Mathematical Music Theory—Status Quo 2000 Guerino Mazzola www.encyclospace.org, ETH Zu¨rich, Departement GESS, and Universit¨at Zu¨rich, Institut fu¨r Informatik 15th October 2001 Abstract Wegiveanoverviewofmathematicalmusictheoryasithasbeende- velopedinthepasttwentyyears. Thepresenttheoryincludesaformal languageformusicalandmusicologicalobjectsandrelations. Thislan- guage is built upon topos theory and its logic. Various models of mu- sical phenomena have been developed. They include harmony (func- tion theory, cadences, and modulations), classical counterpoint (Fux rules), rhythm, motif theory, and the theory of musical performance. Most of these models have also been implemented and evaluated in computer applications. Some models have been tested empirically in neuro-musicology and the cognitive science of music. The mathemat- ical nature of this modeling process canonically embedds the given historical music theories in a variety of fictitious theories and thereby enables a qualification of historical reality against potential variants. Asaresult,thehistoricalrealizationsoftenturnouttobesomekindof “best possible world” and thus reveals a type of “anthropic principle” in music. Thesemodelsusedifferenttypesofmathematicalapproaches,such as—for instance—enumeration combinatorics, group and module the- ory,algebraicgeometryandtopology,vectorfieldsandnumericalsolu- tions of differential equations, Grothendieck topologies, topos theory, and statistics. The results lead to good simulations of classical results of music and performance theory. There is a number of classifiaction theorems of determined categories of musical structures. The overview concludes by a discussion of mathematical and mu- sicological challenges which issue from the investigation of music by mathematics,includingtheprojectof“GrandUnification”ofharmony and counterpoint and the classification of musical performance. 1 Introduction This is the second status quo report on mathematical music theory. The first was written exactly ten years ago for the Deutsche Mathematiker- Vereinigung [37], and ten years after my first steps into mathematical music theory [31]. The former report essentially paralleled with the book Ge- ometrie der To¨ne [34], whose title reflects the theoretical approach of that time: The central concern was not logic but geometry, i.e., the investigation of categories of local and global compositions which formalize the relevant objects and relations for harmony (cadence and modulation), counterpoint (Fux rules), melody, rhythm, large musical forms (in particular the classical sonatatheory),includingtheirclassification,andtheparadigmaticsemiotics of musical structures as described by Ruwet [57] and Nattiez [49]. This approach included satisfactory theorems which model modulation, counterpoint, and string quartet theory in coincidence with the classical knowledgeandtradition,whichyieldclassificationforsomeinterestingglobal structures [34, 37], and which have been operationalized in music compo- sition software [35] and corresponding CDs [5, 6, 7, 36]. It was however incomplete and too narrow in its concept framework for many musical prob- lems. Here are some critical points: • The Yoneda point of view was not properly developed. This defect became virulent after Noll’s reconstruction of Riemann harmony [50]. • The development of the music platform RUBATO(cid:114) for analysis and performance [39, 42] enforced a critical review of music data models for universal purposes from score representation to performance [38] and the definition of an extended concept framework whose elements were described in [38, 62] and implemented in RUBATO(cid:114)’s PrediBase DBMS. • The complexity of musical performance asked for concepts and meth- ods from differential geometry, such as vector fields and their integra- tion (ordinary differential equations and associated numerical Runge- Kutta-Fehlbergmethods),Liederivatives,andcharacteristicsmethods in partial differential equations. • Thedifferenciationbetweenmathematicalfictionandmusicalfacticity had to be explicated and led to the concept of textual and paratextual predicates [43, 44]. At this point, logical and geometric perspectives wereforcedtounite. Thisapproachiscenteredaroundtopos-theoretic 2 construction of musical predicates by means of logical and geomet- ric operations and also targets at the design of universal composition tools. Presently, several research groups (e.g. TU Berlin, IRCAM Paris, U Osnabru¨ck, UNAM Mexico, ETH Zu¨rich, U Zu¨rich) are col- laborating in the theoretical and software design of these extensions. So this second status report will center around the most important im- provementsandextensionsofthetheorysincetheearlynineties. Again, this report is paralleled by an upcomping book The Topos of Music [45] with ex- tensive discussions of the old and new topics. A history of mathematical music theory has however not been written, and this report is just a flash on the ongoing