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Tonality and functional equivalence: A multi-level model for the cognition of triadic progressions in 19th century

Fabian C. Moss University of Music and Dance Cologne, Germany [email protected]

In: Jakubowski, K., Farrugia, N., Floridou, G.A., & Gagen, J. (Eds.) Proceedings of the 7th International Conference of Students of Systematic (SysMus14) London, UK, 18-20 September 2014, http://www.musicmindbrain.com/#!sysmus-2014/cfmp This paper is released under a CC BY-NC-ND Creative Commons License (http://creativecommons.org/licenses/).

The subject of this paper is the cognition of triadic progressions in 19th century tonal music. Music psychological research concerning the cognition of harmonic progressions mainly relies on diatonic music in which triads are easily relatable to a key. Triadic distance is therefore measured in terms of relationships to the tonal center, the tonic (Krumhansl & Kessler, 1982). This conception is not directly applicable to chromatic music where musical coherence is not only obtained by common a key. Transformational puts strong emphasis on voice-leading parsimony as a measure of distance. The most efficient transformations between triads are P (parallel), R (relative) and L (leading-tone exchange), which is also in accordance with empirical findings of diatonic triadic relatedness (Krumhansl, 1998). Notably, P and R generate an containing eight major and minor triads which are claimed to be functionally equivalent. Transformational analyses result in sequential patterns of triadic progressions and an overarching key is not required. Based on an extended notion of function and acknowledging that there are compelling arguments for both hierarchical and sequential representations of the cognition of harmonic progressions a multi-level model is proposed that combines both approaches, adopting features of the generative model by Rohrmeier (2011). The two main components of the model are the concept of functional equivalence and the distinction between the hierarchic-syntactic cognition of functional progressions and the schematic cognition of functional values.

The subject of this paper is the representation marizes important empirical research on to- of the cognition of triadic progressions in 19th nality perception and reveals difficulties when century tonal music. The proposed model is applying the results to 19th century . supported both by music theoretic literature Special attention is given to the debate (historic as well as contemporary) and music whether tonal music has syntactic features psychological research. Despite the fact that comparable to language or not. In the different are interrelated by dis- and main section a cognitive multi-level mod- tinct physical ratios of frequencies, tonality is el for triadic progressions is outlined. The a cognitive concept that cannot be found in model attempts to reconcile the syntax- vs. the external musical stimuli but in the human schema-based accounts in assuming that mind. Cognitive models of tonality should both cognitive processes take part in the cog- reflect the fact that this concept evolved and nition of 19th century . It is also changed its meaning over time and that it is mentioned how this model can inspire further possible for contemporary listeners to distin- empirical research and open questions are guish between different kinds of . In addressed. A final short conclusion points out other words, “the perception of musical struc- the main statements and results of this pa- ture depends on the processing of pitch in- per. formation with reference to a system of If not mentioned otherwise, uppercase letters knowledge about the conventional uses of (e.g., C, Db, F#) indicate major triads and pitches within the musical tradition” lowercase letters (e.g., c, db, f#) indicate (Krumhansl & Kessler, 1982, p. 334). This minor triads. Italic abbreviations (e.g., P, paper attempts to present such a model. auth, Id) stand for transformations. The first section provides the theoretical background from transformational music the- A Riemannian motivation ory and introduces the concept of functional The chord sequence in Fig. 1 is presented by equivalence. The second section briefly sum- Hugo Riemann in the final chapter on tonality

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transformation (L) that lowers the root of triad and raises the of a minor triad by a . It turns out that these voice-leadings coincide with inversions of the triad at one of its constituting intervals. The left side of Fig. 2 shows those inversions of C. The black solid lines visualize the inverted in- Figure 1. Chord sequence presented by Riemann terval. P inverts the fifth, R inverts the major (1880). third and L inverts the minor third. The right side of Fig. 2 shows the triads embedded into of his “Outline of a new method of harmonic theory”1 (Riemann, 1880, p. 67). This se- quence will be employed as a representative for triadic progressions in 19th century tonali- ty. Riemann observes:

