A common misconception about music theory is that it is there to tell musicians what they can and cannot do: that it is prescriptive. In reality, music always leads the way, with music theory scurrying behind to describe what music is doing.
That being said, there is nothing wrong with learning a bit of obscure music theory for the purpose of developing some new compositional ideas.
Whether the following brief introduction to Neo-Riemannian Theory describes music you are already making or inspires you to make your music a little differently, there is no wrong way to add tools to your theoretical toolkit.
What Is Neo-Riemannian Theory?
Put most simply, Neo-Riemannian Theory characterizes harmonic progressions not based on tonal function but instead on parsimonious voice leading. It describes movements from chord to chord as “transformations” and names these transformations according to the pitches that are preserved from one chord to the next and to what degree the changed pitches are changed.
Neo-Riemannian Theory introduces (or, in some cases, reintroduces) some visual tools to music theory, the most recognizable being the tonnetz.
Neo-Riemannian Theory is named after the German musicologist Hugo Riemann, who was the leading music theorist of his time. Riemann contributed the concept of harmonic dualism, whereby a major triad is an upside-down minor triad and vice versa.
His theoretical principles, spanning vast swathes of musical concepts, are known as Riemannian Theory. In contrast, Neo-Riemannian Theory grows out of the loose association of writings by later theoreticians David Lewin, Brian Hyer, Richard Cohn, and Henry Klumpenhouwer.
From The Top
Before anyone has a chance to get lost, let’s define some terms. Harmony is the relationship of multiple pitches. The theory of harmony is a quasi-mathematical study and description of how pitches relate to each other. Chords are made of three or more pitches in relation to one another.
Triads are a form of chord made of three pitches: a root (the pitch for which the triad is named), a third above the root, and a fifth above the root (which is also a third above the third).
Triads can be stacked exactly as just described (also known as root position) or inverted so that a pitch other than the root is at the bottom of the triad. A triad with the third as the lowest pitch is in the first inversion, and a triad with the fifth as the lowest pitch is in the second inversion.
Major triads comprise a major third between the root and the third and a minor third between the third and the fifth. Minor triads reverse the order of the thirds, so a minor third is found between the root and third and a major third between the third and fifth. This relationship is why Riemann characterized one triad as being the “upside down” version of the other.
Diatonic harmony is harmony that centers on a particular key. Chord progressions viewed through the lens of diatonic harmony are on a path that orbits a particular tonic.
They may move towards pre-dominant chords and dominant chords but one way or another they are going to find their way back to the tonic chord or they will be making a major statement by not finding their way back to the tonic chord.
This brings us to Neo-Riemannian Theory. Neo-Riemannian Theory, as a descriptive tool, can be used as an alternative lens to view chord progressions that otherwise fit comfortably in the realm of diatonic harmony, but it is also able to deal with music that does not fit the mold of diatonic harmony.
Music theory, after all, is neither prescriptive nor universal, so new theoretical tools always have to follow new musical developments, and the result is a sort of patchwork music theory constantly expanding to cover music that, itself, is constantly expanding into uncharted territory.
The Tonnetz
The tonnetz, German for “tone network,” predates Hugo Riemann by over a century, as it was originally introduced by Leonhard Euler in 1739. That said, it did not get much use until Ernst Naumann dug it up in the mid-19th century, and Arthur von Oettingen and Hugo Riemann began using it a decade later to characterize harmonic motion.
By the end of the 19th century and into the 20th century, quite a few theorists had reached for the tonnetz in order to explain harmonic ideas.
The tonnetz arrays pitches in a lattice structure such that horizontal lines moving from left to right connect a pitch to the pitch a fifth above. Diagonally ascending lines from left to right connect a pitch to the pitch a major third above. Diagonally descending lines from left to right connect a pitch to the pitch a minor third above.
In this way, a triangle pointing upwards contains the pitches (vertices) and intervals (line segments) of a major triad. A triangle pointing downwards contains the pitches and intervals of a minor third.
