x1, x2, x3) := f(x1) = x1 g(x1) = x1 h(x1) = x2 i(x3) = x2 j(x2) = x3

The idealized dogmatist would therefore limit the scope of his interestfrom all possible F123-Denotators to those for which the predicate Pis true and therefore aquires the ability to filter them out - even thosethat practically would never have been observed by a hermeneutician. Heturns the predicate P into a system of equations for Forms

11This is reflected in Mazzolas convention for the notation of Coordinators of Limit andColimit type: Form Diagrams without arrows are written as a lists of Forms. Their positionsin the list represent the nodes of the diagram. The 12 Cofactors of the TwelveTone-Formcorrespond to nodes of a diagram without arrows, each being loaded with the SingleTone-Form.

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X

c

dd

dd

p1 p2 p3

TT'&%

F1 F2 F3E E'

fg

h i

j

The variable Form X of this system of equations involves three variableset-maps pi : FS(X) Ai i = 1, 2, 3 from the FrameSet FS(X) of Xinto the AmbientSets of F1, F2 and F3 and the equations read as follows:

f p1 = p1 g p1 = p1

h p1 = p2 i p3 = p2 j p2 = p3

A Limit-Form for the diagram D is an optimal12 solution of thisequation system. One such optimal solution L is explicitly given as follows:

F S(L) := {(f1, f2, f3) FS(F123) | P(f1, f2, f3)},

where the maps p1, p2 and p3 are the natural projections from the carte-sian product FS(F123) = A1 A2 A3 to its three factors.

In our heuristic we associate the dual construction of a Colimit-Formfor a diagram like D with the activity of an idealized modelist. His mainactivity consists in gluing ob jects. He may do so on the denotator-levelas well as on the formal level. The global object obtained from the fourEuler-Tone-Net-Maps (cf. section 1) is a typical example for such anactivity on the denotator level.13

Another type of gluing things is classification. This is what happens ina Colimit-Form construction. Our idealized modelist starts by studyingthe Coproduct

F123 Colimit(F1, F2, F3) = Colimit(D0).Identity

Its FrameSet is the disjoint union

FS(F123) = A1 A2 A3

In his further activity he aquires the ability to identify those F123-denotators with each other that are connected by one of the set-maps inthe diagram D. He thus turns the predicate P into a system of equationsfor Forms

12The optimality is expressed in the universality property for Limits.13this is actually a Colimit-construction cf. [9], chapter 13

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Y

T

dd

dds

q1 q2 q3

TT'&%

F1 F2 F3E E'

fg

h i

j

The variable Form Y of this system of equations involves three variableset-maps qi : Ai F S(Y) i = 1, 2, 3 from the AmbientSets of F1, F2and F3 into the FrameSet F S(Y) of Y and the equations read as follows:

q1 f = q1 q1 g = q1

q2 h = q1 q2 i = q3 q3 j = q2

A Colimit-Form for the diagram D is an optimal solution for this systemof equations. One such optimal solution C is explicitly given in terms ofthe FrameSetFS(C) - being the set of equivalence classes generated fromthe five graphs of the set-maps f,g,h,i,j within FS(F123) F S(F123).The reader may imagine chains of dominos that provide equivalencesbetween their two ends. The dominos themselves are elements fromthe five graphs (x1, f(x1)), (x1, g(x1)), (x2, h(x2)), (x3, i(x3)), (x2, j(x2))(xi Ai) and can be turned into their mirror images as well, i.e., into(f(x1), x1), ..., (j(x2), (x2)). The three maps q1, q2 and q3 of this solutionare induced by the injections ei : Ai FS(F

123).In order to inspect a music-theoretical example, we study a much

simpler diagram M3, whose graph consists of just one node and one arrow.The node is loaded with the TwelveToneChord-Form and the arrow is

loaded with the Minor-Third-Transposition for chords: t{}3 : 2{0,...11}

2{0,...11}. The transposition t{}3 for chords is defined by lifting the Minor-Third-Transposition for tones

t3 : {0,...11} {0,...11}, t3(i) := i+3mod 12

to chords: t{}3 (X) := {t3(x) | x X}. For simplicity of notation, from

now on, we use the same symbol t3 instead of t{}3 .

