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Society for Music Theory

Functional Harmony Revisited: A Prototype-Theoretic ApproachAuthor(s): Eytan AgmonSource: Music Theory Spectrum, Vol. 17, No. 2 (Autumn, 1995), pp. 196-214Published by: University of California Press on behalf of the Society for Music TheoryStable URL: http://www.jstor.org/stable/745871

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Functional Harmony RevisitedPrototype TheoreticpproachEytanAgmon

Functional Harmony RevisitedPrototype TheoreticpproachEytanAgmonI. PRELIMINARIES

"How do we sort the objects, people, events, and ideas inthe world into their proper categories? What transforms the'booming, buzzing confusion' that enters our eyes and earsat birth into that orderly world we ultimately experience andinteract with?" With these questions Stevan Harnad intro-duces an important collections of essays entitled CategoricalPerception: The Groundwork of Cognition.' As Harnad im-mediately explains,Categorical perception occurs when the continuous, variable, andconfusablestimulation hat reaches the sense organs s sorted out bythe mindinto discrete, distinctcategorieswhose memberssomehowcome to resemble one another more than they resemble membersof other categories. The best-known example is color categories:Physically speaking, colors differ only in their wavelengths, whichgraduallyget shorteracross the spectrumof visible colors. What wesee, however, are qualitativechanges, from red to orangeto yellowto green, and so forth.The same is true of musicalpitches:Graduallyincreasingfrequenciescan come to be heard as categoricalchangesfrom C to C-sharp o D to E-flat. A lesser-knownexample is "stop-consonants":(synthesized) "ba," "da," and "ga"also vary along a

An early version of this paper was presented at the Annual Meeting ofthe Society for Music Theory, Kansas City, 1992.'Ed. StevanHarnad(Cambridge:CambridgeUniversityPress, 1987), ix.

I. PRELIMINARIES"How do we sort the objects, people, events, and ideas inthe world into their proper categories? What transforms the

'booming, buzzing confusion' that enters our eyes and earsat birth into that orderly world we ultimately experience andinteract with?" With these questions Stevan Harnad intro-duces an important collections of essays entitled CategoricalPerception: The Groundwork of Cognition.' As Harnad im-mediately explains,Categorical perception occurs when the continuous, variable, andconfusablestimulation hat reaches the sense organs s sorted out bythe mindinto discrete, distinctcategorieswhose memberssomehowcome to resemble one another more than they resemble membersof other categories. The best-known example is color categories:Physically speaking, colors differ only in their wavelengths, whichgraduallyget shorteracross the spectrumof visible colors. What wesee, however, are qualitativechanges, from red to orangeto yellowto green, and so forth.The same is true of musicalpitches:Graduallyincreasingfrequenciescan come to be heard as categoricalchangesfrom C to C-sharp o D to E-flat. A lesser-knownexample is "stop-consonants":(synthesized) "ba," "da," and "ga"also vary along a

An early version of this paper was presented at the Annual Meeting ofthe Society for Music Theory, Kansas City, 1992.'Ed. StevanHarnad(Cambridge:CambridgeUniversityPress, 1987), ix.

physicalcontinuum,yet we hear them as three qualitativelydistinctand discrete categories. In all three cases, perceptual boundarieshave somehow arisenalong the physicalcontinuum,dividing t intodiscreteregions, with qualitativeresemblanceswithineach categoryandqualitativedifferences between hem. These boundedcategoriesmay provide the groundworkfor higher-ordercognition and lan-guage.Harnad's example of color categories aptly introduces theideas of prototype and prototypicality, quite often associatedwith categorical perception. Consider the color category"red." The perceptual class of all reds consists of a varietyof different hues, the "redness" of which, intuitively, comesin varying strengths (indeed, there are borderline caseswhich can be cross-classified as "orange" or "purple"); pro-

totypicality is a technical term used to refer to these varyingstrengths of redness. When ordering all reds on a prototyp-icality scale, an apex is reached in the form of a particularred (or perhaps a narrow band of reds) that exemplify "sheerredness"; prototype is a technical term used to refer to sucha red. Prototype theory is a theory of categorical perceptioncast in prototype-structural terms. All three examples of cat-egorical perception cited by Harnad in this passage-that isto say, not only color perception, but also pitch perceptionand speech perception-have received prototype-theoreticaccounts. Prototype theory has also been applied to higher-

physicalcontinuum,yet we hear them as three qualitativelydistinctand discrete categories. In all three cases, perceptual boundarieshave somehow arisenalong the physicalcontinuum,dividing t intodiscreteregions, with qualitativeresemblanceswithineach categoryandqualitativedifferences between hem. These boundedcategoriesmay provide the groundworkfor higher-ordercognition and lan-guage.Harnad's example of color categories aptly introduces theideas of prototype and prototypicality, quite often associatedwith categorical perception. Consider the color category"red." The perceptual class of all reds consists of a varietyof different hues, the "redness" of which, intuitively, comesin varying strengths (indeed, there are borderline caseswhich can be cross-classified as "orange" or "purple"); pro-

totypicality is a technical term used to refer to these varyingstrengths of redness. When ordering all reds on a prototyp-icality scale, an apex is reached in the form of a particularred (or perhaps a narrow band of reds) that exemplify "sheerredness"; prototype is a technical term used to refer to sucha red. Prototype theory is a theory of categorical perceptioncast in prototype-structural terms. All three examples of cat-egorical perception cited by Harnad in this passage-that isto say, not only color perception, but also pitch perceptionand speech perception-have received prototype-theoreticaccounts. Prototype theory has also been applied to higher-

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Functional Harmony Revisited 197unctional Harmony Revisited 197

level mental processes involving semantic analysis, memoryorganization, and abstract thought.2Music cognition is a relatively young discipline, formed inthe wake of the cognitive revolution of the late fifties. Likeall branches of cognitive science, music cognition is inter-disciplinary in nature: philosophy, psychology, neurology,linguistics, computer science, and of course, music theory,have all contributed to shaping the field. Given that prototypetheories have 6een highly influential in cognitive psychologyin the last two and a half decades, it is not surprising that suchtheories have found their way into music-cognitive discourse.A recent issue of Psychomusicology, for example, is devotedin its entirety to the idea of music prototypes. According toMari Riess Jones, the guest editor, the issue shows that theidea "is useful in developmental approaches to music per-ception, in generating new approaches to rhythm perceptionand production, in understanding aspects of tonality, as wellas in expanding our understanding of complex music struc-ture."3 Among the researchers who have referred to proto-type theory in a musical context even before the appearanceof this issue of Psychomusicology are Carol Krumhansl, FredLerdahl, and Anna Unyk.4

2Fora prototype-theoreticaccountof colorperceptionsee Eleanor RoschHeider, "Universals n ColorNamingandMemory,"Journalof ExperimentalPsychology93 (1972): 10-20; for a prototype-theoreticaccount of pitchper-ception see Eytan Agmon, "Towardsa Theory of Diatonic Intonation,"In-terface22 (1993): 151-63; and for a prototype-theoreticaccount of speechperception see Gregg Oden and Dominic Massaro, "Integrationof FeaturalInformation n Speech Perception," PsychologicalReview85 (1978): 172-91.Eleanor Rosch has been highlyinfluential n prototype-theoreticaccountsofhigher-level mental processes; e.g., "Classificationof Real-WorldObjects:Origins and Representations in Cognition,"in Thinking:Readings in Cog-nitiveScience, ed. P. N. Johnson-Lairdand P. C. Wason(Cambridge:Cam-bridge University Press, 1977), 212-22.3Mari Riess Jones, "Preface,"Psychomusicology10 (1991): 71.4CarolKrumhansl,"The PsychologicalRepresentationof Musical Pitchin a Tonal Context," CognitivePsychology 11 (1979): 349; Fred Lerdahl,

level mental processes involving semantic analysis, memoryorganization, and abstract thought.2Music cognition is a relatively young discipline, formed inthe wake of the cognitive revolution of the late fifties. Likeall branches of cognitive science, music cognition is inter-disciplinary in nature: philosophy, psychology, neurology,linguistics, computer science, and of course, music theory,have all contributed to shaping the field. Given that prototypetheories have 6een highly influential in cognitive psychologyin the last two and a half decades, it is not surprising that suchtheories have found their way into music-cognitive discourse.A recent issue of Psychomusicology, for example, is devotedin its entirety to the idea of music prototypes. According toMari Riess Jones, the guest editor, the issue shows that theidea "is useful in developmental approaches to music per-ception, in generating new approaches to rhythm perceptionand production, in understanding aspects of tonality, as wellas in expanding our understanding of complex music struc-ture."3 Among the researchers who have referred to proto-type theory in a musical context even before the appearanceof this issue of Psychomusicology are Carol Krumhansl, FredLerdahl, and Anna Unyk.4

2Fora prototype-theoreticaccountof colorperceptionsee Eleanor RoschHeider, "Universals n ColorNamingandMemory,"Journalof ExperimentalPsychology93 (1972): 10-20; for a prototype-theoreticaccount of pitchper-ception see Eytan Agmon, "Towardsa Theory of Diatonic Intonation,"In-terface22 (1993): 151-63; and for a prototype-theoreticaccount of speechperception see Gregg Oden and Dominic Massaro, "Integrationof FeaturalInformation n Speech Perception," PsychologicalReview85 (1978): 172-91.Eleanor Rosch has been highlyinfluential n prototype-theoreticaccountsofhigher-level mental processes; e.g., "Classificationof Real-WorldObjects:Origins and Representations in Cognition,"in Thinking:Readings in Cog-nitiveScience, ed. P. N. Johnson-Lairdand P. C. Wason(Cambridge:Cam-bridge University Press, 1977), 212-22.3Mari Riess Jones, "Preface,"Psychomusicology10 (1991): 71.4CarolKrumhansl,"The PsychologicalRepresentationof Musical Pitchin a Tonal Context," CognitivePsychology 11 (1979): 349; Fred Lerdahl,

