To teach undergraduate music theory is to recount over and over and over, year after year without variation, facts, terminology, and principles that haven't changed since well before I was born. But Wednesday I managed to teach something new, and got a real kick out of it: Neo-Riemannian theory (named for the German musicologist Hugo Riemann, 1849-1919). I had never heard of the subject until the 2007 Minimalism conference in Wales, where Scott Alexander Cook applied the methodology to the music of Gavin Bryars. The idea is that relationships between triads can be characterized by variously close or distant pitch replacements, categorized by the three functions P (parallel), L (leading tone), and R (relative). The P function moves the third of a triad to change it from major to minor, or vice versa:

The L function lowers the root of a major triad a half-step, or raises the fifth of a minor triad a half-step, creating a new triad of the opposite modality in either case. The R function raises the fifth of a major triad a whole-step to produce the triad of the relative minor, or lowers the root of a minor triad a whole-step. And there are some other derived functions, such as D, which does from a triad to its dominant, or vice versa. And functions can be combined, so you get PL, RP, LPL, and so on. 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:

Now instead of calling that i-VI in E minor and wondering where in hell the B-flat major came from, I could label the sequence L, RLRL (or DD, since C to B-flat is two dominant jumps), PR, D, PR. It doesn't really explain the music - though the PR 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 own music. In 1983, I 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:

(Interestingly, in the immediate repeat of this passage Bruckner replaces the RP with a PR, and ends up in D major instead of A-flat.) 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 1983:

The chord changes, if I have parsed them correctly with my amateur notions of Neo-Riemannian terminology, are PLP, RPR, LPL, RPR, LPL, LPL, RPRP, LPL. Note that no chord change requires fewer than three transformations - that was my conscious discipline for the piece, but of course I wasn't thinking of it in Neo-Riemannian terms, which hadn't been invented yet. This was pretty much my default harmonic language from Baptism of 1983 to my "Last Chance" Sonata of 1999. In 2000 I studied jazz harmony with John Esposito, and switched over to bebop chords. My microtonal music uses a process of micro-interval voice-leading that Harry Partch called "Tonality Flux," related to Neo-Riemannian transformations, but of course not reducible to them. However, the large-scale tonal structure of my 2002 microtonal chamber opera Cinderella's Bad Magic consists of a chain of simple Riemannian pitch shifts, all tuned to pure triads and evolving from E-flat major to C double-sharp minor, through single note-shifts R P R P L P R P L:

These Neo-Riemannian labels neither explain nor justify my music, but they do give me simple ways to refer to my voice-leading methods as classes of harmonic transforms. And, as papers at the last minimalism conference proved, Gavin Bryars and John Adams were using a similar kind of non-functional harmonic consistency from the early 1980s on as well. This terminology can make it easier to point out how the harmonic practices of Liszt, Bruckner, and Reger returned, following the period of widespread atonality, to form a new common practice in minimalist and especially postminimalist music - starting, I think we'd have to say, with Einstein on the Beach, whose "Spaceship" scene may succumb to only this kind of analysis. Neo-Riemannian theory may fill in some cracks in our analysis of Romantic music, when dealing with composers who couldn't abide within the Germanic idea of a centralized tonality. For postminimalism, it may prove to be the analytical technique that fits the music like a glove.

COMMENTS:

Dmitri says: Honestly, I think Neo-Riemannian language is not quite right for you, or for Gavin Bryars or the other minimalists. The details are technical, but perhaps interesting.
The basic point is that what motivates a lot of chromatic tonality is voice leading -- when E minor moves to G# minor in Wagner, this is (typically, in general) because of the nice voice leading relationships between the two chords.
NR-theory initially looks like a way of talking about voice leading, but it really isn't. From an NR perspective F major is closer to C major than F minor is (LR vs PLR), whereas contrapuntally the opposite is true. This means NR techniques are not able to explain simple instances of chromaticism such as IV-iv-I.
The other big problem is that NR theory is fundamentally dualist. From an NR perspective, C major->E major is "the same" as C minor->Ab minor (PL in both cases). I really doubt that chromatic composers of any century (Wagner, Adams, whoever) thought this way. C major->E major is more likely to be associated with C minor->E minor, preserving the direction of root motion.
In any case, I think there more interesting and revealing ways of talking about this music ... NR theory (in my view) was a good first step in the right direction, but we've gone a long way since then.
KG replies: I'm sure you know more about it than I do, but I can't say I find your caveats convincing, or even entirely relevant. In my music, C major->E major *is* "the same" as C minor->Ab minor - that's exactly the way I was thinking in the '80s and '90s.

