Slash notation

Jason Yust · December 2019

Last week I told my grad. students that it violates the RN notational system to write anything other than V or viio over a slash, and that even though it is tempting to write something like "ii/IV" in certain situations, you need to use a workaround like bracket notation or a key change. Two of them said that they were taught to write things like "ii/IV" by their undergrad. teachers, which surprised me. I'm curious: Does anyone actually teach this, or have you encountered it?

Nathan Baker · December 2019

I would think this type of labeling would be fairly common among jazz theorists, given the centrality of ii-V-I progressions to that analytical lens. I also think a vi/x makes perfect sense when a secondary dominant resolves deceptively.

Eric Isaacson · December 2019

I've always thought that the first chord of the Waldstein Sonata should be heard of—and thus labeled—as IV/V.

Nathan Baker · December 2019

Also, if I recall correctly, the commonly used Musician's Guide to Theory and Analysis textbook includes material on secondary predominants. I'll double check that when I'm in my office tomorrow and edit this post if I'm incorrect in my recollection.

Richard Cohn · December 2019

I teach it, although not systematically; that is, as an option that students should be aware of, and might seek to use creatively. IV/IV (especially in minor, iv/iv) and iv/ii are particularly useful. For the former, consider the Ab-minor chord at the end of bar 2 of the "Denn alles Fleisch" movement of the Brahms Deutsches Requiem, For the latter, think of Lewin's analysis of the G-minor 6/3 chord in Morgengruss. And isn't IV/VI a better analysis than bII for the D-flat major chord in bar 2 of the Chopin C-minor Prelude? Riemann functionalism is way ahead of American Roman-numeralism in this respect.

The slash notation is a notationally weird stand-in for functional notation in the mathematical sense. To write V/vi and say "five of six" is equivalent to the mathematical expression V(vi). There is no reason that any scale-degree function could not substitute for V. If you have a function that adds 2 to every number in a series, why would you disallow a function that adds 3 to every number in a series, etc.? BTW I am not suggesting that Riemann took the term "function" from mathematics, an article by Trevor Pearce about ten years ago suggested some reasons not to jump to this otherwise attractive inference.

There is also no reason not to have recursive functions such as bII(VI(iv(ii)). (In C major, that's F-flat major, in case you're curious). This can be particularly useful if you want to reserve modulations for tonics that get cadenced; in some development sections, we go a long time without a cadence. Some years ago I worked out a convincing four-level-deep applied-chord analysis of the first half of the development of the Schubert D. 960 B-flat major piano sonata (1st mvt).

Richard Cohn · December 2019

Addendum, I believe that Piston advocated for a more general use of secondary functions .

Mike Rogers · December 2019

I have only ever taught V and viio as secondary functions with slashes, and teach tonal levels for very brief (3-6chords) modulations

Frank Lehman · December 2019

I'm amenable to this style of harmonic analysis and suppose I have taught it to undergrads now and then, though only briefly and informally. It's easily abused, but I do think it has a quite a few practical and conceptual advantages over bracket notation for tonicizations.

1) It expands the universe of "applicable" functions beyond dominant (V, vii^o) in a way that I think makes sense for a lot of idioms, particularly popular & non-Common-Practice ones. If you think dominant-function is uniquely privileged and that's why it & it alone can garner the x/y notation, that's fine; but that's a major theoretical assumption that needs to be articulated.

2) It alerts students to harmonic routines and patterns that they might otherwise miss. Take a simple blues progression in C major. like:

C-F-C
F-B♭-F

Standard Roman Numeric analysis would treat this as a I-IV-I-IV-♭VII-IV. This is all well and good. But the slash-version, I-IV-I-IV-(IV/IV)-IV shows something deeper. That is, it recognizes that B♭ major as having an analogous reltationship to F major as that chord had w/r/t C major: a subdominant of the subdominant. A pattern-sensistive analyst, or someone who is deeply familiar with blues tonal procedues, may already intuit this, but the slash notation makes it explicit. (As would bracket-tonicization notation, albeit with a different emphasis.)

3) It can make for some fun, if nonsensical puzzles, for students to figure out. One I like to assign is vii^o/vi/V/IV/iii/ii/C.

In general, it's easy to get carried away with over-elaborate, vanishingly musical labels like, say, ii/V/♭III/♭II. And I prefer bracket notation in most cases. Nevertheless, I see value in even the most extravagant slash-descriptions. It makes the "functionality" part of harmonic function, which is really easy to under-theorize in an undergraduate curriculum context, more vivid.