TTwelveToneChordt3

The reader might try to determine its Limit and Colimit before he or shecontinues reading.

The diagram M3 has only one node, hence its Limit is a filter ofthe TwelveToneChord-Form. It passes exactly those TwelveToneChordswhich are invariant under the Minor-Third-Transposition t3. Such trans-position invariant chords are known as MessiaenChords.

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TTwelveToneChordMessiaen3Chord

E

t3

Concrete examples of Messiaen3Chord-Denotators are written as:

Example 1.1 : Messiaen3Chord(0, 3, 6, 9)Example 1.2 : Messiaen3Chord(0, 1, 3, 4, 6, 7, 9, 10)

The ColimitofM3 classifies those TwelveToneChord-Denotators as equiv-alent which can be transformed into one another through recursive minor-third-transposition. The resulting Formcan be named Trans3ChordClass.

T

TwelveToneChord Trans3ChordClassE

t3

Concrete examples of Trans3ChordClass-Denotators are written as:

Example 2.1 : Trans3ChordClass(0, 4, 7)Example 2.2 : Trans3ChordClass(0, 4, 7, 10)

Note that any representative of a Trans3ChordClass provides suitablecoordinates of the Trans3ChordClass-Denotator, i.e., one may alterna-tively write:

Example 2.1 : Trans3ChordClass(3, 7, 10)Example 2.2 : Trans3ChordClass(1, 3, 7, 10)

Usually one classifies TwelveToneChords with respect to the Fifth-Trans-position t7, because by recursion one reaches all twelve transpositions.

The Colimit of the corresponding M7-Diagram yields a coarser classifica-tion than the M3-Diagram:

TTwelveToneChord TransChordClass

E

t7

In order to obtain a full chord classification with respect to the 48-elemented symmetry group of the TwelveToneSystem, one has to addtwo suitable arrows to the M7-Diagram, loaded with the inversion m11(multiplication of the Cofactor indices by -1 mod 12) and fifth circle trans-formation m7 (multiplication of the of the Cofactor indices by 7 mod 12):

TT'&%T987TwelveToneChord ChordClassE

t7m11

m7

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At this point we stop working within the pointing ontology, in orderto compare it with Mazzolas one. Readers who are not familiar with

category theory may skip the rest of this section and may continue withsection 4, such as if

1. the Forms and Denotators would still have the prefix

2. the Form PiMod12 would still be the Form TwelveTone

While we have been dealing with the category Sets (having sets as its ob-jects and set-maps as its morphisms), there is another category of majorimportance for Mathematical Music Theory, representing musical param-eters and their transformations. The category of modules Mod, whichsuits for this purposes, shows a different behaviour with respect to theuniversal constructions Limit and Colimit. The Power-construction doesnot work at all in this category. A natural way out of this problem isthe consideration of functors F : Mod Sets yielding structure pre-serving models of the category Mod. Mazzola is concerned with the

contravariant functor-category

Mod@ := SetsModop

having contravariant functors as its objects and natural transformationsas its morphisms. We explain how these functors are related to Forms.We revisit the pointing ontology by saying that it is concerned with thecontravariant functor-category

@ := Sets = Sets.

The Pointer Category consits of one object and no arrows besides itsidentity arrow (which we identify with ). The evaluation of the corre-sponding representable functor @ Sets at this one and only objectyields @ = {} = . Recall that Simple Forms are coordinated by and