Dmitri says: Well, if you were thinking that C major->E major is the same as C minor->Ab minor, then something like NR-theory may be for you.
The issue here is whether we think of major and minor as related by (diatonic) transposition or by inversion. In the late twentieth-century, after Schoenberg, we might've started to become more sensitive to the inversional relationship between C->E and c->ab. So maybe this is an instance of your indebtedness to 12-tone music!
I still have my doubts about whether earlier composers thought this way; it's hard for me to think that Liszt (say) thought G major->Ab major is the same as G minor->F# minor, but of course I could be wrong.
One other small point: there's actually no combination of L, P, and R that produces "D". (Any combination of LPR that takes C major to G major will take C minor to F minor.) Again, this is because L, P, and R act oppositely on major and minor chords.
KG replies: "The D transformation can also be written as a combination of R and L transformations (e.g. applying the L transformation to CEG produces BEG, that is, EGB; then applying the R transformation to EGB produces DGB, that is, GBD). Thus, the D transformation might be redundant." - Carol L. Krumhansl, "Perceived Triad Distance: Evidence Supporting the Psychological Reality of Neo-Riemannian Transformations," Journal of Music Theory, Vol. 42, No. 2, Neo-Riemannian Theory (Autumn, 1998), pp. 265-281
"What is to be made of the fact that, applied to a major triad, D has the same effect as R followed by L, but applied to a minor triad, D has the same effect as L followed by R? Is D superfluous?" - Julian Hook, "Uniform Triadic Transformations," Journal of Music Theory, Vol. 46, No. 1/2 (Spring - Autumn, 2002), pp. 57-126
I'll believe anything I read, but you guys need to get your story straight. I'm taking this stuff straight out of articles in JSTOR. I assume the Journal of Music Theory is peer-reviewed? Perhaps Princeton has an entirely different system than Yale?

Dmitri: What Krumhansl says is incomplete and a bit misleading. What Julian Hook says is right, and is consistent with what I was saying: for major triads, D is equal to LR but for minor triads it is RL.
In other words, there is no single (ordered) combination of R and L that is equivalent to "D" for both major and minor triads. No single sequence of L, P, and R both takes C major to G major and C minor to G minor (or C minor to G major). Any sequence of LPR that moves a major chord up by X semitones will also move a minor triad down by X semitones.
KG replies: Then we agree, though I don't believe I had implied that there was any single combination of R and L that would, in all cases, produce D.

Richard Cohn: This is a belated response to Dmitri's initial response Kyle's initial post on this thread, both of which I only just read for the first time.
Dmitri has lately been arguing that neo-Riemannian theory is 1) fundamentally dualist, and 2) is not about voice leading. He is fundamentally wrong on both counts. He also repeats these claims in his re-writing of the neo-Riemannian article for Wikipedia. Let's take them one at a time
1. What is the evidence that neo-Riemannian theory is fundamentally dualist? Surely it is not that its primary objects are major and minor triads, which are inversions of each other. One can derive these objects dualistically, as Riemann did. But there are alternatives that don't invoke that sort of dualism. One can simply say that major and minor triads are the sonorities that you get when you minimally perturb an augmented triad (i.e. a trisection of the octave) in one direction of the other. I suppose the "one direction or the other" might be construed as dualism, of a sort: it buys into the belief that pitch is organized on a linear or cyclical continuum, and there are two directions to a line or cycle In music we think of those directions in terms of "up" and "down" . But I don't think this is the kind of dualism to which Dmitri is referrin. Or rather, it's not a kind of dualism that Dmitri can afford to object to, since it is fundamental to his work (as it has been to mine since the early 1990's). So I doubt that he wants to make the argument on that basis.
The evidence that Dmitri offers here, as in the Wikipedia page, is that PL takes C major to E major, but takes C minor to Ab minor in the other direction. For Riemann, this was a result of dualist action of his Schritte; but again, it need not be so for us. One can think of these two as equivalent by virtue of both carrying one voice from C to B, one voice from G to Ab, and holding the third voice stationary. No dualism there. What's there, in fact, is voice leading.
2) Which leads us to Dmitri's second claim, which is that neo-Riemannian theory is not about voice leading. Dmitri is correct that LPR proximity is not a perfect correlation with voice-leading proximity. It's a pretty good measure, but there are some mis-matches, and Dmitri identifies them. But the LPR transformations are only one aspect of neo-Riemannian theory, only one stage in its development. I emphasized them in an article that I published in 1997, but already in 1998 (in my article "Square Dances with Cubes") I was viewing chromatic transformations in a different framework that overcomes the imperfections that Dmitri identifies.
So those who are attracted to neo-Riemannian theory should not be swayed by Dmitri's telling you to stay away from it, and pay attention to his work instead. By all means, pay attention to Dmitri's work, which will be published soon by Oxford University Press. It's completely brilliant and mind blowing. (Confession of subject position/potential conflict of interest: I edit the series on which his book will appear.) But don't be deterred from pursuing your initia attraction on the basis of what Dmitri says here. There's much more to neo-Riemannian theory than meets Dmitri's eye.
As a postscript, Kyle asks about neo-Riemannian work that is not larded with mathematics, and is written for musicians rather than technical music theorists. I'd be happy to supply some references of things currently in print. I'm completing a book on 19th-century harmony that draws together my twenty years of thinking about these matters. It will have lots of analysis, lots of geometry, but nothing that will cause technical stumbles for anyone reads this blog.
KG replies: Thanks. It's not the mathematics in the articles that bothers me. It's that some of the articles seem more interested in figuring out inherent properties of the transformations (that may or may not be compositionally relevant) than describing actual musical processes.

