Neo-Riemannian theory is a loose collection of ideas present in the writings of music theorists such as David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer. What binds these ideas is a central commitment to relating harmonies directly to each other, without necessary 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 major and E minor triads are close by virtue of requiring only a single semitonal shift to move from one 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 by an "L" transformation. Extended progressions of harmonies are characteristically displayed on a geometric plane, or map, which portrays the entire system of harmonic relations. Where consensus is lacking is on the question of what is most central to the theory: smooth voice leading, transformations, or the system of relations that is mapped by the geometries. 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) In the 1880s, Riemann proposed a system of transformations that related triads directly to each other [2] The revival of this aspect of Riemann's writings, independently of the dualist premises under which they were initially conceived, 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]
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 need 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.
Neo-Riemannian music theory's PLR operations applied to a minor chordQ.
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 third down a semitone (C major to C minor), in a Minor Triad move the third up a semitone (C minor 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 C major)
The L transformation exchanges a triad for its Leading-Tone Exchange. In a Major Triad the root moves down by a semitone (C major to E minor), in a Minor Triad the fifth moves up by a semitone (E minor to C major)
Observe that P preserves the perfect fifth interval (so given say C and G there are only two candidates for the third note: E and E♭), L preserves the minor third interval (given E and G our candidates are C and B) and R preserves the major third interval (given C and E our candidates are G and A).
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.[3]
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 in that order.[4]
The H relation (LPL) exchanges a triad for its hexatonic pole (C major and A♭ minor)[5]
Any combination of the L, P, and R transformations will act inversely on major and minor triads: 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 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.[6][7] 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 transformations described above.[6] (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).[8] 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.[7]
Graphical representations
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 geometric structures. The Riemannian Tonnetz ("tonal grid," shown on the 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♯ = A♭), which wraps the planar graph into a torus.
David Bulger's 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).[6] Another geometric figure, Cube Dance, was invented by Jack Douthett; it features the geometric dual of the Tonnetz, where triads are vertices instead of triangles (Douthett and Steinbach, 1998) and are interspersed with augmented triads, allowing smoother voice-leadings.
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.[9] 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").[7][10] Finally, Callender, Quinn, and Tymoczko together proposed a unified framework connecting these and many other geometrical spaces representing diverse range of music-theoretical properties.[11]
The Harmonic table note layout is a modern day realisation of this graphical representation to create a musical interface.
Planet-4D model embeds the traditional Tonnetz onto the surface of a Hypersphere
In 2011, Gilles Baroin presented the Planet-4D model,[12] a new visualisation system based on graph theory that embeds the traditional Tonnetz on a 4D Hypersphere. Another recent continuous version of the Tonnetz — simultaneously in original and dual form — is the Torus of phases[13] which enables even finer analyses, for instance in early romantic music.[14]
Criticism
Neo-Riemannian theorists often analyze chord progressions as combinations of the three basic LPR transformations, the only ones that preserve two common tones. 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 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.[8] 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 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 (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 the voice-leading efficiency of the IV-iv-I progression, one of the basic routines of nineteenth-century harmony.[8] 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 is maximized 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 N or S transformation, two voices move by semitone. When common-tone maximization is prioritized, R is more efficient; when voice-leading efficiency is measured by summing the motions 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-tone preservation. Accordingly, the distinction between "primary" and "secondary" transformations becomes problematized. 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".[15] 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.[8] In this line of research, the triadic transformations lose the foundational status that they held in the early phases of neo-Riemannian theory. The geometries to which voice-leading proximity give rise attain central status, and the transformations become heuristic labels for certain kinds of standard routines, rather than their defining property.