have as their FrameSet. The key to Mazzolas ontology is to consider @as a variable FrameFunctor instead of its only value and to replace Am-bientSets by their corresponding functors with repect to the isomorpy ofcategories Sets = Sets. A new phenomenon in Mazzolas ontology is thepossibility of Adress variation. Modules play a double role: Each ModuleA Mod provides a different viewpoint into a variable Form-FunctorF un(F) Mod@ and gives access to a local AmbientSet A@F un(F) of aForm F. Mazzola calls these functors FrameSpaces and AmbientSpaceshighlighting the geometrical nature of his approach. Simple Forms are co-ordinated by Modules M and have the corresponding representable func-tors @M as their FrameSpaces. Identifiers are supposed to be naturalfunctor monomorphsims. Limits, Colimits and Power - constructions aredefined with respect to the functor-category Mod@. The Coordinates ofan A-adressed Denotatorof a Form F are defined as an element of the SetA@F un(F). The category Mod@ is a Topos, i.e., it has good propertiesthat allow to built Logics on it. On a metalevel of Metalanguage-Modelingwe may consider the only functor ! : Mod sending all modules to thepointer . It induces a natural transformation !@ : @ Mod@ which isan faithfull embedding of the pointer ontology into Mazzolas one. Am-bientSets in the pointer ontology correspond to constant AmbientSpaces,

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i.e., to constant functors in Mazzolas ontology. The FrameSet for Simple@-Forms corresponds to the constant functor sending each module M to

, it is isomorphic to the representable functor of the Zero Module. Henceall regular, i.e., non-circular, -Forms correspond to Forms having onlyone simple coordinator in their recursive construction: the Zero-Module.

Now recall the Form TwelveTone. With regard to some problems inthe context of American Set Theory one might want to work with thisForm. But note that the Minor-Third-Transposition t3 is qualitatively notdistingiushed from any other p ermutation of the 12 Cofactors. Hence,specific arithmetic operations on denotators are not supported by thepointer ontology.

Instead of the compound TwelveTone-Formone can build denotatorson the basis of a Simple Form PiMod12 in Mazzolas ontology (= Pitchmodulo 12 cf. [9], section 6.4):

PiMod12 Simple(Z12)

Id

4 A Music-theoretical Example

In his monograph on late romanic harmony Daniel Harrison presents thefollowing table of dual correspondences (cf. [4], p. 27), which he calles adual network of harmonic concepts:

Major Minor

7-8 6-5Dominant SubdominantAuthentic cadence Plagal cadenceAscending 5th semicadence Descending 5th semicadence

Sharp Flat

dd

d4 7 6 2

We discuss this table as example for a rich structure of inheritance in anetwork of denotators, starting from a natural endomorphism of SimpleForm PiMod12 and demonstrate how Limit- and Colimit constructionscan suitably explain the correspondences described by Harrison. After atechnical preparation we will revisit them in detail.

We consider the affine symmetry e711 : Z12 Z12, e711(z) := z + 7.

It induces a natural transformation @e711 : @Z12 @Z12 of the corre-sponding functor, which is the AmbientSpace of the Form PiMod12.

Now consider the following diagram DTone having two nodes loadedwith the Form PiMod12 and one arrow between them loaded with @e

711:

PiMod12 PiMod12E@e

711

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In addition, we consider the following Forms:

PiMod12Step Limit(PiMod12,PiMod12)Id

PiMod12Set Power(PiMod12)fin

PiMod12SetStep Limit(PiMod12Set,PiMod12Set)Id

The affine symmetry e711 induces natural transformations of all threeforms:

@e711Step, @e711Set , @e

711SetStep

In the case of the Power type Form PiMod12Set one has two natural

choices to define @e7

11Set - namely to choose either the image or the pre-image of a set under @e711, but in this case both coincide because e711 isself-inverse. The same graph with two nodes and one arrow can be loadedin three other ways, namely each time with one of the three compoundForms at both nodes and the corresponding natural transformation at itsarrow. The resulting diagrams are called:

DStep, DSet , DSetStep

Defining the following Limit-Forms - having these Diagrams as their Co-ordinators - enables us to discuss Harrisons dual network of harmonicconcepts and express it in terms of suitable Denotators.

PiMod12DualTones Limit(DTone)Id

PiMod12DualSteps Limit(DStep)Id

PiMod12DualSets Limit(DSet)Id

PiMod12DualSetSteps Limit(DSetStep)Id

1. Modal Duality: Major / Minor. The initial inversion e711 hasbeen chosen in such a way that the underlying diatonic scales ofC-Major and C-Minor are exchanged. This is expressed in terms ofthe PiMod12DualSets-Denotator DualScales:

DualScales : PiMod12DualSets({0, 2, 4, 5, 7, 9, 11}, {0, 2, 3, 5, 7, 8, 10})

2. Agents Discharge: 7 - 8 / 6 - 5: The typical leading tone-motion 7 - 8 is characterized in Harrisons system as a dischargeof the Dominant agent into the Tonic base. Dually he considers

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the discharge 6 - 5 of the Subdominant agent into the Tonic asso-ciate. Both discharges are highly significant for key determination.

