Neo-Riemannian theory

Neo-Riemannian theory refers to a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. Drawing on the work of Hugo Riemann (1849-1919), these theorists grouped together inversionally related chord progressions: thus, from a neo-Riemannian perspective, the progressions C major->E major and C minor->Ab minor belong to the same category ("Terzschritt" or L-then-P) (see counter parallel). The first of these moves a major triad up by major third, while the second moves a minor triad down by major third, with the switch from ascending to descending motion accompanying the change from major to minor. The basic transformations of neo-Riemannian theory, discussed below, all associate changes in direction with the switch from major to minor. The theory is often invoked when analyzing harmonic practices within the Late Romantic period characterized by a high degree of chromaticism, including work of Schubert, Liszt, Wagner and Bruckner.^[1]

Illustration of Riemann's 'dualist' system: minor as upside down major.

Neo-Riemannian theory is named after Hugo Riemann (1849-1919), whose "dualist" system for relating triads was adapted from earlier 19th-century harmonic theorists. (The term "dualism" refers to the emphasis on the inversional relationship between major and minor, with minor triads being considered "upside down" versions of major triads; this "dualism" is what produces the change-in-direction described above. See also: Utonality) The revival of this aspect of Riemann's writings originated with David Lewin (1933-2003), particularly in his article "Amfortas's Prayer to Titurel and the Role of D in Parsifal" (1984) and his influential book, Generalized Musical Intervals and Transformations (1987). Subsequent development in the 1990s and 2000s has expanded the scope of Neo-Riemannian theory considerably, with further mathematical systematization to its basic tenets, as well as inroads into 20th century repertoires and music psychology.^[1]

Dualistic triadic transformations and voice leading

The principal transformations of Neo-Riemannian triadic theory connect triads of different species (major and minor), and are their own inverses (a second application undoes the first). These transformations are purely harmonic, and do not require any particular voice leading between chords: all instances of motion from a C major to a C minor triad represent the same neo-Riemannian transformation, no matter how the voices are distributed in register.

• The P transformation exchanges a triad for its Parallel, In a Major Triad move the middle down a semitone (C major to C minor), in a Minor Triad move the middle note up a semitone (C minor to C Major)
• The R transformation exchanges a triad for its Relative, In a Major Triad move the upper note up a tone (C major to A minor), in a Minor Triad move the bottom note down a tone (A minor to C Major)
• The L transformation exchanges a triad for its Leading-Tone Exchange, In a Major Triad the bottom note moves down by a semitone (C major to E minor), in a Minor Triad the top note moves up by a semitone (A minor to F Major)

Secondary operations can be constructed by combining these basic operations:

• The N (or Nebenverwandt) relation exchanges a major triad for its minor subdominant, and a minor triad for its major dominant (C major and F minor). The "N" transformation can be obtained by applying R, L, and P successively.^[2]
• The S (or Slide) relation exchanges two triads that share a third (C major and C# minor); it can be obtained by applying L, P, and R successively.^[3]
• The H relation (LPL) exchanges a triad for its hexatonic pole (C major and Ab Minor)^[4]

Any combination of the L, P, and R transformations will act inversely on major and minor triads: for instance, R-then-P sends C major down a minor third, to A major, while moving C minor up a minor third, to Eb minor.

Initial work in neo-Riemannian theory treated these transformations in a largely harmonic manner, without explicit attention to voice leading. Later, Cohn pointed out that neo-Riemannian concepts arise naturally when thinking about certain problems in voice leading.^[5][6] For example, two triads (major or minor) share two common tones and can be connected by stepwise voice leading the third voice if and only if they are linked by one of the L, P, R transpositions described above.^[5] (This property of stepwise voice leading in a single voice is called voice-leading parsimony.) Note that here the emphasis on inversional relationships arises naturally, as a byproduct of interest in "parsimonious" voice leading, rather than being a fundamental theoretical postulate, as it was in Riemann's work.

More recently, Dmitri Tymoczko has argued that the connection between neo-Riemannian operations and voice leading is only approximate (see below).^[7] Furthermore, the formalism of neo-Riemannian theory treats voice leading in a somewhat oblique manner: "neo-Riemannian transformations," as defined above, are purely harmonic relationships that do not necessarily involve any particular mapping between the chords' notes.^[6]

Graphical representations

On the neo-Riemmanian Tonnetz, pitches are connected by lines if they are separated by minor third, major third, or perfect fifth.