The present article applies prototype theory to the domainof harmonic theory. Specifically, it presents a prototype-structural account of the music-theoretic construct knownas harmonic functions. The hallmarks of functionalism are:(1) the characterization of individual chords as tonic (T),subdominant (S), or dominant (D) in function; and (2) thenotion that the so-called primary triads I, IV, and V somehowembody the essence of each of these functional categories.5A theory of functions as such was first announced by HugoRiemann, who openly acknowledged his debts to other the-orists. Riemannian Funktionstheorie engendered consider-able controversy early on, and continues to do so even today.Yet its significance is borne out by the many modern accountsof traditional harmony which incorporate one version or an-other of functionalism as an essential component. Possibleexamples are the harmony textbooks of William Mitchell,Allen Forte, and Joel Lester, from the last of which Example1 is reproduced.6 One conspicuous counterexample would"TimbralHierarchies,"ContemporaryMusicReview 2 (1987): 144-45; AnnaUnyk, "An Information-ProcessingAnalysis of Expectancy in Music Cog-nition," Psychomusicology9 (1990): 236-37.5For empirical studies of harmonic representation incorporating aprototype-theoreticapproachsee CarolKrumhansl,JamshedBharucha,andEdwardKessler, "PerceivedHarmonicStructureof Chords n Three RelatedMusicalKeys,"Journalof ExperimentalPsychology:HumanPerceptionandPerformance8 (1982):24-36, andJamshedBharuchaand CarolKrumhansl,"TheRepresentationof HarmonicStructure n Music:Hierarchiesof Stabilityas a Function of Context," Cognition13 (1983): 63-102. These studies dif-ferentiatebetween a harmoniccore consistingof the triadsI, IV, and V, andthe remainingfour diatonictriads, but do not otherwise partitionthe groupof seven diatonic triads into three functionalcategories.6WilliamMitchell, ElementaryHarmony,2nd ed. (New Jersey:Prentice-Hall, 1948), 65-66; Allen Forte, Tonal Harmony in Conceptand Practice(New York: Holt, Rinehartand Winston, 1962), 120;Joel Lester, Harmonyin TonalMusic,vol. 1 (New York:Knopf, 1982),21, 25, 251. See also MarionGuck, "TheFunctionalRelationsof Chords:A Theoryof MusicalIntuitions,"In TheoryOnly 4 (1978): 29-42; Charles J. Smith, "Prolongationsand Pro-gressionsas MusicalSyntax," n MusicTheory:SpecialTopics,ed. Richmond

The present article applies prototype theory to the domainof harmonic theory. Specifically, it presents a prototype-structural account of the music-theoretic construct knownas harmonic functions. The hallmarks of functionalism are:(1) the characterization of individual chords as tonic (T),subdominant (S), or dominant (D) in function; and (2) thenotion that the so-called primary triads I, IV, and V somehowembody the essence of each of these functional categories.5A theory of functions as such was first announced by HugoRiemann, who openly acknowledged his debts to other the-orists. Riemannian Funktionstheorie engendered consider-able controversy early on, and continues to do so even today.Yet its significance is borne out by the many modern accountsof traditional harmony which incorporate one version or an-other of functionalism as an essential component. Possibleexamples are the harmony textbooks of William Mitchell,Allen Forte, and Joel Lester, from the last of which Example1 is reproduced.6 One conspicuous counterexample would"TimbralHierarchies,"ContemporaryMusicReview 2 (1987): 144-45; AnnaUnyk, "An Information-ProcessingAnalysis of Expectancy in Music Cog-nition," Psychomusicology9 (1990): 236-37.5For empirical studies of harmonic representation incorporating aprototype-theoreticapproachsee CarolKrumhansl,JamshedBharucha,andEdwardKessler, "PerceivedHarmonicStructureof Chords n Three RelatedMusicalKeys,"Journalof ExperimentalPsychology:HumanPerceptionandPerformance8 (1982):24-36, andJamshedBharuchaand CarolKrumhansl,"TheRepresentationof HarmonicStructure n Music:Hierarchiesof Stabilityas a Function of Context," Cognition13 (1983): 63-102. These studies dif-ferentiatebetween a harmoniccore consistingof the triadsI, IV, and V, andthe remainingfour diatonictriads, but do not otherwise partitionthe groupof seven diatonic triads into three functionalcategories.6WilliamMitchell, ElementaryHarmony,2nd ed. (New Jersey:Prentice-Hall, 1948), 65-66; Allen Forte, Tonal Harmony in Conceptand Practice(New York: Holt, Rinehartand Winston, 1962), 120;Joel Lester, Harmonyin TonalMusic,vol. 1 (New York:Knopf, 1982),21, 25, 251. See also MarionGuck, "TheFunctionalRelationsof Chords:A Theoryof MusicalIntuitions,"In TheoryOnly 4 (1978): 29-42; Charles J. Smith, "Prolongationsand Pro-gressionsas MusicalSyntax," n MusicTheory:SpecialTopics,ed. Richmond

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It is a central-albeit unproven-thesis in the present articlethat harmonictheory in general is best conceived in terms oftwo independent, interacting components, namely a theoryof functions (which describes the three harmoniccategories)and a theory of chord progression (which characterizes thevarious root progressions,ascendingthird, descendingfifth,etc.). Indeed, one way of statingthe core idea of the presentarticle is: given a separationof chord progressionfrom har-monic function, the notions function and primary triad arefully reducible to categoryand prototype, respectively.

It is my firmbelief that musictheory, althougha latecomerinto the cognitive-scientificarena, is no second-rateplayerinthat field. The basis for my belief is twofold. First, musictheory has had a rich and impressivehistory-probably un-paralleledin the theoryof the arts;thishistoryis replete withsignificantcognitiveundertones. Second, the ontologicalsta-tus of music is possibly unique, being a product of humancognition minimallyconstrainedby the structureof the outerworld. By this I mean that music seems to fulfill Claude Levi-Strauss'sdictum that "when the mind is left to communewithitself and no longer has to come to terms with objects, it isin a sense reduced to imitatingitself as object."9Music, inother words, offers a unique and invaluable window intothe inner workings of the human mind. It follows that aprototype-theoretic approach to harmonic functions couldserve as an importantimpetusto theories of the mind whichuphold the existence of categories and prototypes.No approachto functionalharmony,however innovative,can afford to ignore the controversywhich has surroundedthe idea since its inception. Part 3 of this article, therefore,is a critical discussion of the functionalcontroversy;the main

9ClaudeLevi-Strauss,TheRaw and the Cooked, trans. J. and D. Weight-man (New York:Harper& Row, 1969), 10;quoted in HowardGardner, TheMind's New Science (New York: Basic Books, 1985), 240-41.

It is a central-albeit unproven-thesis in the present articlethat harmonictheory in general is best conceived in terms oftwo independent, interacting components, namely a theoryof functions (which describes the three harmoniccategories)and a theory of chord progression (which characterizes thevarious root progressions,ascendingthird, descendingfifth,etc.). Indeed, one way of statingthe core idea of the presentarticle is: given a separationof chord progressionfrom har-monic function, the notions function and primary triad arefully reducible to categoryand prototype, respectively.

It is my firmbelief that musictheory, althougha latecomerinto the cognitive-scientificarena, is no second-rateplayerinthat field. The basis for my belief is twofold. First, musictheory has had a rich and impressivehistory-probably un-paralleledin the theoryof the arts;thishistoryis replete withsignificantcognitiveundertones. Second, the ontologicalsta-tus of music is possibly unique, being a product of humancognition minimallyconstrainedby the structureof the outerworld. By this I mean that music seems to fulfill Claude Levi-Strauss'sdictum that "when the mind is left to communewithitself and no longer has to come to terms with objects, it isin a sense reduced to imitatingitself as object."9Music, inother words, offers a unique and invaluable window intothe inner workings of the human mind. It follows that aprototype-theoretic approach to harmonic functions couldserve as an importantimpetusto theories of the mind whichuphold the existence of categories and prototypes.No approachto functionalharmony,however innovative,can afford to ignore the controversywhich has surroundedthe idea since its inception. Part 3 of this article, therefore,is a critical discussion of the functionalcontroversy;the main

9ClaudeLevi-Strauss,TheRaw and the Cooked, trans. J. and D. Weight-man (New York:Harper& Row, 1969), 10;quoted in HowardGardner, TheMind's New Science (New York: Basic Books, 1985), 240-41.

issues in dispute are considered in light of the theory as pro-posed in the present article. Although a clear line is drawnbetween the notions of harmonic function and chord pro-gression, some tentative suggestionsas to how the two com-ponents might possibly interact to produce a richer and in-tuitively more satisfying harmonic theory are included inParts4 and 5 of the article, which complete the theoreticaldiscussionbegun in Part 2.

II. A PROTOTYPE-THEORETIC ACCOUNT OF HARMONIC FUNCTIONS

Translatingfunctionalism into prototype-theoretictermsnecessarily involves three operations: (1) selecting as pro-totypes the three primary riadsI, IV, and V; (2) associatingwith each prototype a group of triads, corresponding(to-gether with the prototype) to a conventional functionalcategory,as in T= {I,VI,III} (see Example 1); (3) associatingwith each categorized triad a prototypicalityindex corre-spondingto the triad'sfunctionalstrength (e.g., the sense inwhich I is the strongesttonic triad, followed by both VI andIII in the second place). First, however, some preliminariesare in order.Let the notion degreeof triadic imilaritybe defined for alldistinctpairsof triads formed from tones of a given diatoniccollection;in particular, et two (distinct)diatonic triads hav-ing two tones, one tone, or no tones in common be termedrespectively maximally similar, intermediatelysimilar, andminimallysimilar.In Figure 1 the seven diatonic triads are orderedcyclicallyby third. In this representation,maximallysimilar triads oc-cupy adjacentpositions on the circle's circumference; f theshortest distance along the circumference of the circle con-necting two triads passes through one or two interveningpositions, the triads are intermediatelysimilar or minimallysimilar, respectively.

issues in dispute are considered in light of the theory as pro-posed in the present article. Although a clear line is drawnbetween the notions of harmonic function and chord pro-gression, some tentative suggestionsas to how the two com-ponents might possibly interact to produce a richer and in-tuitively more satisfying harmonic theory are included inParts4 and 5 of the article, which complete the theoreticaldiscussionbegun in Part 2.

