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F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 Computer Analysis of Jazz Chord Sequences:
Is Solar a Blues ?
François Pachet
Abstract:This chapter investigates the issue of the role of the computer in musical analysis. Starting with asurvey of the main approaches in computer analysis, we focus on the particular problem of Jazz chordsequences harmonic analysis. We propose a theory of chord sequence analysis, based on an explicitconceptual hierarchy of analysis objects. We discuss the implementation of the theory and its results ona typical example (Blues for Alice, by Charlie Parker), for which the system produces an analysiswhich conforms exactly to human interpretation. We also exhibit a chord sequence, Solar (by MilesDavis), for which the results of the system do not conform to human perception, i.e. it does not find it isa Blues. We conclude on the issue of the role for the computer in musical analysis.
Key words: harmonic analysis, Jazz, Blues, computer analysis, production rules.
There is a fascinating force governing musical analytical processes: the pleasure of possession. Sincemusical analysis became a pursuit in its own right, i.e. approximately the end of the XIXth century,various analytical methods have been devised and, to a large extent, formalized: formal, chordal,functional, Schenkerian, see e.g. (Bent & Drabkin, 1987) or (Cook, 1987). These methods differ in thenature of the musical material under study, and in the form of their output, but they are similar in theirgoal and operating mode: they consist in chopping the musical material into pieces, comparing thesepieces and classifying them, in order to eventually reformulate the original material with the terms of anestablished corpus of concepts. Through such a reconstruction, a successful analysis may eventuallyprovide a sense of possession, an intimate feeling of appropriation of the analyzed material which iscomparable to the feeling the composer has for his own creation. Regardless of the prominent placeanalysis holds in musical aesthetics and compositional theory, analysis, seen as a way of understandingmusic by reformulation, is an enjoyable process. This pleasure of analysis certainly accounts for a largepart in the leading role of analysis in musical studies.
The use of the computer as a partner in musical analysis is as old as the computer itself. There hasundoubtedly been a dream behind the use of computers in musical analysis: the dream of entirelyautomating the analytical task, to analyze quickly large corpuses of musical material. However, theresult is not clearly in favor of the computer. In 1980, Bo Alphonce argued that musical theory is notmature enough to be used as a basis for computer programs (Alphonce, 1980). This pessimisticstatement is somewhat confirmed by the absence of analysis programs in the Humdrum general-purpose software tool developed at University of Waterloo (Huron, 1994). Humdrum is one of the mostambitious attempts so far at providing computational power to musicians in order to perform complexanalysis of musical pieces. A laconic sentence may be found in the introductory documents: "Programsto do automatic functional analysis are not sufficiently reliable to be used in music scholarship".
However, lots of work have been done in the field of computer analysis and these works did produceinteresting and insightful results, if not fulfilling the dream of an autonomous, complete and automatic F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 analyzer. Indeed, the computer has assumed virtually any possible role in the analytical game: from asimple passive tool to a simulation of an intelligent analyzer agent. Artificial intelligence techniques inparticular have been used extensively to provide various kinds of frameworks in which analyticalprocesses may be investigated, studied, and highlighted in various ways.
This chapter investigates the issue of the role of the computer in musical analysis. Starting with asurvey of the main approaches in computer analysis, we focus on the particular analytical problem ofJazz chord sequences. We exhibit a model that allows to analyze automatically such sequences. Wediscuss its implementation and results on a typical example (Blues for Alice, composed by CharlieParker), for which the system produces an analysis which conforms satisfactorily to humaninterpretation. We also exhibit a chord sequence, Solar (composed by Miles Davis), for which theresults of the system do not conform to human perception - it does not find it is a Blues. We concludeon an emerging role for the computer in musical analysis.
Artificial Intelligence has traditionally been split into two categories of pursuits: developing techniquesto produce sophisticated artifacts, or building computational models of human cognition. These twocategories correspond to two main roles that have been assigned to the computer in musical analysis: 1)the computer is used as tool for musicologists, and 2) the computer is used as a simulator of a modeldevised by a cognitician to account for the analytical process. A third category of experiments is whenthe computer itself is the subject of study, and musical analysis becomes an experimental framework inwhich knowledge representation techniques are evaluated, and musical theories are empiricallyvalidated.
One of the most straightforward uses of the computer for musical analysis is software designed toperform various "low-level" tasks in order to help musicologists in their routine work. These tasks aretypically computations of frequency distribution for pitch classes or intervals in melodic lines. Variousforms of such statistical techniques were used in the 70s (Lincoln, 1970) to perform style analysis.
These techniques are appealing because they apply to all kinds of musical corpuses, including ethnicalmusic. The more recent work by (Mason, 1985) is in the same vein, using vectorial analysis andcomplex number notations and representation. However, the abstract musical concepts which arenecessary for fully-fledged analysis (e.g. cadences), are out of reach of purely numerical techniques,thus limit the scope of these systems (see e.g. a critical comment by (Rothgeb, 1971)).
In reaction to this numerically oriented trend of work, research was conducted using some symbolicrepresentation of music. Several software packages were developed to provide computer-aidedanalysis of tonal music (Kassler, 1975), (Byrd, 1977), (Brinkman, 1980) i.e. sets of programs to be usedby a human that perform limited, well defined analytical actions, but that do not propose a completeanalysis. In the same spirit, (Smoliar, 1980) developed tools for computer-aided Schenkerian analysis.
These tools were designed with the goal of understanding the intricacies of the theory - in this respect,they could also be classified in the third category, following section - but concluded with theimpossibility of building a totally automated analyzer (Frankel et al., 1978). More recent works by(Camilleri et al., 1987) attempted to combine the brute force of numerical analysis with smarter typesof hierarchical parsing in a single software package, also to be controlled by human (students ormusicologists). The work of Pierre-Yves Rolland (see his chapter in this issue) on pattern induction inJazz corpuses may be seen as a continuation of this trend, using more advanced numerical techniques F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 (dynamic programming), together with a rich representation of musical objects (the MusES system,(Pachet, 1994a)).