H Gilligan: In reply to Richard Cohn.
Richard Cohn, I would be interested in finding out about those references you mention. I am looking for sources that my undergraduate seniors will be able to understand, with the help of some fruitful class discussion and explanation on my part. Any suggestions?

Stephanie Lind: In reply to H Gilligan.
Sorry, this is a really late reply as I've just read this blog (good points all around, by the way!). I taught some Neo-Riemannian theory in an upper-year undergraduate seminar last year, and used Nora Engebretsen and Per Broman's "Transformational Theory in the Undergraduate Curriculum: A Case for Teaching the Neo-Riemannian Approach". You can find it in the Journal of Music Theory Pedagogy, 2007. It's not entirely comprehensive (for example, it doesn't discuss harmonic dualism all that much), but it covers most of the basics without getting too bogged down in technical specifics. I ended up using it as a start point to introduce my students to basic concepts, and we moved from there into more advanced readings.
KG replies: Thanks, I'll take a look at it.

Tom Bergeron: Wait a minute. Dmitri says: "C major->E major is "the same" as C minor->Ab minor (PL in both cases)." When I apply PL to C, I get Ab (C*P=c; c*L=ab) and then c*PL=E. Unless I'm misunderstanding the explanation, to get from C to E or from c to ab requires LP, not PL.
KG replies: I think you're right.

Dustin Soiseth: I second Eric's comments. As a conductor I've struggled to find good analyses of my favorite minimalist and post-minimalist works to supplement my own observations, and am glad to know that there are theorists doing the work.
In addition to the journals listed above, where else could one find analysis like this? Thanks.
KG replies: I'd be really curious to hear what one conductor's favorite postminimalist works are. I'm working on a book myself, poring through mountains of material - right now I'm working on an article analyzing John Luther Adams's orchestra works, for a proposed book about his music (edited by someone else). If you have access to JSTOR (something that perhaps only academics get), I've just been putting in terms like Neo-Reimannian and LPL transformations. Some of the articles are too technical and mathematical for my needs, some devoted to specific repertoires or problems, some more general. If anyone else knows of particularly helpful articles or books, I'd be glad to hear about them

Dustin Soiseth: I do love JLA's In The White Silence.

Eric Grunin: I can't thank you enough for posting this.
It suggests a way of filling in an important gap-one that has bothered me for years-in my understanding of modulatory practice in tonal and neo-tonal music.