II. A PROTOTYPE-THEORETIC ACCOUNT OF HARMONIC FUNCTIONS

Translatingfunctionalism into prototype-theoretictermsnecessarily involves three operations: (1) selecting as pro-totypes the three primary riadsI, IV, and V; (2) associatingwith each prototype a group of triads, corresponding(to-gether with the prototype) to a conventional functionalcategory,as in T= {I,VI,III} (see Example 1); (3) associatingwith each categorized triad a prototypicalityindex corre-spondingto the triad'sfunctionalstrength (e.g., the sense inwhich I is the strongesttonic triad, followed by both VI andIII in the second place). First, however, some preliminariesare in order.Let the notion degreeof triadic imilaritybe defined for alldistinctpairsof triads formed from tones of a given diatoniccollection;in particular, et two (distinct)diatonic triads hav-ing two tones, one tone, or no tones in common be termedrespectively maximally similar, intermediatelysimilar, andminimallysimilar.In Figure 1 the seven diatonic triads are orderedcyclicallyby third. In this representation,maximallysimilar triads oc-cupy adjacentpositions on the circle's circumference; f theshortest distance along the circumference of the circle con-necting two triads passes through one or two interveningpositions, the triads are intermediatelysimilar or minimallysimilar, respectively.

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200 MusicTheorySpectrum00 MusicTheorySpectrum

Figure 1. The circle of thirds,representingdegrees of triadic Figure2.similarity a) Degrees of triadicsimilarityn relation to the tonic triad

III maximal i imilarity VIV IV

VII II \ > m

intermediate ; similarityV-\ /IV

VII 11ii minimal similarityVII II

b) The selectionof the three harmonicprototypesOne might object that common tones are not the onlypossible criterion by which similaritybetween triads mightbe assessed; for example, triads may also be compared onthe basis of their intervallicstructure, n which case all triadsof a given quality (major, minor, etc.) would be regardedas / \maximally similar. Yet there are compelling theoreticalgroundsfor grantingthe common-tone criteriona privilegedstatus in this case. After all, what makes a specific diatonictriad (say, IV) a unique object is its pitch content, not itsintervallicstructure.In other words, pitch content is the de-fining property of specific diatonic triads; it follows thatpitch content must be the main criterionby which similaritybetween such triads is assessed. vnFigure2a adds an importantassumptionto the content ofFigure1, namely, that the tonic triadis referentialwithin thediatonictriadicset. Thus, if Figure1 depictsdegreesof triadicsimilarity n general, Figure 2a depicts, in addition, degreesof triadic similarityin relation to the tonic triad. There are

Figure 1. The circle of thirds,representingdegrees of triadic Figure2.similarity a) Degrees of triadicsimilarityn relation to the tonic triad

III maximal i imilarity VIV IV

VII II \ > m

intermediate ; similarityV-\ /IV

VII 11ii minimal similarityVII II

b) The selectionof the three harmonicprototypesOne might object that common tones are not the onlypossible criterion by which similaritybetween triads mightbe assessed; for example, triads may also be compared onthe basis of their intervallicstructure, n which case all triadsof a given quality (major, minor, etc.) would be regardedas / \maximally similar. Yet there are compelling theoreticalgroundsfor grantingthe common-tone criteriona privilegedstatus in this case. After all, what makes a specific diatonictriad (say, IV) a unique object is its pitch content, not itsintervallicstructure.In other words, pitch content is the de-fining property of specific diatonic triads; it follows thatpitch content must be the main criterionby which similaritybetween such triads is assessed. vnFigure2a adds an importantassumptionto the content ofFigure1, namely, that the tonic triadis referentialwithin thediatonictriadicset. Thus, if Figure1 depictsdegreesof triadicsimilarity n general, Figure 2a depicts, in addition, degreesof triadic similarityin relation to the tonic triad. There are

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FunctionalHarmonyRevisited 201unctionalHarmonyRevisited 201

c) The delineationof categories

III I/\/ \TONIC/ \

c) The delineationof categories

III I/\/ \TONIC/ \

three suchdegrees, namelymaximallysimilar o the tonic (VIand III), intermediatelysimilarto the tonic (IV and V), andminimallysimilarto the tonic (II and VII). The obvious sym-metry of these relationshipsis highlightedin the figure.10There are three principleson the basis of which the pri-marytriadsI, IV, and V may be selected as prototypes. Thefirstprincipleis self-evident: the tonic triad, by virtue of itsspecial, referential status, is a prototype. (Referentialandprototypicalare synonymousfor all practical purposes.) Thesecond principlemay be termed theprincipleof symmetry.Itstates that the symmetricalstructuredepicted in Figure 2amust not be violated. In other words, if one selects any triadas prototype, one must also select its mirror mage (e.g., theselection of VI entails the selection of III and vice versa, IV

m'Concerninghe criteriaby which the tonicis selected, see Eytan Agmon,"Tonicityand the Tritone:Beyond the RarityIssue," Proceedingsof the FirstInternationalConferenceon CognitiveMusicology (Jyviskyla, Finland:Uni-versityof Jyvaskyla,1993),74-87. Commontones also playan importantrolein Riemann's theory, and indeed, in many earlier as well as later harmonictheories.

three suchdegrees, namelymaximallysimilar o the tonic (VIand III), intermediatelysimilarto the tonic (IV and V), andminimallysimilarto the tonic (II and VII). The obvious sym-metry of these relationshipsis highlightedin the figure.10There are three principleson the basis of which the pri-marytriadsI, IV, and V may be selected as prototypes. Thefirstprincipleis self-evident: the tonic triad, by virtue of itsspecial, referential status, is a prototype. (Referentialandprototypicalare synonymousfor all practical purposes.) Thesecond principlemay be termed theprincipleof symmetry.Itstates that the symmetricalstructuredepicted in Figure 2amust not be violated. In other words, if one selects any triadas prototype, one must also select its mirror mage (e.g., theselection of VI entails the selection of III and vice versa, IV

m'Concerninghe criteriaby which the tonicis selected, see Eytan Agmon,"Tonicityand the Tritone:Beyond the RarityIssue," Proceedingsof the FirstInternationalConferenceon CognitiveMusicology (Jyviskyla, Finland:Uni-versityof Jyvaskyla,1993),74-87. Commontones also playan importantrolein Riemann's theory, and indeed, in many earlier as well as later harmonictheories.

entails V). The principleof symmetryfollows from the ref-erentialstatus of the tonic triad.A triad selected as prototypeis compared to the tonic triad; the comparisonnecessarilyevokes exactly one other triad, namely, the triadwhich cor-respondsto the selected prototype in terms of its degree ofsimilarityto the tonic triad.The third and lastprinciplestates that two maximallysim-ilar triads cannot, simultaneously,be selected as harmonicprototypes (in other words, no two prototypes may occupyadjacentpositions on the circle of thirds depicted in Figure1). Unlike the first two principles,whichpertainsolely to thetheory of harmony, this third principle is more universal.Clearly, prototypes must be maximally dissimilar to eachother if categorizationin general is to serve any useful pur-pose;how couldsubjectsbenefitfromcategorization f it weredifficult to differentiate one prototype from another? In thecolor domain, for example, to select both "yellow"and "yel-low with a slight orange tinge"asprototypesis inconceivable;similarly n the harmonicdomain, it is inconceivableto selectas prototypes, say, both II and IV.11These three principlesuniquelyselect the triadsI, IV, andV as prototypes of three harmonic categories (Figure 2b).Two self-evident propositions(P1 and P2, below) determinethe additional members of each categoryand theirrespectiveprototypicalities:P1. Additional riadscan onlybe admitted nto a given categoryin an orderwhichcorrespondso theirdegreeof similarityo the

"A "principleof maximal differentiation"has been proposed by AndreMartinet for the domain of phonemicstructure:"ifby accident a given pho-neme is not as much differentiatedfrom its neighbours n the system as theorgans could achieve, one might expect the articulationof the phoneme inquestion to be modifieduntil such maximaldifferentiation s obtained"(El-ements of General Linguistics, trans. Elisabeth Palmer [London: Faber &Faber, 1964], 191). See also Eytan Agmon, "Towardsa Theory of DiatonicIntonation," 154, for an applicationof a similarprincipleto the domain ofpitch perception.

entails V). The principleof symmetryfollows from the ref-erentialstatus of the tonic triad.A triad selected as prototypeis compared to the tonic triad; the comparisonnecessarilyevokes exactly one other triad, namely, the triadwhich cor-respondsto the selected prototype in terms of its degree ofsimilarityto the tonic triad.The third and lastprinciplestates that two maximallysim-ilar triads cannot, simultaneously,be selected as harmonicprototypes (in other words, no two prototypes may occupyadjacentpositions on the circle of thirds depicted in Figure1). Unlike the first two principles,whichpertainsolely to thetheory of harmony, this third principle is more universal.Clearly, prototypes must be maximally dissimilar to eachother if categorizationin general is to serve any useful pur-pose;how couldsubjectsbenefitfromcategorization f it weredifficult to differentiate one prototype from another? In thecolor domain, for example, to select both "yellow"and "yel-low with a slight orange tinge"asprototypesis inconceivable;similarly n the harmonicdomain, it is inconceivableto selectas prototypes, say, both II and IV.11These three principlesuniquelyselect the triadsI, IV, andV as prototypes of three harmonic categories (Figure 2b).Two self-evident propositions(P1 and P2, below) determinethe additional members of each categoryand theirrespectiveprototypicalities:P1. Additional riadscan onlybe admitted nto a given categoryin an orderwhichcorrespondso theirdegreeof similarityo the

"A "principleof maximal differentiation"has been proposed by AndreMartinet for the domain of phonemicstructure:"ifby accident a given pho-neme is not as much differentiatedfrom its neighbours n the system as theorgans could achieve, one might expect the articulationof the phoneme inquestion to be modifieduntil such maximaldifferentiation s obtained"(El-ements of General Linguistics, trans. Elisabeth Palmer [London: Faber &Faber, 1964], 191). See also Eytan Agmon, "Towardsa Theory of DiatonicIntonation," 154, for an applicationof a similarprincipleto the domain ofpitch perception.

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category's rototypethat s, triadsmaximallyimilaro the givenprototypeare admittedirst,etc.); moreover,a triad'sdegreeofsimilarityo thegivenprototypes synonymouso that triad'spro-totypicality.P2.Theprocessof admittance escribedn P1 muststop justshortof another rototype,inceonlyoneprototypeanbe a member fany givencategory.