Extensions
Beyond its application to triadic chord progressions, neo-Riemannian theory has inspired numerous subsequent investigations. These include
Voice-leading proximity among chords with more than three tones- among species of hexachords, such as the Mystic chord (Callender, 1998)[16]
Common-tone proximity among dissonant trichords [17]
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).[18]
Transformations among all possible triads, not necessarily strict mode-shifting involutions (Hook, 2002).[19]
Transformations between chords of differing cardinality, called cross-type transformations (Hook, 2007).[20]
Some of these extensions share neo-Riemannian theory's concern with non-traditional relations among familiar tonal chords; others apply voice-leading proximity or harmonic transformation to characteristically atonal chords.
The major scale is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at double its frequency so that it is called a higher octave of the same note.
In jazz, the altered scale, altered dominant scale, Palamidian Scale, or Super Locrian scale is a seven-note scale that is a dominant scale where all non-essential tones have been altered. This means that it comprises the three irreducibly essential tones that define a dominant seventh chord, which are root, major third, and minor seventh and that all other chord tones have been altered. These are:
In music, modulation is the change from one tonality to another. This may or may not be accompanied by a change in key signature. Modulations articulate or create the structure or form of many pieces, as well as add interest. Treatment of a chord as the tonic for less than a phrase is considered tonicization.
Modulation is the essential part of the art. Without it there is little music, for a piece derives its true beauty not from the large number of fixed modes which it embraces but rather from the subtle fabric of its modulation.
Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is called the tonic. The root of the tonic chord forms the name given to the key, so in the key of C major, the note C is both the tonic of the scale and the root of the tonic chord. Simple folk music songs often start and end with the tonic note. The most common use of the term "is to designate the arrangement of musical phenomena around a referential tonic in European music from about 1600 to about 1910". Contemporary classical music from 1910 to the 2000s may practice or avoid any sort of tonality—but harmony in almost all Western popular music remains tonal. Harmony in jazz includes many but not all tonal characteristics of the European common practice period, usually known as "classical music".
A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent notes in a 12-tone scale. For example, C is adjacent to C♯; the interval between them is a semitone.
In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance, was further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.
Karl Wilhelm Julius Hugo Riemann was a German musicologist and composer who was among the founders of modern musicology. The leading European music scholar of his time, he was active and influential as both a music theorist and music historian. Many of his contributions are now termed as Riemannian theory, a variety of related ideas on many aspects of music theory.
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships under consideration, the models may be multidimensional. Models of pitch space are often graphs, groups, lattices, or geometrical figures such as helixes. Pitch spaces distinguish octave-related pitches. When octave-related pitches are not distinguished, we have instead pitch class spaces, which represent relationships between pitch classes. Chordal spaces model relationships between chords.
Voice leading is the linear progression of individual melodic lines and their interaction with one another to create harmonies, typically in accordance with the principles of common-practice harmony and counterpoint.
In scale (music) theory, a maximally even set (scale) is one in which every generic interval has either one or two consecutive integers specific intervals-in other words a scale whose notes (pcs) are "spread out as much as possible." This property was first described by John Clough and Jack Douthett. Clough and Douthett also introduced the maximally even algorithm. For a chromatic cardinality c and pc-set cardinality d a maximally even set is
In Western music and music theory, augmentation is the lengthening of a note or the widening of an interval.
In music theory, the harmonic major scale is a musical scale found in some music from the common practice era and now used occasionally, most often in jazz. In George Russell's Lydian Chromatic Concept it is the fifth mode (V) of the Lydian Diminished scale. It corresponds to the Raga Sarasangi in Indian Carnatic music.
Music theorists have often used graphs, tilings, and geometrical spaces to represent the relationship between chords. We can describe these spaces as chord spaces or chordal spaces, though the terms are relatively recent in origin.
In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. Various visual representations of the Tonnetz can be used to show traditional harmonic relationships in European classical music.
Transformational theory is a branch of music theory developed by David Lewin in the 1980s, and formally introduced in his 1987 work, Generalized Musical Intervals and Transformations. The theory—which models musical transformations as elements of a mathematical group—can be used to analyze both tonal and atonal music.
