Neo-Riemannian theory From Wikipedia, the free encyclopedia
Neo-Riemannian theory is theory is a loose collection of ideas present in the writings of music of music theorists such theorists such as David Lewin, Lewin, Brian Hyer, Richard Cohn, Cohn, and Henry lumpenhouwer ! What "inds these ideas is a central commitment to relating harmonies harmonies directly directly to each other, without necessary reference to a tonic tonic!! #nitially, those harmonies were ma$or ma$or and and minor triads% triads% su"se&uently, neo'Riemannian theory was e(tended to standard dissonant dissonant sonorities sonorities as well! Harmonic pro(imity is characteristically gauged "y efficiency of voice leading! leading! )hus, C ma$or and * minor triads are close "y virtue of re&uiring only a single semitonal semitonal shift shift to move from one to the other! +otion "etween pro(imate harmonies is descri"ed "y simple transformations! For e(ample, motion "etween a C ma$or and * minor triad, in either direction, is e(ecuted "y an L transformation! *(tended progressions of harmonies are characteristically displayed on a geometric plane, or map, which portrays the entire s ystem of harmonic relations! Where consensus is lacking is on the &uestion of what is most central to the theory- smooth voice leading, transformations, or the system of relations that is mapped "y the geometries! )he theory is often invoked when analy.ing harmonic practices within the Late Romantic period Romantic period characteri.ed "y a high degree of chromaticism of chromaticism,, including work 012 of /chu"ert /chu"ert,, Lis.t Lis.t,, Wagner and and Bruckner !
#llustration of Riemann3s 3dualist3 system- minor as upside down ma$or!
4eo'Riemannian theory is named after Hugo Riemann 5167891818:, Riemann 5167891818:, whose dualist system for relating triads was adapted from earlier 18th'century harmonic theorists! 5)he term dualism dualism refers to the emphasis on the inversional relationship "etween ma$or and minor, with minor triads "eing considered upside down versions of ma$or triads% this dualism is what produces the change'in'direction descri"ed a"ove! /ee also- ;tonality ;tonality:: #n the 166
>9=<<>:, Lewin 518>>9=<<>:, particularly in his article ?mfortas3s @rayer to )iturel )iturel and the Role of D in @arsifal 51867: and his influential "ook, Generalized Musical Intervals and Transformations 5186A:! Transformations 5186A:! /u"se&uent development in the 188
1)riadic transformations transformations and voice leading
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=raphical representations
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>Criticism
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7*(tensions
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References
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A*(ternal Links
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6Further reading
Triadic transformations and voice leading 0edit2 )he principal transformations of neo'Riemannian triadic theory connect triads of different species 5ma$or and minor:, and are their own inverses 5a second application undoes the first:! )hese transformations are purely harmonic, and do not need any particular voice leading "etween chords- all instances of motion from a C ma$or to a C minor triad represent the same neo'Riemannian transformation, no matter how the voices are distri"uted in register! )he three transformations move one of the three notes of the triad to produce a different triad•