In applying these two propositions, it remains to be de-cided whether P2 is the sole constraint on admittance into acategory accordingto P1, or whether the process of admit-tance might possibly stop before P2 becomes applicable.Al-though one could conceivably formulate some principle tostop admitting triads into a given category earlier than P2dictates, parsimonyurges us not to do so. Why burden thetheory with an additional assumption, one whose rationalemay be difficult to establish, if P1 and P2, which are self-evident, suffice?Thus, I proposethatP1 and P2 complete thetheory, as shown in Figure 2c.A comparisonof Figure2c withExample1 reveals that theproposedtheoryaccordsverywell with the conventionalviewof harmonic functions.Yet the decision not to invoke a con-strainingprinciple on category membership other than P2results in a somewhat curious feature: not only VI and III,but also II and VII, have dual citizenship.That is to say, inaddition to the primary and conventionallywell-established)classification of II and VII as subdominant and dominanttriads, respectively, the function of II emerges as weaklydominant, and correspondingly,that of VII as weakly sub-dominant(weakly in this context refers to the triad'sproto-typicality, for a relation of intermediate similarity, ratherthan maximal similarity, obtains between the triad and itscategory'sprototype).The dual functionalstatus of II and VIIsurelycalls for separatediscussion; however, since the issuesinvolved are rather specialized, and moreover touch upon anumber of intricate side issues, such discussion appears inPart4 of this article. A more pressingquestion concerns the

category's rototypethat s, triadsmaximallyimilaro the givenprototypeare admittedirst,etc.); moreover,a triad'sdegreeofsimilarityo thegivenprototypes synonymouso that triad'spro-totypicality.P2.Theprocessof admittance escribedn P1 muststop justshortof another rototype,inceonlyoneprototypeanbe a member fany givencategory.

In applying these two propositions, it remains to be de-cided whether P2 is the sole constraint on admittance into acategory accordingto P1, or whether the process of admit-tance might possibly stop before P2 becomes applicable.Al-though one could conceivably formulate some principle tostop admitting triads into a given category earlier than P2dictates, parsimonyurges us not to do so. Why burden thetheory with an additional assumption, one whose rationalemay be difficult to establish, if P1 and P2, which are self-evident, suffice?Thus, I proposethatP1 and P2 complete thetheory, as shown in Figure 2c.A comparisonof Figure2c withExample1 reveals that theproposedtheoryaccordsverywell with the conventionalviewof harmonic functions.Yet the decision not to invoke a con-strainingprinciple on category membership other than P2results in a somewhat curious feature: not only VI and III,but also II and VII, have dual citizenship.That is to say, inaddition to the primary and conventionallywell-established)classification of II and VII as subdominant and dominanttriads, respectively, the function of II emerges as weaklydominant, and correspondingly,that of VII as weakly sub-dominant(weakly in this context refers to the triad'sproto-typicality, for a relation of intermediate similarity, ratherthan maximal similarity, obtains between the triad and itscategory'sprototype).The dual functionalstatus of II and VIIsurelycalls for separatediscussion; however, since the issuesinvolved are rather specialized, and moreover touch upon anumber of intricate side issues, such discussion appears inPart4 of this article. A more pressingquestion concerns the

validityof the idea of harmonicfunctions in the firstplace,a question which has generated, as the reader is probablyaware, some rather heated debate.

III. HARMONIC FUNCTIONS: EMPIRICAL FACTOR THEORETICAL FICTION?

It should be clear by now that the term "theoryof har-monic functions"is used in the present article with a veryspecific meaning. "Function,""primarytriad," and "func-tional strength"are reduced to "category,""prototype,"and"prototypicality,"respectively. Moreover, the theory isstrippedof any chord-progressionalconnotations, and thushas no standingwith regardto how the three functionalcat-egories may follow each other in time. Nonetheless, signif-icant connections do exist between the proposed theory andthe historical Funktionstheorie,usually attributedto HugoRiemann. Riemann'stheory has been severely criticizedonseveral accounts, even to the point of questioning the ex-istence of harmonic unctionsaltogether.The presentsectionaddresses two of the most prominentcriticismsof Riemann:one which stems from a rival approachto harmony knownas Stufentheorie,and another which stems from a theorywhichpurports o redefinealtogetherthe notion of harmony,namely the Schenkerianapproachwith its emphasison hier-archicalstructureand voice leading.Stufentheorieversus Funktionstheorie.A historicallyim-portant alternative to Riemann'sFunktionstheorie s an ap-proachto harmonyknown as Stufentheorieliterally,"theoryof harmonic degrees," but more appropriately, "theory ofdegree progression").l2 Stufentheoriestresses the individu-

'2SimonSechter is usuallyconsidered the most prominentproponent ofStufentheorie. ee RobertWason,VienneseHarmonicTheory romAlbrechts-berger o Schenkerand Schoenberg Ann Arbor:UMI ResearchPress, 1985),33; CarlDahlhaus,Studieson theOriginof HarmonicTonality, rans.RobertGjerdingen (Princeton: Princeton University Press, 1990), 33 ff.; William

validityof the idea of harmonicfunctions in the firstplace,a question which has generated, as the reader is probablyaware, some rather heated debate.

III. HARMONIC FUNCTIONS: EMPIRICAL FACTOR THEORETICAL FICTION?

It should be clear by now that the term "theoryof har-monic functions"is used in the present article with a veryspecific meaning. "Function,""primarytriad," and "func-tional strength"are reduced to "category,""prototype,"and"prototypicality,"respectively. Moreover, the theory isstrippedof any chord-progressionalconnotations, and thushas no standingwith regardto how the three functionalcat-egories may follow each other in time. Nonetheless, signif-icant connections do exist between the proposed theory andthe historical Funktionstheorie,usually attributedto HugoRiemann. Riemann'stheory has been severely criticizedonseveral accounts, even to the point of questioning the ex-istence of harmonic unctionsaltogether.The presentsectionaddresses two of the most prominentcriticismsof Riemann:one which stems from a rival approachto harmony knownas Stufentheorie,and another which stems from a theorywhichpurports o redefinealtogetherthe notion of harmony,namely the Schenkerianapproachwith its emphasison hier-archicalstructureand voice leading.Stufentheorieversus Funktionstheorie.A historicallyim-portant alternative to Riemann'sFunktionstheorie s an ap-proachto harmonyknown as Stufentheorieliterally,"theoryof harmonic degrees," but more appropriately, "theory ofdegree progression").l2 Stufentheoriestresses the individu-

'2SimonSechter is usuallyconsidered the most prominentproponent ofStufentheorie. ee RobertWason,VienneseHarmonicTheory romAlbrechts-berger o Schenkerand Schoenberg Ann Arbor:UMI ResearchPress, 1985),33; CarlDahlhaus,Studieson theOriginof HarmonicTonality, rans.RobertGjerdingen (Princeton: Princeton University Press, 1990), 33 ff.; William

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alityandindependenceof the seven harmonicdegrees. More-over, unlike Funktionstheorie,where the primaryharmonicmodel is the I-IV-V-I progression, Stufentheorie leansheavily on the cycle of descendingfifthsI-IV-VII-III-VI-II-V-I.Both aspectsof Stufentheorie igure prominently n Schen-ker's acrimoniousattack on functionalism n the first volumeof Counterpoint,echoes of which reverberate in the writingsof some of his followers.13For example, Schenkerquestionsthe explanatory value of an account of the major systemwhere the "individual cale degrees, except for I, IV, and V,aredeprivedof theirindependenceand thus of theirattractivecapabilityof assumingvarious functions";and in discussingharmonicprogressionsby descending fifths, he asks: "Con-sidering that none of these passages manifests 'sequential

Caplin,"Harmonyand Meter in the Theories of Simon Sechter,"Music The-ory Spectrum2 (1980): 74-89.Ernst Kurth, who used the term Fundamenttheorie o refer to Sechter'ssystem, seems to have been the first scholar to conceive of Sechter and Rie-mann as two major opposing figures in the theory of harmony. See ErnstKurth, Die Voraussetzungender theoretischenHarmonik und der tonalenDarstellungssystemeBern:MaxDrechsel, 1913),6 ff. and89 ff. See also LeeRothfarb, "ErnstKurth's Die Voraussetzungen er theoretischenHarmonikand the Beginningsof MusicPsychology,"Theoria4 (1989): 19-20. Needlessto say, Stufentheorie s well as Funktionstheorie re, to a considerableextent,abstractions,for neither exists in the work of a single theorist in as pure aform as their hypothesized rivalrysuggests.13Seeespecially Oswald Jonas, Introductionto the Theory of HeinrichSchenker, rans. and ed. JohnRothgeb (New York:Longman,1982), 127-28,and Rothgeb's review of Hellmut Federhofer,Akkord und Stimmfiihrungnden musiktheoretischen ystemenvon Hugo Riemann,Ernst Kurthund Hein-rich Schenker(Vienna: Verlag der OsterreichischenAkademie der Wissen-schaften, 1981)in Music TheorySpectrum4 (1982): 131-37, where "the rec-ognition of only three principalfunctions" s dubbed "specious"(132). Seealso Federhofer, Beitrage zur musikalischenGestaltanalyse Graz: Akade-mische Druck, 1950) and Akkord und Stimmfiihrung; nd MatthewBrown,"A Rational Reconstructionof SchenkerianTheory" (Ph.D. diss., CornellUniversity, 1989), 192-99. Both Brown and Wason (Viennese HarmonicTheory, 134-35) present synopses of Schenker'sarguments.

alityandindependenceof the seven harmonicdegrees. More-over, unlike Funktionstheorie,where the primaryharmonicmodel is the I-IV-V-I progression, Stufentheorie leansheavily on the cycle of descendingfifthsI-IV-VII-III-VI-II-V-I.Both aspectsof Stufentheorie igure prominently n Schen-ker's acrimoniousattack on functionalism n the first volumeof Counterpoint,echoes of which reverberate in the writingsof some of his followers.13For example, Schenkerquestionsthe explanatory value of an account of the major systemwhere the "individual cale degrees, except for I, IV, and V,aredeprivedof theirindependenceand thus of theirattractivecapabilityof assumingvarious functions";and in discussingharmonicprogressionsby descending fifths, he asks: "Con-sidering that none of these passages manifests 'sequential

