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Neo-Riemannian theoryNeo-Riemannian theory is a loose collection of ideas present in the writings of  musictheorists such as David Lewin, Brian Hyer, Richard Cohn,  and Henry Klumpenhouwer. Whatbinds these ideas is a central commitment to relatingharmonies directly to each other, withoutnecessary reference to a tonic.  Initially, those harmonies were major and minor triads; subsequently, neo-Riemannian theory was extended to standard dissonant sonorities as well.Harmonic proximity is characteristically gauged by efficiency of  voice leading. Thus, C majorand E minor triads are close by virtue of requiring only a single semitonal shift to move fromone to the other. Motion between proximate harmonies is described by simple transformations.For example, motion between a C major and E minor triad, in either direction, is executed byan "L" transformation. Extended progressions of harmonies are characteristically displayed ona geometric plane, or map, which portrays the entire system of harmonic relations. Whereconsensus is lacking is on the question of what is most central to the theory: smooth voiceleading, transformations, or the system of relations that is mapped by the geometries. Thetheory is often invoked when analyzing harmonic practices within the Late Romantic periodcharacterized by a high degree ofchromaticism,  including workof  Schubert, Liszt, Wagner and Bruckner. 

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

Neo-Riemannian theory is named after Hugo Riemann (1849 – 

1919), whose "dualist" system forrelating 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 minortriads being considered "upside down" versions of major triads; this "dualism" is what producesthe change-in-direction described above. See also: Utonality) In the 1880s, Riemann proposed asystem of transformations that related triads directly to each other [2]  The revival of this aspectof Riemann's writings, independently of the dualist premises under which they were initiallyconceived, originated with David Lewin (1933 – 2003), particularly in his article "Amfortas'sPrayer to Titurel and the Role of D in Parsifal" (1984) and his influential book, GeneralizedMusical Intervals and Transformations  (1987). Subsequent development in the 1990s and2000s has expanded the scope of neo-Riemannian theory considerably, with furthermathematical systematization to its basic tenets, as well as inroads into 20th centuryrepertoires and music psychology.

 Triadic transformations and voice leading

 The principal transformations of neo-Riemannian triadic theory connect triads of differentspecies (major and minor), and are their own inverses (a second application undoes the first).

 These transformations are purely harmonic, and do not need any particular voice leadingbetween chords: all instances of motion from a C major to a C minor triad represent the sameneo-Riemannian transformation, no matter how the voices are distributed in register.

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 The three transformations move one of the three notes of the triad to produce a different triad:

   The P  transformation exchanges a triad for its Parallel.  In a Major Triad move the thirddown a semitone (C major to C minor), in a Minor Triad move the third up a semitone (Cminor to C major)

   The R transformation exchanges a triad for its Relative. In a Major Triad move the fifth up

a tone (C major to A minor), in a Minor Triad move the root down a tone (A minor to Cmajor)

   The L   transformation exchanges a triad for its Leading-Tone Exchange. In a Major Triadthe root moves down by a semitone (C major to E minor), in a Minor Triad the fifth movesup 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 majordominant (C major and F minor). The "N" transformation can beobtained by applying R, L, and P successively.

   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.

 

 The H relation (LPL) exchanges a triad for its hexatonic pole (C major and A♭ minor)

Any combination of the L, P, and R transformations will act inversely on major and minortriads: for instance, R-then-P transposes C major down a minor third, to A major via A minor,

whilst transposing C minor to E♭ minor up a minor 3rd via E♭ major.

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

More recently, Dmitri Tymoczko has argued that the connection between neo-Riemannianoperations and voice leading is only approximate (see below). Furthermore, the formalism ofneo-Riemannian theory treats voice leading in a somewhat oblique manner: "neo-Riemanniantransformations," as defined above, are purely harmonic relationships that do not necessarilyinvolve any particular mapping between the chords' notes.

Graphical representations 

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Pitches in the Tonnetz are connected by lines if they are separated by minor third, major third, or perfect

fifth. Interpreted as a torus the Tonnetz has 12 nodes (pitches) and 24 triangles (triads).

