Aesthetics No.23-24 (2021): 1-13

The Japanese Society for Aesthetics

Neo-Riemannian and Schenkerian Theories:
Priority of Interpretation or Method*
NISHIDA Hiroko

Kyushu University, Fukuoka

1. Neo-Riemannian and Schenkerian Theories
Since the latter half of the 20th century in America, various kinds of music theory have beendeveloped. Along such development, studies on the tendencies and changes within the Americanmusic theory have been published since the 1970s (e.g. Rahn 1980, McCreless 1998, Christensen
2002), promoted by the establishment of the Society for Music Theory. These studies prompted
the publication of several specific journals by various institutions and, as a result, diversified thetheoretical methods. This work attempts to understand how these diverse spreading theories relate
among each other.
As a case study, this work selects Neo-Riemannian theory, which is one of the recent
mainstream theories, and reveals the collaborative or competitive relationship of such theory to
an existing one. Specifically, it examines the effects of methodological arguments proposed by
Neo-Riemannian theory on Schenkerian, which is an established theoretical empire.
Neo-Riemannian theory is derived from the transformational theory proposed by David
Lewin’s Generalized Musical Intervals and Transformation (1987); it is among the music
theoretical clans represented by a series of discussions by Brian Hyer, Henry Klumpenhouwer,
and Richard Cohn beginning in the 1980s. In this work, transformational and Neo-Riemannian
theories are regarded as almost synonymous. 1
Schenkerian theory concerns mainly the tonal music, whereas Neo-Riemannian theory, thepost-tonal and the late-Romantic chromatic music. In this way, as seen from the analytic objects,
Neo-Riemannian and Schenkerian theories cannot be compared in the same horizon. Although
Julian Hook emphasized the fundamental difference of both theories, Hook pointed out ‘[the]* This paper is based on the Japanese version printed in Bigaku 68/1 (2017): 121-132, published by the JapaneseSociety for Aesthetics.
1
The first use of the term ‘transformation’ was seen in Lewin’s Generalized Musical Intervals andTransformation in 1987, whereas that of the term ‘Neo-Riemannian’ was in Cohn’s article in 1996. (In the eighth
chapter of Generalized Musical Intervals and Transformation, Riemann’s concepts are introduced). Transformational
theory is often called Neo-Riemannian theory. Combinations, such as ‘Transformational Riemannian theory’ (e.g.
Klumpenhouwer 2002, 473) or ‘Riemann transformation’ (e.g. Klumpenhouwer 1994) are also often employed. In
most cases, Neo-Riemannian theory is named as a subdivision of theories by certain theorists among transformational
theorists (e.g. Nolan 2002, 296). In Journal of Music Theory Vol. 42 No. 2 in 1998, which is a special volume for
Neo-Riemannian theory, Cohn stated that ‘the papers collected in this issue represent an emerging species oftransformational theory drawn together under the Neo-Riemannian designation’ (Cohn 1998, 167). From thisdescription, Cohn used both terms almost without distinguishing them. Although ‘Neo-Riemannian theory’ is usedin this work in principle, the context and terminology of the terms Neo-Riemannian and transformational theories
are used.

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NISHIDA Hiroko

mistake to regard “transformation” and “prolongation” as antithetical conceptions’ (Hook 2007,
168). Thus, several studies have either integrated or differentiated them, referring to the boundary
pieces.2
This work focuses on the studies that examined the relationship of the two theories. By
referring to these studies, this work examines the effects of methodological arguments, whichwere proposed by the new theory against the existing one, on the development of the relevant
discipline.
The term ‘Neo-Riemannian’ is derived from German music theorist Hugo Riemann’s(1849–1919) conceptions, which were shown in his harmonic theory (cf. Riemann 1880 & 1916).
His main conceptions refer to the three kinds of transformation, namely, ‘parallel’ (‘P’), ‘relative’(‘R’), and ‘Leittonwechsel’ (L).3 In Example 1, the P transformation transforms major triad tominor and minor triad to major by moving the third tone by semitone. The R transformationtransforms major triad to minor by moving the fifth whole tone up, and minor triad to major, bymoving the root whole tone down. The L transformation transforms major triad to minor by
moving the root semitone down, and minor triad to major, by moving the fifth tone semitone up.