Caplin,"Harmonyand Meter in the Theories of Simon Sechter,"Music The-ory Spectrum2 (1980): 74-89.Ernst Kurth, who used the term Fundamenttheorie o refer to Sechter'ssystem, seems to have been the first scholar to conceive of Sechter and Rie-mann as two major opposing figures in the theory of harmony. See ErnstKurth, Die Voraussetzungender theoretischenHarmonik und der tonalenDarstellungssystemeBern:MaxDrechsel, 1913),6 ff. and89 ff. See also LeeRothfarb, "ErnstKurth's Die Voraussetzungen er theoretischenHarmonikand the Beginningsof MusicPsychology,"Theoria4 (1989): 19-20. Needlessto say, Stufentheorie s well as Funktionstheorie re, to a considerableextent,abstractions,for neither exists in the work of a single theorist in as pure aform as their hypothesized rivalrysuggests.13Seeespecially Oswald Jonas, Introductionto the Theory of HeinrichSchenker, rans. and ed. JohnRothgeb (New York:Longman,1982), 127-28,and Rothgeb's review of Hellmut Federhofer,Akkord und Stimmfiihrungnden musiktheoretischen ystemenvon Hugo Riemann,Ernst Kurthund Hein-rich Schenker(Vienna: Verlag der OsterreichischenAkademie der Wissen-schaften, 1981)in Music TheorySpectrum4 (1982): 131-37, where "the rec-ognition of only three principalfunctions" s dubbed "specious"(132). Seealso Federhofer, Beitrage zur musikalischenGestaltanalyse Graz: Akade-mische Druck, 1950) and Akkord und Stimmfiihrung; nd MatthewBrown,"A Rational Reconstructionof SchenkerianTheory" (Ph.D. diss., CornellUniversity, 1989), 192-99. Both Brown and Wason (Viennese HarmonicTheory, 134-35) present synopses of Schenker'sarguments.

character,' does it not follow necessarily that all scale-degrees-not only I, IV, andV-must be recognizedas scale-degrees in their own right?"14The Schenkerianapproach:hierarchyand voice leading.While Schenker's 1910critiqueof functionalism s of a clear,Stufentheorieorigin, the subsequent development of histheories, with its well-knownemphasison voice leading andhierarchicalstructure,has inevitablyled to friction of quitea different sort. Jonas, for example, states thatfunctionalism"had o fail, because it neglectedthe fact that two occurrencesof the same chord could be worldsapart n meaning,and thateverything depended on context."15 In a footnote to thatstatement, John Rothgeb concedes that functionalismallowsa secondarychord, such as III, to assumedifferentfunctionsin different contexts, but nevertheless insists that "thetheory fell far short of a recognitionof the phenomenon ofcomposing-out, and, as a result, failed to discriminate be-tween verticallyand horizontallygenerated chords."My response to all of these importantobjections rests onthree major points: (1) the proposed version of functionaltheory is not Riemann's;(2) the theory is not meant to ex-haust the harmonicdomain;and (3) the proposed theory iscompatiblewith a hierarchicalapproach.DeparturesromRiemann. CertaincontroversialaspectsofRiemann'stheoryhave no counterparts n the proposedthe-ory. Most notably, the proposedtheory abandonsRiemann's

14Heinrich chenker, Counterpoint, ol. 1, trans.John Rothgeb andJiir-gen Thym, ed. John Rothgeb (New York: Schirmer,1987), 23, 27. As JohnRothgeb explains n a footnote on p. 348, "theapplication .. [of the theoryof tonal functions] is especially problematical in passages involving 'se-quences' by descending fifths. This led Riemann to declare that 'as wasrecognized first of all by Fetis, however, the true harmonic movement-the cadential progression-remains stationary for the duration of thesequence....' " The reference is to Hugo Riemann, Handbuch der Har-monielehre,7th ed. (Leipzig: Max Hesse, n.d.), 202.15Jonas, ntroduction,127.

character,' does it not follow necessarily that all scale-degrees-not only I, IV, andV-must be recognizedas scale-degrees in their own right?"14The Schenkerianapproach:hierarchyand voice leading.While Schenker's 1910critiqueof functionalism s of a clear,Stufentheorieorigin, the subsequent development of histheories, with its well-knownemphasison voice leading andhierarchicalstructure,has inevitablyled to friction of quitea different sort. Jonas, for example, states thatfunctionalism"had o fail, because it neglectedthe fact that two occurrencesof the same chord could be worldsapart n meaning,and thateverything depended on context."15 In a footnote to thatstatement, John Rothgeb concedes that functionalismallowsa secondarychord, such as III, to assumedifferentfunctionsin different contexts, but nevertheless insists that "thetheory fell far short of a recognitionof the phenomenon ofcomposing-out, and, as a result, failed to discriminate be-tween verticallyand horizontallygenerated chords."My response to all of these importantobjections rests onthree major points: (1) the proposed version of functionaltheory is not Riemann's;(2) the theory is not meant to ex-haust the harmonicdomain;and (3) the proposed theory iscompatiblewith a hierarchicalapproach.DeparturesromRiemann. CertaincontroversialaspectsofRiemann'stheoryhave no counterparts n the proposedthe-ory. Most notably, the proposedtheory abandonsRiemann's

14Heinrich chenker, Counterpoint, ol. 1, trans.John Rothgeb andJiir-gen Thym, ed. John Rothgeb (New York: Schirmer,1987), 23, 27. As JohnRothgeb explains n a footnote on p. 348, "theapplication .. [of the theoryof tonal functions] is especially problematical in passages involving 'se-quences' by descending fifths. This led Riemann to declare that 'as wasrecognized first of all by Fetis, however, the true harmonic movement-the cadential progression-remains stationary for the duration of thesequence....' " The reference is to Hugo Riemann, Handbuch der Har-monielehre,7th ed. (Leipzig: Max Hesse, n.d.), 202.15Jonas, ntroduction,127.

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notion of "apparent consonance" (Scheinkonsonanz), whichtruly deprives the so-called secondary triads, right from thestart, of any potential independence.16 More generally, theRiemannian sense in which a secondary degree is said torepresent or substitute for a primary degree does not exist inthe proposed theory. Unlike Riemann's somewhat ambigu-ous use of tonic, subdominant, and dominant to refer to bothfunctions and/or chords, in the proposed theory these termsare used exclusively to refer to functions, that is, categoriesof chords; reference to individual chords is made by Romannumerals.17 II, for example, represents an abstract category(namely the subdominant), which is also represented by IV;at the same time, IV is of course the more prototypicalrepresentative of that category.The proposed theory largely bypasses another highly con-troversial aspect of Riemann's theory, namely dualism.18 Af-ter all, there is no necessary connection between function-alism and dualism, however essential it might have seemedto Riemann to view the rules of harmony and the major/minor polarity as stemming from a single source. Indeed,there is no lack of evidence that functionalism can flourish-

16SeeDahlhaus, Studies,38: "the concept of functionscan be separatedfrom Riemann'smethodof demotingsecondarydegrees to dissonantvariantsof primarydegrees, so that one can retain the concept of fundamentalpro-gressionswithout giving up the concept of functions." For Dahlhaus'selab-oration of this statement, see especially pp. 57-59.

17Concerning the Riemannian chord/functionambiguity, see Dahlhaus,Studies, 50. In a paper entitled "An Idea and its Politics: Hugo Riemann'sTreatment of HarmonicFunction,"delivered at the 1992 annualmeeting ofthe Society for Music Theory, Daniel Harrisonhas argued that the chord/functionambiguity s not so muchinherent n Riemann'stheory(as Dahlhausimplies), as it is an unfortunate result of Riemann's attempts to make histheory more generally accessible.8sThe losest counterpart n the proposed theory to Riemann's dualismis the principleof symmetry, discussed in connection with Figure 2.

notion of "apparent consonance" (Scheinkonsonanz), whichtruly deprives the so-called secondary triads, right from thestart, of any potential independence.16 More generally, theRiemannian sense in which a secondary degree is said torepresent or substitute for a primary degree does not exist inthe proposed theory. Unlike Riemann's somewhat ambigu-ous use of tonic, subdominant, and dominant to refer to bothfunctions and/or chords, in the proposed theory these termsare used exclusively to refer to functions, that is, categoriesof chords; reference to individual chords is made by Romannumerals.17 II, for example, represents an abstract category(namely the subdominant), which is also represented by IV;at the same time, IV is of course the more prototypicalrepresentative of that category.The proposed theory largely bypasses another highly con-troversial aspect of Riemann's theory, namely dualism.18 Af-ter all, there is no necessary connection between function-alism and dualism, however essential it might have seemedto Riemann to view the rules of harmony and the major/minor polarity as stemming from a single source. Indeed,there is no lack of evidence that functionalism can flourish-

16SeeDahlhaus, Studies,38: "the concept of functionscan be separatedfrom Riemann'smethodof demotingsecondarydegrees to dissonantvariantsof primarydegrees, so that one can retain the concept of fundamentalpro-gressionswithout giving up the concept of functions." For Dahlhaus'selab-oration of this statement, see especially pp. 57-59.