The work mentioned above usually applies to tonal music. As far as post-tonal music is concerned, letus mention two kinds of efforts in opposite directions. The theoretical work of Chemillier showed thatserial music is, in some sense, rational (Chemillier, 1990), but this result did not lead to actualimplementations of serial music analyzers. On the other hand pitch-set theory was primarily designedwith implementation constraints in mind. A tremendous amount of work has been devoted to buildingcomputer implementation of pitch-set theory, e.g. (Forte, 1973a; Forte, 1973b), (Rahn, 1980a), (Harris& Brinkman, 1986). The most recent attempt to provide a complete set of tools is the ContemporaryMusic Package by (Castine, 1994) which acquired the status of a real working system, as opposed tothe myriad of prototypes designed so far. Finally, the case of electronic music is special, since no stableanalytic theory is available, which poses new problems as yet unaddressed (Stroppa, 1984).
A more elaborate use of the computer as a tool for analysis is to validate specific, user defined theoriesof a musical piece. Here the computer is used as a simulator of a model, carrying its own semantics,rather than as a simple neutral tool. This is typically the case of the Morphoscope project, in which thecomputer is used to rebuild a score (Mesnage, 1993), with numerous and convincing applications(Mesnage & Riotte, 1990), (Rokita, 1996). An alternative and interesting work in the same spirit is thereconstruction of a fragment of Ligeti's Melodien by (Chemillier, 1995), using a model specificallydesigned for the material studied. The model is implemented with the Patchwork system (Laurson &Duthen, 1989). Through an explicit reconstruction of the entire score, these studies emphasize the ideathat analysis and composition have strong, organic relations (Mesnage, 1995).
The computer as a simulator of the analytic process The second main role of Artificial Intelligence is to model human cognition. This is an ambitious goalsince such models in principle, require experiments in cognitive psychology to be validated. Musicalanalysis is then seen as a particular form of a general, typically human ability. Using the computer as asimulator of such a model is one of the goals of Minsky's model of K-lines (Minsky, 1986), which hasbeen applied to the simulation of analytical processes in music (Minsky, 1985), including Jazz(Horowitz, 1995). In the same spirit, Greussay's Beethoven graphs (Greussay, 1973; Greussay, 1985)is a model for analyzing Beethoven's Diabelli variations. The idea here is not so much to analyze thematerial itself, but rather to validate a conception of the analytical process, seen as a kind ofcooperation between various autonomous agents. The models in this category are designed to becomputationally tractable, and therefore may also have practical applications, especially in the contextof performance-oriented analysis. The Cypher system for instance (Rowe, 1993), contains a real-timeanalyzer that draws on Minsky's model, in which agents essentially operate frequency computations(see chapter by Rowe in this issue).
More generally, the segmentation of musical data was identified as a general issue for cognitivesciences. Grouping and segmentation was addressed by Baker (Baker, 1989a; Baker, 1989b), whocompared two techniques applied to the same segmentation problem, and (Camilleri et al., 1990), whouse an expert system approach on the same issue. Quite appealing computational models weredeveloped to find boundaries in atonal music, e.g. Forte (Forte, 1973a) using pattern recognitionprocedures, or Polansky's hierarchical analysis inspired by the gestalt theory (Polansky, 1979), (Tenney& Polansky, 1980). Other interesting models were developed with the goal of modeling the perceptionof the human analyst but without direct corresponding implementations, e.g. (Hasty, 1978) or (Chouvel,1990).
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 The third category of studies corresponds to a radical shift in focus, compared to the two previousones: the computer becomes the actual subject-matter, instead of being used as a passive tool orsimulator. Moreover, music analysis is considered as a rich experimental field, a source of well posedproblems. The main goal is not so much to study musical corpuses or theories as such, but rather tostudy how certain types of knowledge can be put into a computer, and hence belong to the field ofknowledge representation at large.
Almost all AI techniques have been applied to some kind of analytical problem. Proceduralapproaches were used by (Ulrich, 1977), to analyze Jazz chord sequences, and various algorithms weredeveloped to realize specific forms of analysis (Brinkman, 1986; Brinkman, 1990), (Elliott, 1987),(Mouton, 1995). The work of (Winograd, 1968) for analyzing musical scores using systemic grammarspaved the way for a whole generation of studies on grammars. Grammar-based approaches becamevery popular in the 70s and were applied to virtually all available musical corpuses (see e.g. (Baroni,1984) and a review on the use of grammars for musical analysis in (Roads, 1988)), culminating in thereference work of (Lerdahl & Jackendoff, 1983). More classical AI mechanisms such as productionrules were used to develop analytical programs of all sorts: (Bein & Winold, 1983), (Maxwell, 1984;Maxwell, 1992). Languages from the area of logic programming were also put to work. For instance,Schenkerian analysis is performed as one of the viewpoint of Ebcioglu's choral generator (Ebcioglu,1992). His system is based on a particular logic programming language (BSL) developed specially forthe purpose of the application, and uses heuristics to control bottom-up parsers (Ebcioglu, 1987). Theformalism of conceptual dependency (Schank, 1973) was used by (Meehan, 1980) to implement theimplication/realization theory of Narmour. Representation of general-purpose musical structures foranalysis and composition were developed by (Smaill & Wiggins, 1990), who use an adaptation ofRuwet's analytic theory (Ruwet, 1972) to analyze pieces of Debussy's Syrinx.
This third category is large: it contains most experimental work in AI and Music. It is also the subjectof much litigation, as its objectives are not always clearly stated. Indeed, these approaches have all incommon a tendency to turn into an "exercice de style", with a limited impact: applying a sophisticatedtechnique to a complex problem may not show much more than the virtuosity of the technician. As(Smoliar, 1992) convincingly argues: "each technological advance becomes a new temptation to put the(musical) data into the computer again", and strong critiques were issued against this trend of work(see e.g. (Rahn, 1980b)).