Example 1. Three kinds of transformation in Neo-Riemannian theory (Mason 2013, 7)

Riemann’s conceptions have been employed as a method to analyse chromatic progressions
and progressions based on mediant relations, which cannot be grasped in the framework of
traditional functional theory and atonal music.4 By using the conceptions derived from Riemann,the ‘Tonnetz’ 5 (cf. Klumpenhouwer 1994; Cohn 1998), 6 Neo-Riemannian theory has beendeveloped. Neo-Riemannian theory is not as explicitly systematic as Schenkerian theory.
In contrast, Schenkerian theory is derived from the conceptions of Austrian theorist Heinrich
Schenker (1868–1935), and has been applied since the end of the Second World War. It is ananalytic method to extract structurally important tones in a musical piece and, finally, reduce thetone to a fundamental structure called ‘Ursatz’. Schenkerian theory’s history is a half century
2
As seen in Hook’s study in the first and second sections of this work, Neo-Riemannian and Schenkerian
theories tend to be contrasted as ‘transformation’ and ‘prolongation’. However, the characteristics of the latter theory
cannot be reduced only to the concept of prolongation. One must keep in mind that, in the process of contrasting boththeories, Schenkerian theory is simplified.
3
Although ‘Leittonwechsel’ is translated into English as ‘leading-tone exchange’ or ‘leading-tone change’, theGerman term is also frequently used. For this reason, this work uses the German term. ‘Leitton’ means chromatic
progression, and ‘Wechsel’ means the exchange of major and minor. P and R correspond to Riemann’s ‘Variante’ and‘Parallele’ (Riemann 1880).
4
For example, in Figure 1 in the second section of this work, one can see ‘mm.154–59 (=PL)’ with an arrow.
The note b means the triad b d f-sharp. Its P is b d-sharp f-sharp. Moreover, its L is b-flat d-sharp f-sharp. If this istransformed enharmonically, it becomes b-flat e-flat g-flat, which is designated as e-flat in Figure 1.
5
It is simply called ‘network’.
6
On the misinterpretations of Hyer and Lewin, and Klumpenhouwer’s application and explanations of concepts,
see Klumpenhouwer 1994.

Neo-Riemannian and Schenkerian Theories

3

longer than Neo-Riemannian’s.
Individual discussions on the expansion of analytic objects and methodological refinements
and applications in both theories have been active. Especially, Neo-Riemannian theory, whichappeared later than Schenkerian theory, has recently been argued in Journal of Music Theory Vol.
4 No. 2 (1998), a publication dedicated to the discussion on such theory, and The Oxford
Handbook of Neo-Riemannian Music Theories (2011). The historical aspects of both theories
have also been examined (e.g. Christensen [ed.] 2002).
As individual arguments have been promoted, studies on the relationship of the two theories
have emerged sporadically. The best example is the Journal of Schenkerian Studies Vol. 2 (2007),
titled ‘Special Volume on Schenkerian and Neo-Riemannian Methodologies’. This work
deciphers the characteristics of the arguments on theoretical methodologies observed in thesestudies, which focused on the relationship between the two theories.

2. Arguments on Neo-Riemannian and Schenkerian Theories
In this section, the arguments on the relationship between the two theories are examined. As
certain studies did not compare the two explicitly, this work selects only those that focused clearlyon the comparison (Cohn 1999, Samarotto 2003, Hook 2007, Goldenberg 2007, Rings 2007,
Baker 2008). 7 In most cases, these arguments were carried out by Neo-Riemannian theorists;
because it is more recent than Schenkerian theory,8 its difference from the existing theories is,naturally, emphasized.
This work classifies the arguments into two standpoints:
1. Studies that consider the integration of both theories impossible (2.1)(1) those that attempt to segregate them methodologically and do not attempt to integratethem (through a specific analysis)(2) those that attempt to integrate through a concrete analysis and then demonstrate theincompatibility of both theories
2. Studies that integrate both theories through a concrete analysis (2.2.)
2.1. Studies that do not integrate the methods
First, Hook’s study (2007) is selected as one that attempts a methodological segregation.
This study regarded the comparison of the two theories as impossible, as their characters andanalytic methods are fundamentally different. This standpoint does not attempt at integration atall, because its argument starts from the premise of the difference of both theories.
Neo-Riemannian theorist Hook has three points. The first one is the difference in nature ofboth theories:
7
Excluded in this work are Warren Darcy’s study in 2005, Hyer’s in 1995, and David Kopp’s in 2002, in which
the comparison is not explicit. Steven Rings’ dissertation in 2006 [2011] is used as a complement of his study in2007.
8Among a series of studies covered by this work, the scholars who can be regarded as Schenkerian are onlyFrank Samarotto and Michael Baker. However, other scholars can be considered to have fundamental knowledge andanalytic practice of Schenkerian theory.