17Concerning the Riemannian chord/functionambiguity, see Dahlhaus,Studies, 50. In a paper entitled "An Idea and its Politics: Hugo Riemann'sTreatment of HarmonicFunction,"delivered at the 1992 annualmeeting ofthe Society for Music Theory, Daniel Harrisonhas argued that the chord/functionambiguity s not so muchinherent n Riemann'stheory(as Dahlhausimplies), as it is an unfortunate result of Riemann's attempts to make histheory more generally accessible.8sThe losest counterpart n the proposed theory to Riemann's dualismis the principleof symmetry, discussed in connection with Figure 2.

quite apart from any attendant commitment to a dualisticdoctrine.19The scope of the theory. As already indicated, harmonictheory is viewed in the present article as a composite oftwo main subtheories: a theory of functions and a theory ofchord progression. This view is indebted to the historicalFunktionstheorielStufentheorie opposition; however, the cor-respondence is far from exact. Most notably, the proposedfunctional theory is static, in the sense that it merely describesthe three harmonic categories and their internal structure; theT-S-D-T paradigm of functional succession (a venerablecomponent of Funktionstheorie) is not within its scope. Theexclusion of functional succession from functional theorymight seem a bit odd. However, although excluded fromfunctional theory, the T-S-D-T paradigm remains a part ofharmonic theory in general-that is, the interaction of thetheory of functions with a theory of chord progression.Once the scope of functional theory is reduced, it cannotbe faulted for failing to deal convincingly with the cycle ofdescending fifths. The priority of the descending-fifth rela-tionship is clearly an aspect of the theory of chord progres-sion, not the theory of functions. When Schenker and hisfollowers point out, for example, that the II in II-V is notmerely a IV-substitute because a descending-fifth relationshipobtains between the II and the V, they are certainly correct;but their argument concerns the theory of chord progression(as it interacts with the theory of functions), not the theoryof functions per se. It is surely senseless to reject a theory

19Sincea numberof American theorists have taken renewed interest inRiemannin general and dualismin particular, t might be clarified hat theintention here is not to rekindle he dualismcontroversy; ather,the intentionis merely to rebut criticismsof Riemann's dualism that might be addressedtoward the proposed theory. An example of American Neo-Riemannism sDavid Lewin, "A FormalTheory of Generalized Tonal Functions,"Journalof MusicTheory 6 (1982):23-60.

quite apart from any attendant commitment to a dualisticdoctrine.19The scope of the theory. As already indicated, harmonictheory is viewed in the present article as a composite oftwo main subtheories: a theory of functions and a theory ofchord progression. This view is indebted to the historicalFunktionstheorielStufentheorie opposition; however, the cor-respondence is far from exact. Most notably, the proposedfunctional theory is static, in the sense that it merely describesthe three harmonic categories and their internal structure; theT-S-D-T paradigm of functional succession (a venerablecomponent of Funktionstheorie) is not within its scope. Theexclusion of functional succession from functional theorymight seem a bit odd. However, although excluded fromfunctional theory, the T-S-D-T paradigm remains a part ofharmonic theory in general-that is, the interaction of thetheory of functions with a theory of chord progression.Once the scope of functional theory is reduced, it cannotbe faulted for failing to deal convincingly with the cycle ofdescending fifths. The priority of the descending-fifth rela-tionship is clearly an aspect of the theory of chord progres-sion, not the theory of functions. When Schenker and hisfollowers point out, for example, that the II in II-V is notmerely a IV-substitute because a descending-fifth relationshipobtains between the II and the V, they are certainly correct;but their argument concerns the theory of chord progression(as it interacts with the theory of functions), not the theoryof functions per se. It is surely senseless to reject a theory

19Sincea numberof American theorists have taken renewed interest inRiemannin general and dualismin particular, t might be clarified hat theintention here is not to rekindle he dualismcontroversy; ather,the intentionis merely to rebut criticismsof Riemann's dualism that might be addressedtoward the proposed theory. An example of American Neo-Riemannism sDavid Lewin, "A FormalTheory of Generalized Tonal Functions,"Journalof MusicTheory 6 (1982):23-60.

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theory where dominant and subdominant are seen, in Dahl-haus's words, as "antithetical extremes."28 Yet the dominantfunction of II, I believe, may be felt in certain contexts whereII (or II6) is followed by I (or I6). An example in major, fromthe opening of a well-known piano sonata by Haydn, appearsin Example 2a. Interestingly, at the beginning of the devel-opment section the opening idea is restated (in the dominantkey) with VII6 in the place of II6 (Example 2b). One couldargue, of course, that Haydn's reharmonization, necessitatedby the transfer of the thematic material to the bass, amountsto a functional reinterpretation (that is, D-T instead of S-T).

theory where dominant and subdominant are seen, in Dahl-haus's words, as "antithetical extremes."28 Yet the dominantfunction of II, I believe, may be felt in certain contexts whereII (or II6) is followed by I (or I6). An example in major, fromthe opening of a well-known piano sonata by Haydn, appearsin Example 2a. Interestingly, at the beginning of the devel-opment section the opening idea is restated (in the dominantkey) with VII6 in the place of II6 (Example 2b). One couldargue, of course, that Haydn's reharmonization, necessitatedby the transfer of the thematic material to the bass, amountsto a functional reinterpretation (that is, D-T instead of S-T).

Example 2. II as dominant in major: Joseph Haydn, Piano So-nata in D major, Hob. XVI:37a) m. 1

Example 2. II as dominant in major: Joseph Haydn, Piano So-nata in D major, Hob. XVI:37a) m. 1

b) m. 41) m. 4128"Thenotion that ii before I should be taken as V9, and vi before V as

ii9, is absurd, since it leads to a confusion of antitheticalextremes-the sub-dominant (subdominantparallel) with the dominant, and the tonic (tonicparallel)with the dominantof the dominant" Dahlhaus, Studies,37). Dahl-haus is responding(in part) to Sechter's nterpretationof II-I(6) as II-V9-I,which, ironicallyperhaps,is a possible precedentto the idea that II is weaklydominant in function. See Wason, Viennese Harmonic Theory, 40-42, andCaplin, "Harmonyand Meter," 76 (especially Example 2).To the best of my knowledge, the closest Riemann ever comes to ac-knowledging a dominant quality in II is when he identifies (possibly afterHauptmann) he root of II or II7 n majorwith the fifth of V. See for examplehis HarmonySimplified,trans. H. Bewerung (London: Augener, 1896), 55.To be sure, the theory of Klangvertretungwhich Riemann credits to Helm-holtz) allows Riemann to state that "each of the secondarytriadsof the keyrepresent simultaneouslytwo of the three primaryharmonies, one primaryharmonybeing comprehendedas the main content (consonance), the other,the foreignaddition(dissonance)" Hugo Riemann,Historyof Music Theory,vol. 3, trans. and ed. William Mickelsen [Lincoln: University of NebraskaPress, 1977], 218). However, except for II in major, the principleof Klang-vertretungdoes not seem to apply to II and VII. For Riemann, II in minoris the majortriad 62-4-6 (the Leittonwechselklang f IV) and VII in majoris the minortriad1-2-#4 (the Leittonwechselklangf V), while VII in minoris the major triad (t)4-2-4 (the Parallelklangof the naturaldominant). Inanyevent, I amnot awarethatRiemann ever actuallyanalyzedII as dominant(even in major)or VII as subdominant.In particular,a VII triad(diminished,of course) is invariablyanalyzed by Riemannas a rootless dominant-seventhchord.

28"Thenotion that ii before I should be taken as V9, and vi before V asii9, is absurd, since it leads to a confusion of antitheticalextremes-the sub-dominant (subdominantparallel) with the dominant, and the tonic (tonicparallel)with the dominantof the dominant" Dahlhaus, Studies,37). Dahl-haus is responding(in part) to Sechter's nterpretationof II-I(6) as II-V9-I,which, ironicallyperhaps,is a possible precedentto the idea that II is weaklydominant in function. See Wason, Viennese Harmonic Theory, 40-42, andCaplin, "Harmonyand Meter," 76 (especially Example 2).To the best of my knowledge, the closest Riemann ever comes to ac-knowledging a dominant quality in II is when he identifies (possibly afterHauptmann) he root of II or II7 n majorwith the fifth of V. See for examplehis HarmonySimplified,trans. H. Bewerung (London: Augener, 1896), 55.To be sure, the theory of Klangvertretungwhich Riemann credits to Helm-holtz) allows Riemann to state that "each of the secondarytriadsof the keyrepresent simultaneouslytwo of the three primaryharmonies, one primaryharmonybeing comprehendedas the main content (consonance), the other,the foreignaddition(dissonance)" Hugo Riemann,Historyof Music Theory,vol. 3, trans. and ed. William Mickelsen [Lincoln: University of NebraskaPress, 1977], 218). However, except for II in major, the principleof Klang-vertretungdoes not seem to apply to II and VII. For Riemann, II in minoris the majortriad 62-4-6 (the Leittonwechselklang f IV) and VII in majoris the minortriad1-2-#4 (the Leittonwechselklangf V), while VII in minoris the major triad (t)4-2-4 (the Parallelklangof the naturaldominant). Inanyevent, I amnot awarethatRiemann ever actuallyanalyzedII as dominant(even in major)or VII as subdominant.In particular,a VII triad(diminished,of course) is invariablyanalyzed by Riemannas a rootless dominant-seventhchord.

Allego con brio

D--T D--T(?)

n IL_~lI I IT I

D--T D--T

Allego con brio

D--T D--T(?)

n IL_~lI I IT I

D--T D--T

The VII6, however, is so clearly a variant of the II6 chord,that to hear the two chords as contrasting in function seemscounterintuitive.2929Alsorelevant n thisconnectionare mm. 5-6, a variedrepetitionof mm.1-2. Replicatingthe II6 chord would have sounded awkwardwith the newsixteenth-notefiguration n the left hand, and thereforeHaydn departsfrommm. 1-2 harmonicallyby keepingthe inner-voicea' stationary, husimplying

the succession V4-I6 rather than II6-I6.One couldobject thatin comparingparallelpassages n the HaydnSonataI am confusingthe question of functionalparallelismwith the independentquestionof thematicparallelism.Yet I makeno claimswhatsoeverconcerninga necessaryconnectionbetween the two kinds of parallelisms.It is importantto bear in mind that whereas thematicparallelism n the present case mightseem to supportthe dominantinterpretationof II, thematicparallelismcouldhardlysupport, say, a dominantinterpretationof IV, for the simple reasonthat the latter possibilitydoes not exist within functional theory.

The VII6, however, is so clearly a variant of the II6 chord,that to hear the two chords as contrasting in function seemscounterintuitive.2929Alsorelevant n thisconnectionare mm. 5-6, a variedrepetitionof mm.1-2. Replicatingthe II6 chord would have sounded awkwardwith the newsixteenth-notefiguration n the left hand, and thereforeHaydn departsfrommm. 1-2 harmonicallyby keepingthe inner-voicea' stationary, husimplying

the succession V4-I6 rather than II6-I6.One couldobject thatin comparingparallelpassages n the HaydnSonataI am confusingthe question of functionalparallelismwith the independentquestionof thematicparallelism.Yet I makeno claimswhatsoeverconcerninga necessaryconnectionbetween the two kinds of parallelisms.It is importantto bear in mind that whereas thematicparallelism n the present case mightseem to supportthe dominantinterpretationof II, thematicparallelismcouldhardlysupport, say, a dominantinterpretationof IV, for the simple reasonthat the latter possibilitydoes not exist within functional theory.

