Diagrams, gestures and formulae in music Guerino Mazzola, Moreno Andreatta

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Guerino Mazzola, Moreno Andreatta. Diagrams, gestures and formulae in music. Journal of Mathe- matics and Music, Taylor & Francis, 2007, 1 (1), pp.23-46. ￿hal-01161060￿

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Journal of Mathematics and Music Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t741809807 Diagrams, gestures and formulae in music

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[31]. see procedure; combinatorial solved a since methodology, are well-known graphs a fact of in groups is fundamental latter The graphs. of groupoids fundamental gestural tracking or for joysticks inter- as rubettes instrumental such components with also hardware or interaction devices. instruments and for software it hypergestures particular be in faces, gestures, interaction, of and manipulations curves, of visualization 44 hsi ut aifigfrtentokpr ftepeeigter,btfrthe for but theory, preceding the of part network the for satisfying quite is This h prahdsrbdi hspprcudb ae sasatn on o the for point starting a as taken be could paper this in described approach The surprisingly a in unfold branches two these view, of point mathematical a From essentially i.e. gestoids, of calculation the for routines in lies interest particular A for rubettes writing also includes topic this targets, mathematical purely the Besides optrtopology computer † sntytu odt.Tepiti httednttrfraimis formalism denotator the that is point The date. to up yet not is . .MzoaadM Andreatta M. and Mazzola G. Top Top @ ed ob mlmne rmsrtha a as scratch from implemented be to needs , ftplgclsae n otnosmp is maps continuous and spaces topological of optralgebra computer Downloaded By: [Andreatta, Moreno] At: 19:25 26 March 2007 1]Lnr,E n aqi,J-. 05 aeoisi otx:Hsoia,fudtoa,adphilosophical. and foundational, Historical, context: in Categories 2005, J.-P., Marquis, and E. Landry, [12] 1]Gbil . 92 e cate Des 1962, P., Gabriel, [14] 2004, L., Corry, [13] 1]Adeta . 03 Me 2003, M., Andreatta, 1986, [11] S., Lane, Mac [10] etn-tutr sraie o h set the for realized is Heyting-structure g Heyting h rmtcgpbtenpanbdl etclto n btatitliec.We Sub( intelligence. sets abstract sub-digraph and the gesticulation on bodily and algebra plain context) between that observe that gap and dramatic [33] in also the see digraphs etc.; for quivers, [28]. wild literature (synonym and classical tame the quivers theorems, classification of corresponding theory representation oia prtoso Sub( on operations logical have edn ftemnsrp n hi rcossgetoscnenn better a concerning suggestions References precious their and topic. manuscript the of the presentation Andre of Yves thank reading to like would We Acknowledgements embodiment. of Intelligence Artificial New the of problem harmony. musical in logic this of ple ogsue n omleb pligtelgcloeain oterespective the to operations logical the applying by if precisely, formulae More domains. and gestures to applies in have nvrie fteintersection the of vertices on 6 lni . 05 igams&cate & Diagrammes 2005, C., topos Alunni, and categories [6] of case the collision: by ideas mathematical of development The 1989, S., Lane, Mac [5] 3 rso,CA,20,St,ctgre n tutrls.I .Sc (Ed.), Sica G. In structuralism. and categories Sets, 2005, C.A., Drossos, [3] 9 igt . 