Theoretically, the tonnetz extends indefinitely in every direction, but there is a fairly narrow portion of it that is most useful: that in which pitches are limited to a natural, one flat, or one sharp. As you move up and to the right in the tonnetz you start seeing double-sharps, triple-sharps, and so on, and down and to the left reveals double-flats, triple-flats, and so on.
These pitches are enharmonically equivalent to pitches with which we are more familiar (C#### is just an E), so it is not like we are meeting brand new pitches, and we might as well stay in territory that does not require a great deal of conversion to enharmonic equivalents to be useful.
Transformations
As mentioned in the introduction, movements from one chord to another in Neo-Riemannian Theory are called transformations. There are six of them, and each has a particular name and formula. It is worth noting that each transformation toggles between two triads back and forth so that a repetition of the same transformation brings one back to where they started and not to a third triad.
It is also worth noting that each transformation toggles between a major and a minor triad. It is possible, of course, to view a progression from a major triad to another major triad and from a minor triad to another triad using the Neo-Riemannian lens, but these progressions involve combining two transformations
The first three transformations maintain two common tones and shift a single pitch by a semitone or whole-tone. The fourth and fifth transformations maintain a single common tone and shift two pitches by either a semitone or a whole-tone. The final transformation maintains no common tones but shifts each pitch by only a semitone.
Now that we have reviewed terms and met the tonnetz, let’s put them to work and learn how to recognize, use, and write about the six Neo-Riemannian transformations.
Relative
The first Neo-Riemannian transformation is the flipping across a diagonal side between a Major triad and its relative minor (or vice versa). These transformations maintain the two common tones of a shared major third and shift by a whole tone the remaining pitch. This pitch toggles between the fifth of the major triad and the root of the minor triad, depending on which side of the transformation you land.
Parallel
The Parallel transformation flips across the horizontal line at the base of a triad between the upwards-pointing major triad and the downwards-pointing minor triad. The two common tones preserved across the transformation form the fifth of the triad, whereas the pitch that changes is the third, toggling between a major third and a minor third in the triad, a distance of a semitone.
Leading-Tone Exchange
While the Relative transformation toggles between a major triad and the minor triad a minor third below (or vice versa), the Leading-tone exchange transformation toggles between a major triad and the minor triad a major third above (or vice versa).
Thus, the common tones of a shared minor third form the axis across which the transformation takes place and the root of the major triad is exchanged for the fifth of the minor triad, a semitone lower (or the reverse).
Slide
The Slide transformation reflects across a vertex as opposed to a side, which means that the result of the transformation is a single common tone as opposed to two (as is the case when transforming across a side). A slide toggles between a major triad and the minor triad one semitone higher (or vice versa).
Nebenverwandt
The Nebenverwandt, or “secondary relative” in German, transformation toggles between a major triad and the minor triad a fifth below. Again, there is a single common tone, but in this case, it forms the root of the major triad and the fifth of the minor triad.
Hexpole
The Hexpole transformation is the one Neo-Riemannian transformation with no common tones. It toggles between a major triad and the minor triad a major third below.
It is especially clever in that smooth voice leading, in no case exceeding a semitone, transforms the minor third interval of the major triad into the fifth interval of the minor triad, and the root of the major triad becomes the third of the minor triad.
Mapping Transformations
The process of utilizing Neo-Riemannian Theory involves identifying major and minor triads and determining how they are related to one another. The transformations themselves, when used in a score analysis, are typically summarized by first letter: (R)elative, (P)arallel, (L)eading-tone exchange, (S)lide, (N)ebenverwandt, and (H)expole.
Compound transformations simply combine the letters (for example, RP below is a combination of Relative and Parallel transformations).
Conclusion
Neo-Riemannian theory, like the rest of music theory, exists to describe things that sound good to us. It is there to help us understand why things sound good and, in that way, may help us write more things that sound good, but it is by no means here to set boundaries on what can sound good and what can’t.
Neo-Riemannian theory is not a substitute for writing good music, but having more words to describe good music can’t hurt.
As alluded to at the beginning of the article, the patchwork of music theory, like the tonnetz, extends indefinitely in every direction. There is so much to cover that writing a beginner’s guide feels like ending a story before it has even begun.