process. The report first deals with mathematical models in music, discussing the methodological background, then illustrating it by three classical mod- els: modulation, counterpoint, and performance. The second section in- troduces the concept framework of forms and denotators, including their operationalization on the RUBATO(cid:114) workstation and the Galois theory of concepts. Thirdly, we discuss the central category of local and global com- positions with general Yoneda ‘addresses’, i.e., domains of presheaves. This leads to Grothendieck topologies and sheaves of affine functions which are essential for classification purposes. This latter subject is dealt with in the fourth section. We discuss enumeration theory of musical objects and algebraic schemes whose points parametrize isomorphism classes of global compositions. Based on a substantial isomorphism between harmonic and contrapuntalstructures, we give a previewinthe fifthsection ofwhatfuture research in mathematical music theory could (and should) envisage. The fact that this report has been realized under the excellent organi- zation of the Universidad Nacional Auto´noma de M´exico and Emilio Lluis- Puebla, president of the Mathematical Society of M´exico, is also a sign that mathematical music theory has transcended its original Swiss roots and has attended international acceptance. At this point, I would like to acknowl- edge all my collaborators and colleagues for their continuous support and encouragement. 1 Models 1.1 What Are Models? Basically, mathematical models of musical phenomena and their musicolog- ical reflexions are similar to corresponding models of physical phenomena. 3 The difference is that music and musicology are not phenomena of exterior nature, but of interior, human nature. To begin with, there is a status of musicstructuresandcorrepondingconceptualfields, togetherwithcomposi- tions in that area, and the modeler first has to rebuild this data in a precise concept framework of mathematical quality. Next, the historical material selection in music and musicology (scales, interval qualities, for example) has to be paralleled in the mathematical concept framework by a selection of instances. Here, the historical genesis is contrasted by the systematic definition and selection of a priori arbitrari instances. After this positioning act, the musical and/or musicological process type (such as a modulation or cadence or contrapuntal movement) has to be rephrased in terms of the mathematical concept framework. With this in mind, the historically grown construction and analysis rules of that determined process have to be mod- eled on the level of mathematics. This means that the formal process re- statementmustbecompletedbystructuretheorems(includingtheproofs, a strong change of paradigm!), and then, by use of such theorems, the grown rules must be deduced in the mathematical concept framework. The typical property of mathematical models in music is this: To en- able a quasi-automatic generalization to situations where the classical music theory for which the model was constructed has no answer. In the case of modulation which originally was modeled for major scales, the generaliza- tionextendstoarbitrary7-tonescales. Thisisduetotheapriorisystematic concept framework of mathematics. Once a bunch of concepts and struc- tureshasbeensetup, thereisnoreasonwhatsoevertosticktothehistorical material selection, the genericity of precise concepts and theorems enables a broader perspective which pure historicity cannot offer. Thepropertyofextensibilityofamathematicalmodelrelocatestheexist- ing music theory (which it models) in a field of potential, fictitious theories. This puts the historically grown facticity into a relation with the potential ‘worlds of music’. The purely historic justification of existing modulation rules, for example, does not give us reasons for this choice, and this makes the purely historical approach a poor knowledge basis: We know that some- thing is the case, but not why, and why other possibilities are not. In contrast, the mathematical approach gives us a field of potential theories wherein the actual one can be asked for its possible special properties with respect to non-existing variants. This differentia specifica is a remarkable advantage of mathematical methodology against the historical approach of musicologywhichcannotembedthefactsinaviarietyoffictionsandthereby understand the selection of what is against what is not. ThisevokesLeibnizideathattheexistingworldisthebestofallpossible 4 worlds: Istheexistingmusictheorythebestpossiblechoice? Orisitatleast adistinguishedone? Incosmology,thisideahasbeenrestatedunderthetitle of the “anthropic principle” [3]. It says that the physical laws are the best possible for the existence of humans, more precisely (and less radically), it is the theorem stating that a slight variation of the fundamental constants, such as the gravitational constant, or the electric charge of electrons and protons, would make any higher molecular complexity as it is necessary for the carbon-based biochemistry impossible. 