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)he P transformation e(changes a triad for its @arallel! #n a +a$or )riad move the third down a semitone 5C ma$or to C minor:, in a +inor )riad move the third up a semitone 5C minor to C ma$or: )he R transformation e(changes a triad for its Relative! #n a +a$or )riad move the fifth up a tone 5C ma$or to ? minor:, in a +inor )riad move the root down a tone 5? minor to C ma$or: )he L transformation e(changes a triad for its Leading')one *(change! #n a +a$or )riad the root moves down "y a semitone 5C ma$or to * minor:, in a +inor )riad the fifth moves up "y a semitone 5* minor to C ma$or:
E"serve that P preserves the perfect fifth interval 5so given say C and there are only two candidates for the third note* and *♭:, L preserves the minor third interval 5given * and our candidates are C and B: and R preserves the major third interval 5given C and * our candidates are and ?:! /econdary operations can "e constructed "y com"ining these "asic operations•
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)he N 5or Nebenverwandt : relation e(changes a ma$or triad for its minor su"dominant, and a minor triad for its ma$or dominant 5C ma$or and F minor:! )he 4 transformation can "e o"tained "y applying R, L, and @ successively!0>2 )he S 5or Slide: relation e(changes two triads that share a third 5C ma$or and C ♯ minor:% it can "e o"tained "y applying L, @, and R successively in that order! 072 )he H relation 5L@L: e(changes a triad for its he(atonic pole 5C ma$or and ? ♭ minor:02
?ny com"ination of the L, @, and R transformations will act inversely on ma$or and minor triads- for instance, R'then'@ transposes C ma$or down a minor third, to ? ma$or via ? minor, whilst transposing C minor to * ♭ minor up a minor >rd via *♭ma$or! #nitial work in neo'Riemannian theory treated these transformations in a largely harmonic manner, without e(plicit attention to voice leading! Later, Cohn pointed out that neo'Riemannian concepts arise naturally when thinking a"out certain pro"lems in voice leading! 020A2 For e(ample, two triads 5ma$or or minor: share two common tones and can " e connected "y stepwise voice leading the third voice if and only if they are linked "y one of the L, @, R transformations descri"ed a"ove!025)his property of stepwise voice leading in a single voice is called voice'leading parsimony!: 4ote that here the emphasis on inversional relationships arises naturally, as a "yproduct of interest in parsimonious voice leading, rather than "eing a fundamental theoretical postulate, as it was in Riemann3s work! +ore recently, Dmitri )ymoc.ko has argued that the connection "etween neo'Riemannian operations and voice leading is only appro(imate 5see "elow:! 062 Furthermore, the formalism of neo'Riemannian theory treats voice leading in a somewhat o"li&ue manner- neo'Riemannian transformations, as defined a"ove, are purely harmonic relationships that do not necessarily involve any particular mapping "etween the chords3 notes! 0A2
Graphical representations 0edit2
@itches in the )onnet. are connected "y lines if they are separated "y minor third, ma$or third, or perfect fifth! #nterpreted as a torus the )onnet. has 1= nodes 5pitches: and =7 triangles 5triads:!
4eo'Riemannian transformations can "e modeled with several interrelated geometric structures! )he Riemannian )onnet. 5tonal grid, shown on the right: is a planar array of pitches along three simplicial a(es, corresponding to the three consonant intervals! +a$or and minor triads are represented "y triangles which tile the plane of the )onnet.! *dge'ad$acent triads share two common pitches, and so the principal transformations are e(pressed as minimal motion of the )onnet.! ;nlike the historical theorist for which it is named, neo'Riemannian theory typically assumes enharmonic e&uivalence 5♯ ?♭:, which wraps the planar graph into a torus!
Ene toroidal view of the neo'Riemannian )onnet.!