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An example in minor (from the opening of a Brahmsin-termezzo) appears n Example3a. For reasons to be clarifiedshortlyin connection with VII, the dominant function of IIin minor(a dissonant,diminishedtriad)is even more evidentthan that of its major counterpart. (It is probably needlessto remind the reader that the function of II, in minoras wellas major, is first and foremost subdominant.)Note that inthe consequent phrase, whichbegins in m. 9, the opening II6chord concludes the dominantprolongation which begins inm. 6 (Example 3b).30

Withrespectto the leading-tonetriad,the questionof dualfunctionality s even more delicate. A subdominant eading-tone triad is difficult to find.31Yet the leading-toneseventhchord (VII7) is known to assume (in certain contexts) sub-dominantfunction. In what follows I suggest that althoughboth VII and VII7possess a weak subdominantpotential, incontextonly the latterchord's subdominantpotentialis likelyto be realized. My explanation involves equally the succes-sional paradigmT-S-D-T and the special status of VII asthe only dissonanttriad in the diatonicsystem. It should beemphasized that the T-S-D-T paradigmis evoked tenta-tively, by way of suggesting how function and chord pro-gressionmightinteractin a more complete harmonictheory.VII is a dissonant, diminished triad (in minor, the har-monic form of the scale is assumed), and as such demandsresolution.In addition,the dominantpotentialof the leading-tone triad is strongerthan its subdominantpotential. Thesetwo inherentattributesof VII are mutuallysupportive,sinceboth the resolutionof dissonanceand the paradigmaticD-Tfunctionalsuccessionare strongly goal-oriented. As a result,

30Notealsothatin the intermezzo'scontrastingmiddle section the openingmotive is restated(in D, major)with V7-I in the place of II6-I6 (e.g., mm.22-23).31See,however, Yizhak Sadai, Harmony in its Systematicand Phenom-enological Aspects, trans. J. Davis and M. Shlesinger (Jerusalem: Yanetz,1980), 149.

An example in minor (from the opening of a Brahmsin-termezzo) appears n Example3a. For reasons to be clarifiedshortlyin connection with VII, the dominant function of IIin minor(a dissonant,diminishedtriad)is even more evidentthan that of its major counterpart. (It is probably needlessto remind the reader that the function of II, in minoras wellas major, is first and foremost subdominant.)Note that inthe consequent phrase, whichbegins in m. 9, the opening II6chord concludes the dominantprolongation which begins inm. 6 (Example 3b).30

Withrespectto the leading-tonetriad,the questionof dualfunctionality s even more delicate. A subdominant eading-tone triad is difficult to find.31Yet the leading-toneseventhchord (VII7) is known to assume (in certain contexts) sub-dominantfunction. In what follows I suggest that althoughboth VII and VII7possess a weak subdominantpotential, incontextonly the latterchord's subdominantpotentialis likelyto be realized. My explanation involves equally the succes-sional paradigmT-S-D-T and the special status of VII asthe only dissonanttriad in the diatonicsystem. It should beemphasized that the T-S-D-T paradigmis evoked tenta-tively, by way of suggesting how function and chord pro-gressionmightinteractin a more complete harmonictheory.VII is a dissonant, diminished triad (in minor, the har-monic form of the scale is assumed), and as such demandsresolution.In addition,the dominantpotentialof the leading-tone triad is strongerthan its subdominantpotential. Thesetwo inherentattributesof VII are mutuallysupportive,sinceboth the resolutionof dissonanceand the paradigmaticD-Tfunctionalsuccessionare strongly goal-oriented. As a result,

30Notealsothatin the intermezzo'scontrastingmiddle section the openingmotive is restated(in D, major)with V7-I in the place of II6-I6 (e.g., mm.22-23).31See,however, Yizhak Sadai, Harmony in its Systematicand Phenom-enological Aspects, trans. J. Davis and M. Shlesinger (Jerusalem: Yanetz,1980), 149.

in context the dominantpotentialof the leading-tonetriadismuch more likely to be manifest than the weaker, subdom-inant potential.32Nonetheless, the leading-tone chord canassume subdominant unction in the form of a seventhchord;this is readily explained, for the seventh contributes to thechord another element in common with IV or II. Variouscontextual factors suchas inversion,doubling,and chordsuc-cession, can help in this respect, as the excerptsin Example4 illustrate.V. TWO ADDITIONAL THEORETICAL CONSIDERATIONS

Even thoughthe proposedtheorypromotesan essentiallysymmetricalfunctional structure, the previous section sug-gests that contextual considerations can modifythat symme-try to some extent; for example, the subdominant functionof VII is considerablyless prominentthan is the dominantfunction of II. The present section considers two additionalfactors which interfere with the system's underlying sym-metry.Theeffect of roots. The so-called root of any triad or sev-enth chord is a tone of privileged status. The existence ofroots disturbsthe balance among the so-called secondarytri-ads of each functionalcategory, since those secondarytriadsin each categorywhich contain the prototype'sroot possessan obvious advantageover those which do not. For example,VI has an advantageover III withinthe tonic category, andsimilarly,II over VI within the subdominantcategory. Thedominantcategory is an exception, since VII is obviously a

32I do not assumethat the dissonant ritone contained n the leading-tonetriadmustresolve by stepwisecontrarymotion, or that the leadingtone mustproceedto i. As argued n the appendix o thisarticle,to assume the necessityof these linear behaviorsis to assumea D-T functionalsuccession. It is notdifficultto see that the same factors which suppressthe weak subdominantpotential of VII enhance the (weak) dominantpotential of II in minor, asargued in connection with the Brahms example.

in context the dominantpotentialof the leading-tonetriadismuch more likely to be manifest than the weaker, subdom-inant potential.32Nonetheless, the leading-tone chord canassume subdominant unction in the form of a seventhchord;this is readily explained, for the seventh contributes to thechord another element in common with IV or II. Variouscontextual factors suchas inversion,doubling,and chordsuc-cession, can help in this respect, as the excerptsin Example4 illustrate.V. TWO ADDITIONAL THEORETICAL CONSIDERATIONS

Even thoughthe proposedtheorypromotesan essentiallysymmetricalfunctional structure, the previous section sug-gests that contextual considerations can modifythat symme-try to some extent; for example, the subdominant functionof VII is considerablyless prominentthan is the dominantfunction of II. The present section considers two additionalfactors which interfere with the system's underlying sym-metry.Theeffect of roots. The so-called root of any triad or sev-enth chord is a tone of privileged status. The existence ofroots disturbsthe balance among the so-called secondarytri-ads of each functionalcategory, since those secondarytriadsin each categorywhich contain the prototype'sroot possessan obvious advantageover those which do not. For example,VI has an advantageover III withinthe tonic category, andsimilarly,II over VI within the subdominantcategory. Thedominantcategory is an exception, since VII is obviously a

32I do not assumethat the dissonant ritone contained n the leading-tonetriadmustresolve by stepwisecontrarymotion, or that the leadingtone mustproceedto i. As argued n the appendix o thisarticle,to assume the necessityof these linear behaviorsis to assumea D-T functionalsuccession. It is notdifficultto see that the same factors which suppressthe weak subdominantpotential of VII enhance the (weak) dominantpotential of II in minor, asargued in connection with the Brahms example.

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Example3. II as dominant n minor: JohannesBrahms,Intermezzo,op. 117 no. 2xample3. II as dominant n minor: JohannesBrahms,Intermezzo,op. 117 no. 2a) m. 1) m. 1

Andanteon roppo conmoltompreione

1bbbbbp dolce

col v.D--------T (?)b) mm. 6-10

D------------------------------------------------------------------------------------------------T

Andanteon roppo conmoltompreione

1bbbbbp dolce

col v.D--------T (?)b) mm. 6-10

D------------------------------------------------------------------------------------------------T

more characteristicdominant triadthan III, even though III,and not VII, contains the prototype's root. As discussedabove, context often enhances the dominant function of VII.Moreover, in comparisonto VII-I, the progressionIII-I asD-T is not particularly ffective, sinceIII andI aremaximallysimilar triads, and thus offer very little contrast.3333It ollows that III is functionallythe most underprivilegedof all triads:it is not only less of a tonic than VI, but also less of a dominantthan VII.This conclusion, no doubt, is empiricallywell grounded. Lester (Harmony,vol. 1, 251), for example, has commented on the "degree of functional am-biguity inherent in III," noting that it is "greaterthan is the case with anyother diatonic chord in the key system. As a result, [Lester continues,] III

more characteristicdominant triadthan III, even though III,and not VII, contains the prototype's root. As discussedabove, context often enhances the dominant function of VII.Moreover, in comparisonto VII-I, the progressionIII-I asD-T is not particularly ffective, sinceIII andI aremaximallysimilar triads, and thus offer very little contrast.3333It ollows that III is functionallythe most underprivilegedof all triads:it is not only less of a tonic than VI, but also less of a dominantthan VII.This conclusion, no doubt, is empiricallywell grounded. Lester (Harmony,vol. 1, 251), for example, has commented on the "degree of functional am-biguity inherent in III," noting that it is "greaterthan is the case with anyother diatonic chord in the key system. As a result, [Lester continues,] III

The norms of doubling in four-partharmony-e.g., thenormsby whichone doublesthe third of the secondarytriadsis the diatonic triad east frequentlyused. Withthe exception of a few clearlydefined circumstances,most of its usages are dependent upon context to agreaterextent than any other diatonic chord."In a similarvein, Smith("TheFunctionalExtravagance,"113) states that "the mediant triad is notorious,especially n major,where it hasa leading-tone,but soundsperhaps oo muchlike a tonicto be clearlyheard as a dominant n mostcontexts.Mediantsrarelyhave anystraightforwardunction;sometimesthey are followed by dominantpreparation hords, sometimesby dominants,andsometimesby tonics." Thedominantfunction of III can be enhanced throughinversion and doubling,as in the progressionIII6-I (as discussedbelow). In minor, the use of #1 shighly effective in enhancingthe mediant'sdominantfunction.