1990, J., Piaget, [9] abendla der Grundlagen Morphologische 1995, T., Noll, [8] 1987, D., Lewin, [7] limit as K-nets view: of point categorical a From 2006, Summer M., Andreatta, and G. Mazzola, [4] functors. Adjoint 1958, D., Kan, [2] 2002, (Ed.), G. Mazzola, [1] neisteHyigsrcueo t domain its on Heyting-structure the inherits ia odo h dao introducing of idea the the on word of final A application the evokes approach this of branch algebraic the Evidently, hsfraimmyb sda oia eiefrsedn ih ntefundamental the on light shedding for device logical a as used be may formalism This hsHyiglgci dniidwt h otaain uco @ functor contravariant the with identified is logic Heyting This V ol cetfi) 1 Scientific), World In: theory. hlspi Mathematica Philosophia the Press). Vincennes). pense de Universitaires la dans diagrammatiquement denotators. Polimetrica). (Monza: Birkha (Basel: Universita D G V oius nltqe tcmoiines h hss Ircam/EHESS. thesis, PhD compositionnels. et analytiques ´oriques, G ffl / [ V ihvle ntecategory the in values with / S S G ial all finally ,  ¨  Berlin. t aeoia oooyadisRlto oAayi,AgbaadCombinatorics and Algebra Analysis, to Relation its and Topology Categorical Digraph / G Perspecti / ( V ¨ S user). D opimse aeois oprre Transformer et Comparer Categories. et Morphismes / oenAgbaadteRs fMteaia Structures Mathematical of Rise the and Algebra Modern S eeaie uia Inter Musical Generalized and Á / V v 9. ahmtc,Fr n Function and Form Mathematics, so e Music New of es G satps oeeydigraph every so topos, a is D ) @ G arw ewe etcsin vertices between -arrows , greabe ´gorie h oo fMusic of Topos The toe alge ´thodes 13 / – V D igas etrsadfrua nmusic in formulae and gestures Diagrams, ,1 S g / r sflos If follows. as are ) S h rosaetee hyinclude they these: are arrows The . : Á V  G 43. rnatoso h mrcnMteaia Society Mathematical American the of Transactions G / ´liennes. G 0 S e n .Bt (Ed.), Batt N. In: ´e? gre om prole comme ´gories @ biuse uiu tmsclged XXe du musicologie et musique en ´briques , / V / X 44 0 S n h nnmu eiwr o hi accurate their for reviewers anonymous the and ´ S h implication The . . sagsue hntesto subgestures of set the then gesture, a is htaentin not are that Heyting v D ultnd aSocie la de Bulletin l n Transformations and als Sub ) *  etrllogic gestural emti oi fCnet,Ter,adPerformance and Theory, Concepts, of Logic Geometric / ( D c @ faformula a of ) G fHyigagba.Tefntralso functor The algebras. Heyting of V ¨ NwYr:Springer-Verlag). York: (New , dshnHroi.Dcoa hss Technische thesis, Doctoral Harmonik. ndischen ´gome o h uojc classifier subobject the for , D S G . a aoia oi,ie Heyting a i.e. logic, canonical a has r w u-irpsof sub-digraphs two are V esrprl Diagramme le par Penser uai mutandis Mutatis G nsa `nes ´te G ute all further , Mathe ´ S G sa prahfrovercoming for approach an as / [ aqeto:q’s-eqes’orienter que qu’est-ce question: la ` V S S (Neucha NwHvn T aeUniversity Yale CT: Haven, (New c n etcsin vertices and mtqed France de ´matique sabtmr novd we involved: more bit a is e 3]fra example an for [34] See . 2den (Birkha edn) (2nd G tl eahu tNiestle et Delachaux ˆtel: hti aeoyTheory? Category is What D S arw nvertices on -arrows / S hnalarrows all then , h analogous the , m sie `me , V 87 : Prs Presses (Paris: 294 , Taek NJ: (Teaneck, ¨ , D V Digraph ce aspects `cle: srVerlag). user 90 Sub hnwe then , D . Á V 329. : ( / V g The )of G . 