#1
No no, not that Riemann, Ean, this Riemann.

So I was bored and thought to myself, "Zach, what piece of obscure music theory can I talk about?" Then I remembered this one thing I sat through a lecture about but completely forgot. It took me a good 15 minutes of Googling just to remember the name of it, so at that point I knew it was perfect.

Although Riemann lived in the late 19th Century, Neo-Riemannian theory (from now on I'm calling it NR) is relatively cutting edge, only really being developed in the last 20 years or so. It's built on the idea of Riemann's "Dualist" system where he viewed a major triad as an upside down minor triad. Incidentally, set theory bears this out with the idea of inversional equivalence, whereby sets that are inversions of one another are in fact the same set. Both the major and minor triads are represented as (037) in set theory (if that's at all interesting to you I did a huge primer on set theory). The most important connection to Riemann is his system of relating triads to one another.

Now, if you know virtually any theory at all, whether you know it or not you've learnt "chord grammar." Chord grammar is the way that chords interrelate with one another and is the basis for harmonic function. Certain chords function as a dominant, some as a tonic, others as a predominant, etc. The binding concept of tonal chord grammar is that everything is relating to a central tonic pitch. Predominant chords move away from the tonic and towards the dominant, which in turn moves towards the tonic. With Riemann we ditch that concept and chords are related purely through common tones, not in relation to a tonic.

This idea becomes very useful when analyzing certain harmonies that are triadic, but don't necessarily follow the expected chord grammar that roman numeral analysis is built for. It's come to be used most commonly for late Romantic composers like Liszt, Mahler and Bruckner, but also has applications to 20th Century composers, specifically Post-Minimalists like John Adams.

With that said, as with all theory, its beauty is in its abstraction not its application (yeah I just said that).

There are three basic relationships/transformations in NR that show us how chords are interrelated. They're abbreviated PRL (or LPR or LRP or PLR or RPL or RLP).

Parallel: this is altering the third of a triad by a half step so as to move to its parallel species. That is, a major triad becomes a minor triad and vice versa. In this case, species refers to major or minor.

Relative: this is either moving the fifth of a major triad up a whole step or the root of a minor triad down a whole step. The resulting triad is the tonic triad of the relative key.

Leading tone exchange: this is moving the root of a major triad down a half step or the fifth of a minor triad up a half step. This results in a move by third and the inverse species.



These three transformations all have certain things in common. First, they all alter only one note of the triad. Second, they all invert the species of the triad (minor becomes major and major becomes minor). Third, they invert themselves, meaning if you perform two of the same PLR transformations in a row you end up with the same chord.

Intuitively, it's hardest to see that third point with the Leading Tone Exchange, but it's there:



Note that I've been using the smoothest voiceleading up to this point, but there's nothing explicitly in NR theory that says this is required. The important aspect is the idea of connecting chords with common tones, not how the tones moves.

So those transformations aren't hella exciting, but we can combine those transformations into more exciting progressions. Here are some examples of dual transformations (I've included the intermediate chord, but it's not necessary):



With these double transformations we get what chord buttgrammar might call chromatic mediants, which are considered to be relatively distant harmonically. To us, though, they're just two quick transformations and are quite closely related. These all result in one common tone between chords.

To create even more interesting (and from the perspective of chord grammar, complex) relationships we can do triple transformations. Obviously there are many available. There are six possibilities if we don't repeat any transformations, but remember that the order of transformations matters, so RPR, LPL, RLR, etc. are all available as well, which leads to a lot of options.

There are three common ones that I'll mention.

N (Nebenverwandt) is an RLP transformation, which results in an inverse subdominant or dominant relationship. A major triad transforms to its minor subdominant and a minor triad transforms to its major dominant.

S (Slide) is an LPR transformation, which results in a triad that shares a third. A major triad transforms to the minor triad a half step above and a minor triad transforms to the major triad a half step below.

H (hexatonic pole) is an LPL transformation, which results in another type of chromatic mediant (although the actual relationship is way more complicated from a NR theory standpoint). A major triad transforms to a minor triad a major third below and a minor triad transforms to a major triad a major third above.



With these triple transformations we're not really in an area where chord grammar is fruitful anymore, especially if we do what is well within our rights and start chaining some of these transformations together. Look what happens when we do a slide and then an LPL and then a Nebenverwandt:



I've put the roman numerals you might use to analyze this underneath. Those roman numerals are effectively worthless at describing anything meaningful about that progression, but with our simple transformations we can see that there is in fact the concept of common tones and smooth voiceleading chaining the chords together.

Neo-Riemannian theory can get a whole lot more esoteric than this, but I doubt I'll bother going into it. I just think it's kind of cool because from these three elegant transformations (step-wise voiceleading like this has been termed voiceleading parsimony) we can get very dramatic chord progressions that really don't reflect even some of the most chromatic of tonal harmony at all.

I'll leave you with a touch of Liszt. This is Sposalizio from Années de Pelerinage.



The chords aren't complicated. The basic progression is:

Emin - Cmaj - Bbmaj - Dbmaj - Ebmaj - Abmaj - Bmaj

Simple triads all. I fucking dare you to try and put roman numerals underneath those chords. We're in I-don't-even-know-what key, C# minor overall I guess, but that doesn't really help us. The article I took this piece from called it "recalcitrant to roman numeral analysis" which is a pretty hilarious understatement.

But what if we look at it from the perspective of RPL? Emin to Cmaj is easy, that's just an L transform.

Cmaj to Bbmaj is an RLRL (since RL results in a dominant jump, you can also call this a DD transform because it's two dominant jumps (Cmaj RL -> Fmaj RL -> Bbmaj).