However, there are often misconceptions about the objectives of these experiments. Although acomputer implementation per se may not prove anything on the music model from which it is inspired, itmay nevertheless provide two kinds of benefits. First, these achievements may actually be used tovalidate general knowledge representations technique, in a complex domain. In this context, musictheories are interesting for AI only to the extent that they are computationally tractable. These issuesare not purely technical, though. They embody an experimental view of Artificial Intelligence, in whichthe study of music is seen from a broad perspective of knowledge engineering, and where the goal is toelicit knowledge with practical objectives in mind, thereby raising important issues in knowledgerepresentation.
Second, even if the epistemological status of theories in analysis is not clear (after all, what is a theorywithout theorems ?), there is a growing community of analysts who look for musical theories havingpredictive capacities, to the point of becoming falsifiable (Kunst, 1987). In this regard, work in our thirdcategory may reveal the incompleteness of a theory, as it was for instance the case with the study of(Rothgeb, 1968) on unfigured bass, which revealed inconsistencies and omissions in official treatises ofharmony.
We will illustrate these two issues in the next section, on a simple and practical analytical problem, andanalyze its strengths and limitations.
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 The problem of harmonic analysis of Jazz chord sequences perfectly illustrates the third role of thecomputer in musical analysis. It is a well posed problem, of relative simplicity, and it may be used toillustrate our two arguments stated above. We will first state the problem in musical terms, reviewthree approaches on the problem, and discuss their limitations. We will then propose an analyticalmodel and its implementation using Artificial Intelligence techniques. The model will be illustrated ontwo examples, one successful (Blues for Alice), and one problematic (Solar). Finally, we will discusstwo issues raised by experimenting with such a system; one issue concerning musicology, oneconcerning Artificial Intelligence. We will finally conclude on the role of Artificial Intelligence incomputer analysis.
Like classical harmony, tonal Jazz harmony is a well studied domain, as one can see by the largenumber of books on this subject. This profusion of literature is directly related to the size of the Jazzmusical corpus, typically illustrated by books such as (Fake, 1983; NewReal, 1991; Real, 1981). Suchcorpuses contain approximately 2000 Jazz chord sequences, referred to by experts as "standards".
Most of these tunes were composed by Jazz musicians in the 50s (the be-bop period, including CharlieParker, Dizzie Gillespie, Miles Davis) and later (the so-called hard-bop period, and more recently and tosome extent the Jazz-rock period).
We will state here the problem of Jazz chord analysis informally, and propose a more rigorous theory inthe following sections. The problem consists in analyzing harmonic functions in chords in Jazz chordsequences, i.e. finding, for each chord in the sequence, its underlying tonality and its function in thistonality. The input of the problem is a chord sequence as found in the literature (see example Figure1). The output is an annotation of the sequence, in which each chord is labeled with its harmonicfunction. The harmonic function is usually represented by a degree (a number, written as a Romannumeral) and a tonality, itself consisting in a root (a pitch class) and a scale type, e.g. I of Cb major, IVof G minor, etc.
Figure 1. Chord sequence Blues for Alice composed by Charlie Parker, as found in the Real
Book (Real, 1981). Each square lasts four beats. Starting beats are indicated in small font on
the top left of each chord.
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 There is a distinctive feature of Jazz chord sequence analysis, which makes it rather unique in the fieldof analytical studies in general: the fact that the output of the analysis has an unusually precise andpractical goal. These analyses are typically performed prior to improvisation because they yield validscales the soloist may use to build his chorus. Indeed, the starting point of improvisation is the fact thatthe harmonic function (e.g. IV of G minor) contains enough information to deduce the set of "valid"notes that can be used, as well as, for each of them, their relative importance (the notes of the tonality,starting from the given degree, and alternatively considered as strong and weak notes). We do notaddress here the problem of how precisely improvisation may be built from such an analysis, which hasreceived much attention (see e.g. (Ulrich, 1977), (Fry, 1980), (Giomi & Ligabue, 1991), (Johnson-Laird,1991), (Walker et al., 1992), (Baggi, 1992)). But in all cases, a good improvisation relies on a preciseharmonic analysis of the chord sequence. Moreover, not only tonalities and harmonic functions areused, but also more abstract harmonic patterns, such as two-fives, or turnaround, which are directlyassociated with memorized "licks" from which the improviser builds his melodic lines (see, e.g. themodel of improvisation developed by (Ramalho & Ganascia, 1994), based on the association of suchlicks with such harmonic patterns using a case-based representation of musical memory). Becausethese patterns play a central role in the improvisation, their recognition in Jazz chord sequence is one ofthe main duties of analysis.
This practical aspect of analysis has an important consequence concerning its hierarchical nature: atune may be globally in C major, but some parts may be in F (modulation), and these parts maythemselves contain sub parts that modulate, and so forth. Firstly, the level of embedded modulationsmay be high in a typical Jazz chord sequence, as opposed to the usual levels of embedded modulationfound in e.g. Baroque music. This situation arises mainly because of chord substitutions, which areused systematically and recursively. Second, contrarily to classical music in which only the surfacelevel of embedded modulations are usually interesting, in Jazz all levels may potentially be relevant,because the improvisation will build musical phrases of varying size, and thus will be based on tonalitiesof varying depth, as was shown experimentally by (Järinen, 1995). A chord therefore is not reallyanalyzed in one single tonality, and several tonalities may be considered, depending on the depth of theanalysis considered. In example of Figure 2, the first chord (A7) may be considered in the tonality of D,from a short-sighted viewpoint, or G, from a higher viewpoint, or, even here, C. In each case differentscales would be used for improvisation, and the choice of the tonality depends on the nature of themusical improvisation being conducted. Note that we consider the problem of choosing a "best" tonalityfor chords to the responsibility of the improviser, and not to the responsibility of our analyzer.
Figure 2. The hierarchical nature of the analysis of a group of chords.
We will now review three typical attempts in building systems that perform such a chordal analysis taskautomatically.