Neo-Riemannian transformations can be modeled with several interrelated geometricstructures. The Riemannian Tonnetz ("tonal grid," shown on the right) is a planar array ofpitches along three simplicial axes, corresponding to the three consonant intervals. Major andminor triads are represented by triangles which tile the plane of the Tonnetz. Edge-adjacenttriads share two common pitches, and so the principal transformations are expressed asminimal motion of the Tonnetz. Unlike the historical theorist for which it is named, neo-

Riemannian theory typically assumes enharmonic equivalence (G♯ = A♭), 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 orexpand upon certain features of the classical Tonnetz. Richard Cohndeveloped theHyper Hexatonic system to describe motion within and between separate major third cycles, allof which exhibit what he formulates as "maximal smoothness." (Cohn, 1996).   Anothergeometric figure, Cube Dance, was invented by Jack Douthett; it is the geometric dual of the

 Tonnetz, and represents triads as named points rather than as triangles (Douthett andSteinbach, 1998).

Many of the geometrical representations associated with neo-Riemannian theory are unifiedinto a more general framework by the continuous voice-leading spaces explored by CliftonCallender, Ian Quinn, and Dmitri Tymoczko. This work originates in 2004, when Callenderdescribed 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 slidcontinuously from one note to another. Later, Tymoczko showed that paths in Callender'sspace were isomorphic to certain classes of voice leadings (the  "individually T related" voiceleadings discussed in Tymoczko 2008) and developed a family of spaces more closely analogous

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to those of neo-Riemannian theory. In Tymoczko's spaces, points represent particular chords ofany size (such as "C major") rather than more general chord types (such as "major triad").Finally, Callender, Quinn, and Tymoczko together proposed a unified framework connecting

these and many other geometrical spaces representing diverse range of music-theoreticalproperties.

 The Harmonic table note layout is a modern day realisation of this graphical representation to

create a musical interface.

In 2011, Gilles Baroin presented the Planet-4D model. a new vizualisation system based ongraph theory that embeds the traditional Tonnetz on a 4DHypersphere. 

Criticism 

Neo-Riemannian theorists often analyze chord progressions as combinations of the threebasic LPR transformations, the only ones that preserve two common tones. Thus theprogression 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 A♭ minor,

since L of C minor is A♭ major, while P of A♭ major is A♭ minor.) These distances reflect

voice-leading only imperfectly. For example, according to strains of neo-Riemannian theory

that prioritize common-tone preservation, the C major triad is closer to F major than to Fminor, since C major can be transformed into F major by R-then-L, while it takes three movesto get from C major to F minor (R-then-L-then-P). However, from a chromatic voice-leadingperspective 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 (A♭->G and F->E) whereas it takes three semitones

to transform F major into C major. Thus LPR transformations are unable to account for thevoice-leading efficiency of the IV-iv-I progression, one of the basic routines of nineteenth-century harmony. Note that similar points can be made about common tones: on the Tonnetz,

F minor and E♭ minor are both three steps from C major, even though F minor and C major

have one common tone, while E♭ minor and C major have none.

Underlying these discrepancies are different ideas about whether harmonic proximity ismaximized when two common tones are shared, or when the total voice-leading distance is

minimized. For example, in the R transformation, a single voice moves by whole step; in the Nor S transformation, two voices move by semitone. When common-tone maximization isprioritized, R is more efficient; when voice-leading efficiency is measured by summing themotions of the individual voices, the transformations are equivalently efficient. Early neo-Riemannian theory conflated these two conceptions. More recent work has disentangled them,and measures distance unilaterally by voice-leading proximity independently of common-tonepreservation. Accordingly, the distinction between "primary" and "secondary" transformationsbecomes problematized. As early as 1992, Jack Douthett created an exact geometric model ofinter-triadic voice-leading by interpolating augmented triads between R-related triads, which hecalled "Cube Dance". Though Douthett's figure was published in 1998, its superiority as amodel of voice leading was not fully appreciated until much later, in the wake of thegeometrical 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. In this line of research, the triadic transformationslose the foundational status that they held in the early phases of neo-Riemannian theory. Thegeometries to which voice-leading proximity give rise attain central status, and thetransformations become heuristic labels for certain kinds of standard routines, rather thantheir defining property. 

Extensions 

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Beyond its application to triadic chord progressions, neo-Riemannian theory has inspirednumerous subsequent investigations. These include

  Voice-leading proximity among chords with more than three tones- among speciesof  hexachords, such as the Mystic chord (Callender, 1998)

  Common-tone proximity among dissonant trichords

 

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).

   Transformations among all possible triads, not necessarily strict mode-shifting involutions (Hook, 2002).

   Transformations between chords of differing cardinality, called cross-typetransformations  (Hook, 2007).

  Applicability to pop music

  Applicability to film music. 

Some of these extensions share neo-Riemannian theory's concern with non-traditionalrelations among familiar tonal chords; others apply voice-leading proximity or harmonictransformation to characteristically atonal chords.