“The old conception of key, meaning a harmony based on [diatonic] scales is by now faltering be- cause of the ‘altered chords’; the categories of al- tered chords in the old sense of ‘chords with tones not belonging to a scale’ had to be extended too disproportionately, if one wanted to hold onto the scale-key, facing today’s liberal harmony. Pro- gressions like [Fig. 1] mock any old-fashioned schematization; even if one would ridiculously em- ploy three or four different keys to clarify the mat- ter, it wouldn’t fit” (Riemann, 1880, p. 67, highlighting original, translation by the author). Figure 2. Effects of the transformations P, R and L on C {0, 4, 7} (light grey). The dashed line marks the mirror Subsequently, an analysis in terms of key axis, the arrows mark parsimonious voice-leading; T and E are abbreviations for 10 and 11, respectively. scale-degrees is an inadequate attempt and other explications are needed2. the pitch-class circle. The black double- Music theory headed arrows show the voice-leading and the grey dashed lines visualize the corre- 5 Transformational music theory sponding inversional mirror axis . Being a subdiscipline of mathematical music The dominant transformation D that moves a theory, transformational theory3 uses mathe- triad by a fifth downwards is not parsimoni- matical concepts for , espe- ous but musically of great importance. The cially from group theory4. Strong emphasis is identity transformation Id doesn’t change put on voice-leading parsimony between tri- anything but is necessary for formal reasons. ads or other pitch collections as a measure of Transformational analyses often employ an distance (Cohn, 1997; Douthett & Steinbach, abstract map to depict tonal relations, the 1998; Lewin, 1980). The smaller the amount “” (tone net, Fig. 3), most commonly of voice-leading steps, the closer the rela- spanned between a fifth axis (horizontal) and tionship. Thus, the most efficient transfor- a major third axis (lower left to upper right). mations between major and minor triads re- garding parsimony are those that raise or lower only a single voice by a tone or a semi- tone. These are the relative transformation (R) that raises the fifth of a major triad and lowers the root of a minor triad by a full step; the parallel transformation (P) that lowers or raises the third of a major or minor triad by a half step, respectively; and the leading-tone

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Figure 5. A function decomposed as minor third related fifths, as a scale and as a succession of P and R related triads.

Figure 3. The “Tonnetz”, reprinted from Cohn (1998). According to music theorists of the Hungarian Through concatenation6 of the triadic trans- tradition (B. Haas, 2004; Lendvai, 1971, formations it is possible to “navigate” on the 1995; Schild, 2010) triads on this path share Tonnetz through paths like on a map. An the same function, they are functionally 9 overarching key to which each member of the equivalent . Following this theoretical para- sequence can be related is not required. This digm, “function” subsequently denotes the fact predestines this method for the analysis membership to the octatonic scale as well as of 19th century music where a key is some- its manifold triadic instantiations. Fig. 5 times difficult to identify. The transformation- shows a function, decomposed into minor al analysis of the C–Ab–D– third separated fifths (top left), an octatonic G–C is shown in Fig. 47. The initial C (top right) and eight major and minor triad is marked with an asterisk (*). triads (bottom). There are three transposi- tions of a function which can in a concrete musical context be identified as tonic (T), dominant (D) and (S)10. This guarantees that each of the 24 minor or ma- jor triads can be uniquely assigned to a func- tion. In Fig. 4 and 6 the three functions are colored blue, red and green, respectively. The inter- action of the three functions (their mutually shared pitch classes) is depicted in Fig. 6. There are two functional progressions: au- Figure 4. Transformational analysis of the triadic se- 11 quence C–Ab–D–G–C. The asterisk (*) marks the initial thentic and plagal, denoted auth and plag . and final triad C. They are indicated by arrows on the left and the right of Fig. 6. Maintaining the same func- tion is called functional prolongation12, denot- Function and functional equivalence ed prol, which is not shown in Fig. 6. The function of a chord in diatonic contexts is usually identified with its root's position in the scale of a given key. But according to Hugo Riemann (1880, 1893) different chords can have the same function or, at least, substitute another functionally. Traditionally, P and R re- lated chords are understood as being func- Figure 6. Interaction of the three functions tonic (T), tionally very close8. Applying P and R alter- dominant (D) and subdominant (S). Uppercase letters stand for pitch classes. The arrows on the left and right nately on a triad generates an octatonic scale indicate functional progressions (auth and plag). containing eight major and minor triads (see Fig. 5) that corresponds to a diagonal path on the Tonnetz from upper left to lower right (colored differently in Fig. 4).

