Harmonic Space

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Harmonic Space

Oxford Handbooks Online
Harmonic Space
The Oxford Handbook of Neo-Riemannian Music Theories
Edited by Edward Gollin and Alexander Rehding
Print Publication Date: Dec 2011 Subject: Music
Online Publication Date: Sep
2012


No aspect of Riemann's theoretical work has been more central to the technology of neoRiemannian theory than Riemann's system of contextual harmonic relationships—his
Harmonieschritte. And yet the idea of categorizing triadic relations according to the directed
intervals between their dually conceived Haupttöne was not Riemann's, but rather one first devised
by Arthur von Oettingen. Oettingen, in his 1866 Harmoniesystem in dualer Entwickelung, classified
triadic relationships according to the directed intervals between Haupttöne and whether they
preserved or changed mode (calling the former homonomic, and the latter antinomic relations).
Riemann first adopted Oettingen's relational system in his 1875 Die Hülfsmittel der
Modulationslehre and his 1877 Musikalische Syntaxis, retaining Oettingen's homonomic/antinomic
terminology, but proposing a further categorical distinction: Riemann posited homologic relations, in
which the intervals between triadic Haupttöne were oriented in the direction of chord components;
and antilogic relations, in which intervals between triadic Haupttöne proceeded in a direction
contrary to chord components. Riemann's breakthrough in his 1880 Skizze eine neuen Methode der
Harmonielehre, the treatise most commonly associated with his exposition of the system of
Harmonieschritte, was ironically not one of technology, but of terminology: Riemann replaced what
he himself acknowledged to be his “impractical Greek terminology” (Skizze, 89) with the relatively
simpler German Schritt/Wechsel, schlichter/gegen categories. Thereby, what had formerly (and
cumbersomely) been named the homolog-antinomer Terzschritt became the Terzwechsel, the
antilog-homonomer Doppelquintschritt became the Gegenganztonschritt, and so on.
(p. 350) The Harmonieschritte, either explicitly or implicitly (in the guise of the functional variants
L, P, and R), form the group-theoretical foundation of neo-Riemannian triadic relationships,
constituting the elements of the canonical group of contextual triadic transformations. Yet despite
the importance of the Skizze for its exposition of the Harmonieschritte, the work is known in the
neo-Riemannian literature primarily through secondary sources. Nora Engebretsen's contribution
seeks to rectify this state of affairs. The chapter examines Riemann's discussion of the
Harmonieschritte within the Skizze, locating that discussion within a nineteenth-century
combinatorial tradition shaped by Riemann's conception of key. Engebretsen explores how
Riemann presents two distinct expositions of the Harmonieschritte: one circumscribed by a
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Harmonic Space
“diatonic” conception of key, bounded by the triadic content of Hauptmannian tone schemes, and
an unbounded, “chromatic” conception of key that embraces chromatic third relations. The chapter
concludes with a translation of Riemann's Systematik der Harmonieschritte, the summary of the
complete “chromatic” family of triadic relations from the Skizze.
Edward Gollin's contribution to this section explores a less examined aspect of Riemann's Skizze, its
Dissonanzlehre and in particular, Riemann's short-lived category of Doppelklang dissonances
presented therein. Riemann's Doppelklänge are dissonances formed through the union of two
distinct Klänge in a coordinate relationship—a relationship, moreover, that Riemann defines through
recourse to the contextual relationships of his Harmonieschritte. Riemann's Dissonanzlehre was an
integral component of his Harmonielehre, illustrating how complex, dissonant sonorities could be
understood to derive from simple, consonant Klänge, which could accordingly participate in the
relationships set forth in Riemann's expositions of Harmonieschritte. But the inherent duality of the
Doppelklänge, their projection of distinct but equal Haupttöne, posed a challenge to Riemann's
relational harmonic system (a many-to-one or one-to-many problem), and Riemann consequently
purged all subsequent editions of the Skizze (renamed the Handbuch der Harmonielehre) of the
Doppelklang theory. Gollin nevertheless explores how an explicitly transformational reinterpretation
of Riemann's Doppelklänge, viewing them as structures that can project their internal contextual
relationships through their participation in external transformational progressions, can offer
analytical insights into tonal, posttonal and transitional musical repertoires. Gollin presents as an
appendix to the chapter a translation of Riemann's Schematisirung der Dissonanzen from the
Skizze.
Finally, David Kopp looks critically at the role of key and function as a component of Riemann's
relational harmonic system. Kopp argues that the neo-Riemannian abstraction of Riemann's
Harmonieschritte—understanding them as products of parsimonious voice-leading or as
mathematical transformations acting on a family of harmonic triads—while offering certain insights
into the nature of chromatic relations in nineteenth-century music, has also resulted in a view of
harmonic relations uncomfortably divorced from the tonal and functional contexts in which they were
conceived. Kopp suggests how neo-Riemannian analysis can benefit by reconnecting Riemannian
harmonic relations to the functional tonal contexts in which they arose, illustrating a recovered and
renewed nineteenth-century perspective with analyses of music by Beethoven, Schubert, and Wolf.

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PRINTED FROM OXFORD HANDBOOKS ONLINE (www.oxfordhandbooks.com). (c) Oxford University Press, 2015. All Rights
Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a title in Oxford
Handbooks Online for personal use (for details see Privacy Policy).
Subscriber: Istanbul Teknik University; date: 06 December 2015

Harmonic Space

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PRINTED FROM OXFORD HANDBOOKS ONLINE (www.oxfordhandbooks.com). (c) Oxford University Press, 2015. All Rights
Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a title in Oxford
Handbooks Online for personal use (for details see Privacy Policy).
Subscriber: Istanbul Teknik University; date: 06 December 2015

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