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 Previous work on automatic chordal analysis We review briefly three exemplar systems that propose a solution to the chord sequence analysis,emphasizing their main advantages and drawbacks. The first two are operational systems, the third oneis a model with no implementation.
Maxwell's system (Maxwell, 1984; Maxwell, 1992) produces chord function analysis of Baroque piecesusing a knowledge-based, expert system approach. Based on the preceding work of (Bein & Winold,1983) in the automatic analysis of Bach chorales, Maxwell considers two main tasks for chord functionanalysis:1 - Determine which vertical sonorities constitute a chord worthy of a function label (this problem isirrelevant in our case since chords are already explicitly mentioned on the score).
2 - Determine the tonal regions in which chords should be analyzed.
The chord analyzer produces an annotation of the score similar to a human annotation, i.e. consisting inchord functions for each chord of the sequence. These functions are roman numeral labels togetherwith tonalities. The output of the analysis is therefore flat, which is justified by the nature of the corpusbeing analyzed (Baroque music), in which the hierarchical aspect of analysis is not of centralimportance. One of the main goal of the system is to minimize modulations. The reasoning consistsmainly in detecting cadences, and then interpreting the rest of the chords according to these cadences.
The system is implemented by a set of production rules associated with numerical priorities, andhandled by a prioritized agenda to control conflict resolution. It proceeds from left-to-right, and decidesto modulate only when some numeric threshold of functional weakness is exceeded. The most obviouscritique that can be issued is the intensive use of numerical values in rules. For example, rule 43 states:"if the root motion from pre-cadence chord to goal chord of a p-cadence is a descending perfect fifthAND the pre-cadence chord contains a major third above the root, THEN the p-cadence is authentic,and its strength should be increased by 20". The drawbacks of using numerical values in rules has nowlong been established: they are difficult to justify, difficult to maintain, and have poor explanatorycapacity. Also, the proposed approach has a strong procedural component and the left-to-right schemeis counter-intuitive. To reduce that rigidity, rules such as rule 41 are introduced (and used extensively):"If the chord cannot be analyzed in the key of the previous chord OR the analysis is "doing badly"AND enough is known about future keys to look for a better key THEN determine which key willprovide the strongest analysis of the new few chords or measures, modulate to that key, analyze thepresent chord in the new key, and assign pivot functions to some previous chords". However,Maxwell's system produces interesting and occasionally insightful results for musicologists.
The system described by (Ulrich, 1977) solves the harmonic analysis problem for simple Jazz chordsequences, as a part of a general system for building Jazz improvisation. Similarly to Maxwell's system,Ulrich's system produces a flat, one-to-one labeling of chords, in which each chord is assigned to a keyenter (a tonality) and a function, such as "tonic", "subdominant", or "transition". As in Maxwell'ssystem, the analysis system tries to minimize the modulations, i.e. changes in tonality, but uses asymbolic rather than a numeric tactic. Its implementation is also based on a left-to-right algorithm thatparses the sequence. Chords are progressively "eaten" when they can be analyzed in the tonality of thealready analyzed sequence. The system has an appealing organic quality: it is based on an "island-growing" mechanism in which isolated groups of chords (islands) try to grow as much as possible toencompass adjacent chords. Ulrich concludes his presentation by noting that "Jazz encodes so much F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 harmony in the local structure of the music that global considerations can be ignored". Indeed, hissystem may be used to analyze simple chord sequences (How high the moon, by Morgan Lewis, not aterribly difficult one), and Ulrich proposes a simple algorithm for building improvisations asjuxtapositions of motifs, adapted to the harmonic functions found by the analysis system. Expectedly,the quality of the system's improvisation is poor (as stated by the author himself). One reason is thatthe analysis system provides only a small part of the information required to build a more elaboratemelody. It does not detect the specific harmonic patterns of Jazz harmony, and cannot detectembedded modulations. It is therefore unable to provide a hierarchical, semantically rich view of theglobal tonality.
The model proposed by (Steedman, 1984) aims at describing 12-bar Blues, a particular, and important,subset of Jazz chord sequences. One important characteristic of the model is that it explicitly takes intoaccount the hierarchical nature of Jazz chord sequences. The model is particularly representative of thetrend of grammar-based research: it only describes the corpus, and is not intended to be directlyimplemented.
Steedman identifies a set of 6 transformation rules that model the 12-bar Blues, by applying,recursively, various transformations to an initial, simple chord sequence, considered as an "essential"Blues. These rules are showed to be sufficient to generate a large set of complex 12-bar Blues tunes.
The output of Steedman's model (produced manually) is a derivation tree that displays all thetransformation rules to apply to the initial axiom of the Blues to reconstruct the given input chordsequence.
A spectacular aspect of this work is the fact that a small set of rules captures a large amount ofpossible variations from the original 12-bar template. Steedman's model is, in a way, validated by theexistence of a large quantity of 12-bar Blues, that all fit nicely in this formalism, i.e. can all begenerated by his set of rules. It is difficult indeed to resist the appeal of this model, and not see it as asort of "Maxwell equations" (no relation with the author mentioned in the preceding section) or"abstract truth" of the Blues, as quoted by Steedman from a song by O. Nelson (Nelson, 1961).
However this model suffers a number of drawbacks. First, the model is not implementable. This comesessentially from the presence of so-called context-dependent rules of the model. Theoretical work ongrammar parsing concluded that context free grammars can be parsed using automatic tools, butcontext dependent grammars are much more problematic (see e.g. (Roads, 1988)). For instance, rule(4) is as follows: Dx7 x(7) → bStx(7) x (7). This rule states that any seventh chord that resolves (thex(7)) can be substituted by the seventh chord of the tritone (bStx stands for "flattened supertonic of x").
