Reddit reviews Audacious Euphony: Chromatic Harmony and the Triad's Second Nature (Oxford Studies in Music Theory)

Schenkerian analysis takes everything you learned in harmony and makes it useful. It helped me make sense of counterpoint too. I link some sources in this thread.

Then there's neo-Riemannian theory, which is more or less about chords that don't behave well under traditional analysis. I learned from bashing my head against really dense academic articles, but I hear Richard Cohn's Audacious Euphony is supposed to be good. What you want to do is learn PLR-family transformations and the triadic Tonnetz, learn the concept behind compound transformations, then jump into Cohn's writing on cyclical progressions and wrap your head around Douthett & Steinbach's graphs.

Form is super important, especially Caplin's theory of formal functions. This stuff meshes well with Schenkerian theory, in my opinion. Schmalfeldt's book is also a very useful for the study of form. I always feel a little strange recommending Hepokoski and Darcy, because that book is so dense and I don't want to push anyone into purchasing a 600-page book on the analysis of sonata forms that's just going to end up collecting dust on their shelf. However, it's easily one of the most important music theory treatises of the 21st century so far (and the other ones I'm mentioning are right up there too), so I feel I should mention it.

Question is, what are you into? What do you want to learn? What do you hope to do with this knowledge?

IV. Voice-Leading Parsimony

("Parsimony" means "thriftiness, frugality; unwillingness to spend money.")

One interesting fact about P, L, and R: they leave 2 notes untouched, and the voice that does move only moves by a step. P and L only move one note by a half step, and R is a little more extravagant by moving a voice by a whole step. So these transformations are "parsimonious" (frugal) in the sense that they can get you new chords for very little effort (motion). It turns out that the triad is pretty cool for being able to do this: very few other chord types in the world can. (For example, you can't get from one French 6th chord or fully diminished 7th to another just by moving one voice a tiny amount.)

The next thing that Neo-Riemannian theory asks is "What happens if I chain a bunch of transformations together?" For example, what happens if I make a sequence by alternating P's and L's? Each step along the way changes only 1 half-step, but how many different notes does it use total? How long before I get back to my starting chord? (Will I go through all 12 major and all 12 minor triads? Or do I only use a fraction of the total?) Neo-Riemannian theory maps out the possibilities and describes them using a concept from modern algebra known as an algebraic "group." The transformations P, L, and R form a "group" of things that you can combine to make new things (e.g. imagine considering L-then-R to be a single transformation of its own). Group theory is used to explore the structure of the possibilities there.

V. Enharmonic Equivalence

(That is, the assumption that there are only 12 notes and that spelling doesn't matter, so G# = Ab.)

This doesn't sound very exciting, because we're pretty used to it by now. But it was a radical notion early in the Romantic period, and composers like Schubert got some cool effects out of exploiting it.

Earlier I asked "What happens if I make a sequence out of alternating P's and L's?" Well, it turns out that I go through 6 different chords, like this: CM - Cm - AbM - Abm - EM - Em (then back to CM). Every L takes me to a chord with a root a M3 lower, so that after 6 steps I've gone down by 3 major thirds and end up back where I started. This needs enharmonic equivalence to work, because without it I'd go C - Ab - Fb - Dbb... so that, in some weird conceptual world I'm actually not where I started. We're used to making that enharmonic shift, but it was relatively unfamiliar at the time. Partially that had to do with tuning, but also it had to do with the fundamental role of the diatonic scale at the time. Every interval had a meaning within a major or minor scale, and there were some combinations of intervals (like 3 M3's in a row) that couldn't be accomplished in any single scale. So shoving them all together like that, and forcing enharmonic equivalence on you, came very close to being a moment of atonality within tonal music!

This, again, is why the Neo-Riemannian approach of ignoring tonality and diatonic scales is useful: because there are pieces that do just that, in order to combine triads in weird ways (like the P-L sequence) that require enharmonic equivalence to make sense.

VI. The Tonnetz

In order to visualize the universe of possibilities that we've opened up with all this theorizing, Neo-Riemannian theory likes to create visual maps of the chord layouts that are possible. This kind of map is called a Tonnetz (German for "tone network"). Here's an example of a Tonnetz. Each letter represents a note (not a chord). Horizontal lines connect notes by perfect 5ths; diagonals that go up-right (or down-left) connect minor thirds; diagonals that go up-left (or down-right) connect major thirds.

The triangles that are formed in this picture represent triads: triangles pointing up are minor triads and triangles pointing down are major triads. So you can see the triangle framed by C, Eb, and G bolded in the picture, which of course is a C minor triad. Below it is the C,E,G of C major.

The nice thing about a Tonnetz like this is that it can also show our transformations. Consider the C major triad (just below the bolded triangle). Now look for the triangles that share a side with C major: they turn out to be exactly the 3 triangles that I can transform C major into via P, L, or R. So we can imagine those transformations as ways of flipping one triangle onto another inside the Tonnetz; we can make analyses of pieces by tracing out their chord progressions as if on a map.

