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Chapter 3 Topological Structures in Computer-Aided Music Analysis

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Chapter 3 Topological Structures in Computer-Aided Music Analysis Chapter 3 Topological Structures in Computer-Aided Music Analysis Louis Bigo and Moreno Andreatta Abstract We propose a spatial approach to musical analysis based on the notion of a chord complex.Achord complex is a labelled simplicial complex which represents a set of chords. The dimension of the elements of the complex and their neighbourhood relationships highlight the size of the chords and their intersections. Following a well-established tradition in set-theoretical and neo-Riemannian music analysis, we present the family of T/I complexes which represent classes of chords which are transpositionally and inversionally equivalent and which relate to the notion of Generalized Tonnetze. A musical piece is represented by a trajectory within a given chord complex. We propose a method to compute the compactness of a trajectory in any chord complex. Calculating the trajectory compactness of a piece in T/I complexes provides valuable information for music analysis and classification. We introduce different geometrical transformations on trajectories that correspond to different musical transformations. Finally, we present HexaChord, a software package dedicated to computer-aided music analysis with chord complexes, which implements most of the concepts discussed in this chapter. Louis Bigo Department of Computer Science and Artificial Intelligence, University of the Basque Country UPV/EHU, San Sebastian,´ Spain e-mail: [email protected] Moreno Andreatta CNRS, France IRCAM, Paris, France Universite´ Pierre et Marie Curie, Paris, France e-mail: [email protected] 57 REPORTS connect the elements of one chord type (or set to 0. This transforms ℝn into ℝn−1,creatinga tients. Any ordered pair of points in any quotient class) to those of another; these are OPT (or barycentric coordinate system in the quotient space represents an equivalence class of progres- OPTI) voice-leading classes, resulting from the (basis vectors pointing from the barycenter of a sions related individually by the relevant combi- application of uniform OP (or OPI) and individ- regular n-simplex to its vertices). Permutation nation of OPTIC equivalences. The image of a ual T (1, 5)(Fig.1).Theseequivalencerelations equivalence identifies points in ℝn with their line segment in ℝn [a “line segment” in the quo- can reveal connections within and across musical reflections in the hyperplanes containing chords tient, although it may “bounce off” asingularity works and can simplify the analysis of voice with duplicate notes. Musical inversion is (1, 9)] can be identified with an equivalence class leading by grouping the large number of possi- represented by geometric inversion through the of progressions related uniformly by the relevant bilities into more manageable categories. origin. Permutation and inversion create singular combination of OPIC and individually by T. (This Geometrically, a musical object can be quotient spaces (orbifolds) not locally Euclidean is because T acts by orthogonal projection.) represented as a point in ℝn.ThefourOPTI at their fixed points. C equivalence associates Intuitively, pairs of points represent successions equivalences create quotient spaces by identify- points in spaces of different dimension: The between equivalence classes, considered as in- ing (or “gluing together”)pointsinℝn (fig. S3). result is the infinite-dimensional union of a series divisible harmonic wholes; line segments repre- Octave equivalence identifies pitches p and p + of finite subset spaces (6–8). sent specific connections between their elements. n n 12, transforming ℝ into the n-torus T . One can apply any combination of the OPTI Music theorists have proposed numerous Transpositional equivalence identifies points in equivalences to ℝn,yielding24 =16quotient geometrical models of musical structure (fig. S4 ℝn with their (Euclidean) orthogonal projections spaces for each dimension (Table 1); applying C and table S3), many of which are regions of the onto the hyperplane containing chords summing produces 16 additional infinite-dimensional quo- spaces described in this report. These models have often been incomplete, displaying only a portion of the available chords or chord types and omitting ABCDEF singularities and other nontrivial geometrical and topological features. Furthermore, they have been explored in isolation, without an explanation of how they are derived or how they relate (10). Our model resolves these issues by describing the com- plete family of continuous n-note spaces cor- responding to the 32 OPTIC equivalence relations. Of these, the most useful are the OP,OPT, and on December 28, 2010 OPTI spaces, representing voice-leading rela- tions among chords, chord types, and set classes, Fig. 1. Progressions belonging to the same OPT and OPTI voice-leading classes. Each group exhibits the n respectively (4). The OP spaces T = n (n-tori same underlying voice-leading structure: Analogous elements in the first chord are connected to analogous S elements in the second, and the distances moved by the voices are equal up to an additive constant. (A)A modulo the symmetric group) have been iv6-V7 progression from Mozart’sCminorfantasy,Köchelcatalognumber(K.)457,measures13and14. described previously (9). The OPT space n−1 (B)Aprogressionfrommm.15-16ofthesamepiece,individuallyT-relatedto(A).