In recent years Hugo Riemann's ideas have thoroughly captured the music-theoretical imagination, both in the United States and abroad. Neo-Riemannian. Neo-Riemannian theory originated as a response to the analytical issues surrounding Romantic music that was both chromatic and triadic while not “functionally. 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. Initially, those harmonies were major and minor triads; subsequently, neo-Riemannian theory was extended to standard dissonant sonorities as well.Triadic transformations · Graphical representations · Criticism · Extensions.
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More recently, Dmitri Tymoczko has argued that the connection between neo-Riemannian operations and voice leading is only approximate see below.
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 Neo riemannian theory "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. One 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 neo riemannian theory classical Tonnetz.
Richard Neo riemannian theory developed the Hyper Hexatonic system to describe motion within and between separate major third cycles, all of which exhibit what he formulates as neo riemannian theory smoothness. 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 inwhen 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 neo riemannian theory note to another.
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". Planet-4D model embeds the traditional Tonnetz onto the surface of a Hypersphere InNeo riemannian theory Baroin presented the Planet-4D model,  a new vizualisation 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.
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. These distances reflect neo riemannian theory only imperfectly. 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.
Postclassic: So I'm Neo-Riemannian: Who Knew?
Underlying these discrepancies are different ideas neo riemannian theory 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. And there are some other derived functions, such as D, which does from a triad to its dominant, or vice versa.
If I'm oversimplifying or getting it wrong, some kind reader will correct me. The occasion of my bringing this up was my biennial analysis of a gorgeous passage from Liszt's Annees de Pelerinage, the beginning of "Sposalizio," which is hardly complicated, but notoriously recalcitrant to Roman numeral analysis: It doesn't really explain the music - though the Neo riemannian theory does make clear the equivalent jumps from B-flat to D-flat and A-flat to B, which the enharmonic pitch notation hides - it just gives me a way to label Liszt's key jumps, more radical in their unconcern for tonality than Chopin's or Schumann's.
And I think the students enjoyed learning a theoretical technique that wasn't from the musty, immemorial past, but evolved during their lifetimes. The even greater interest for me is the potential application to my neo riemannian theory music.
Neo-Riemannian theory - Wikipedia
InI switched over, with some trepidation, to writing in a triadic style, though not at all functionally tonal. I had been studying Bruckner and taking tips from passages like this wonderful one from his Eighth Symphony: I soon found myself rather obsessed with what I now learn are called LPL and RPR transformations the latter yielding a tritone transposition.
Here's Baptism, from Note that no chord change requires fewer than neo riemannian theory transformations - that was my conscious discipline for the piece, but of course I wasn't thinking of it in Neo-Riemannian neo riemannian theory, which hadn't been invented yet.
- Film Music and Neo-Riemannian Theory - Oxford Handbooks
- Oxford Handbook of Neo-Riemannian Music Theories - Oxford Handbooks
- Film Music and Neo-Riemannian Theory
- Frank Lehman