And there's something latently transformational to it too. It's a small step from describing D♭ in Cmin as ♭VI/iv to calling it an L of Fm. Or, from calling A♭♭m in the key of G♭ major a iv/♭VI to calling it an N transformation of the flattened submediant. (And, yes, of course, Roman Numerals have completely different functional commitments than the tonic-neutral triadic transformation labels; but the analytical thought-process, based on relationality rather than fixed identiy, is not all that different.)

Scott Murphy · December 2019

IV/IV (instead of, or in addition to (Biamonte 2010) its interpretation as bVII) comes up in the context of "double-plagal" motion, as in Open Music Theory's discussion of plagal progressions.

John Paul Ito · December 2019

This strikes me as a great place to apply the prototype category, with its ability to talk about some things as overwhelmingly more prevalent, entrenched, referential, etc., while leaving the door open to all kinds of unusual possibilities.

So from my perspective, I'd say yes, V/x and vii/x are overwhelmingly the most prevalent options.

Within common practice music, VI/vi comes next, though it also very much retains a fundamental identity as IV (as secondary chords retain fundamental diatonic identity in the great majority of cases).

I would be very interested to hear about it if any others are consistently enough patterned to rise above the level of individual occurences.

For popular musics, IV/IV as a way of hearing bVII comes to mind for me first, as it does for some others, though I don't know the repertoire well enough to claim that there aren't others that occur even more commonly.

But that's the basic direction I take. If the human lifespan were twice what it is, I'd be excited about undertaking a study of the development of 19thC chromaticism from this perspective, as practice that gradually expands on itself, with things that begin as minor extensions gradually becoming well enough entrenched to serve as the basis for new extensions. The development of common-tone diminished chords from the late 18thC to the late 19thC is for me a model of the utility of this way of looking at things.

Aaron Grant · December 2019

IV/IV (instead of, or in addition to (Biamonte 2010) its interpretation as bVII) comes up in the context of "double-plagal" motion, as in Open Music Theory's discussion of plagal progressions

Another example can be seen in Scott Burnham's 2017 19th-Century Music article. Burnham invites us to hear the DM harmony in m. 3 of Lizst's "Über allen Gipfeln ist Ruh" as both a subtonic harmony in the key of EM and IV/IV—"a doubly inward subdominant of the subdominant." In fact, he goes a step further in a footnote, calling the GM sonority at the end of the song (m. 42) IV/IV/IV!

Jason Yust · December 2019

Wow: lots of interesting comments and I'm interested to see they are all some form of support for the idea, and no one yet to come out and say "I'm horrified by this" or something to that effect. (And it is especially interesting that most comments refer to popular music idioms.) This is making me really rethink how I teach RN analysis (even though I'm usually coming in after they presumably have already learned it, with a wide range of possible approaches). John Ito and Rick Cohn's comments for instance really make me consider, for the purpose of understanding 19th century chromaticism better, how it might be beneficial to go all in on this sort of thing rather than fight it.

Here's where I come from: As a general principle of doing harmonic analysis, that I am constantly trying to impart, putting a label down that accurately describes what notes are there is only a precondition for analysis. The point is that the label somehow explain the harmony. This is what I spend the most time helping students with (e.g., V7/V is not "II7").

The RN system is one based mainly on recognizing conventionalized harmonic routines. While the system is based on some generalized principles (scale degrees, inversions, and, with the slash notation, embedded referencing), the usual undergrad textbook teaches harmony as a series of chord types, discussing each one in turn, such that the few principles underlying the notational system are of little importance. (You could just as well call I, V, and IV, "Betty," "Derek," and "Lily," and it wouldn't change the whole procedure in substance.) In fact, most time and effort seems to be put into fighting the implications of the systematic elements of the notational system (c.f. 6/4 chords), which goes to show how inherently appealing to many students concepts and systems are, put against rules and conventions. This is where I come to the "ii/IV" thing: I wonder if the students have developed a concept of secondary supertonic function which they are expressing, or if they simply see that this label is systematically available. I suspect that they have been taught a theory of types that does not include secondary supertonics, and that they are responding to the implicit transformational system of the slash label. The idea of teaching the system as such, the Riemannian system as a conceptual approach to harmony, is very appealing -- if that is happening in their undergrad programs I should be reinforcing and building upon it. I am mindful, though, that most undergrad harmony teaching is not done by professional theorists who regularly deal with the concepts and systems underlying analysis.

Nathan Baker · December 2019

The RN system is one based mainly on recognizing conventionalized harmonic routines. While the system is based on some generalized principles (scale degrees, inversions, and, with the slash notation, embedded referencing), the usual undergrad textbook teaches harmony as a series of chord types, discussing each one in turn, such that the few principles underlying the notational system are of little importance. (You could just as well call I, V, and IV, "Betty," "Derek," and "Lily," and it wouldn't change the whole procedure in substance.) In fact, most time and effort seems to be put into fighting the implications of the systematic elements of the notational system (c.f. 6/4 chords), which goes to show how inherently appealing to many students concepts and systems are, put against rules and conventions.