Neo-Riemannian transformations can be modeled with several interrelated geometric structures. The Riemannian Tonnetz ("tonal grid," shown at right) is a planar array of pitches along three simplicial axes, corresponding to the three consonant intervals. Major and minor triads are represented by triangles which tile the plane of the Tonnetz. Edge-adjacent triads share two common pitches, and so the principal transformations are expressed as minimal motion of the Tonnetz. Unlike the historical theorist for which it is named, neo-Riemannian theory typically assumes enharmonic equivalence (G# = Ab), which wraps the planar graph into a torus.

One toroidal view of the neo-Riemannian Tonnetz.

Alternate tonal geometries have been described in Neo-Riemannian theory that isolate or expand upon certain features of the classical Tonnetz. Richard Cohn developed the Hyper Hexatonic system to describe motion within and between separate major third cycles, all of which exhibit what he formulates as "maximal smoothness." (Cohn, 1996).^[5] Another geometric figure, the Chicken Wire Torus, was invented by Jack Douthett; it is the geometric dual of the Tonnetz, and represents triads as named points rather than as triangles (Douthett and Steinbach, 1998).

Many of the geometrical representations associated with neo-Riemannian theory are unified into a more general framework by the continuous voice-leading spaces explored by Clifton Callender, Ian Quinn, and Dmitri Tymoczko. This work originates in 2004, when Callender described a continuous space in which points represented three-note "chord types" (such as "major triad"), using the space to model "continuous transformations" in which voices slid continuously from one note to another.^[8] Later, Tymoczko showed that paths in Callender's space were isomorphic to certain classes of voice leadings (the "individually T related" voice leadings discussed in Tymoczko 2008) and developed a family of spaces more closely analogous to those of neo-Riemannian theory. In Tymoczko's spaces, points represent particular chords of any size (such as "C major") rather than more general chord types (such as "major triad").^{[6] [9]}. Finally, Callender, Quinn, and Tymoczko together proposed a unified framework connecting these and many other geometrical spaces representing diverse range of music-theoretical properties.^[10]

The Harmonic table note layout is a modern day realisation of this graphical representation to create a musical interface.

Criticism

Neo-Riemannian theorists often analyze chord progressions as combinations of the three basic LPR transformations described above. Thus the progression from C major to E major might be analyzed as L-then-P, which is a 2-unit motion since it involves two transformations. (This same transformation sends C minor to Ab minor, since L of C minor is Ab major, while P of Ab major is Ab minor.) These distances reflect voice-leading only imperfectly.^[7] For example, according to neo-Riemannian theory the C major triad is closer to F major than to F minor, since C major can be transformed into F major by R-then-L, while it takes three moves to get from C major to F minor (R-then-L-then-P). However, from a chromatic voice-leading perspective F minor is closer to C major than F major is, since it takes just two semitones of motion to transform F minor into C major (Ab->G and F->E) whereas it takes three semitones to transform F major into C major. Thus LPR transformations are unable to account for the voice-leading efficiency of the IV-iv-I progression, one of the basic routines of nineteenth-century harmony.^[7] Note that similar points can be made about common tones: on the Tonnetz, F minor and Eb minor are both three steps from C major, even though F minor and C major have one common tone, while Eb minor and C major have none.

In the case of voice leading, these limitations were eventually overcome through the combined work of several theorists, producing a more general theory of voice leading that goes beyond "neo-Riemannian theory" in the strict sense. As early as 1992, Jack Douthett created an exact geometric model of inter-triadic voice-leading by interpolating augmented triads between R-related triads, which he called "Cube Dance". ^[11] Though Douthett's figure was published in 1998, its superiority as a model of voice leading was not fully appreciated until much later, in the wake of the geometrical work of Callender, Quinn, and Tymoczko; indeed, the first detailed comparison of "Cube Dance" to the neo-Riemannian "Tonnetz" appeared in 2009, more than fifteen years after Douthett's initial discovery of his figure.^[7]

Extensions

Beyond its application to triadic chord progressions, Neo-Riemannian theory has inspired numerous subsequent investigations. These include

• Transformations involving various more complex sonorities - among species of hexachords, such as the Mystic chord (Callender, 1998)^[12]
• Progressions among triads within diatonic rather than chromatic space.^{[citation needed]}
• Transformations among scales of various sizes and species (in the work of Dmitri Tymoczko).^[13]
• Transformations among all possible triads, not necessarily strict mode-shifting involutions (Hook, 2002).^[14]
• Transformations between chords of differing cardinality, called cross-type transformations (Hook, 2007).^[15]
• Applicability to pop music.^[16]
• Applicability to film music.^[17]

What these extensions hold in common with neo-Riemannian theory is a concern with non-traditional relations among familiar tonal objects.