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NISHIDA Hiroko

Transformation theory is a large and varied toolbox; there are only some minimal
instructions for using the tools, and no designs at all for what one can build with them.
Schenkerian theory, in contrast, is a smaller and more specialized toolbox, but it comeswith more instructions and even some partial sets of blueprints. If we assign a piece to aclass and instruct them to “do a Schenkerian analysis,” the students will not all produce
identical (or equally good) graphs, but we have a reasonable idea in advance of the steps
they will go through and the results they will get, because they will all be using the same
tools and reading from the same plans. If we give instead the instruction “do atransformational analysis,” we can have no such preconceptions: the results will be as
diverse as the students’ imaginations. (Hook 2007, 166)
Using the toolbox as an analogy to the nature of the theories and the amount of instructions
for their uses, the difference of the theories is highlighted; thus, when one uses transformation
theory, the analytic result depends highly on the individual. Moreover, according to Hook, theSchenkerian theory is based on rigorous guidelines in advance, whereas the Neo-Riemannian
theory lacks such guidelines:
While Schenkerian analysts have (fairly) rigorous guidelines and well-honed intuitions
for making the required decisions, transformational analysts may not. We may be able toarrive at such guidelines and intuitions, but only after making some more fundamental
decisions about what types of objects and transformations will be used in a network, andprobably only after acquiring some experience at working with these objects andtransformations in a variety of situations. An analytical approach in which each analysis
must write its own rules may seem inefficient ...; indeed, such an approach may well be
imperative in those broad swaths of the post-tonal repertoire in which it seems that eachpiece writes its own rules. (Hook 2007, 166)
Based on the difference in their nature, difference of analytic objects in both theories isexpressed by Hook. After differentiating both theories in this way, Hook employed thetransformational graph of Beethoven’s First Symphony, the first movement into ‘quasiSchenkerian graph’, and then pointed out its incompatibility:… as a Schenkerian graph, Example 4 [Example 2] is ill-formed. The C-C slur marks theintervening G as structurally subordinate, while the G-G slur marks the second C as
subordinate. Either slur is possible by itself, but the two are incompatible. Intransformational theory, such associations are perfectly permissible; it is up to the analyst
to decide when they are meaningful. In this way, at least, the transformational analysis is
more flexible and less prescriptive than the Schenkerian; the transformational approachsuggests a reading to which a rigidly Schenkerian approach may have blinded us, and iscertainly more amendable to the possibility of accommodating multiple interpretations
within a single analysis. (Hook 2007, 167)

Neo-Riemannian and Schenkerian Theories

5

Example 2. Displacement of transformational graph into quasi-Schenkerian graph (Hook 2007, 167)

In the Schenkerian theory, the hierarchy of tones must have a logical consistency; in contrast,
the transformational theory does not depend on the hierarchical relationship, that is, the context.
This is Hook’s third point. The latter stresses not the hierarchical consistency but themeaningfulness that an analyst finds. The statement that the transformational theory
accommodates more multiple interpretations can be considered part of the difference intheoretical nature (Hook’s first point).
Next, this work focuses on Richard Cohn’s article in 1999 and Frank Samarotto’s oralpresentation in 2003, 9 as examples of studies that provided a concrete analysis to show themethodological incompatibility of the two theories.
Cohn is a leading figure that has developed Neo-Riemannian theory widely after Lewin had
set up transformational theory. He used the first movement of Schubert’s piano sonata b-flat
major, demonstrated the transformational analysis by a Schenkerian graph, and pointed out its
insufficiency, as follows:… if a Schenkerian model were to label harmonies, it would simply measure distances
between roots along a diatonic spectrum. To do so here [in Schubert’ sonata] would be
problematic, since … the exposition is diatonically paradoxical. The relatively
unintepreted labels [by Neo-Riemannian theory] liberate the triads from theircommitment to a particular diatonic context. (Cohn 1999, 220)
Schenkerian theory is an inappropriate model for the analysis of pieces that contain diatonic
contradictions, as it performs the harmony analysis in the diatonic framework. In contrast, thediatonic contextualization is unnecessary in employing the Neo-Riemannian theory. In this sense,
Cohn insisted that the latter is suitable for analysing this kind of pieces. Cohn’s first point of viewis identical with Hook’s third point on the ‘contextual dependence’, which also highlights thedifference in the objects to be analysed in both theories, that is, Hook’s second point.
Moreover, in the conclusion of his article, Cohn shattered the ‘organicist vision’ of eachlayer’s (foreground, middle ground, and background) organic coherence in Schenkerian theory:
It is typical of much of Schubert’s music that diatonically indeterminate progressions
governed by efficient voice leading at the middleground are structurally “sandwiched” inbetween monotonal prolongations at the background and diatonically based
prolongations, cadentially articulated, at the foreground… Such images dissonate with
9

The author expresses her appreciation to Professor Frank Samarotto for sending his presentation material.