1.2 Modulation The historically first model in mathematical music theory dealt with tonal modulation, more precisely: with Arnold Scho¨nberg’s model of a tripartite modulation process from tonality X to tonality Y, as it is described in the classical treatise on harmony [58]. The process parts are the following, exemplified for X = C-major, Y = F-major: A. Neutralization of the old tonality X, neutral degrees of X are presented, for example I ,VI . C C B. The pivotal root progression degrees (German: “Fundamentschritte der Modulation”)areplayedtoenforcetheturningmovementtowardsthe new tonality, for example degrees II ,IV ,VII . F F F C. The new tonality (F-major in our example) is evidenced by a set of cadence degrees, for example II ,V . F F In [58], such transition processes are described for a set of tonality couples, but not for all possible couples: These omitted couples are dealt with by a chainofatleasttwosuccessivemodulationsthroughintermediatetonalities. Also is the construction of the core steps, i.e., the pivotal degrees, not inde- pendent of the specific constellation, it is rather an ad hoc argumentation. Moreover, the concepts are quite fuzzy, as usual in musicology. Finally, one cannot infer, how such an argumentation should deal with non-European tonalities. So there is the mathematical modeling enterprise as described above, on the level of musicological theory. Besides that, the model must alsobetestedonthecorporaofcompositionswherethereisacertainchance to recognize such modulation processes. But let us get off on the theoretical level first and comment on the experimental work later. In the first steps, one makes the concepts of “tonality”, “degree”, “ca- dence” precise. Then, one should model the modulation mechanism, and last, one has to prove theorems which yield the pivotal degrees in process 5 part B. Since this model has been described on several occasions [31, 32, 34, 48],weshallbeverysketchyandonlymarkthecornerstonesofthemodeling operation1. For the tonalities, one takes a seven-element scale S ⊂ Z of 12 pitch classes and covers S by seven triadic degrees I ,II ,...VII which S S S are three-element subsets with each an intermediate pitch class between the first and second, and between the second and third degree pitch. For the C- major scale S = C, this gives us the classical triadic degrees. By definition, a tonality S(3) is a scale S, together with its covering (3) by triadic degrees. For the given modulation problem, we consider the translation orbit Dia(3) of the C-major tonality C(3). For a given couple S(3),T(3), the modula- tion mechanism is the datum of a symmetry S(3) → T(3), i.e., a translation or an inversion on the ambient space Z which carries the first tonality 12 onto the second. The cadence concept is grasped by minimal subsets of triadic coverings such that only the respective scales contain these degrees as their degree subsets. In Dia(3), there are five such minimal cadential sets, i.e., {II ,III },{III ,IV },{IV ,V },{II ,V },{VII }. So finally, S S S S S S S S S amodulationfromS(3) toT(3) inDia(3) isaquatruple(S(3),T(3),g,c)where g : S(3) → T(3) is a modulation symmetry, and c is one of the five minimal cadential sets for the target tonality. The last point of this model is the calculation of the pivotal degrees. This is achieved by what we call a “modulation quantum”. This is a subset M ⊂ Z such that 12 1. g is an inner symmetry of the quantum; 2. the quantum contains all degrees of the cadence c; 3. M ∩T is rigid, i.e., has no translation or inversion symmetry as inner symmetry and is covered by degrees of T(3); 4. M is minimal with properties 1. and 2. So a modulation quantum ‘materializes’ the modulation symmetry (much like quanta in physics materialize forces), contains enough elements to ex- press a cadence for the target tonality, has its trace M∩T covered by target tonality degrees and determines uniquely its associated symmetry (this fol- lows from rigidity) and is a minimal such candidate (economical condition). Ifsuchaquantumexists,weshall(bydefinition!) recoverthepivotaldegrees from the triadic covering (M ∩T)(3) of the trace M ∩T by degrees of T(3). 1A detailed and mathematically generalized discussion is also contained in [45]. 