The DualDischarge-Denotator of the Form PiMod12DualStepsexpresses this duality:

DualDischarge : PiMod12DualSteps(((11, 0), (8, 7))

3. Function: Dominant - Subdominant. The initial inversion e711maps the G-Major-Triad onto the F-Minor-Triad and vice versa(see DualFunction1), and it maps the G-Minor-Triad onto the F-Major-Triad and vice versa (see DualFunction3). Furthermore itexchanges the G-Dominant-Seventh-Chord and the F-Minor-Chordwith added Sixth (see DualFunction2).

DualFunction1 : PiMod12DualSets({7, 11, 2}, {5, 8, 0})DualFunction2 : PiMod12DualSets({7, 11, 2, 5}, {5, 8, 0, 2})

DualFunction3 : PiMod12DualSets({7, 10, 2}, {5, 9, 0})

4. Cadence: Authentic / Plagal. With respect to the functional du-alism between Dominant and Subdominant and the modal dualismbetween the two Tonic variants as well as the two dominants thereresults a dualism between certain authentic and plagal cadences,namely those which differ modally in both chords:

DualCadence1 : PiMod12DualSetSteps(({7, 11, 2}, {0, 4, 7}), ({5, 8, 0}, {0, 3, 7}))

DualCadence2 : PiMod12DualSetSteps(({7, 11, 2, 5}, {0, 4, 7}), ({5, 8, 0, 2}, {0, 3, 7}))

DualCadence3 : PiMod12DualSetSteps(({11, 2}, {0, 4, 7}), ({5, 8}, {0, 3, 7}))

DualCadence4 : PiMod12DualSetSteps(({7, 10, 2}, {0, 3, 7}), ({1, 5, 9}, {0, 4, 7}))

5. Semicadences: These are obtained as retrograde versions of thecadence steps. It is clear that the resulting denotators fit into thePiMod12DualSetSteps-Form. We omit the details. The inner sym-metry of all four Dual-Forms makes clear, that it is not possible toformally decide upon what is left and what is right in Harrisonstable.

6. Alteration: Sharpen / Flatten. Harrison lists two correspondingpairs of alterations:

the flattening of the 7th scale degree in C-Major B Bb to-gether with the sharpening of the 6th scale degree Ab A inC-Minor. In modulations these alterations produce, for exam-

ple, a 4th scale degree in F-Major and a 2nd scale degree inG-Minor, respectively. The alteration-pair as well as b oth re-sulting Scales are explained by suitable denotators Alteration1and DualAlteredScales1.

the sharpening of the 4th scale degree F F# in C-Majortogether with the flattening of the 2nd scale degree D Db

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in C-Minor. In modulations these alterations produce, for ex-ample, a 7th scale degree in G-Major and a 6th scale degree in

F-Minor, respectively. Again, the alteration-pair as well as bothresulting scales are explained by suitable denotators Alteration2and DualAlteredScales2.

Alteration1 : PiMod12DualSteps((8, 9), (11, 10))Alteration2 : PiMod12DualSteps((5, 6), (2, 1))DualAlteredScales1 : PiMod12DualSets(

{0, 2, 4, 5, 7, 9, 10}, {0, 2, 3, 5, 7, 9, 10})DualAlteredScales2 : PiMod12DualSets(

{0, 2, 4, 6, 7, 9, 11}, {0, 1, 3, 5, 7, 8, 10})

Finally, we inspect two mutually dual sequences discussed in Harrisonsbook:

Sequence 1 (Bach)

Figure 2: Sequence from Bachs G-Minor Fantasy, BWV 542 mm. 31-34 assimplified to Harrison ([4], p. 33