mclaren: The I IV V I and even the I ii IV V I and I ii IV V vii I progressions are all entirely built out of 5ths, while the NR progressions are built from 3rds. Since the 19th century, music theorists have obsessed over progressions by 5th because they can theoretically visit all 12 keys in 12 equal, a feature of endlessly fetishistic monomania to the 19th century music theorists who idolized extreme chromaticism in music in the manner of a deviant who collects and fanatically caresses women's shoes. However, this extreme fetishization of chord progressions by p5ths should not blind us to the obvious reality that composers prior to the 19th century had often used progressions by 3rds. In 12 equal, progression by 3rds does not permit a composer to visit all 12 keys in 12 equal, but this was of little concern in earlier eras when extreme chromaticism did not represent the end-all and be-all of music theory that it did in the late 19th century.
I IV V I can be constructed entirely in p5ths so: tonic, p5th down from tonic, p5th up from tonic, tonic. I ii IV V I can be constructed entirely in p5th as: tonic, pth + pth up, p5th up from tonic, tonic. I ii IV V vii[o - ed.] I can be built from p5ths as: tonic, p5th + p5th up from tonic, p5th down from tonic, pth up from tonic, p5th + p5th + p5th up from tonic, tonic. (The vii should have a superscript circle appended to indicate that it's a diminished triad but that's not possible in these comments, so sue me.)
We find root progressions by thirds throughout the Elizabethan virginal composers and constantly in music of the 17th century, particularly Purcell and Buxtehude. Since the music of these eras had no particular interest in extreme lush chromaticism and a peripatetic vistation of all 12 keys, that was not a problem. However, by the late 19th century, when the incessant use of the dim 7th chord had earned it the nickname "l'accord d'ennui" by French and Italian operagoers, progressions by 3rds had fallen out of use. Hindemith's music and that of other composers who combined both major and minor modes in the same chord progressions brought back the use of progressions by 3rds.
Leaving aside P, R and L are really root progressions by thirds. That can have a big emotional impact, but limits the number of other keys to which a composer can traverse. With the advent of minimalism and other 20th century styles in which values other than relentless extreme chromaticism came to the fore, worries about the number of different keys a composer could visit by means of chord progressions began to wane.
It's worth noting that the limitations imposed by root movements of 3rds only apply in 12 equal because 4 scale steps (major third) and 3 scale steps (minor third) are both factors of the composite number 12. In a tuning which breaks up the octave into a different number of logarithmically equal scale steps, progressions by 5th limit the number of keys to which you can move, while progressions of a 3rd now allow the composer to chromatically visit all keys in the scale. A good example involves 10 equal, in which the p5th limits you to a single circle of 5 perfect fifths, only half the keys possible in 10 equal. By contrast, root progressions by a 3rd let you visit all 10 keys in 10 equal. You can hear a musically effective example of this kind of chromatic progression by 3rds in one of William Sethares' compositions from his CD Xentonality in 10 equal.
In equal divisions of the octave which are prime, both progressions by 3rds and progressions by p5ths enjoy equal status in term of opening up chromatic possibilities for modulation, which offers a good reason for composing in equal tunings with primes numbers of steps per octave: viz, 5, 7, 11, 13, 17, 19, 23, 29, 31 equal steps per octave, etc. Of these 11 and 13 and 23 equal do not permit recognizable p5ths, however, while the others do.
The artificial limitation to progressions by p5th also constitutes a pitfall of traditional (narrow-minded) just intonation, in which the p5th is always a 3/2. In such limited and ultraconservative JI tunings, progressions by 5ths tend to dominate. In more enlightened JI tunings such as non-octave JI tunings or tunings which use p5ths quite different from the 3/2 but still within the empirical range for recognizable perfect fifths (680 cents to 720 cents) determined by Moran and Pratt in their well known 1926 article in the American Journal of Psychology, progressions by 3rds can take precedence over progressions by p5ths, adding vividness and variety to JI music and breaking JI compositions out of the starchy stuffy stale same-old same-old 3/2 p5th David-Doty-rut of early crude pre-Erv-Wilson JI composition.
Of course you're talking about P and L and R in terms of voice-leading rather than root progression. I would submit that they represent two facets of the same entity, inasmuch as you can view P as bimodality (major and minor modes combined into one scale) and L and R as bitonality (two keys melded together and used as a single scale). Bimodality and bitonality both expand explosively in musical interest and in music-theoretic fascination when we break free of the conventional 12 equal tuning, since in both JI and equal tunings as well as in NJ NET tunings, bimodality now allows neutral as well as major and minor modes, while bitonality permits inclusion of multiple distinct circles of fifths, as in multiples of 5 equal up to 35, and multiples of 7 equal up to 35.

Richard Friedman: Gee, for a minute there I thought you were referring to Bernhard Riemann, the mathematician.
http://en.wikipedia.org/wiki/Riemann
My mind boggled with the concept of Riemannian harmony, based on differential geometry and the famous Riemann Zeta function.
Hmmmm. Sound velly interestink. Something worth looking into.

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