The Harmonic Table note-layout, or tonal array, is a key layout for musical instruments that offers interesting advantages over the traditional keyboard layout.
In music theory, Terzschritt is the progression from one major chord to another major chord, or a minor chord to another minor chord by major third root movement. Additionally, and more specifically, it is a dualistic major third relationship, in which the ascending progression from a major tonic triad to major mediant triad is equivalent to the descending one between a major tonic triad and a flat subdominant minor triad. The major chord on the mediant is itself the Terzklang.
Carl Friedrich Weitzmann was a German music theorist and musician.
"Riemannian theory" in general refers to the musical theories of German theorist Hugo Riemann (1849–1919). His theoretical writings cover many topics, including musical logic, notation, harmony, melody, phraseology, the history of music theory, etc. More particularly, the term Riemannian theory often refers to his theory of harmony, characterized mainly by its dualism and by a concept of harmonic functions.
Parallel and counter parallel chords are terms derived from the German to denote what is more often called in English the "relative", and possibly the "counter relative" chords. In Hugo Riemann's theory, and in German theory more generally, these chords share the function of the chord to which they link: subdominant parallel, dominant parallel, and tonic parallel. Riemann defines the relation in terms of the movement of one single note:
The substitution of the major sixth for the perfect fifth above in the major triad and below in the minor triad results in the parallel of a given triad. In C major thence arises an apparent A minor triad, D minor triad (Sp), and E minor triad (Dp).
↑ Baroin, Gilles (2011). "The planet-4D model: An original hypersymmetric music space based on graph theory". In Agon, C.; Andreatta, M.; Assayag, G.; Amiot, E.; Bresson, J.; Mandereau, J. (eds.). Mathematics and Computation in Music. MCM 2011. Lecture Notes in Computer Science. Vol.6726. Berlin, Heidelberg: Springer. pp.326–329. doi:10.1007/978-3-642-21590-2_25. ISBN9783642215896.
↑ Amiot, Emmanuel (2013). "The Torii of phases". In Yust, J.; Wild, J.; Burgoyne, J.A. (eds.). Mathematics and Computation in Music. MCM 2013. Lecture Notes in Computer Science. Vol.7937. Berlin, Heidelberg: Springer Berlin Heidelberg. pp.1–18. arXiv:1208.4774. doi:10.1007/978-3-642-39357-0_1. ISBN9783642393563.
↑ Callender, Clifton, "Voice-Leading Parsimony in the Music of Alexander Scriabin", Journal of Music Theory 42/2 (1998), 219–233
↑ Siciliano, Michael (October 2005). "Toggling Cycles, Hexatonic Systems, and Some Analysis of Early Atonal Music". Music Theory Spectrum. 27 (2): 221–248. doi:10.1525/mts.2005.27.2.221.
↑ Tymoczko, Dmitri. "Scale Networks and Debussy," Journal of Music Theory 48/2 (2004):: 215–92.
↑ Hook, Julian, "Uniform Triadic Transformations", Journal of Music Theory 46/1–2 (2002), 57–126
↑ Hook, Julian, "Cross-Type Transformations and the Path Consistency Condition", Music Theory Spectrum (2007)
External links
TouchTonnetz – an interactive mobile app to explore Neo-Riemannian Theory – Android or iPhone
Further reading
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). ISBN978-0-300-03493-6.
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). ISBN978-0-19-505834-5.
Kopp, David. Chromatic Transformations in Nineteenth-century Music (Cambridge University Press, 2002). ISBN978-0-521-80463-9.
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.
Cohn, Richard. Audacious Euphony: Chromaticism and the Triad's Second Nature (New York: Oxford University Press, 2012). ISBN978-0-19-977269-8.
Gollin, Edward and Alexander Rehding, Oxford Handbook of Neo-Riemannian Music Theories (New York: Oxford University Press, 2011). ISBN978-0-19-532133-3.
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