?lternate tonal geometries have "een descri"ed in neo'Riemannian theory that isolate or e(pand upon certain features of the classical )onnet.! Richard Cohn developed the Hyper He(atonic system to descri"e motion within and "etween separate ma$or third cycles, all of which e(hi"it what he formulates as ma(imal smoothness! 5Cohn, 188:! 02 ?nother geometric figure, Cu"e Dance, was invented "y Gack Douthett% it features the geometric dual of the )onnet., where triads are vertices instead of triangles 5Douthett and /tein"ach, 1886: a nd are interspersed with augmented triads, allowing smoother voice'leadings! +any of the geometrical representations associated with neo'Riemannian theory are unified into a more general framework "y the continuous voice'leading spaces e(plored "y Clifton Callender, #an uinn, and Dmitri )ymoc.ko! )his work originates in =<<7, when Callender descri"ed a continuous space in which points represented three'note chord types 5such as ma$or triad:, using the space to model continuous transformations in which voices slid continuously from one note to another!082 Later, )ymoc.ko showed that paths in Callender3s space were isomorphic to certain classes of voice leadings 5the individually ) related voice leadin gs discussed in )ymoc.ko =<<6: and developed a family of spaces more closely analogous to those of neo'Riemannian theory! #n )ymoc.ko3s spaces, points represent particular chords of any si.e 5such as C ma$or: rather than more general chord types 5such as ma$or triad:! 0A201<2 Finally, Callender, uinn, and )ymoc.ko together proposed a unified framework connecting these and many other geometrical spaces representing diverse range of music'theoretical properties! 0112 )he Harmonic ta"le note layout is a modern day realisation of this graphical representation to create a musical interface! @lanet'7D model em"eds the traditional )onnet. onto the s urface of a Hypersphere
#n =<11, illes Baroin presented the @lanet'7D model,01=2 a new vi.ualisation system "ased on graph theory that em"eds the traditional )onnet. on a 7D Hypersphere! ?nother recent continuous version of the )onnet. I simultaneously in original and dual form I is the )orus of phases01>2 which ena"les even finer analyses, for instance in early romantic music!0172
Criticism0edit2 4eo'Riemannian theorists often analy.e chord progressions as com"inations of the three "asic L@R transformations, the only ones that preserve two common tones! )hus the progression from C ma$or to * ma$or might "e analy.ed as L' then'@, which is a ='unit motion since it involves two transformations! 5)his same transformation sends C minor to ?♭ minor, since L of C minor is ? ♭ ma$or, while @ of ? ♭ ma$or is ?♭ minor!: )hese distances reflect voice'leading only imperfectly!062 For e(ample, according to strains of n eo'Riemannian theory that prioriti.e common'tone preservation, the C ma$or triad is closer to F ma$or than to F minor, since C ma$or can "e transformed into F ma$or "y R'then'L, while it takes three moves to get from C ma$or to F minor 5R'then'L'then'@:! However, from a chromatic voice'leading perspective F minor is closer to C ma$or than F ma$or is, since it takes $ust two semitones of motion to transform F minor into C ma$or 5? ♭'J and F'J*: whereas it takes three semitones to transform F ma$or into C ma$or! )hus L@R transformations are una"le to account for the voice'leading efficiency of the #K'iv'# progression, one of the "asic routines of nineteenth'century harmony!062 4ote that similar points can "e made a"out common tones- on the )onnet., F minor and *♭ minor are "oth three steps from C ma$or, even though F minor and C ma$or have one common tone, while *♭ minor and C ma$or have none! ;nderlying these discrepancies are different ideas a"out whether harmonic pro(imity is ma(imi.ed when two common tones are shared, or when the total voice'leading distance is minimi.ed! For e(ample, in the R transformation, a single voice moves "y whole step% in the 4 or / transformation, two voices move "y semitone! W hen common'tone ma(imi.ation is prioriti.ed, R is more efficient% when voice'leading efficiency is measured "y summing the motions of the individual voices, the transformations are e&uivalently efficient! *arly neo'Riemannian theory conflated these two conceptions! +ore recent work has disentangled them, and measures distance unilaterally "y voice'leading pro(imity independently of common'tone preservation! ?ccordingly, the distinction "etween primary and secondary transformations "ecomes pro"lemati.ed! ?s early as 188=, Gack Douthett created an e(act geometric model of inter' triadic voice'leading "y interpolating augmented triads "etween R'related triads, which he called Cu"e Dance! 