Bbmaj to Dbmaj is PR

Dbmaj to Ebmaj is RPD (RPRL)

Ebmaj to Abmaj is D (RL)

Abmaj to Bmaj is PR



The progression is nothing arcane to us now that we have Riemann on our side. In fact, the transformations are hardly extreme; the most one chord to the next does is a quadruple transform.

This weird little theory actually gives us a pretty powerful way to label complex chromatic music. And in a way that, arguably, could be more reflective of how Liszt was thinking than roman numerals are.

This isn't even to mention all the interesting progressions and combinations you could come up with for your own music.

Go forth armed with neo-Riemannian theory and create.
#2
Big ass thumbs up.

For the pop music guys, you often see a similar (but unrelated in many ways) concept known as "constant structure."

While not the same concept as the one demonstrated by Zach, that might be as close as that field gets to this concept, in the sense that you are now relating harmonies solely to each other instead of some tonic center.

Also an outlandish suggestion: Take the OTHER Riemann and apply his work to serial music...
"There are two styles of music. Good music and bad music." -Duke Ellington

"If you really think about it, the guitar is a pointless instrument." - Robert Fripp
#3
that's literally why i clicked on this thread
i don't know why i feel so dry
#5
yes my song is the contour integral along a rectifiable counter-clockwise of any holomorphic function. i call it 4'33"

HAHA GET IT BECAUSE THAT INTEGRAL ALWAYS EVALUATES ZERO HAHAHAHAHA

later.
i don't know why i feel so dry
#6
But 4'33" isn't silence....people always miss the point of that piece...sigh.

To be fair, I did laugh reasonably hard at your post though.
"There are two styles of music. Good music and bad music." -Duke Ellington

"If you really think about it, the guitar is a pointless instrument." - Robert Fripp
#7
This was pretty cool.

And possibly useful. I just came up with a nonfunctional chromatic progression from noodling around. Maybe I'll try to apply this to it. Although it might be a bit over my head.

A6 A#-7 G6/B B/C Dm/C# A/D

So it's the first inversion minor chord shape alternating with the 2nd inversion maj chord shape down to A with a rising bass line.

F#m P F#Maj L A#m...but how does this work with 4 note chords?
#8
This is cool. It certainly makes sense to relate the chords to each other without regard to a tonic in many cases. Although, I can't help but feel that there might be a better way to relate chords to each other when there are no common tones between them - multiple transformations are the logical explanation within the theory, but that seems more like making the harmonic progression fit the theory rather than the theory truly illuminating the harmonic progression. But this theory is still being worked out, right?

Are there any competing theories for addressing the same thing that NR does? Do Schenkerians like/dislike NR?
#9
Quote by Duaneclapdrix
This was pretty cool.

And possibly useful. I just came up with a nonfunctional chromatic progression from noodling around. Maybe I'll try to apply this to it. Although it might be a bit over my head.

A6 A#-7 G6/B B/C Dm/C# A/D

So it's the first inversion minor chord shape alternating with the 2nd inversion maj chord shape down to A with a rising bass line.

F#m P F#Maj L A#m...but how does this work with 4 note chords?

With anything other than triads it becomes trickier because it's built on that idea, but I think people have worked something out with dissonant sonorities. I'm not really familiar with that. You could attempt to adapt something for four note chords though. Even if it's just by saying that the seventh doesn't matter and just dealing with the triad underneath.

So for your progression you might say A6 to A#-7 is a slide (Amaj to A#min) then to Gmaj would be PLP etc.

So even though you have these dissonances along for the ride, the underlying triads are still defined by the transformations. That's less satisfying to me, but maybe you can think of something better.

Quote by Harmosis
This is cool. It certainly makes sense to relate the chords to each other without regard to a tonic in many cases. Although, I can't help but feel that there might be a better way to relate chords to each other when there are no common tones between them - multiple transformations are the logical explanation within the theory, but that seems more like making the harmonic progression fit the theory rather than the theory truly illuminating the harmonic progression. But this theory is still being worked out, right?

Are there any competing theories for addressing the same thing that NR does? Do Schenkerians like/dislike NR?


I'd generally agree. I don't think this theory necessarily sheds light on what a progression sounds like, but as I said it can make an elegant way to look at and label progressions. I mean, any analytical theory has some element of fitting the music to the theory and Schenker is certainly no exception to it. There are a lot of pieces that don't really jive with Schenkerian theory but can have excuses made for them so they do. The most obvious one is when a piece ends 7-8 even though Schenker's entire precept is the falling action from 2-1. He might call the 7 just an embellishment of that falling action or something, but whether you want to buy that or not is a different question.

And NR is, like I said, pretty obscure and there's a lot of contention even amongst advocates of it and people who study it how relevant it is in certain cases. I'm not sure if I'd say there are really competing theories. The unique thing about it is it really exists outside of a tonal paradigm and looks at triads as pure and functionless. Every other analysis method tends to focus on how we can fit chords into a functional tonal idiom.

Schenkerians would likely have a conniption on this. To them it would be entirely missing the point of everything. But I really think NR exists alongside, not in competition with, other analysis methods.
#10
Yes, that's my problem as well (focus on how we can fit chords into a functional tonal idiom)!

That's a good point, though - using NR in conjunction with other analyses.

Thanks JRF.
#11
Yeah but those crazy Schenkerians are a bunch of weirdos anyway. Supposedly Schenker intended it as an alternate theory/analysis for PERFORMERS....

The other thing you could do is say the roots don't matter and use the upper structure triads, gluing them together with a sweet bassline. Might not be pure Riemann, but it works alright.

Either way its certainly a good way to start looking at non-functional harmony, by relating the chords to each other.

Also 0/10 no Tonnetz.
"There are two styles of music. Good music and bad music." -Duke Ellington

"If you really think about it, the guitar is a pointless instrument." - Robert Fripp