This rule is inherently context-dependent because the application of the substitution may occur onlywithin a particular context (x(7)). Other rules are even more problematic from a computational point ofview. Rule number 3 writes as follows: w x7 → Dx7 xm7, where w may match any chord. However,to avoid problems of infinite loops, it is necessary to impose the constraint that w should not match achord that has had its root changed by the previous application of a substitution rule! The second problem is that the model only accounts for "well-formed" sequences, and therefore is verysensible to perturbations, idiomatic progressions and other harmonic "mistakes". Similarly, while themodel captures some 12-bar Blues, there are a lot of other "initial" chord sequences from which Jazzchord sequences can be derived. Moreover, interesting Jazz chord sequences often do not derive froma particular, known, initial chord sequence (e.g. Nardis by Miles Davis). Finally, the model produces anoutput that is only a small part of the analysis. When the tune is analyzed as a Blues, the modelprovides a derivation tree from which is not clear how to find the underlying tonalitie s (e.g. to be usedfor improvising). When the tune is not analyzed as a Blues, the model simply answers "no", even if F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 portions of the tune could have been correctly analyzed. Note, however that such a grammar can beused without problem in a generative mode, to produce variations of existing chord sequences, asproposed e.g. by (Johnson-Laird, 1991). In short, the proposed model accounts convincingly for therecursive nature of bluesy substitutions in chord sequences, but does only a small part of the analyticaljob.
We will describe now our solution to the problem of automatic Jazz chord sequence analysis.
Conversely to the systems presented here, our system produces a hierarchical, complete analysis ofJazz chord sequences that is robust to harmonic mistakes. It uses an entirely symbolic approach (nonumeric values), and can be seen as an extension of Ulrich's island growing mechanism. Finally, itsoutput may directly interpreted to produce tonalities to be used for improvisation. We will firstemphasize the role of analysis objects in the reasoning, and propose an ontology for these objects. Thenwe will propose a model for the reasoning process based on this ontology. Finally we outline theimplementation of the theory using AI techniques and discuss its results.
As we said in the introduction of this chapter, the very act of analysis consists in recreating the chordsequence. This re-creation involves the reconstitution of a imaginary process of composition thatproduced the sequence. However, in the preceding approaches to the chord analysis problem, thisreconstitution is not made explicit, since there is no real language for expressing it. Ulrich's andMaxwells's systems use a scarce representation for the chord labeling ('dominant', 'subdominant',etc.). There is no representation of the abstract entities that are manipulated by the analysis process, sothere is no real reconstitution in this sense. Steedman's model produces a tree of rule derivations whichcan be seen as a kind of reconstitution, since it expands an initial, essential axiom of the Blues down tothe complete chord sequence. But this reconstitution is effected in a abstract, uniform world, in whichthe only reconstituted action is the anonymous application of a generative rule (in technical terms, thetree is not an abstract syntax tree, i.e. nodes in the tree are not labeled with harmonic functions.) In the particular context of Jazz, the output of the chord analysis plays, as we have seen, a mostimportant role. Not only is the analysis inherently hierarchical, to account for the many tonal levels, butits expression must coincide with the patterns identified by Jazz musicians, on which they can build theirmelodic lines. The flat outputs, as well as the semantically neutral derivation tree are not adapted to thisaim. We claim that a relevant representation of the analysis should, in this case, be a hierarchicalstructure (a tree), whose nodes denote meaningful analytical abstractions.
These abstractions can be classified into two main categories: basic building blocks that representpartial chord sequences, and harmonic operations, that manipulate these building blocks to producechord sequences. The main idea of our approach is to explicitly represent both building blocks andharmonic operations, by turning them into abstract concepts with which the analysis, in fine, will beformulated. More precisely, we propose to classify all the objects making up the analysis under acommon concept: the shape. A shape is a temporal object, describing a collection of chords. The basicbuilding blocks as well as the reified harmonic operations are represented as special kinds of shapes.
The basic building blocks are simply the chords themselves, considered as atomic entities, as well as asmall number of fixed idioms belonging to the corpus under study. These idioms are typical, cliché,sequences of chords which bear harmonic meaning in themselves, such as turnarounds, two-fives, ortwo-five-ones. At a higher level, global macro forms such as Blues or AABA are also considered ashigh-level idioms, to which we add, for the sake of completeness, shapes describing chord sequenceswith less prominent structure (MonoTonalShape, BiTonalShape, etc.).
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 The harmonic operations consist in producing chord sequences by combining or modifying existingchord sequences, according to some combination rules. Typical harmonic operations are chordsubstitutions, whereby a chord in a certain context is substituted by another chord. More generalforms of harmonic operations include, for instance, ExtendedShape, i.e. shapes obtained by extendinga shape with a chord analysable in the tonality of the shape. Particular cases of shapes such asModalBorrowing are also classified in this category (see below). This classification results in thehierarchy of concepts illustrated in Figure 4 (the inheritance relation represents a generalization relationbetween types of shape).
An important characteristic of shapes is that they manifest themselves through well identified patternsof chords, or of other shapes. To each shape in the catalog of shapes corresponds such a pattern.
These patterns can be found in most books on Jazz harmony (Coker, 1964) (Beaudoin, 1990). Forinstance, a turnaround shape such as (C maj7 / Eb7 / Dmin7 / Db7) is perfectly identified by a set ofharmonic relations between adjacent chords. Here the relations could be abstracted by: (a chord of rootX, a chord of root (X + minor third) seventh, a chord of root (X + second minor ) seventh, a chord ofroot (X + diminished second 7). Similarly, a Blues shape may be described as three adjacent shapescovering the whole sequence, such that the first and last one are in the same tonality, and the middleone is analyzed in the fourth of the others.
More complex shapes such as "modal borrowing" follow similar patterns. A modal borrowing is a localmodulation which may be considered as non significant, when it comes in between two shapesanalyzable in the same tonality, and when this local perturbation may be analyzed in the relative minortonality of the adjacent shapes. This pattern is illustrated in Figure 3.
Figure 3. Modal borrowing configuration. Here, the local perturbation is a Ab major chord.
Ab major may be analyzed in C minor (VIth degree) and therefore be considered as a modal
borrowing in C major.
