Neo-Riemannian theory, named after music theorist Hugo Riemann, provides a means of rationalizing triadic progressions that involve sharing common tones, more so than staying within one key.

Every Neo-Riemannian transformation toggles between one major and one minor triad.

The most basic Neo-Riemannian transformations shift one note and keep two common tones.

Relative relates a major triad and the minor triad a minor third lower, such as C and Ami.

Parallel relates a major triad and the minor triad sharing the same root, such as C and Cmi.

Leading-tone exchange relates a major triad and the minor triad a major third higher, such as C and Emi.

Other important Neo-Riemannian transformations keep one common tone and shift two notes.

Slide relates a major triad and the minor triad one semitone higher, such as C and C♯mi.

Nebenverwandt relates a major triad and the minor triad a fifth below, such as C and Fmi.

Hexpole relates a major triad and the minor triad a major third below, such as C and A♭mi. This transformation is unique because it does not keep any common tones; instead, each note is shifted by a half step to get to the new chord.

Neo-Riemannian transformations are abbreviated to one letter each: R, P, L, S, N, and H.

Theorists have come up with several networks of transformations that help visualize these relationships, such as the Tonnetz, the Weitzmann regions, and the Cube Dance.

A one-page summary of transformations is available as an interactive score and as a PDF.

In the late 19th century, composers often used triadic progressions that confound conventional Roman numeral analysis. Consider the following excerpt from Brahms's concerto for violin and cello (Example 1):

[](https://viva.pressbooks.pub/app/uploads/sites/12/2019/07/brahms-concerto-for-vln-vc-268-79.png) Example 1. Brahms, Concerto for Violin and Cello, I, mm. 268–79 (10:22).

A reduction of the chord progression in the excerpt above can be found in Example 2. The passage connects two A♭ major triads; however, the chords in between those triads do not belong to A♭ major in any useful way, nor do they follow any of the conventions of functional harmony.

[](https://viva.pressbooks.pub/app/uploads/sites/12/2019/07/brahms_vc_vln_concerto_reduction.png) Example 2. Brahms, Concerto for Violin and Cello, I, mm. 270–76, reduction.

One might dismiss the passage altogether as "non-functional harmony," but when you listen to it, it follows a certain kind of logic. As Richard Cohn writes, "if this music is not fully coherent according to the principles of diatonic tonality, by what other principles might it cohere?" (1998, 169).

Neo-Riemannian Transformations

Neo-Riemannian theory describes a way of connecting major and minor triads without a tonal context. Example 3 shows the three basic Neo-Riemannian operations. Notice that each operation preserves two common tones in a triad and changes its mode.

The Relative transformation (R) preserves the major third in the triad and moves the remaining note by whole tone.

The Parallel transformation (P) preserves the perfect fifth in the triad and moves the remaining note by semitone.

The Leading-Tone Exchange transformation (L) preserves the minor third in the triad and moves the remaining note by semitone.

[](https://viva.pressbooks.pub/app/uploads/sites/12/2019/07/NROs_-_basic_voice_leading_examples-PLR.png) Example 3. The three basic Neo-Riemannian operations: Relative, Parallel, and Leading-Tone Exchange.

It's important to note that each transformation is a toggle between two chords. As an analogy, think of the caps lock key on your keyboard, which toggles between two cases of letters. If you are typing with lowercase letters and then press caps lock, this tells your keyboard to begin typing in uppercase letters. But if you press caps lock again, it toggles back to lowercase—it doesn't go to some third case.

Similarly, if you do the L transformation on a C major triad, you get an E minor triad. If you do the L transformation on an E minor triad, you return to a C major triad. Put another way, successive repetitions of the same transformation alternate between two chords, as shown in Example 4.

th-century composers didn't write progressions using the same transformation over and over again, you will find this technique used in twentieth-century works like "O Superman" by Laurie Anderson, which uses successive L transformations throughout.

The Tonnetz

Example 5 shows a Tonnetz. A Tonnetz is a visual representation of pitches arranged such that perfect fifths are read from left to right, major thirds are read diagonally from the top left to the bottom right, and minor thirds are read diagonally from the bottom left to the top right. Any three pitches in a triangle form a major or minor triad. Neo-Riemannian transformations can be visualized by flipping a triangle along one of its three edges.

[](https://viva.pressbooks.pub/app/uploads/sites/12/2019/07/tonnetz.png) Example 5. The Tonnetz.

The (P)arallel transformation flips the triangle along the edge belonging to the line of perfect fifths (left to right).

The (R)elative transformation flips the triangle along the edge belonging to the line of major thirds (top left to bottom right).

The (L)eading-tone transformation flips the triangle along the edge belonging to the line of minor thirds (top right to bottom left).

[](https://viva.pressbooks.pub/app/uploads/sites/12/2019/07/tonnetz-with-NROs-in-color.png) Example 6. Tonnetz showing the three Neo-Riemannian operations (P, L, and R) performed on a C major triad and a G minor triad.

Example 6 shows each of the transformations on the Tonnetz. If you perform a P transformation on the C major triad, highlighted