0 45 ´). / Downloaded By: [Andreatta, Moreno] At: 19:25 26 March 2007 2]McLn,S,1971, S., Lane, Mac [26] 2]Nl,T,19,Webla Wie 1998, T., Noll, [22] 2]Gbil . 92 ersnain ffiiedmninlagba.I:AI otii n ..Shafarevich I.R. and Kostrikin A.I. In: algebras. finite-dimensional of Representations 1992, P., Gabriel, [28] 2]LgsLm,E,2003, E., Lima, Lages [27] 2]Mzoa . 2004, 2002, G., J., Mazzola, Uhde, and [25] R. Wieland, [24] Mu [23] 2005, D., McNeill, [21] 3]Bno,D,1998, D., Benson, [33] 2000, J.M., Lee, [31] 1980, J.-P., Serre, [30] 2]Lwn . 90 lmehue ewrsadsm sgahe htivlethem. involve that isographies some and networks Klumpenhouwer 1990, D., Lewin, [29] Pavic Milorad to key A gesture: Anthropoetic 2006, R., Katsman, [20] 3]Mzoa G., Mazzola, [32] 3]Mzoa . 05 uia etrsadterdarmai oi.CgiieSineSmnr ILab, AI Seminar, Science Cognitive logic. diagrammatic their and gestures Musical 2005, G., Mazzola, [34] 1990, J.-C., Schmitt, [19] 2004, R., Hatten, [18] 1]Sikr . ac 98 a btatoi e ei Taylor. Cecil bei abstraktmotiv Das 1998, March V., Spicker, [17] 1992, (Eds), H. Danuser, and C. Dahlhaus, 2005, [16] (Ed.), N. Batt, [15] 46 Verlag). (Eds.), (Eds.), CC03 ICMC n Associati and 2006). November 26 (accessed oebr2006). November Spectrum Tea). Interpretation Universita Press). University ¨ lr .adMzoa . 03 osritbsdsaigo etrlperformance. gestural of shaping Constraint-based 2003, G., Mazzola, and S. ller, cdmcEetoi ora nSla in Journal Electronic Academic Ko nylpei fMteaia Sciences Mathematical of Encyclopaedia ¨ , ¨ preeugnudir Bedeutungen ihre und rperbewegungen tZu AnAbr I ICMA). MI: Arbor, (Ann 12 . v Algebras e Lae:Laaber-Verlag). (Laaber: tal. et ¨ ih vial niea:ht:/w.nylsaeogtlsdao.p acse 26 (accessed http://www.encyclospace.org/talks/dialog.ppt at: online Available rich. nrdcint oooia Manifolds Topological to Introduction Trees ersnain n Cohomology and Representations Comprehensi etr n Thought and Gesture 96f. UAOo h nent vial niea:http://www.rubato.org at: online Available Internet. the on RUBATO ff., 1996 , nepeigMsclGsue,Tpc,adTropes and Topics, Gestures, Musical Interpreting aeoisfrteWrigMathematician Working the for Categories aRio e etsdn cietMe Occident l dans Gestes des Raison La ¨ esrprl Diagramme le par Penser tr a nenmhpreilnGeba hypermedialen einem in man ttert udmna rusadCo and Groups Fundamental NwYr:Springer-Verlag). York: (New Cmrde K abig nvriyPress). University Cambridge UK: (Cambridge, v ahmtc o optrScientists Computer for Mathematics e oshne U Forschendes .MzoaadM Andreatta M. and Mazzola G. v Ciao L nvriyo hcg Press). Chicago of University IL: (Chicago, cStudies ic ee aduhdrMusikwissenschaft der Handbuch Neues ¨ Prs rse nvriarsd Vincennes). de Universitaires Presses (Paris: o.7 Bri:Springer-Verlag). (Berlin: 73 Vol. , ben Bri:Bri-elgAn Spitz). Arno Berlin-Verlag (Berlin: o.1: Vol. , o 6 nvriyo Toronto. of University 16, No. , v Kse:Ba (Kassel: rn Spaces ering Hiebr:Springer-Verlag). (Heidelberg: ai ersnainTer fFnt Groups Finite of Theory Representation Basic ¨ dneio?I:C cmue n .Noll T. and Schmauser C. In: rdenlexikon? ´die ¨ MusikTexte renreiter). Hiebr:Springer-Verlag). (Heidelberg: ´ v Ntc,M:Peters). MA: (Natick, al ’ otc LnsaePitdwith Painted (Landscape poetics ´’s Prs Gallimard). (Paris: osI I(edleg Springer- (Heidelberg: II I, Vols , , Bomntn N Indiana IN: (Bloomington, 73/74 . o.11: Vol. , rceig fthe of Proceedings ui Theory Music Musikalische