TurnAround1TurnAround2TwoFiveTwoFiveOneResolv ingSeventh BluesShapeAABAShapeABABShapeMonoTonalShapeBiTonalShapeTriTonalShapeQuadriTonalShapePentaTonalShape F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 ComposedShapeExtendedShapeExtendedIntersectedShapeTransitionShape Figure 4. The ontology of analysis objects. Indentation denotes inheritance.
Based on our ontology of analysis objects, we propose to specify formally the problem around the threefollowing points: A) Basic principlesThe theory is based on two basic principles: legality and minimization.
This principle says that each chord, out of any context, can be analyzed in a fixed set of possibletonalities. For instance, a given C major chord may be analyzed as: I degree of (the tonality of) Cmajor, IV of G major, V of F major, VI of E harmonic minor, and so forth. This legal set may be simplycomputed, once for all, by extracting scale-tone chords from all possible scales (Pachet, 1994b).
In the context of a sequence, the choice of the "good" tonality for a chord will of course depend on itsmetrical location, and on its relation with adjacent chords. The idea here is that the best tonality will bethe one that minimizes modulations, i.e. the one that is common to the greatest number of adjacentchords. For instance, the chord sequence (C / F / E min / A min) has only one tonality that is commonto all chords: C major.
B) Shape identification and analysisAs we saw, a central hypothesis in the analysis is that to each shape in the ontology of analysis objectscorresponds a configuration of chords that perfectly identifies the shape. Moreover, recognised shapeshave specific tonalities, which can be computed directly from their structure. For instance, aturnaround such as (C maj7 / Eb7 / Dmin7 / Db7) should always be analyzed in the tonality of its firstchord, here C major. The main problem - and exciting part - of analysis comes from the fact that theanalysis of a shape may violate the legality principle for some of its chords. The turnaround (C maj7 /Eb7 / Dmin7 / Db7) should be analyzed in C major, regardless of the fact that C major does not belongto the legal set of Db7 and Eb7. These chords in abstracto may not be analyzed in C major, but theycan be within a given shape. In other words, idioms are configurations of chords that bear harmonicmeanings in themselves.
C) RecursionFinally, our analysis is recursive: any recognized shape may itself be considered as atomic for a higherlevel of analysis. This recursive nature accounts for the hierarchical nature of the analysis, and is of theutmost importance in Jazz as we argued in the Problem Statement section. For instance, resolvingseventh chords may be considered as preparations, and therefore may be integrated in their resolvingchord, as illustrated in Figure 2.
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 The output of the system and its interpretation Let us put our theory of analysis to work on the sequence Blues for Alice of Figure 1. The output ofour system is illustrated in Figure 5, in which the tree is to be interpreted as follows. Each linecorresponds to a node in the analysis tree. The node is labeled by the lapse (starting and ending beats),and the name of the shape identified. Here, the chord sequence in itself is recognized as a Blues in Fmajor, as indicated by the root of the tree (1-48 BluesShape in F MajorScale). The three subshapesmaking up the Blues are 1) 45-4 ResolvingSeventh in F MajorScale, 2) 5-20 ChordSubstitution inBb MajorScale and 3) 17-48 ExtendedShape in F MajorScale. Each of these three shapes in turnis decomposed into various shapes, until the chords are reached. The explanation for the starting beatof the first shape (45-4) is that the identified shape starts at the end of the sequence (beat 25) and endsat the beginning (beat 4). More details on this aspect are given in the section on Circularity.
5-14 ChordSubstitution in C HarmonicMinor 5-10 ChordSubstitution in D HarmonicMinor F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 Figure 5. An analysis of the chord sequence of Figure 1. Leaves are between brackets and
correspond to the chords of the sequence. Intermediary nodes correspond to recognized
shapes (e.g. two-five, chord substitution), preceded by their respective start beat and last
beat, and followed by their tonality.
Let us mention here again the fact that such a tree actually fills the need of the improviser, which is themain driving force behind this model. The tonality of a chord depends on the width of the musicalphrase being played, as mentioned in C) of the preceding section. This tonality is given by the tonalityof the corresponding segment in the chord sequence. Here the tonality of a segment is determined bychoosing the smallest shape in the tree that contains the segment. For instance, the analysis of chord(Bb min 7) in measure 21, seen with a short segment of two bars, would be Ab major. Seen from alarger segment starting at beat 21 and ending at beat 28, it would be D major (shape 21-28ChordSubstitution in D Major). Considered from an even higher level, say beat 21-40 it would be Cmajor. A very long segment from beat 21 to, say, 44 would consider the chord as being in F, and soforth. Four different tonalities may thus be considered for this single chord, depending on the span ofthe musical segment considered for improvising. The tree not only gives a variable tonality for a givensegment, but also yields a corresponding shape type, which can then be associated to various musicalidioms (see again the work of P.-Y. Rolland, in this issue, on the detection of such formulaic patterns).
These two aspects make the output of the analysis system directly usable for improvisation.
The reasoning process as described in our theory is represented by 1) an object-oriented representationof the analysis objects, 2) a representation of the reasoning by first-order forward-chaining productionrules, and 3) a model of circular time.
The first aspect does not pose any problem. Object-oriented languages implement conceptualhierarchies by the mechanism of class inheritance, which is, in our case, perfectly adapted forrepresenting the simple tree-like hierarchy of our musical concepts.