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That's pretty much all I've got stamina for, tonight. I've left a bunch out, so I'd be happy to get corrections/additions (or questions!), but I hope this has been a plausible overview of the basics of Neo-Riemannian theory.

If this stuff piques your interest, here are two books that are very much worth taking a look at:

Audacious Euphony by Richard Cohn, who is one of the founders of the theory, and who explores its possibilities through many nice analyses in this book.

A Geometry of Music by Dmitri Tymoczko, who is critical of standard Neo-Riemannian theory in many ways. His book (which builds two articles he helped write for Science in, I think, 2006 and 2008) offers another perspective on some of the same issues, drawing on geometry rather than algebra for his underlying mathematics.

Sure thing! This is going to be a bit of a doozy length wise because there's some background I should give first. You'll find some pictures in this link that will help you visualize some of the stuff I'm talking about here.

So let's start with the three basic transformations of Neo-Riemannian theory. We can use these transformations to turn some triads into others. A parallel transformation (P) will preserve a chord's root and fifth while swapping the quality of the third. So P applied to a C major triad will create a C minor triad and applied to an E minor triad will create an E major triad.

Then there's the leading tone transformation (L). When L is applied to a major chord, it moves the root down by half step to the leading tone (e.g. C major becomes E minor) and when applied to a minor chord, it moves the chordal fifth up a half step as if that chordal fifth was a leading tone (e.g. Bb minor becomes Gb major).

Finally, there's the relative transformation (R) which moves the chordal fifth of a major chord up by whole step (e.g. C major becomes A minor) or the root of a minor chord down by whole step (e.g. E minor becomes G major). This transformation relates relative major and minor keys.

Now notice, that the transformations all have something in common; no matter what triad we apply them to, we get a triad of opposing modality. If we use them on a major chord, we know a minor chord is the result and vice versa. Also notice that each transformation requires very minimal voice leading; the "biggest" transformation moves only one triad member by a whole step.

We can actually map all twelve chromatic pitches so that any equilateral triangle formed by immediately adjacent pitches is a triad. When we do this, we can arrange our map such that any two triangles that share an edge are related by one of our three transformations. Look at the first image here to see what I'm talking about. Technically, however, this pitch space is better thought of as a torus (Image 2) but I'm not trying to go too left field here.

Alternatively, we can just map all major and minor triads such that any two adjacent triads are related by one of our three transformations. Doing so gives us this hexagonal, chicken-wire fence shape that charts paths between chords via our transformations (Image 3).

Either/both of these representations help us visualize musical geometry, tonal relations, and voice leading in a very clear way. Before going on though, I should say that other maps are possible. For example, Allen Forte, in his Structure of Atonal Music, creates a neat map for trichordal set space. And tangentially, Klumpenhower and his networks operate like spiritual siblings to the same idea. But let's just worry about triadic and tonal spaces for now.

We can play around with these transformations and spaces a bit to see if we can't create a cycle. Cycles are any pattern of repeated transformations that (eventually) start and end with the same chord. Let's see what I mean. For this, you'll probably want to follow along on either the Neo-Riemannian pitch space or triad space maps.

Start with an E minor chord. Apply the P operation and get an E major chord. Apply the R transformation to get a C# minor chord. Apply P again, C# major. Apply R, Bb minor. Apply P, Bb major. Apply R, G minor. Apply P, G major. Apply, R, E minor.

So by just chaining P and R transformations back to back, we've managed to wind up back where we started. Hey, wait a minute! All of the pitches of these eight chords fit neatly into an octatonic scale!!! Because of this, we can call this P-R cycle an octatonic cycle because this chain of transformations produces an octatonic collection. You can see this more clearly in Images 4-6.

We can do the same thing to create the hexatonic collection, just by using a different set of operations. If we instead chain P and L operations together applied to any triad, we'll end up with a hexatonic cycle because again, we'll end on the same chord where we began. I'll leave it to you to map out all the changes for yourself but check out Images 7-9 to see what I mean.

I'm naturally skipping over a lot of juicy stuff in this discussion but I hope it at least sheds light on the basics of what I mean when I say crazy sh%t like "hexatonic cycles." There's this really nifty youtube video here that does a nice job of introducing plenty of the same concepts; please watch it! One of our Eternally Luminous Theory Monarchs has collected some resources for Neo-Riemannian theory that you can check out here and here.

There's also tons of lovely books and articles on the topic. Here are links to three; I would start with the Mason because it's designed to be a beginner's textbook in the field.

Cohn's Audacious Euphony

Tymoczko's A Geometry of Music

Mason's Essential Neo-Riemannian Theory for Today's Musician

Finally, there are some sweet videos on youtube that model chord progressions from real music on the tonnetz as the music plays. It shows just how audible this stuff is and it's also just cool to look at and listen to.

Adams: https://youtu.be/edyM_iH0jJc

Satie: https://youtu.be/nidHgLA2UB0

Chopin: https://youtu.be/c-HDDiWWWTU

I hope this helps and take care!