(C)Aprogressionfrom T = n is the quotient of an (n – 1)-simplex, S Beethoven’sNinthSymphony,movementI,measure102,relatedto(A)byindividualTanduniformOPI. whose boundary is singular, by the rigid (D)Acommonvoiceleadingbetweenfifth-relateddominant-seventhchords.(E)Acommonvoiceleading transformation cyclically permuting its vertices 1. Le Tonnetz : évolution, applications et dérivations 13 n−1 www.sciencemag.org 58 Louisbetween Bigo tritone-related and dominant-seventh Moreno chords, Andreatta related to (D) by individual T. (F)Avoiceleading (4). The OPTI space T = n Z2 is the between tritone-related half-diminished sevenths, related to (D) by individual T and uniform I. quotient of the resulting spaceðS by an additionalÞ AB048 major key Downloaded from TM (k) dominant chordV 037 036 CM (k) tonic 027 026 025 024 chordI P(k) 016 015 014 013 012 subdominant IV chord tonic 006 005 004 003 002 001 000 ! !"#$%&$!"#$%&'!()*%&'!#*+!#,-(.*/.+!/&)#+0!#0!/1.2!#33.#&!)*!/1.!%&4)5%6+!&.3&.0.*/)*-!/1&..7 Figure 6 – La spiral array organisant les hauteurs suivant l’intervalle de quinte juste le *%/.!81%&+09!!:.&.'!/&)#+0!#&.!3#&/)8,6#&62!86%0.!/%!/1.)&!(#$%&7/1)&+!/&#*03%0)/)%*09! long d’une spirale verticale. L’étoile représente une tonalité dont la position est donnée ! ;1)*-0! 4.8%(.! 0%(.<1#/! (%&.! 8%(36)8#/.+'! 1%<.=.&'! <1.*! <.! 8%*0)+.&! /1.! par le barycentre de ses accords de degrésThree I, IV symbolic et V représentés spaces par des dedicated triangles pleins. to musical representations: the2 spiral2 array (left), the Fig. 3.1 +)08&./.!6#//)8.0!/1#/!&.3&.0.*/!=%)8.76.#+)*-!&.6#/)%*01)30!#(%*-!5#()6)#&!+)#/%*)8!%&!Fig. 2. (A) T / 3 is a cone. (B) T /( 3 ℤ2)isatriangle.Numbersrefertopitchclasses,with0=C,1=C♯,etc.Pointsrepresentequivalenceclassesof Tonnetz (centre) and a voice leading space (right) 81&%(#/)8! 81%&+09!transpositionally ! >%&! .?#(36.'!S (A) or>)-,&.! transpositionally @! &.8%&+0!S and /1.! inversionally (%0/! .55)8).*/! (B) related=%)8.! chords. Thus, 6.#+)*-0! (C, D, E) and (D, E, F♯)arebothinstancesof024. #(%*-!+)#/%*)8!5)5/10A<1)81!8#*!4.!&.3&.0.*/.+!,0)*-!#*!)&&.-,6#&'!%*.7+)(.*0)%*#6! D B)-7B#-!*.#&!/1.!8.*/.&!%5!/1.!"C4),0!0/&)3!! E!D9!!F;1.!B)-7B#-!0..(0!/%!4.!)&&.-,6#&!www.sciencemag.org SCIENCE VOL 320 18 APRIL 2008 347 Tonnetz comme un complexe simplicial 6 résultant de l’assemblage des triades représentées 4.8#,0.!/1.!5)-,&.!)0!+&#<*!,0)*-!/1.!81&%(#/)8!0.()/%*.!#0!#!,*)/G!<.&.!<.!/%!,0.!/1.! +)#/%*)8!0/.3'!)/!<%,6+!4.!&.-,6#&9H!!I40/&#8/62'!/1.0.!=%)8.!6.#+)*-0!5%&(!/1.!8)&86.! comme des 2-simplexes (i.e., des triangles) [Catanzaro 2011]. Cette formalisation du Ton- 01%<*! %*! /1.! &)-1/! %5! >)-,&.! @9! ! ;1.! 5)-,&.! +.(%*0/&#/.0! /1#/! /1.&.! #&.! 3,&.62! netz permet à Catanzaro d’appliquer cette construction aux 11 autres familles d’accords à 8%*/&#3,*/#6! &.#0%*0! /%! #00%8)#/.! 5)5/17&.6#/.+! +)#/%*)8! 5)5/10J! 5&%(! /1)0! 3.&03.8/)=.! 3 sons équivalents par transposition3.1 Introduction et inversion 7 et d’en étudier les aspects topologiques. KL'!MN!)0!86%0.!/%!KM'!ON'!*%/!4.8#,0.!%5!#8%,0/)80'!4,/!4.8#,0.!/1.!5)&0/!+2#+!8#*!4.! /&#*05%&(.+! )*/%! /1.! 0.8%*+! 42! (%=)*-! /1.! *%/.! L! ,3! 42! %*.! +)#/%*)8! 0/.39! ! P*.! Le chapitre 5 propose une approche simpliciale semblable pour la construction d’es- 5#08)*#/)*-!3%00)4)6)/2A<1)81!<.!,*5%&/,*#/.62!8#**%/!3,&0,.!1.&.A)0!/1#/!#8%,0/)8! #55)*)/).0!#8/,#662!!"#$%"&'#()!=%)8.76.#+)*-!5#8/0J!)/!)0!3%00)46.!/1#/!/1.!.#&!#00%8)#/.0! paces d’accords, en fournissantThis en work plus une concentrates méthode permettant on lathe généralisation harmonic et la aspect/1.!/1)&+!1#&(%*)8!%5!#!8%(36.?!/%*.!<)/1!/1.!0.8%*+!1#&(%*)8!%5!#*%/1.&!/%*.!#!5)5/1! of musical sequences and introduces systématisation de la construction de complexes, associés à tout type d’accord. #4%=.! )/'! #*+! /1.! 5%,&/1! 1#&(%*)8! %5! /1.! 6%<.&! *%/.! <)/1! /1.! /1)&+! %5! /1.! ,33.&'! )*! a spatial approach for its analysis. Musical analysis.55.8/!/&#8Q)*-!=%)8.76.#+)*-!&.6#/)%*01)30!#(%*-!/1.!3#&/)#609! often requires specific tools >)-,&.0! R7S! 3&.0.*/! /1&..! #*#6%-%,0! 0/&,8/,&.0J! >)-,&.! R! 8%**.8/0! /&)#+0! )*! /1.! 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