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Empirical research on tonality they play significant functional roles15, em- phasizing again the prominence of the parsi- Probe-tone ratings and key-profiles monious transformations. Empirical research on tonality attempts to This study has inspired a great deal of subse- find out which cognitive processes underlie quent research on tonality (e.g. Bharucha & the perception of harmony in tonal music13. Krumhansl, 1983; Temperley, 2001) though One of the most influential publications on remarkably there have been no serious at- tonality perception was the study of tempts to leave the key-bound diatonic con- Krumhansl and Kessler (1982). By asking lis- text so far. Even if chromatic music (i.e. not teners to rate how well a probe-tone fits into diatonic music) is considered in research ei- a given harmonic context (established by ther diatonic scales are the point of departure previous listening to different types of ca- or any tonal context is forsaken (e.g. atonal dences) they deduced stability values for music) (Woolhouse & Cross, 2010; each member of the with ref- Woolhouse, 2012). However, Riemann’s (Fig. 1) seems to challenge the tonal found by Krumhansl and Kessler. Ab is rather distant from the C-major key and Ab and D are even further apart (see Fig. 7)16. Also recall that a transformational analysis requested only a few steps to generate this sequence17 and no modulation occurs. The overall impression is that of a tonally coherent cadence. Consider- ing that empirical research strongly points to a hierarchic structure of tonal harmony but acknowledging that music theory suggests sequential analyses of 19th century triadic phrases, there is a gap that needs to be Figure 7. The representation of key distances obtained bridged. with multidimensional scaling, reprinted from Krumhansl & Kessler (1982). The letters stand for major and minor Tonal syntax or harmonic schemata? keys. Syntactic models of tonality trace back at erence to the tonal center. Those values were least to the groundbreaking publication of “A used to generate key-profiles for the major Generative Theory of Tonal Music” (Lerdahl & and minor mode, and a four-dimensional map Jackendoff, 1983) that incorporates linguistic of interkey distances was produced using conceptions and presumes that music’s syn- multidimensional scaling (Fig. 7). On this map tactic features are comparable, if not essen- perceived distances are represented by geo- tially identical, to linguistic ones. It has be- metric distances. As can be seen the closest come evident that a great deal of empirical perceived relationships between keys are in findings account for a hierarchical representa- astonishing accordance with parsimonious tion of harmonic structure18. More recently, voice-leadings between triads. This means, there has also been support from neurosci- Krumhansl and Kessler’s representation is ence for the existence of musical hierarchies qualitatively equivalent to the Tonnetz14. Both (Fitch & Martins, 2014; Koelsch, Rohrmeier, depictions have the shape of a torus, mean- Torrecuso, & Jentschke, 2013; Patel, 2008, ing that opposing sides coincide. 2012). The next step was to measure a chord’s func- Critics of a parallelism between language and tion by determining the distance of its root in music state that “tones and chords are not reference to the tonal center of a key. Not really syntactic at all” (London, 2012, p. 242) very surprisingly, it turned out that triads are and that musical function is better understood perceived most closely to those keys in which in terms of schemata. Schemata model “how

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Figure 8. The multi-level model. On top, hierarchic functional progression is given by auth, by prol. Default values are assigned by inst. The default value for S is altered by PR and RP, respectively. The resulting triadic progression reflects the musical surface. humans organize experience into structured A multi-level model categories. [They have] particular variable and default values, and can be intricately or- Based on the extended notion of function that ganized in ways unique to the domain” assumes functional equivalence for triads (Lerdahl, 1991, p. 273). This was already contained in an octatonic scale, and having in suggested by Krumhansl and Kessler: mind that there are compelling arguments for both hierarchical and schematic representa- “Both the construction and perception of pitch se- tions of harmonic progressions a multi-level quences around one or a few tones may reflect a approach is proposed. general cognitive phenomenon that has been shown to apply to a wide variety of perceptual and Hierarchic functional progressions and semantic domains […]. According to this view there schematic functional values are certain members of natural categories that function as cognitive reference points, or proto- The main components of the model are the types, for the category as a whole. These elements concept of functional equivalence and the are described as the most representative of the distinction between the hierarchic-syntactic categories, in relation to which all other category members are seen” (Krumhansl & Kessler, 1982, cognition of functional harmonic progressions p. 363). and the schematic cognition of tonal function- al values, following Rohrmeier (2007, Despite the fact that hierarchic and schematic 2011)19. He distinguishes between a function- cognition are two distinct cognitive mecha- al and a scale degree level20, tearing apart nisms, the present approach suggests that functionality and diatonic . This is they are tightly intertwined and may both worth mentioning because a great deal of underlie the cognition of triadic progressions empirical research on tonality fails to make th of 19 century music. this important differentiation. The psychologi- cal independence of the dominant transfor- mation and the parsimonious transformations (Krumhansl, 1998) might also support the