We consider four Denotators of the Form PiMod12Set denoting the3rd and the 2nd chord of this sequence (a minor tonic third t and a Major

Dominant D with respect to C-Tonality), as well as the union of 3rd and4rd chord (the sharpening set #) and the union of the 2nd and 3rdchord (the authentic cadence set A)

t : PiMod12Set({0, 3})D : PiMod12Set({7, 2, 11})# : PiMod12Set({0, 3, 4, 7})A : PiMod12Set({7, 2, 11, 0, 3})

The entire sequence can be modeled as a global composition by gluingseveral copies of these four charts. In fact, it is the Colimitof the followingdiagram of Denotators(!):

# A # A # A # A

D t D t D t D

dd dd dd dd dd dd dd

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Sequence 2 (Schubert)

Figure 3: Sequence from Schuberts D-Minor String Quartet, D. 810, 4th movt.mm. 16 - 25 as simplified by Harrison ([4], p. 33)

Again we consider four Denotators of the Form PiMod12Set denotingthe 3rd and the 2nd chord of this sequence (a Major tonic triad t and aMinor14 Subdominant s with respect to C-Tonality), as well as the unionof 3rd and 4th chord (the flattening set b) and the union of the 2nd and3rd chord (the plagal cadence set A)

T : PiMod12Set({0, 4, 7})s : PiMod12Set({5, 8})b : PiMod12Set({0, 3, 4, 7})P : PiMod12Set({5, 8, 0, 4, 7})

(b) P b P b P b P

(s) T s T s T s

dd dd dd dd dd dd dd

The duality of these two sequences is expressed trough two facts:1. The sequences are isomorphic as local as well as global compositions

if suitably extended to the sides or glued to a circle of 24 overlap-ping maps (12 copys of # and 12 copys of A). Note that they areretrogrades of one another.

2. The pairings of # and b as well as A and P are dual:

#/b : PiMod12DualSets({0, 3, 4, 7}, {7, 4, 3, 0})A/P : PiMod12DualSets({7, 2, 11, 0, 3}, {0, 5, 8, 7, 4})

14According to the entire logic of this sequence we dogmatically assume the note Ab in the2nd chord instead ofA, see also [4] Footnote 21.

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References

[1] Fleischer, A., Mazzola, G., Noll, Th: ComputergestutzteMusiktheorie, Musiktheorie, 4(2000), 314-325 .

[2] Fleischer, A.: Die analytische Interpretation - Schritte zur Er-schlieung eines Forschungsfeldes am Beispiel der Metrik, PhD-Manuscript.

[3] Garbers, J.: Konzept eines Musiktheorie-Servers, same edition.

[4] Harrison, D. (1994): Harmonic Function in Cromatic Music, TheUniversity of Chicago Press, Chicago and London.

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

[6] Mazzola, G. (1990): Geometrie der Tone, Birkhauser, Basel.

[7] Mazzola, G. (1997): Music@EncycloSpace - Virtuelle Naviga-

tionsrume. In: Enders, Bernd and Joachim Stange-Elbe (eds.),KlangArt97-Proceedings. Osnabrck. University of Osnabrck.

[8] Mazzola, G. (1998): Humatities@EncycloSpace. Swiss ScienceCouncil, Bern (for downdoad see www.encyclospace.org).

[9] Mazzola, G. (2002): The Topos of Music, Birkhauser, Basel.

[10] Nestke, A. (2002)Paradigmatic Motivic Analysis, Electronic Bul-letin of the Mexican Mathematical Society, Mexico City.

[11] Noll, Th. (2002): Geometry of Chords, Electronic Bulletin of theMexican Mathematical Society, Mexico City.

[12] Noll, Th., Nestke, A. (2002): Die Apperzeption von Tonen, Elek-tronische Zeitschrift der Deutschen Gesellschaft fur Musiktheorie(www.gmth.de).

[13] Posner, R. (1989): What is Culture? Toward a semiotic explicationof anthropological concepts, In: Walter Koch (ed.) The Nature ofCulture: Brockmeyer, Bochum. 240 295.

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NOLL, Thomas - Global Music Theory