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NISHIDA Hiroko

structural expectations conditioned by the organicist vision of a seamless transition
through a series of levels, all unified by a uniform set of principles. (Cohn 1999, 232)
Contrary to organicists who use Schenkerian theory, Cohn, a Neo-Riemannian theorist, does
not find any premise in the coherence among hierarchical layers. Such Cohn’s attitude is among
the characteristics of Neo-Riemannian theory, that is, pieces must not be reduced to coherence.
This is Cohn’s third point.
Following Cohn’s argument in 1998, Schenkerian scholar Samarotto argued thattransformational and prolongational approaches are in conflict:
It seems clear that this arises because transformational and prolongational theories
exemplify two fundamentally different sorts of generative structures. Transformational
theories can be conceptualized as a series of operations applied to Klänge that occur
successively without necessarily being governed by an overall referential Klang or tonal
center. (Samarotto 2003, 2)
Samarotto’s point in his statement above, which is similar to Cohn’s third point, is thatmultiple chords within a piece are not dominated by a single chord; in other words, thetransformational theory does not premise the coherence.
Samarotto selected Brahms’s pieces to illustrate how transformational process does not
reduce pieces to the diatonic whole. Samarotto explained its reason as follows:First, I would stress that there are more conventional Schenkerian explanations available
for these pieces, but the particularly vivid way that these conflicts [among layers] areactualized argues against these; this necessarily a matter of close reading of individual
works, and not of a priori assumptions about what should happen. … This is not a matterof finding certain types of chord successions, however well-defined, but of paying carefulattention to the context that surrounds them. Thirdly, I think it is certainly valuable tofind new ways of discussing the special characteristics of nineteenth century music, but
not if it always carries the conceptual baggage of post-tonal theory, as, it seems to me,that Cohn would have it. (Samarotto 2003, 10)
It is possible to apply both Schenkerian and transformational explanation to one piece.
However, the conflict between the diatonic model and transformational chords cannot be
explained only by the framework of Schenkerian theory. This second point overlaps with Cohn’s
first issue. However, contrary to Hook’s and Cohn’s second point, which indicates the differencein analytic objects, Samarotto insisted that the Neo-Riemannian theory can become meaningful
in not the post-tonal music but the 19th-century. Samarotto asserted that both theories can deal
with the same objects. Moreover, one cannot overlook his third point that accompanies his second
point: the application of Neo-Riemannian theory to the 19th-century music does not come into
effect as an analytic action that is based on a priori assumption, but it is conditioned that theanalytic action becomes possible only by considering the context of the relevant piece.

Neo-Riemannian and Schenkerian Theories

7

Finally, Samarotto contrasted the Neo-Riemannian heuristic nature with the robustness of
Schenkerian theory:
I don’t think that we are helped by broad generalizations about brief passages taken outof context. Let us not forget that the robustness of Schenker’s theory came from his
having spent decades of detailed analysis of the tonal repertoire before venturing theconcepts that are landmarks in the history of theory. (Samarotto 2003, 11)
The viewpoint that one cannot generalize the theory, which functions for only particularpassages under a certain context, can be considered similar to Hook’s first point: the issue on the
difference in nature.
2.2. Studies that integrate the methods
Regarding studies that integrate methods of both theories through concrete analyses, thearticles of Yosef Goldenberg in 2007, Steven Rings in 2007, and Michael Baker in 2008 were
written as a counterargument to Samarotto’s case above.10
First, Goldenberg, who specializes in both Schenkerian and Neo-Riemannian theories,
regarded the biggest difference between two theories is the approaches to chromaticism. However,he stated that ‘the separation is not absolute’ (Goldenberg 2007, 66). Starting from thisassumption, he merged Neo-Riemannian transformation and Schenkerian voice leading to thesame graph in his analysis, based on approaches of the third relations in both theories. Moreover,he added Neo-Riemannian analysis to Schenker’s own analytic graph of the first movement of
Eroica symphony (Schenker 1935, Fig. 62), and concluded as follows:
Analytical integration of Schenkerian voice-leading procedures and neo-Riemannian
operations may be fertile … However, the shared analytical application does not meanthat both tools can be reconciled or integrated into a single theoretical method.
(Goldenberg 2007, 84)Goldenberg distinguished theory from analysis, and warned that although integrating theanalytic procedures of both theories is productive, it does not mean the construction of a newtheory.
Rings, as a transformational theorist, attempted to integrate transformational and
Schenkerian theories consistently within the former framework. Among the pieces he consideredis Schubert’s No. 2 of 4 Impromptus D899 E-flat major:
Part III [in the article] then lays the groundwork for a third analytical perspective on
Schubert’s Impromptu, one that integrates neo-Riemannian operations into a more
traditional tonal-functional context; the integration occurs within the discourse of
10
However, Lewin already noted that the network scheme of transformational theory and the graph of
Schenkerian theory can be integrated (Lewin 1987, 216–218).

