Neo-Riemannian theory
Encyclopedia
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 progression
s: 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
.
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.
(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.
Secondary operations can be constructed by combining these basic operations:
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. 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. (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). 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.
("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
.
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). 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. 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"). . Finally, Callender, Quinn, and Tymoczko together proposed a unified framework connecting these and many other geometrical spaces representing diverse range of music-theoretical properties.
The Harmonic table note layout
is a modern day realisation of this graphical representation to create a musical interface.
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". 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.
What these extensions hold in common with neo-Riemannian theory is a concern with non-traditional relations among familiar tonal objects.
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
such as David Lewin
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
, Brian Hyer, Richard Cohn
Richard Cohn
Richard Cohn is a music theorist and Battell Professor of Music Theory at Yale. Early in his career, he specialized in the music of Béla Bartók, but more recently has written about Neo-Riemannian theory as well as metric dissonance.-External links:*...
, and Henry Klumpenhouwer
Henry Klumpenhouwer
Henry Klumpenhouwer is a musicologist and professor at the University of Alberta. A former PhD student of David Lewin and the inventor of Klumpenhouwer networks, which are named after him. He is the editor of Music Theory Spectrum.-Bibliography:...
. Drawing on the work of Hugo Riemann
Hugo Riemann
Karl Wilhelm Julius Hugo Riemann was a German music theorist.-Biography:Riemann was born at Grossmehlra, Schwarzburg-Sondershausen. He was educated in theory by Frankenberger, studied the piano with Barthel and Ratzenberger, studied law, and finally philosophy and history at Berlin and Tübingen...
(1849-1919), these theorists grouped together inversionally
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
related chord progression
Chord progression
A chord progression is a series of musical chords, or chord changes that "aims for a definite goal" of establishing a tonality founded on a key, root or tonic chord. In other words, the succession of root relationships...
s: thus, from a neo-Riemannian perspective, the progressions C major->E major and C minor->Ab minor belong to the same category ("Terzschritt
Terzschritt
In music theory, Terzschritt 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...
" or L-then-P) (see counter parallel
Counter parallel
In music theory, the counter parallel is terminology used in German theory derived mainly from Hugo Riemann to refer to relative diatonic functions and is abbreviated Tcp in major and tCp in minor...
). The first of these moves a major triad up by major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
, 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
Chromaticism
Chromaticism is a compositional technique interspersing the primary diatonic pitches and chords with other pitches of the chromatic scale. Chromaticism is in contrast or addition to tonality or diatonicism...
, including work of Schubert
Franz Schubert
Franz Peter Schubert was an Austrian composer.Although he died at an early age, Schubert was tremendously prolific. He wrote some 600 Lieder, nine symphonies , liturgical music, operas, some incidental music, and a large body of chamber and solo piano music...
, Liszt
Franz Liszt
Franz Liszt ; ), was a 19th-century Hungarian composer, pianist, conductor, and teacher.Liszt became renowned in Europe during the nineteenth century for his virtuosic skill as a pianist. He was said by his contemporaries to have been the most technically advanced pianist of his age...
, Wagner
Richard Wagner
Wilhelm Richard Wagner was a German composer, conductor, theatre director, philosopher, music theorist, poet, essayist and writer primarily known for his operas...
and Bruckner
Anton Bruckner
Anton Bruckner was an Austrian composer known for his symphonies, masses, and motets. The first are considered emblematic of the final stage of Austro-German Romanticism because of their rich harmonic language, complex polyphony, and considerable length...
.
Neo-Riemannian theory is named after Hugo Riemann
Hugo Riemann
Karl Wilhelm Julius Hugo Riemann was a German music theorist.-Biography:Riemann was born at Grossmehlra, Schwarzburg-Sondershausen. He was educated in theory by Frankenberger, studied the piano with Barthel and Ratzenberger, studied law, and finally philosophy and history at Berlin and Tübingen...
(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
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
(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.
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 inversesInversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
(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 subdominantSubdominantIn music, the subdominant is the technical name for the fourth tonal degree of the diatonic scale. It is so called because it is the same distance "below" the tonic as the dominant is above the tonic - in other words, the tonic is the dominant of the subdominant. It is also the note immediately...
, and a minor triad for its major dominantDominant (music)In music, the dominant is the fifth scale degree of the diatonic scale, called "dominant" because it is next in importance to the tonic,and a dominant chord is any chord built upon that pitch, using the notes of the same diatonic scale...
(C major and F minor). The "N" transformation can be obtained 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 poleHexatonic scaleIn music and music theory, a hexatonic scale is a scale with six pitches or notes per octave. Famous examples include the whole tone scale, C D E F G A C; the augmented scale, C D E G A B C; the Prometheus scale, C D E F A B C; and what some jazz theorists call the "blues scale", C E F F G B...
(C major and Ab Minor)
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. 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. (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
Dmitri Tymoczko
Dmitri Tymoczko is a composer and music theorist. His music, which draws on rock, jazz, and romanticism, has been performed by ensembles such as the Ansermet Quartet, the Brentano Quartet, Janus, Newspeak, the San Francisco Contemporary Players, the Pacifica Quartet, and Ursula Opens...
has argued that the connection between neo-Riemannian operations and voice leading is only approximate (see below). 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.
Graphical representations
Neo-Riemannian transformations can be modeled with several interrelated geometric structures. The Riemannian TonnetzTonnetz
In musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739....
("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
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
.
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
Hexatonic scale
In music and music theory, a hexatonic scale is a scale with six pitches or notes per octave. Famous examples include the whole tone scale, C D E F G A C; the augmented scale, C D E G A B C; the Prometheus scale, C D E F A B C; and what some jazz theorists call the "blues scale", C E F F G B...
system to describe motion within and between separate major third cycles, all of which exhibit what he formulates as "maximal smoothness." (Cohn, 1996). 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. 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"). . Finally, Callender, Quinn, and Tymoczko together proposed a unified framework connecting these and many other geometrical spaces representing diverse range of music-theoretical properties.
The Harmonic table note layout
Harmonic table note layout
The Harmonic Table note-layout, or tonal array, is a key layout for musical instruments that offers interesting advantages over the traditional keyboard 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. 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. 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". 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.
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 hexachordHexachordIn music, a hexachord is a collection of six pitch classes including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.-Middle Ages:...
s, such as the Mystic chord (Callender, 1998) - Progressions among triads within diatonic rather than chromatic space.
- 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-type transformations (Hook, 2007).
- Applicability to pop musicPopular musicPopular music belongs to any of a number of musical genres "having wide appeal" and is typically distributed to large audiences through the music industry. It stands in contrast to both art music and traditional music, which are typically disseminated academically or orally to smaller, local...
. - Applicability to film music.
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