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 The second aspect is more interesting, since the problem is to simulate the process by which theanalysis tree is built. Nicholas Cook in his treatise (Cook, 1987), p. 16, says: "there are essentially twoanalytical acts: the act of omission and the act of relation". In our context, this metaphorical statementturns out to be unexpectedly operational. Inspired by the work on temporal reasoning of (Dojat &Sayettat, 1994), we used a model of analytical reasoning based on two main analytical actions:aggregation and forgetting. The reasoning process per se is represented by a set of rule bases whichperform two kinds of tasks, by observing the initial chord sequence (in no particular order):• a "pattern recognition" task in which shapes are built by aggregating configurations of already • a "forgetting" task, in which irrelevant or redundant shapes created in the preceding task are Pattern recognition (or aggregation) rules all follow the same pattern: they consist in detectingconfigurations of adjacent shapes (the IF part, consisting in sequences of logical assertions separatedby periods), and in building larger shapes by aggregation (the THEN part). The semantics of a rule isgiven by the types of shapes that are detected, the harmonic relations between them, and the type ofshape that is created. For instance, Rule 1 below recognizes a two-five in major (such as Dmin7/G7), inan english-like syntax: recognize major Two-FiveFOR any c1 c2 instances of IsolatedChordIF c1 is minor.
c1 has no flat fifth.
c2 is after c1.
c2 is major.
c2 has a minor seventh.
c2 root = fourth of the root of c1.
THEN
Create a TwoFive object, covering durations of c1 and c2, and analyzed in the tonality: fourth of the root of c2,
major scale.
Rule 1. A rule to detect a "two-five" shape. The rule is executed for all couple of objects
matching the IF part. These objects are instances of any subclass of Shape.
Other rules describe shapes such as resolutions (A7 / D), turnarounds, and substitutions, as well asmore comple x shapes such as modal borrowing. Aggregation rules are also used to describe macroshapes, covering the whole chord sequence. For instance, the pattern identifying a Blues shape may beexpressed by Rule 2: recognize BluesFOR any s1 s2 s3 instances of ShapeIF s1 is analyzed in X.
s2 is analyzed in Y, where Y = fourth of X.
s3 is analyzed in X.
s2 is after s1. s3 is after s2.
Begin beat of s1 = 1.
End beat of s3 = length of the chord sequence.
THEN
Create a BluesShape object, covering the whole chord sequence and analyzed in X.
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 Rule 2. A rule to recognize a Blues by a succession of three shapes.
The second type of rules describe typical situations in which recognized shapes may be safelyforgotten, to speed up the reasoning process, and avoid combinatorial explosion. Such rules embodyknowledge on "omission" according to the vocabulary of (Cook, 1987), i.e. situations in whichanalyzable shapes may be safely forgotten, and removed from the working memory. A typical exampleis Rule 3, that allows safely forgetting a shape without loosing information, when it is subsumed byanother shape of the same tonality: removeSubsumedShapes
FOR s1 s2, instances of Shape
IF
s1 subsumes s2.
s1 is different from s2.
s2 is analyzed. s1 is analyzed.
Tonality of s1 = tonality of s2.
Rule 3. A rule to forget irrelevant shapes.
The system as it is presented here suffers from a problem related to the harmonic stability of startingchords. Because the system favors longer shapes rather than small ones, it may happen that smallshapes occurring at the beginning of a tune may be eaten up by larger shapes following them, whenthese shapes are in close tonalities. For instance, in the tune Blues for Alice of Figure 1, the underlyingtonality of the starting chord (F major) could be interpreted in the tonality of the second shape, i.e. inBb major, as a IVth degree of Bb major. This would have the advantage of forming a large shape in Bb(ExtendedShape, in our ontology). However, it is not the right analysis here, the initial chord in F beingclearly a 1st degree of F. Our experiments showed that this correct analysis can be ascertained bysimply remarking that the tune is circular. By linking the starting chord with the unresolving seventhchord (C 7) of the end of the tune, the system is then able to discover a shape of non atomic length inF: the unresolving end of the tune ensures the tonal stability of the beginning of the tune (Cf. Figure 6).
Figure 6. A normal 2-5-1 on the left. A 2-5-1 that wraps around the end/beginning of the song
on the right.
More generally, we frequently need to manipulate abstract temporal shapes that can wrap around thebeginning of a song. Although we could use a purely linear model of time (such as Allen's), this wouldimply a systematic test for each shape to recognize. This led us to introduce a circular representation oftime in our model, described in details in (Pachet et al., 1996). This model allows us to describeconfigurations of shapes in a tune using circular relations, instead of the usual linear relations. Weshowed that such a model allows the reasoning system to correctly recognize the harmonic stability ofinitial shapes.
The overall reasoning is therefore represented as a series of rule bases alternating shape recognitionand shape forgetting, using the conceptual hierarchy and a circular model of time. The precise list ofrules and scheduling of tasks in described in details in (Pachet, 1997). At the end of the reasoning F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 process, a complete analysis tree is produced, such as the one in Figure 5. The system proved capableof analyzing correctly most standard chord sequences found in the reference corpuses, including Bluessequences deemed difficult by (Steedman, 1984), as well as other non-Blues chord sequences (fromAutumn leaves to Nardis or Stella by starlight). Let us mention now one problem that illustrates thelimitation of the approach.
Solar is a tune composed by Miles Davis in the 50's (see Figure 7). This tune is generally recognizedas a Blues in C minor by Jazz musicians. However, to our knowledge, no system, including ours, wouldbe able to classify it under this category, hence the "Solar problem". Maxwell's system would probablyrecognize cadences and resolutions, but would not say anything about the global structure. Ulrich'ssystem does not provide a hierarchical view of the piece either, and would not recognize embeddedmodulations, nor specific harmonic patterns. Steedman's grammar would simply return false, i.e. thetune is not a Blues, because it cannot be derived from its set of rules (we did not actually prove thisimpossibility, but only tried unsuccessfully to find the derivation).
Our system produces an analysis which is not what a human would do, but which is neverthelessinteresting: a Pentatonal shape, globally analyzed in Bb major, and decomposed as follows: 1-12: an ExtendedShape in Bb major.
9-24: an ExtendedSh ape in F.
25-36: a ChordSubstitution in Eb.
37-44: a ChordSubstitution in Db.
45-48: a ChordSubstitution in C major.