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NISHIDA Hirokotransformation theory, not through a coordination of Schenkerian and neo-Riemannian
methods. 11 (Rings 2007, 33)
The integration of both theories proposed by Rings was carried out by not competition orcollaboration but ‘dialogue’:
The most productive way to understand the relationship between Schenkerian andtransformational discourses is thus not through assimilation into a single method, northrough competition, but through dialogue. When the two methods are brought into
dialogue, a rich picture of the music in question can emerge, as we observe the ways inwhich the respective analytical discourses interact and diverse. (Rings 2007, 45)
What is the concrete example of this dialogue? Rings’ opinion regarding the analysis of
mm.154-169 using each theory, as shown in Figure 1 and Example 3, is as follows:
Particularly striking here is the contrast between the violence of the gesture in theharmonic/key sphere – as modeled in Example 16 [Figure 1] – and its smoothness as
regards voice-leading, as sketched in Example 1 and 18 [Example 3]. In general, I findthis a productive way to think about the interaction of the two analyzes – the Schenkerianreading providing a detailed account of voice-leading structure, the networked analysesof Part IV [in the article] providing a model of the music’s striking key peripatetics.
(Rings 2007, 62)

Figure 1. Network analysis of harmony by transformational
theory (Rings 2007, 59)

Example 3 Analysis of voice leading by
Schenkerian theory (Rings 2007, 62)

11
Rings warned that the attempt could be ‘a superficial reconciliation of Schenkerian and neo-Riemannian
methods’ (Rings 2007, 39n).

Neo-Riemannian and Schenkerian Theories

9

That is, only when a ‘contrast’ through the parallel analyses by both theories is revealed, thedialogue can become meaningful. The significance of both theories’ integration depends on the
result of analytic interpretations.
One must keep in mind that Rings’ integration is premised on the incompatibility of boththeories:
Schenkerian structural readings, with their coordination of harmonic and contrapuntal
parameters in a single analytical structure, differ in crucial ways from the esthesic
pluralism of Lewin’s approach. … Underlying the divide is a fundamental difference inunderstanding of not only the analytical act but also of the nature of musical experience.
(Rings 2007, 39n)
The distinction of the structural reading and esthesic pluralism is concerned about thedifference of both theories’ nature. Such concern is the same with Hook’s and Samarotto’s first
and third points. In other words, Rings attempted to not construct the third new theory but
integrate in the form of paralleling both theories.
Moreover, Rings, in his writing in 2006, stated that the interpretation becomes moreproductive by using various analytic methods in the ‘more pragmatic’ way:
If our aims are instead more pragmatic … then this methodological diversity becomes
not something to bemoan, but something to embrace. I am fully aware that such a
conciliatory conclusion will seem disappointingly bland to the dialectician, or to those
who relish disciplinary conflict. It nevertheless has one signal virtue: it is interpretively
productive. (Rings 2006[2011], 40)
This standpoint overlaps partly with Baker’s approach, which this work examines next.
Contrary to Samarotto, who illustrates the incompatibility of methods of both theories basedon Brahms’ pieces, Schenkerian scholar Baker selected Brahms’ vocal piece In der Fremde, andtried the synthesis of the two theories. It is carried out by a new concept called ‘common-tone
prolongation’:However, if the emphasis is shifted from successive chord-to-chord connections to
associations between a single triad and the harmonies most closely related to it by thethree typical Neo-Riemannian transformations [P, R, and L], an interesting synthesis oftransformational theories and prolongational theories will arise. The two common tones
between a given tonic chord and its relative chord could be harmonically supported byboth of these harmonies in a given musical setting. … Such a musical situation would bestrongly analogous to the concept of prolongation, especially the notion that the
prolonged entity (in this case the tonic chord) would not be literally present at every
moment in a given musical setting, despite the fact that two of its members would be.
One might even go so far as to describe this musical effect as “common-tone
prolongation.” (Baker 2008, 72)