012 )hough Douthett3s figure was pu"lished in 1886, its superiority as a model of voice leading was not fully appreciated until much later, in the wake of the geometrical work of Callender, uinn, and )ymoc.ko% indeed, the first detailed comparison of Cu"e Dance to the neo'Riemannian )onnet. appeared in =<<8, more than fifteen years after Douthett3s initial discovery of his figure!062 #n this line of research, the triadic transformations lose the foundational status that they held in the early phases of neo'Riemannian theory! )he geometries to which voice'leading pro(imity give rise attain central status, and the transformations "ecome heuristic la"els for certain kinds of standard routines, rather than their defining property! 4onetheless, among all possi"le sets of the twenty'four Riemannian triadic transformations, the length of com"inations of mem"ers from the set of L, @, and R transformations "etter correlates with chromatic voice'leading distance than nearly every other set of transformations! For e(ample, if only L and R transformations were used to measure transformational distance "etween triads, the num"er of contradictions "etween transformational distance and voice' leading distance like those e(amples a"ove is much greater than when using L, @, and R! )his partially restores some distinction "etween primary and secondary transformations! 012
Extensions0edit2 Beyond its application to triadic chord progressions, neo'Riemannian theory has inspired numerous su"se&uent investigations! )hese include •
Koice'leading pro(imity among chords with more than three tones' among species of he(achords, such as the +ystic chord 5Callender, 1886:01A2
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Common'tone pro(imity among dissonant trichords
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@rogressions among triads within diatonic rather than chromatic space! 0citation needed 2
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)ransformations among scales of various si.es and species 5in the work of Dmitri )ymoc.ko:!0182
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)ransformations among all possi"le triads, not necessarily strict mode'shifting involutions 5Hook, =<<=:!0=<2
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)ransformations "etween chords of differing cardinality, called cross-type transformations 5Hook, =<
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?pplica"ility to pop music!0==2
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?pplica"ility to film music!0=>20=720=2
/ome of these e(tensions share neo'Riemannian theory3s concern with non'traditional relations among familiar tonal chords% others apply voice'leading pro(imity or harmonic transformation to characteristically atonal chords!
See also0edit2 •
Diatonic function
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+usical set theory
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Riemannian theory
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)ransformational theory
References0edit2 1! Gump up to- Cohn, Richard, ?n #ntr oduction to 4eo'Riemannian )heory- ? /urvey and Historical @erspective, Journal of Music Theory , 7=M= 51886:, 1A916
=! Jump up^ lumpenhouwer, Henry, Some emar!s on the "se of iemann Transformations# +usic )heory Enline ! Jump up^ Cohn, Richard, $eitzmann%s e&ions# My 'ycles# and (outhett%s (ancin& 'ubes# +usic )heory /pectrum ==M1 5=<<<:, 6891<>! 7! Jump up^ Lewin, David, Generalized Musical Intervals and Transformations# Nale ;niversity @ress- 4ew Haven, C), 186A, pg! 1A6 ! Jump up^ Cohn, Richard, ;ncanny Resem"lances- )onal /ignification in the Freudian ?ge, Journal of the )merican Musicolo&ical Society , AM= 5=<<7:, =69>=> ! Gump up to- Cohn, Richard, Ma*imally Smooth 'ycles# +e*atonic Systems# and the )nalysis of ,ate-omantic Triadic ro&ressions. +usic ?nalysis 1M1 5188:, 897< a b c
A! Gump up to-
)ymoc.ko, Dmitri, /cale )heory, /erial )heory, and Koice Leading, Music )nalysis =AM1 5=<<6:, 1978!