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view that they are perceived on a structurally method? This might be difficult because it can different level. be assumed that the and the related feeling of a tonal center of a key is Functional progressions are modelled as hier- anchored much more deeply in most listeners archic branchings (Lerdahl & Jackendoff, than the sense for functional equivalence in 1983; Lerdahl, 1991) and are denoted by chromatic romantic music. The high degree of auth and plag. Applying auth corresponds to of the octatonic scale itself pre- the highest branching off a predefined T func- vents the existence a tonal center (Messiaen, tion, while plag corresponds to the highest 1944). Admittedly some tones can be under- branching off a D function. Functional prolon- stood as structurally more important, because gation is given by prol which expands a func- they form the basis of a fifth interval (see tion on the hierarchic level. Assigning the first bar of Fig. 5) and the fifth is an interval functional default value is called instantia- both music theoretically and psychoacousti- tion21 (denoted by inst). It serves as a media- cally understood as fundamental (Handschin, tor between the hierarchical level of function- 1948; von Helmholtz, 1863) but it is uncer- al progressions and the schematic level of tain if this suffices to establish a stable tonal functional values. The alteration of a func- context as claimed by music theory (B. Haas, tional value is given by P, R and any of their 2004; Lendvai, 1971, 1995; Messiaen, 1944; mutual concatenations. Schild, 2010). As Krumhansl and Kessler An example note, “the of tonal stability is ac- quired through experience with the structural Fig. 8 shows the multi-level model with refer- relations that obtain in the music itself” ence to Riemann’s cadence (Fig. 1). The (Krumhansl & Kessler, 1982, p. 364). It can overall functional progressions are modelled be assumed that intense exposure to chro- hierarchically revealing an authentic T–S–D–T matic music as well as an apt system of con- cadence. On the next level the subdominant cepts (being both concise and unequivocal) function (S) is expanded by prol. Then, the can contribute to create a solid perception of functional default values are instantiated by functional equivalence. inst. Finally, the default value of the subdom- inant (F) is altered via PR and RP into Ab and Also, because both theoretical and empirical D, respectively, so that Riemann’s cadence scholars prefer a toroidal representation of results as the musical surface. tonal space, one could wonder if this torpe- does the concept of hierarchy because a torus Discussion and open questions has no fixed center whereas a hierarchy needs a highest element. This affects particu- Obviously the multi-level model is to date not larly the initial progression in Fig. 1 from C to elaborated entirely. Very important for the Ab. While C is the highest element of the hi- further development will be its formalization erarchy, functional equivalence implies that it in a unified way. The example above de- is also subordinated to Ab because it is an scribes a top-down process. The model can authentic progression. This contradiction is also be understood as a bottom-up represen- shown by the dashed arrow. tation where starting from the musical surface the model unfolds step by step until a hierar- Conclusion chic functional understanding is achieved. To clarify if a top-down or a bottom-up represen- While I do not fully agree with the statement tation is more appropriate will be the scope of that “sequences of tones or chords are not further research. really syntactic at all” (London, 2012, p. 242), I argue that, while there is an overall hierar- Another issue concerns further empirical re- chical understanding of harmonic progres- search. The concept of functional equivalence sions (and thus also of ), har- is theoretically convincing but not yet verified. monic functions themselves can be viewed as Does the octatonic scale provide an adequate schematic entities that have certain default context to generate a stable feeling of tonality values. These can be substituted with func- for listeners in an experiment? Can functional tionally equivalent chords, achieved via the profiles be generated using the probe-tone