Sources

• Lewin, David. "Amfortas's Prayer to Titurel and the Role of D in 'Parsifal': The Tonal Spaces of the Drama and the Enharmonic Cb/B," 19th Century Music 7/3 (1984), 336-349.
• Lewin, David. Generalized Musical Intervals and Transformations (Yale University Press: New Haven, CT, 1987)
• Cohn, Richard. 'An Introduction to Neo-Riemannian Theory: A Survey and Historical Perspective", Journal of Music Theory, 42/2 (1998), 167-180.
• Lerdahl, Fred. Tonal Pitch Space (Oxford University Press: New York, 2001)
• Hook, Julian. Uniform Triadic Transformations (Ph.D. dissertation, Indiana University, 2002)
• Kopp, David. Chromatic Transformations in Nineteenth-century Music (Cambridge University Press, 2002)
• Hyer, Brian. "Reimag(in)ing Riemann", Journal of Music Theory, 39/1 (1995), 101–138
• Mooney, Michael Kevin. The 'Table of Relations' and Music Psychology in Hugo Riemann's Chromatic Theory (Ph.D. dissertation, Columbia University, 1996)
• Cohn, Richard. "Neo-Riemannian Operations, Parsimonious Trichords, and their Tonnetz Representations", Journal of Music Theory, 41/1 (1997), 1–66

See also

• Diatonic function
• Musical set theory
• Transformational theory

References

1. ^ ^{a b} Cohn, Richard, "An Introduction to Neo-Riemannian Theory: A Survey and Historical Perspective", Journal of Music Theory, 42/2 (1998), 167-180.
2. ^ Cohn, Richard, Weitzmann's Regions, My Cycles, and Douthett's Dancing Cubes, Music Theory Spectrum 22/1 (2000), 89-103.
3. ^ Lewin, David, Generalized Musical Intervals and Transformations, Yale University Press: New Haven, CT, 1987, pg.
4. ^ Cohn, Richard, Uncanny Resemblances: Tonal Signification in the Freudian Age, Journal of the American Musicological Society, 57/2 (2004), 285-323
5. ^ ^{a b c} Cohn, Richard, Maximally Smooth Cycles, Hexatonic Systems, and the Analysis of Late-Romantic Triadic Progressions. Music Analysis 15/1 (1996), 9-40
6. ^ ^{a b c} Tymoczko, Dmitri, Scale Theory, Serial Theory, and Voice Leading," Music Analysis 27/1 (2008), 1-49.
7. ^ ^{a b c d} Tymoczko, Dmitri, "Three Conceptions of Musical Distance," Mathematics and Computation in Music, Eds. Elaine Chew, Adrian Childs, and Ching-Hua Chuan, Heidelberg: Springer (2009), p. 258-273.
8. ^ Callender, Clifton. "Continuous Transformations," Music Theory Online, 10.3 (2004)
9. ^ Tymoczko, Dmitri. "The Geometry of Musical Chords," Science 313 (2006): 72-74.
10. ^ Clifton Callender, Ian Quinn, and Dmitri Tymoczko. "Generalized Voice Leading Spaces," Science 320: 346-348.
11. ^ Douthett, Jack and Steinbach, Peter, Parsimonious Graphs: A Study in Parsimony, Contextual Transformation, and Modes of Limited Transposition, Journal of Music Theory, 42/2 (1998), 241-263
12. ^ Callender, Clifton, Voice-Leading Parsimony in the Music of Alexander Scriabin, Journal of Music Theory, 42/2 (1998), 219-233
13. ^ Tymoczko, Dmitri. "Scale Networks and Debussy," Journal of Music Theory 48.2 (2004): : 215-292.
14. ^ Hook, Julian, Uniform Triadic Transformation, Journal of Music Theory, Vol. 46/1-2 (2002), 57-126
15. ^ Hook, Julian, Cross-Type Transformations and the Path Consistency Condition, Music Theory Spectrum (2007)
16. ^ Capuzzo, Guy, Neo-Riemannian Theory and the Analysis of Pop-Rock Music, Music Theory Spectrum, 26/2 2004), Pages 177–200
17. ^ Murphy, Scott, The Major Tritone Progression in Recent Hollywood Science Fiction Films, Music Theory Online, 12/2 (2006)