6 A modulation which has a quantum is called quantized. The main theo- remnowhastoguaranteetheexistenceofquantizedmodulations. Thisisthe alias of the historically grown rule canon in the mathematical model. This theorem in fact guarantees quantized modulations for all couples in Dia(3), and the pivotal degrees coincide with the pivotal degrees in Scho¨nberg’s treatise wherever he considers direct modulations (see [34, section 5.5.2]). The present mathematical model has the advantage that it can also be performedonanyseven-elementscale,andanytranslationclassofthatscale as a modulation domain. So the modulation model immerges the classical case Dia(3) in a variety of modulation scenarios which have never been dealt with in historical contexts. In [48], this extension has been calculated by computer programs (including explicit lists of modulation quanta and pivotal degrees) and commented. That extension exhibits a very special position of the common scales in European harmony which we summarize as follows (see [48] for complete results): • Among the modulation domains of rigid triadic tonalities, the max- imum of 226 quantized modulations occurs for the harmonic minor scale. • Among modulation domains of non-rigid tonalities, the maximum of 114quantizedmodulationsoccursforthemelodicminorscale. Among those scales with quantized modulations for all couples of their modu- lation domains, the minimum of 26 quantized modulations occurs for the diatonic major scale. Besides this “anthropic principle” for modulation, the model and its extension also apply to just tuning pitch spaces, and there, where the math- ematicsisquitedifferentsinceoneworksinZn,onealsohasgoodresults,see [53,45]. Butthemodelanditsextensionalsoapplytocompositionsoftonal character. Of course, the historical context seems to be a critical point here since not every composer would compose in the framework of Scho¨nberg’s harmony. However, the mathematical model is not a poietic model, i.e., it does not claim that the composer has used its approach to set his/her modulations. The mathematical model is more like a model in physics: The phenomena are there (in our case: the compositions), and we have to de- scribe their structure as well as possible, ignoring whether the creator of the universe has ever used our mathematics, our logic or our conceptual model of physical processes. In this spirit a number of successful interpretations of modulatory processes, among them the hitherto poorely understood mod- ulation architecture of Beethoven’s op.106 (“Hammerklavier”), have been 7 realized, see [34]. A reconstruction of the first movement of Beethoven’s op.106 in terms of analogous structures, replacing the minor seventh chord and its satellite structures in op.106 by the augmented triad and its corre- sponding satellite structures, has been realized in [32]. 1.3 Counterpoint Themathematicalmodelofcounterpoint[34]wasfirstusedinthecontextof neurophysiological investigations via Depth-EEG [40], where we tested the perceptionofconsonancesanddissonancesinlimbicandauditorystructures of the human brain. In that research project, classical European theories— followingJohannJosephFux[23]asatypicalreference—wereourobjectives. However, the model later, with the thesis of Jens Hichert [27], turned out to have a similar extension to other interval dichotomies, and again, it turned out that the European choice was an exemplification of a “anthropic prin- ciple”. We shall only sketch the core structures here to illustrate the model- ing methodology. Some more technical details are given in section 5 below. This counterpoint model starts from a specific 6-by-6-element dichotomy K/D of the twelve interval quantities modulo octave which are modeled as elements of Z , i.e., prime = 0, minor second = 1, etc., major seventh = 12 11. So the classical contrapuntal dichotomy is D = {0,3,4,7,8,9}/K = {1,2,5,6,10,11}. This dichotomy has a unique autocomplementarity sym- metry AC(x) = 5x+2, i.e., AC(K) = D. In this theory, such dichotomies are called strong dichotomies. There are six types (i.e., affine orbits) of strong dichotomies. If we draw the dichotomies as partitions of the discrete ∼ torus Z ×Z → Z given by the Sylow decomposition of Z (in fact the 3 4 12 12 torus of minor and major thirds!), then it turns out that the classical di- chotomy K/D has a maximal separation of its parts on the torus among the six strong dichotomy types. It has a remarkable antipode dichotomy which has its parts mixed up more than any other strong type, this is the major dichotomy I/J = {2,4,5,7,9,11}/{0,1,3,6,8,10} whose first part are exactly the proper intervals of the major scale when measured from the tonic! For each strong dichotomy, the results of Hichert enable a new and his- torically fictitious counterpoint rule set. These six ‘worlds of counterpoint’ are quite fascinating for several reasons, one of which we shall now make more explicit. It deals with the seven-element scale in which the counter- point rules are realized2. If one looks for the diatonic scales (those having 2Moreover,butthisisnotourmainconcernhere,theruleofforbiddenparallesoffifth 8 only semi-tone and whole-tone intervals for successive notes) where the free- dom of choice of a successor interval to a given interval is maximal in Fux counterpoint (dichotomy K/D), then the major scale is best, and it has no cul-de-sac, i.e., it is always possible to proceed from one consonant inter- val to another such interval under the given rules. The latter result is, by the way, a fact which has never been demonstrated in a logically consistent way in