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6! Gump up to )ymoc.ko, Dmitri, )hree Conceptions of +usical Distance, +athematics and Computation in +usic, *ds! *laine Chew, ?drian Childs, and Ching'Hua Chuan, Heidel"erg- /pringer 5=<<8:, pp! =69=A>! a b c d
8! Jump up^ Callender, Clifton! Continuous )ransformations, +usic )heory Enline, 1 5=<<7: 11> 5=<<:- A=9A7! 11! Jump up^ Clifton Callender, #an uinn, and Dmitri )ymoc.ko! enerali.ed Koice Leading /paces, /cience >=<- >79>76! 1=! Jump up^ Baroin, illes, )he planet'7D model- ?n original hypersymmetric music space "ased on graph theory, +athematics and Computation in +usic, Heidel"erg- /pringer 5=<11:, pp! >=9>=8! 1>! Jump up^ ?miot, *mmanuel! )he )orii of phases, +athematics and Computation in +usic- 7th #nternational Conference, +C+ =<1>, /pringer! 17! Jump up^ Nust, Gason! /chu"ert3s Harmonic Language and Fourier @hase /pace, Journal of Music Theory <M=<1% 851:-1=1'161 1! Jump up^ Douthett, Gack and /tein"ach, @eter, @arsimonious raphs- ? /tudy in @arsimony, Conte(tual )ransformation, and +odes of Limited )ransposition, Journal of Music Theory 7=M= 51886:, =719=>
1! Jump up^ +urphy, /cott, Review of the "ook )udacious /uphony0 'hromaticism and the Triad%s Second Nature, "y Richard Cohn, Journal of Music Theory , 6M1 5=<17:, A8'1<1 1A! Jump up^ Callender, Clifton, Koice'Leading @arsimony in the +usic of ?le(ander /cria"in, Journal of Music Theory 7=M= 51886:, =189=>> 16! Jump up^ /iciliano, +ichael, )oggling Cycles, He(atonic /ystems, and /ome ?nalysis of *arly ?tonal +usic, Music Theory Specturm =AM= 5=<<:, ==19=7A 18! Jump up^ )ymoc.ko, Dmitri! /cale 4etworks and De"ussy, Journal of Music Theory 76M= 5=<<7:- - =198=! =! Jump up^ +urphy, /cott, )he +a$or )ritone @rogression in Recent Hollywood /cience Fiction Films, Music Theory 1nline 1=M= 5=<<: =7! Jump up^ Lehman, Frank, )ransformational ?nalysis and the Representation of enius in Film +usic, Music Theory Spectrum, >M1 5=<1>:, 19== =! Jump up^ +urphy, /cott, )ransformational )heory and the ?nalysis of Film +usic, in The 1*ford +andboo! of 2ilm Music Studies, ed! David 4eumeyer, 7A19788! E(ford and 4ew Nork- E(ford ;niversity @ress, =<17!
Further reading0edit2 •
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Lewin, David! ?mfortas3s @rayer to )iturel and the Role of D in 3@arsifal3- )he )onal /paces of the Drama and the *nharmonic C"MB, 34th 'entury Music AM> 51867:, >>9>78! Lewin, David! Generalized Musical Intervals and Transformations 5Nale ;niversity @ress- 4ew Haven, C), 186A:! #/B4 8A6'<'><<'<>78>'! Cohn, Richard! 3?n #ntroduction to 4eo'Riemannian )heory- ? /urvey and Historical @erspective, Journal of Music Theory , 7=M= 51886:, 1A916
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Lerdahl, Fred! Tonal itch Space 5E(ford ;niversity @ress- 4ew Nork, =<<1:! #/B4 8A6'<'18'<6>7'!
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Hook, Gulian! "niform Triadic Transformations 5@h!D! dissertation, #ndiana ;niversity, =<<=:!
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opp, David! 'hromatic Transformations in Nineteenth-century Music 5Cam"ridge ;niversity @ress, =<<=:! #/B4 8A6'<'=1'6<7>'8! Hyer, Brian! Reimag5in:ing Riemann, Journal of Music Theory , >8M1 5188:, 1<191>6! +ooney, +ichael evin! The %Table of elations% and Music sycholo&y in +u&o iemann%s 'hromatic Theory 5@h!D! dissertation, Colum"ia ;niversity, 188:! Cohn, Richard! 4eo'Riemannian Eperations, @arsimonious )richords, and their Tonnetz Representations, Journal of Music Theory , 71M1 5188A:, 19! Cohn, Richard! )udacious /uphony0 'hromaticism and the Triad%s Second Nature 54ew Nork- E(ford ;niversity @ress, =<1=:! #/B4 8A6'<'18'8AA=8'6!
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ollin, *dward and ?le(ander Rehding, 1*ford +andboo! of Neo-iemannian Music Theories 54ew NorkE(ford ;niversity @ress, =<11:! #/B4 8A6'<'18'>=1>>'>!