The norms of doubling in four-partharmony-e.g., thenormsby whichone doublesthe third of the secondarytriadsis the diatonic triad east frequentlyused. Withthe exception of a few clearlydefined circumstances,most of its usages are dependent upon context to agreaterextent than any other diatonic chord."In a similarvein, Smith("TheFunctionalExtravagance,"113) states that "the mediant triad is notorious,especially n major,where it hasa leading-tone,but soundsperhaps oo muchlike a tonicto be clearlyheard as a dominant n mostcontexts.Mediantsrarelyhave anystraightforwardunction;sometimesthey are followed by dominantpreparation hords, sometimesby dominants,andsometimesby tonics." Thedominantfunction of III can be enhanced throughinversion and doubling,as in the progressionIII6-I (as discussedbelow). In minor, the use of #1 shighly effective in enhancingthe mediant'sdominantfunction.

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Example4. VII7as subdominant.After Aldwell andSchachter,Harmonyand VoiceLeading,2nd ed., Exx. 24-24 and 31-15 (pp. 392and 534)a) RobertSchumann,no. 5 from Kreisleriana, p. 16, mm. 51-53

210 MusicTheorySpectrum

Example4. VII7as subdominant.After Aldwell andSchachter,Harmonyand VoiceLeading,2nd ed., Exx. 24-24 and 31-15 (pp. 392and 534)a) RobertSchumann,no. 5 from Kreisleriana, p. 16, mm. 51-53[sehrlebhaft] *

S ----------- T

[sehrlebhaft] *

S ----------- T

b) Franz Schubert, Moment Musical D. 780 no. 3 (op. 94 no. 3), mm. 34-38) Franz Schubert, Moment Musical D. 780 no. 3 (op. 94 no. 3), mm. 34-38*o modtoAllegro moderatol K I ] r*o modtoAllegro moderatol K I ] r L II

1~-':~f -j f -TS------------------------------D-----------.---T1~-':~f -j f -TS------------------------------D-----------.---TII (S), III (D), and VI (T)-are anotherempiricalmanifes-tation of the functional significanceof the prototype'sroot.Given the common belief that these norms are better ac-counted for purely in terms of voice leading, the issue ofdoubling is examined more closely in an appendix to thisarticle.

Seventhchords.The symmetrical tructuredepictedin Fig-ure 2a is disturbedby adding a seventh to all six non-tonictriads,as shownin Table 1.34Particularly uggestivein Table1 vis-a-visFigure 2a is that whereas V7, like V, shares onlya single tone with the tonic triad, IV7, unlikeIV, shares two.

II (S), III (D), and VI (T)-are anotherempiricalmanifes-tation of the functional significanceof the prototype'sroot.Given the common belief that these norms are better ac-counted for purely in terms of voice leading, the issue ofdoubling is examined more closely in an appendix to thisarticle.Seventhchords.The symmetrical tructuredepictedin Fig-ure 2a is disturbedby adding a seventh to all six non-tonictriads,as shownin Table 1.34Particularly uggestivein Table1 vis-a-visFigure 2a is that whereas V7, like V, shares onlya single tone with the tonic triad, IV7, unlikeIV, shares two.34The onic is by nature consonant, and thus would not normallyappearas a seventh chord. See Agmon, "Tonicityand the Tritone."34The onic is by nature consonant, and thus would not normallyappearas a seventh chord. See Agmon, "Tonicityand the Tritone."

Perhaps this helps explain why II7, which has only one tonein common with the tonic triad (cf. V7), almost seems to usurpthe role of subdominant prototype in the domain of seventhchords.

VI. CONCLUSION

"The report of my death was an exaggeration," quippedMark Twain after an American newspaper announced that hewas dying in poverty in London.35 Sometimes it seems as

Perhaps this helps explain why II7, which has only one tonein common with the tonic triad (cf. V7), almost seems to usurpthe role of subdominant prototype in the domain of seventhchords.

VI. CONCLUSION

"The report of my death was an exaggeration," quippedMark Twain after an American newspaper announced that hewas dying in poverty in London.35 Sometimes it seems as

35DeLanceyFerguson,Mark Twain:Manand Legend (New York: Russell& Russell, 1965), 272-73.35DeLanceyFerguson,Mark Twain:Manand Legend (New York: Russell& Russell, 1965), 272-73.

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Table 1. The tonic triad versus the six non-tonic seventhchords: a common-tone hierarchyTable 1. The tonic triad versus the six non-tonic seventhchords: a common-tone hierarchyNon-tonic seventh chord

VI7III7, IV7V7, II7VII7

Non-tonic seventh chordVI7III7, IV7V7, II7VII7

Number of tones in commonwith the tonic triad3210

Number of tones in commonwith the tonic triad3210

though a similarlyunfounded announcementconcerninghar-monic theory has been posted. To be sure, tonal theory ingeneral has undergonea majorreassessmentin the past fiveor six decades, due primarily o the work of Heinrich Schen-ker; however, it is a mistake to believe that harmonyis nec-essarilyone of the main casualties of the process. Harmonyis with us to stay; and with it, so it seems, is the idea ofharmonic functions.Harmonicfunction, in the presentview, is only one of twoessential ingredientswhich ultimately should form part of atheory of tonal harmony;the other ingredientis chord pro-gression. A theory of chord progressionis essential in orderto account for the privileged status of certain root relation-ships, most notably by descending fifth. Although the ques-tion of chordprogression ies outside the scope of the presentarticle,some tentativesuggestionshave been made as to howharmonictheory in general may benefit from the interactionof such a (hypothetical)theory with the proposed theory offunctions.Considerable iberty has been taken here with Riemann'sFunktionstheorie,not only for the sake of greatertheoreticalrigor and the removal of arbitrary eatures, but also for thesake of placing music-theoretic discourse within what I be-lieve should be its proper intellectual context, namely cog-nitive science. Although Riemann himself developed Funk-

though a similarlyunfounded announcementconcerninghar-monic theory has been posted. To be sure, tonal theory ingeneral has undergonea majorreassessmentin the past fiveor six decades, due primarily o the work of Heinrich Schen-ker; however, it is a mistake to believe that harmonyis nec-essarilyone of the main casualties of the process. Harmonyis with us to stay; and with it, so it seems, is the idea ofharmonic functions.Harmonicfunction, in the presentview, is only one of twoessential ingredientswhich ultimately should form part of atheory of tonal harmony;the other ingredientis chord pro-gression. A theory of chord progressionis essential in orderto account for the privileged status of certain root relation-ships, most notably by descending fifth. Although the ques-tion of chordprogression ies outside the scope of the presentarticle,some tentativesuggestionshave been made as to howharmonictheory in general may benefit from the interactionof such a (hypothetical)theory with the proposed theory offunctions.Considerable iberty has been taken here with Riemann'sFunktionstheorie,not only for the sake of greatertheoreticalrigor and the removal of arbitrary eatures, but also for thesake of placing music-theoretic discourse within what I be-lieve should be its proper intellectual context, namely cog-nitive science. Although Riemann himself developed Funk-

tionstheorie roma rationalperspectivefor the most part,latein life his general methodological outlook became overtlymentalistic. It seems only fitting, therefore, to bringthe pro-posed, prototype-theoreticaccountof harmonic functions toa close with one of Riemann's most uncompromisingstate-ments in this regard:The"AlphandOmega" fmusical rtistrysnot ound ntheactual,soundingmusic,butrather xists n the mental mageof musicalrelationshipshatoccurs n thecreative rtist'smagination-amentalimage hat ivesbeforet is transformedntonotationndre-emergesin the imagination of the hearer ... If one has graspedthese fun-damental ideas, then it is clear that the inductive method of tone-physiology nd one-psychologysheadedn thewrongdirectionromtheverybeginningwhen t takesas itspointof departurehe inves-tigation f theelementsf soundingmusic,nstead f theexaminationof the elements f musicas it is imagined.n otherwords:neitheracoustics, ortone-physiologyortone-psychologyangivethekeyto the nnermostssence fmusic,butrathernlya "Theoryf TonalImagination"-aheorywhich, o be sure,hasyet to bepostulated,much essdeveloped ndcompleted.36

APPENDIX

THE NORMS OF DOUBLING IN FOUR-PART HARMONY: A CRITIQUEOF THE VOICE-LEADING ARGUMENT

As is well known, it is customaryto double the third ofII, III, and VI, in their capacityas subdominant,dominant,and tonic triads, respectively. This finding, as already indi-cated, easily lends itself to a functionalexplanation.Indeed,36RobertWason and Elizabeth WestMarvin,"Riemann's Ideen zu einerLehre von den Tonvorstellungen':An Annotated Translation,"Journal ofMusic Theory36 (1992): 82-83.

tionstheorie roma rationalperspectivefor the most part,latein life his general methodological outlook became overtlymentalistic. It seems only fitting, therefore, to bringthe pro-posed, prototype-theoreticaccountof harmonic functions toa close with one of Riemann's most uncompromisingstate-ments in this regard:The"AlphandOmega" fmusical rtistrysnot ound ntheactual,soundingmusic,butrather xists n the mental mageof musicalrelationshipshatoccurs n thecreative rtist'smagination-amentalimage hat ivesbeforet is transformedntonotationndre-emergesin the imagination of the hearer ... If one has graspedthese fun-damental ideas, then it is clear that the inductive method of tone-physiology nd one-psychologysheadedn thewrongdirectionromtheverybeginningwhen t takesas itspointof departurehe inves-tigation f theelementsf soundingmusic,nstead f theexaminationof the elements f musicas it is imagined.n otherwords:neitheracoustics, ortone-physiologyortone-psychologyangivethekeyto the nnermostssence fmusic,butrathernlya "Theoryf TonalImagination"-aheorywhich, o be sure,hasyet to bepostulated,much essdeveloped ndcompleted.36

APPENDIX

THE NORMS OF DOUBLING IN FOUR-PART HARMONY: A CRITIQUEOF THE VOICE-LEADING ARGUMENT

As is well known, it is customaryto double the third ofII, III, and VI, in their capacityas subdominant,dominant,and tonic triads, respectively. This finding, as already indi-cated, easily lends itself to a functionalexplanation.Indeed,36RobertWason and Elizabeth WestMarvin,"Riemann's Ideen zu einerLehre von den Tonvorstellungen':An Annotated Translation,"Journal ofMusic Theory36 (1992): 82-83.

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