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37-441: 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

74-484: A C major to a C minor triad represent the same neo-Riemannian transformation, no matter how the voices are distributed in register. The three transformations move one of the three notes of the triad to produce a different triad: 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

111-518: 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

148-450: 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

185-651: A land owner, bailiff and, to judge from locally surviving listings of his songs and choral works, an active music enthusiast. Hugo Riemann was educated by Heinrich Frankenberger, the Sondershausen Choir Master, in Music theory . He was taught the piano by August Barthel and Theodor Ratzenberger (who had once studied under Liszt ). He graduated from the gymnasiums at Sondershausen and Arnstadt . Riemann studied law and finally philosophy and history at Berlin and Tübingen . After participating in

222-452: 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 transformations described above. (This property of stepwise voice leading in

259-569: A mathematical harmony, that is already there in the human being. Thus, what humans feel when hearing music is alike the acoustical phenomenon of sympathetic resonance. He was an advocate of harmonic dualism , and his theory of harmonic function is the foundation of harmonic theory as it is still taught in Germany. He also elaborated a set of harmonic transformations that was adapted by the American theorist David Lewin , and eventually evolved into

296-410: A piano sonata, six sonatinas, a violin sonata, and a string quartet. 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. Its symmetrical, hexagonal pattern of interval sequences places the notes of the major and minor triads together. It is sometimes called

333-564: A significant strain of neo-Riemannian theory . Another pillar of modern neo-Riemannian theory, the Tonnetz , was not Riemann's own invention, but he played an important role in popularizing and disseminating it. He authored many works on many different branches of music. His pupils included the German composer, pianist, organist, and conductor Max Reger and the musicologist and composer Walter Niemann . He wrote many pieces for piano, songs,

370-475: 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. Dmitri Tymoczko has argued that the connection between neo-Riemannian operations and voice leading is only approximate (see below). Furthermore,

407-399: A theorist, he focused more and more on human nature, using psychology as the basis of his ideas. His philosophical background exposed him to Pitagorean musical philosophy, which states that the natural harmony of the macrocosmos, or the world outside, is reflected in the microcosmos, or the world inside. The feelings generated by sounds and chords are just an intuitive rediscovery of a reality,

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444-689: 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 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 which enables even finer analyses, for instance in early romantic music. Neo-Riemannian theorists often analyze chord progressions as combinations of

481-659: 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,

518-447: Is one of the most influential music theorists. In his publications and lectures he coined various terms which are still in everyday use, such as the harmonic function theory (therein popular terms such as the tonic , the dominant , the subdominant and the parallel ). In addition, the term and theory of the metric and rhythmic phrase , a basic element of today's music education, originate in Riemann. Among his best-known works are

555-676: The Franco-Prussian War he decided to devote his life to music, and studied accordingly at the Leipzig Conservatory . He then went to Bielefeld for some years as a teacher and conductor, but in 1878 returned to Leipzig as a visiting professor ( "Privatdozent" ) at the University . As a much-desired appointment at the Conservatory did not materialize, Riemann went to Bromberg in 1880, but 1881–90 he

592-691: The Melodic Table note-layout, and more rarely the Triad note-layout. It is related to the Wicki-Hayden based keyboards and other isomorphic keyboards , both of which can be utilized on the jammer keyboard musical interface. The structure and properties of the Harmonic Table have been well known since at least the 18th century. Indeed, as a pitch space , the Harmonic Table is topologically equivalent to Euler 's Tonnetz , discovered by

629-726: The Musik-Lexikon (1882; 5th ed. 1899; Eng. trans., 1893–96), a complete dictionary of music and musicians, the Geschichte der musiktheorie im IX.-XIX. jahrhundert (1898), a history of music theory in Europe through the 19th century, the Handbuch der Harmonielehre , a work on the study of harmony , and the Lehrbuch des Contrapunkts , a similar work on counterpoint , all of which have been translated into English. The Geschichte

666-495: The major third interval (given C and E our candidates are G and A). 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 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

703-588: The 1880s, Riemann proposed a system of transformations that related triads directly to each other 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

740-538: 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. 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

777-519: 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

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814-428: The Harmonic Table format is unusual in the musically important intervals it uses: Ergonomically, the harmonic table format is exceptionally compact: all notes of the major and minor scales fall under the fingers, and all common chords can be played with one or two fingers. This key layout has attracted the attention of numerous professional musicians, including Brian May and Jordan Rudess who find that it gives them

851-585: The Swiss mathematician Leonhard Euler in 1739. The two pitch arrays are trivially obtained from each other by direct shear mapping . This note layout created by Euler is utilised in Neo-Riemannian theory to geometrically model its musical ideas. The Harmonic Table keyboard layout was used in a keyboard harmonica called the Harmonetta , invented by Ernst Zacharias and manufactured by Hohner from

888-413: 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". Though Douthett's figure was published in 1998, its superiority as a model of voice leading was not fully appreciated until much later, in

925-402: The early 1950s through the mid-1970s. A similar keyboard was developed by Larry Hanson in 1942 for use with a 53 tone scale but turns the fifth sideways and the major third to the right and up. The modern layout was proposed in 1983 by inventor Peter Davies, who obtained an international patent for its use in instruments in 1990. Davies coined the term Melodic Table to refer to the layout. It

962-402: 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. Neo-Riemannian transformations can be modeled with several interrelated geometric structures. The Riemannian Tonnetz ("tonal grid," shown on

999-480: 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, Cube Dance,

1036-497: 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

1073-539: 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. For example, according to strains of neo-Riemannian theory that prioritize common-tone preservation,

1110-680: The transformations become heuristic labels for certain kinds of standard routines, rather than their defining property. Beyond its application to triadic chord progressions, neo-Riemannian theory has inspired numerous subsequent investigations. These include 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. TouchTonnetz – an interactive mobile app to explore Neo-Riemannian Theory – Android or iPhone Hugo Riemann Karl Wilhelm Julius Hugo Riemann (18 July 1849 – 10 July 1919)

1147-495: 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 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

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1184-474: 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. 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

1221-481: 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. Riemann was born at Grossmehlra , Schwarzburg-Sondershausen . His first musical training came from his father Robert Riemann,

1258-561: Was a teacher of piano and theory at Hamburg Conservatory. After a short time at the Sondershausen Conservatory, he held a post in the conservatory at Wiesbaden (1890–95). He eventually returned to Leipzig University as lecturer in 1895. In 1901, he was appointed professor, and in 1914 he was made Director of the Institute of Musicology. Eight days before he would have turned 70, he died of jaundice . Riemann

1295-486: Was afterwards renamed to Harmonic Table by the first major manufacturer, C-Thru Music and publicized by the company. This layout is used in the sonome family of keyboards, currently commercially manufactured as the Axis and Opal keyboards. Keyboards using this layout can also be emulated on tablet computers like the iPad, such as in the app Musix . There are a large number of isomorphic note-assignments possible; however,

1332-512: 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

1369-402: Was widely used among lecturers of music for around half a century after it was written. In his book, he meticulously documented the development of polyphonic theory from the ancient times to the 16th century. With this, he concluded that common intervals heard in polyphonic music were used intuitively in folk practices due to the ostensibly "natural" element of these intervals. As he developed as

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