NISHIDA Hiroko

10Here, the prolongation of a single chord, which is an essential feature of Schenkerian theory,is abandoned. This abandonment is compensated by the actual condition in Schenkerian theory
in which the prolonged chord is inexistent.
Baker’s attempt is the integration by modifying Schenkerian theory and introducing NeoRiemannian concepts. The integrated theory is the third ‘heuristic method’, as seen in thefollowing description:
In the present essay, I have argues that there is much to be gained by examining therelationship between the two theoretical approaches, and that the similarities between
common tone retention and the notion of prolongation can provide a heuristic method for
explanation of music that falls between the cracks of the analytical machinery for eachapproach when taken individually. (Baker 2008, 81)
Although Rings premised the incompatibility of both theories and pointed out theinterpretive productivity by the dialogue of both methods, Baker regarded the heuristic nature of
the third method as his findings.

3. Interpretation and Method
This work has examined the arguments on Neo-Riemannian and Schenkerian theories fromeach standpoint. In conclusion, the issues on the differences in both theories by scholars are
summarized as follows:
Hook’s first point
Hook’s second point
Hook’s third point
Cohn’s first point
Cohn’s second point
Cohn’s third point
Samarotto’s first point
Samarotto’s second point
Samarotto’s third point
Rings’ first point

difference in the nature of theories
difference in analytic objects
whether to premise the hierarchic coherence (context)
whether it fits into the diatonic framework
difference in analytic objects
whether to premise the coherence
whether to premise the coherence
whether it fits into the diatonic framework
difference in the nature of theories
difference in the nature of theories

Alternatively, the issues raised by Hook, Cohn, Samarotto, and Rings can be summarized
into the following four issues:
1. whether the theory premises the coherence
2. whether the theory depends on the (hierarchical or diatonic) context

Neo-Riemannian and Schenkerian Theories

11

3. difference in the nature of theories
4. difference in analytic objects
The first, second, and fourth points are concerned about the difference in value judgments
as a theory; in contrast, the third point does not compare both theories at the same level and poses
the question against comparing them equally. For Hook, the analysis on Schenkerian theory isbased on the rigorous guidelines, whereas that on Neo-Riemannian theory lacks such guidelines.
For Samarotto, Neo-Riemannian theory is heuristic because it is not based on a priori
assumptions, whereas the nature of Schenkerian theory is robust because it has been systematizedby analysing pieces in a quantitative way. For Rings, the prescriptiveness of Schenkerian theory
and esthesic pluralism of Neo-Riemannian theory are fundamentally different, and suchdifference is highlighted more than the difference as an analytic action. Thus, only the third point
is different from the other three in that it can evoke the arguments on the ‘methodology’, such as‘what is the method of a theory’. Behind such arguments, the lack of methodological robustness
of Neo-Riemannian theory as a systematic method seems to be shared among scholars. However,such lack of robustness is covered by the merits of interpretive pluralism and heuristic nature. To
sum, the robustness as a method and interpretive diversity are weighed.
If one summarized approaches by scholars that attempt the integration of methods of boththeories, Goldenberg employs both analytic tools, keeping the difference of both theories.
Nevertheless, his attempt is not identical with the suggestion of a new theory. Rings also
attempted the quasi-integration through a dialogue on the methods of both theories while keeping
their incompatibility. In contrast, Baker integrated the methods of both theories by modifying
Schenkerian theory through the introduction of Neo-Riemannian concepts.
Amid these various methodological models, future studies must look into the arguments ona systematic nature and the possibility for further development as an analytic theory, when one
integrates two methods. At the same time, the effect of systematic nature for which various
theories, such as Schenkerian theory, has aimed must be reflected and the productive employmentof theories in the field of interpretations. Moreover, the whole discipline of music theory requires
the merits of accommodating diverse interpretations, meta-level arguments that question themethodological consistency, the construction of a new theory by accumulating various cases, andthe refinement of the combined model of theories, through the integrated model of methods.
Through those attempts, further activation of arguments beyond a single theory is expected.

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Acknowledgement
This study was supported by JSPS KAKENHI Grant Number JP16K16718.