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application of the parsimonious P and R trans- of Music Theory, 42(2), 265–281. formation. doi:10.2307/843878 Krumhansl, C. L., & Kessler, E. J. (1982). Tracing Acknowledgments. I would like to thank all the dynamic changes in perceived tonal those who have reviewed this paper and organization in a spatial representation of helped me putting my thoughts together. I musical keys. Psychological Review, 89(4), thank Carol Krumhansl and Richard Cohn for 334–368. doi:10.1037/0033-295X.89.4.334 the permission to reprint parts of their work. Lendvai, E. (1971). Béla Bartók: An analysis of his music. London: Kahn & Averill. Also, I want to thank the Society for Educa- Lendvai, E. (1995). Symmetrien in der Musik. tion, Music and Psychology (SEMPRE) for fi- Einführung in die musikalische Semantik. nancial support regarding the travel and ac- Wien: Universal Edition. commodation expenses for attending the Lerdahl, F. (1991). Underlying musical schemata. SysMus14 conference. In I. Cross & P. Howdell (Eds.), Representing musical structure (pp. 273–290). New York: References Academic. Lerdahl, F. (2001). Tonal Pitch Space. Oxford Bharucha, J. J., & Krumhansl, C. L. (1983). The University Press. representation of harmonic structure in Lerdahl, F., & Jackendoff, R. S. (1983). A music: hierarchies of stability as a function generative theory of tonal music. of context. Cognition, 13(1), 63–102. Cambridge, : MIT Press. Cohn, R. (1997). Neo-riemannian operations, Lewin, D. (1980). On generalized intervals and parsimonious trichords, and their “Tonnetz” transformations. Journal of Music Theory, representations. Journal of Music Theory, 24(2), 243–251. 41(1), 1–66. London, J. (2012). Schemas, not syntax: a reply to Crans, A. S., Fiore, T. M., & Satyendra, R. (2009). Patel. In P. Rebuschat, M. Rohrmeier, J. A. Musical Actions of Dihedral Groups. Hawkins, & I. Cross (Eds.), Language and American Mathematical Monthly, 116(6), Music as Cognitive Systems (pp. 242–247). 479–495. Oxford University Press. De Haas, W. B. (2012). Music information retrieval Messiaen, O. (1944). Technique de mon langage based on tonal harmony. Universiteit musical: 1er Volume. Paris: Alphonse Leduc. Utrecht. Patel, A. D. (2008). Language, music, and the Douthett, J., & Steinbach, P. (1998). Parsimonious brain. Oxford: Oxford University Press. graphs: A study in parsimony, contextual Patel, A. D. (2012). Language, music, and the transformations and modes of limited brain: a resource-sharing framework. In P. transposition. Journal of Music Theory, Rebuschat, M. Rohrmeier, J. A. Hawkins, & I. 42(2), 241–263. Cross (Eds.), Language and Music as Fitch, W. T., & Martins, M. D. (2014). Hierarchical Cognitive Systems (pp. 204–223). Oxford processing in music, language, and action: University Press. Lashley revisited. Annals of the New York Riemann, H. (1880). Skizze einer neuen Methode Academy of Sciences, 1316(1), 87–104. der . Breitkopf und Härtel. doi:10.1111/nyas.12406 Riemann, H. (1893). Vereinfachte Harmonielehre Haas, B. (2004). Die neue Tonalität von Schubert oder die Lehre von den tonalen Funktionen bis Webern: Hören und Analysieren nach der Akkorde. Augener. Albert Simon. Wilhelmshaven: Florian Rohrmeier, M. (2007). A generative grammar Noetzel. approach to diatonic harmonic structure. In Handschin, J. (1948). Der Toncharakter: Eine Proceedings of the 4th Sound and Music Einführung in die Tonpsychologie. Zürich: Computing Conference (pp. 97–100). Atlantis Verlag. Lefkada, Greece. Koelsch, S., Rohrmeier, M., Torrecuso, R., & Rohrmeier, M. (2011). Towards a generative Jentschke, S. (2013). Processing of syntax of tonal harmony. Journal of hierarchical syntactic structure in music. Mathematics and Music, 5(1), 35–53. Proceedings of the National Academy of doi:10.1080/17459737.2011.573676 Sciences of the United States of America, Rohrmeier, M., & Cross, I. (2009). Tacit tonality : 110(38), 15443–8. Implicit learning of context-free harmonic doi:10.1073/pnas.1300272110 structure. In J. Louhivuori, T. Eerola, S. Krumhansl, C. L. (1998). Perceived triad distance: Saarikallio, T. Himberg, & P.-S. Eerola Evidence supporting the psychological reality (Eds.), Proceedings of the 7th Triennial of neo-riemannian transformations. Journal Conference of European Society for the Cognitive Sciences of Music (ESCOM 2009)