This result is surprising at first glance, but looking at it in more detail it is not absurd. The first shape isanalyzed in Bb major, which is false, but harmonically plausible. The mistake here is to try to make alarge shape including the three first chords, and Bb major does allow to do that. C minor - the correctanswer - was actually discovered, but removed in subsequent steps of the reasoning process, becauseit could not resist the weight (in size) of the larger Bb shape. The problem comes from the ending two-five, which is correctly analyzed as a two-five in C major, but which is here a kind of "mistake". Hadthe final two-five been in C minor - for instance by having a (D min 7 flat 5) instead of (D min 7) - theinitial shape in C minor would have acquired the necessary stability to establish itself as the righttonality, thanks to the circular reasoning explained in the preceding section. The culprit is the system'sintolerance on major/minor substitutions which are so prevalent in Jazz tunes. But this is a "minor"mistake, from our point of view, which could be corrected by relaxing the major/minor constraints in theaggregation rules. The rest of the analysis is correct, but of course the Blues structure is notdiscovered. Recall that in order to recognize a Blues shape, the system ought to find three shapes,related by the constraints outlined in Rule 2. The main problem here is the inability of the system to"aggregate" the local streak of eccentricity of measures 37-44 (incidentally correctly analyzed in Db)into the passage in Eb, which itself should be interpreted as a major equivalent of C minor, in order to"see" the underlying, hidden Blues structure. Instead, the system simply sees a PentatonalShape(there are eventually five shapes covering the whole sequence), whose tonality is, by definition, thetonality of its first shape, hence the erroneous global tonality in Bb major.
F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 Figure 7. The tune Solar by Miles Davis is a Blues, but how can the system see it ?
A human analysis of Solar could be the following: the tune is a Blues in C minor, with a set ofsubstitutions that form a logical sequence of 2-5-1. The composer chose to extend this sequence, by a"logical" 2-5-1 (the Eb min 7 / Ab 7 / Db maj7 part in measures 37-44). He did so, however at the costof breaking the initial Blues structure, since the ultimate 2-5-1 is no longer analyzable in the tonality ofEb major (or C minor). But the resulting sequence is a Blues, so a musician would feel, because somekind of essential property of the Blues is still preserved. That this essential quality escapes our system,as well as other approaches on the same problem, does not prevent the musician to see Solar as aBlues, and of the best sort.
We feel that the solution to the problem of Jazz chord analysis as it is described here is a good solutionin the sense that it allows to represent faithfully a large corpus of knowledge related to harmonicanalysis, and that it is validated by the results obtained, especially in comparison with other approaches.
One of the main qualities of the system is its ability to analyze "incorrect" tunes, at least partially, and toproduce results which are directly understood, and usable, by humans.
This experiment raises theoretical and technical issues for computer scientists. The mechanismdescribed here was the source of a larger work on ontologies of reasoning mechanisms, which resultedin a general framework for representing hierarchical temporal reasoning, of which our analysis is aspecial case. The framework has an important application in the medical field, for the automatic controlof respiratory devices in intensive care units (Dojat et al., 1997). Another issue is the comparisonbetween the power of an entirely constructivist approach such as ours, using forgetting rules, withdescriptive methods such as generative grammars, which rely on backtracking mechanisms.
The experiment also raises issues concerning musicologists. In our context, a circular model of timeallowed to solve a technical problem. But the importance of circularity in Jazz tunes remains an issuefor musicologists interested in the harmony of Jazz. The question of whether or not circularity has animportance is posed here in unusually precise terms.
Another issue concerning musicologists is the status of theory, as exemplified by the Solar problem. Inthe context presented here, the Solar problem could be solved easily in two ways, none of which areconvincing. The first solution would be to add more rules to the system (ours or Steedman's). A rule F. Pachet, (1997) Computer Analysis of Jazz Chord Sequences: Is Solar a Blues? in Readings in Music andArtificial Intelligence, Miranda, E. Ed, Harwood Academic Publishers, February 2000 could say for instance, that particular 5-part shapes such as the one our system discovered are indeedBlues. We could also devise a rule that somehow aggregates a part in Db with a shape in Eb,according to some "continuation" principle. In all cases, these rules would clearly play the role of adhoc patches, and would also lead the system to qualify non Blues chord sequences as Blues. The othersolution would be to simply stipulate that Solar is not a Blues, at least in its primary form, and thatconsidering it a Blues is a matter of taste, personal education, or some other kind of irrational humanconduct, which lies beyond the responsibility of a pure analyst. In both cases, the solutions amount togiving up on the explanatory power of the model. This failure simply shows that theories are not strongenough to support the construction of simulators with a degree of sophistication attained by humanexperts. Our Jazz analysis system will probably never be able to produce subtle arguments pro and conwhen asked the question "is Solar a Blues or not ? ", not because the techniques used are bad, butbecause there is no reasonable theory of the Blues available.
However, there are limitations to purely automatic analysis from surface input only, such as, in ourexample, a bare collection of chord names. Two directions of research may prove interesting to escapefrom the world of syntax. One important aspect of musical analysis is the semantics implicitly used forinterpreting musical data. New proposals have been made for instance by Steedman in (Steedman,1995) in this direction, using the two dimensional space of (Longuet-Higgins, 1962).
Another direction concerns the relation between analysis and emotions, as we suggested in theintroduction of this chapter. Automatic analysis could integrate results in Cognitive psychology, bytaking into account emotional aspects of musical perception. (Riecken, 1992) showed how arudimentary model of emotions could be used in a model of musical creativity. Additionally, integratingmodels of musical memory in analytical models (as argued by (Smoliar, 1992)) could help understandingthe inner mechanism of analysis by posing the same questions from a different perspective. Thequestion would then not be about an abstract truth (Is Solar a Blues ?), but rather about a subjectivejudgment (do you think it is a Blues ?). An explicit representation of musical memory, accounting forpast, active, and emotionally rich experiences with numerous Jazz chord sequences could then providean answer such as: "I enjoy Solar as a Blues".
I wish to thank Professor Jean-François Perrot for his careful and patient reading of earlier versions ofthe manuscript, his insightful advice and his continuous support.
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