musicology... And the major scale has cul-de-sacs only for the major dichotomy I/J. Among the scales with seven tones without cul-de-sac for the major dichotomy, no European scales appear! However, there is a scale K∗ = {0,3,4,7,8,9,11} without cul-de-saces for I/J. It is nearly a “mela” (No. 15 = {0,1,3,4,7,8,9}), i.e., a basic scale for Indian ragas. And it is very similar to the consonant half K of the Fux dichotomy. So the counterpoint model not only exhibits a variety of fictitious coun- terpoint theories which could very well yield new, interesting counterpoint compositions. It also relates the existent counterpoint theory of the Fux dichotomy K/D to its antipode, the major dichotomy I/J, through the scales where the counterpoint has to be inserted, and thereby to a far-out music structure such as the melas from Indian raga tradition. It is not clear whether these intercultural relations can be made more realistic or whether they remain fictitious. Here, more research must be done. But it becomes evident that the extension of mathematical models could open not only new perspectivesofhistoricaldevelopments,itcouldalsounfoldnewperspectives of cultural specializations. 1.4 Performance The author’s first steps in performance modeling were made 1989-1994 while programming the commercial musical composition software presto(cid:114) for Atari computers [35]. In presto(cid:114)’s “AgoLogic” subroutine, a hierarchy of polygonal tempo curves can be defined and edited. The program uses the definition of musical tempo as a piecewise continuous map T : R → R on + the positive reals of symbolic time E, measured in quarters Q and with val- uesinthepositivereals,measuringthetempoT(E)atsymbolic(score)time E in units of quarters per minute, Q/Min, say. Mathematically, the tempo is the inverse derivative of the physical time e as a function of symbolic time E, asa functionofsymbolictime, i.e., T(E) = (de/dE)−1(E). Theprogram uses the calculation of physical time via the evident integration of 1/T(E). The hierarchical tempo structure implements the fact that musical tempo is is valid, and the coincidence with the Fux rules is extremely high, statistically speaking, the difference is less than 10−8, see [50, II.4.3] for a precise argumentation. 9 not the same for all notes at a given score time. Rather is the tempo layered in a tree of successive refinements of local tempi. Typically, this looks like this: We are given a ‘mother tempo’ curve T , defined on the closed mother symbolic time interval [E ,E ]. In a homophonic piano piece, this could be 0 1 the global tempo which is played by the left hand. If the right hand should play a Chopin rubato during a subinterval [E ,E ] of [E ,E ], then the 00 01 0 1 tempo of the right hand will deviate from the mother tempo in this interval. However, at the start and end times E ,E , we ask the hands to coincide. 00 01 So the daughter tempo T of the right hand should have the same daughter integral as the left hand with its mother tempo, i.e., E01 E01 (cid:90) 1/Tdaughter = (cid:90) 1/Tmother. E00 E00 By use of adaptation algorithms, the tempo hierarchy subroutine in the presto(cid:114) software enables the graphically-interactive construction of such daughter curves, including an arbitrary number of sisters and of genealog- ical depth for daughters, granddaughters, great-granddaughters etc. This means that interpretative time is encoded in a ramified tree of genealogical refinement of local tempi. This first approach was successful on the time level. Therefore, the SNSF grant (1992-196) for the RUBATO(cid:114) project [38, 42] was designed to extend this approach to other parameters, such as pitch, duration, loudness, glissandi, and crescendi. But the presto(cid:114) approach also had no rationale for shaping the tempo hierarchy, except intuitive graphical interaction. So the RUBATO(cid:114) project had to deal with the question of constructing operators for shaping performance from a more analytical point of view. The basic extension of tempo curves to higher parameter spaces is this: The performance is described by a performance mapping ℘ from the n- dimensional real space REHLD... of n symbolic parameters, onset E, pitch H, loudness L, duration D, etc. to the n-dimensional real space Rehld... of n physical parameters, onset e, pitch h, loudness l, duration d, etc. Locally on the score, we suppose that ℘ is a diffeomorphism on an n-dimensional cube C, applied to a finite number of score events which are contained in this cube. So for a symbolic event X (uppercase), x = ℘(X) (lowercase) denotes the associated physical performance event. The extension of the tempo concept is given by the inverse vector field Z of the constant diagonal field ∆(x) = ∆ = (1,...1) on the physical ℘ space, i.e., Z (X) = (J℘(X))−1(∆) with the Jacobian J℘(X). This defines ℘ a performance field associated with the performance map ℘. The value x = ℘(X) can be calculated as follows (still generalizing the situation for 10