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Jyväskylä, Finland (pp. 443–452). Jyväskylä, Finland. we consider triads as sets. Rohrmeier, M., & Rebuschat, P. (2012). Implicit 6 learning and acquisition of music. Topics in The order of the transformations reflects the order of Cognitive Science, 4(4), 525–53. their application, e.g. PL means “first P, then L”. 7 This is the same sequence as presented by Riemann doi:10.1111/j.1756-8765.2012.01223.x 7 7 Schild, J. (2010). “... zum Raum wird hier die (Fig. 1) except that the sevenths of D and G are Zeit.” Tonfelder in Wagners Parsifal. In B. omitted. The minor of a chord is understood to Haas & B. Haas (Eds.), (pp. 313–373). reinforce the dominant feeling of that chord. This fact Hildesheim, Zürich, New York: Georg Olms comes from contrapuntal rules where the seventh of a Verlag. dominant is supposed to resolve into the major or minor Temperley, D. (2001). The Cognition of Basic third of a following tonic. 8 Musical Structures. MIT Press. P related keys share the same tonic, R related keys Tillmann, B., Bharucha, J. J., & Bigand, E. (2000). share the same scale. 9 Implicit learning of tonality: a self-organizing Functional equivalence does not imply that equivalent approach. Psychological Review, 107(4), triads are indistinguishable. 10 885–913. The cognitive process of tonic-finding is a research Von Helmholtz, H. L. F. (1863). Die Lehre von den area on its own and cannot be reproduced here. 11 Tonempfindungen als physiologische These names are adopted from music theory where Grundlage für die Theorie der Musik. “authentic” (or “perfect”) and “plagal” describe Braunschweig: F. Vieweg und Sohn. that employ falling or rising fifths, respectively. 12 Vos, P. G. (2000). Tonality induction: theoretical Recall that the the dominant (D) and the identity problems and dilemmas. Music Perception: transformation (Id) were defined for triads. Functional An Interdisciplinary Journal, 17(4), 403–416. progressions (plag and auth) and functional prolongation doi:10.2307/40285826 (prol) refer to functions (Lerdahl & Jackendoff, 1983; Woolhouse, M. (2012). Wagner in the Round: Lerdahl, 2001). 13 Using Interval Cycles to Model Chromatic The question of tonality induction, how one learns Harmony. In E. Cambouropoulos, C. structural features of tonality (Rohrmeier & Cross, 2009; Tsougras, P. Mavromatis, & K. Pastiadis Rohrmeier & Rebuschat, 2012; Tillmann, Bharucha, & (Eds.), Proceedings of the 12th International Bigand, 2000; Vos, 2000) is not addressed here. 14 Conference on Music Perception and The L relation is not shown in Fig. 6. 15 Cognition and the 8th Triennial Conference This is a different concept of function as the of the European Society for the Cognitive Riemannian one in the present context. 16 Sciences of Music (pp. 1142–1145). Fig. 6 represents perceived distances of keys not Thessaloniki. chords. Nonetheless, a chord is most closely perceived to Woolhouse, M., & Cross, I. (2010). Using interval the key in which it is the tonic. We thus assume that Ab cycles to model Krumhansl’s tonal and D are the closest to the respective keys. 17 hierarchies. Music Theory Spectrum, 32(1), The shortest way from Ab to D is (RP)²=(LR)²LP. For 60–78. graphical reasons it is not shown in Fig. 4. 18 Krumhansl and Kessler recognized that “a hierarchy of levels is believed to exist in the instantiation of tonal regions” (1982, p. 336). 1 “Skizze einer neuen Methode der Harmonielehre”. 19 Although Rohrmeier’s rule-based model explicitly deals 2 It has to be emphasized that the present approach with diatonic music, it facilitates the extension of its differs from Riemann’s resolution. Nonetheless, his application to chromatic music because it understands argument that traditional key-related root analysis is tonal harmony as a context-free grammar. See also W. incapable to account for this harmonic progression still B. de Haas (W. B. de Haas, 2012). holds. 20 In fact, he distinguishes between four levels: phrase 3 Scholars that focus on the analysis of tonal music level, functional level, scale degree level and surface sometimes prefer the term “Neo-Riemannian Theory”, level. Here, only the distinction between functional and honoring Hugo Riemann. scale degree level is taken into account. 4 A group is a set together with a binary operation that 21 Rohrmeier speaks of “the generation of elementary fulfills distinct features. The exact definition is not functional chord terms from functional regions” important here and can be looked up in any textbook of (Rohrmeier, 2011, p. 40). This is primarily a formal higher algebra. Important for the current discussion is assumption, because every schema is believed to have a that groups can be used to represent symmetric default value. In an abstract context it might be likely structures. For a short introduction, see Crans, Fiore, & that the default values of a function are unambiguously Satyendra (2009). determined, but in a concrete musical context many 5 Subsequently, it will not be distinguished between factors might easily interfere. voice-leading and inversional transformations, because

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