You just hit on a couple of the key issues that I have with Roman numerals. I wasn't going to touch on that matter while just answering your question about secondary chords, but there are many solid reasons I've been moving away from Roman numerals (I do introduce them in the second year as a "second language" so my students can understand them at their transfer institution) in favor of an approach using figured bass and harmonic function. I'm happy to chat more about this at length if you're interested (either start a new thread or send me an email, nbaker@caspercollege.edu).

Dimitar Ninov · December 2019

Dear Jason, besides secondary dominants (V or vii of...), music also uses secondary subdominants (IV or ii of...). Those usually work in combination with the dominants in such formulas as ii and V of x, or iv and V of x., modal mixture fully possible in both major and minor key. This happens a lot in tonal music, from Bach to now, but a huge amount of (classically trained) theorists are not aware of it, and this is why they write books that lack information of secondary S chords (they cannot teach what they do not know, can they?). I also attribute this lack of knowledge to the full and unquestionable subscription to Schenkerian theories which erase the S function as one of the three pillars of the key, leaving only the T and D chords to represent tonality. Thus, these theorists will not be able to identify extended tonicization via secondary S and D chords, not to speak of a secondary S acting alone (which happens more rarely, but does). I have seen ridiculous analyses in some textbooks, due to their authors' unawareness of this phenomenon.

However, you are not to blame! I encourage you to break the mold and start discovering extended tonicizations of local keys through secondary S and D chords...you will be surprised.

Occasionally, secondary S chords may act alone, tonicizing a local tonic without the help of the dominant. One common example is the typical cadence in popular music I-bVII-IV-I, where the second chord in the progression functions as S of S. This is extremely common in rock music; many songs are only constructed on this formula (Sweet Home Alabama among them). The reverse order also happens: I-IV-bVII-I, but now bVII functions as a non-leading tone-dominant resolving into the tonic.

Thanks,

Dimitar

Texas State

Dimitar Ninov · December 2019

I forgot to emphasize that hundreds, if not thousands songs in jazz and popular music are based on the (ii7 and V7 of...) formula, it is extremely common. Take a look at such jazz standards as Bluesette, Satin Doll and There Will Never Be Another You, to immediately realize that. The more you discover extended tonicizations in pop music, the more you will come across them in classic-romantic music. Besides, this ii7-V7 formula is also available disguised under the so-called tritone substitute umbrella. For example, Dm7 - G7 - C (main or local tonic), may be disguised as Abm7 - Db7 (!) - C progression (see Satin Doll). Here the interesting element is that this latter progression is also a true ii7-V7 of Gb major, but it shifts a half step down into its tritone counterpart, C major.

Brett Clement · December 2019

My recent JMT article on tonicization in rock music advocates for an expansion of potential secondary chords, including less familiar ones such as IV#/vi. The issue of slash vs. bracket depends for me on whether the tonicization is extended (bracket) or not.

Nicolas Meeùs · December 2019

The initial question and some of the answers given leave me somewhat perplexed. I have several additional questions.

A first question concerns the usage of lower case Roman numerals. When did this usage develop in North America? We know that Gottfried Weber proposed it, but his proposal met with limited success, in continental Europe at least. Neither Schoenberg nor Schenker used lower case numerals, while e.g. G. W. Chadwick did in the New England Conservatory in 1896.

A second question is what really is meant by V/V. Schenker would have written II^#, and probably Schoenberg as well. Is V/V merely another way to denote the major third in the chord of II? Upper case II would be sufficient to denote the major third if the minor chord would have been labeled ii, but this of course might lead to confusion.

Let's assume that V/V really means the dominant of the dominant. But in what sense are each of these "dominants" to be understood? I can see in what sense the chord so labeled is a dominant – especially if it resolves on V. But in what sense is V/V the dominant of the dominant? Is it the dominant in the key of the dominant or, in other terms, does V/V imply a local modulation? And would something similar be true also in the case of ii/IV? Or does ii/IV merely mean V^b? If the numeral over the slash denotes a function, what is the function of ii?

I see no reason why students should be forbidden to use Roman numerals other than V above a slash, provided that they know what they mean. In other words, I have no problem with ii/IV in itself, but I utterly fail to see what it could possibly mean. It might be justified in some specific situation, say in a progression ii/IV–V/IV, but might it not be better then to write something like ^ii–V/_IV, or to give a double numbering, one in the main key and another one in the key of IV? Is the student aware that ^ii–V/_IV actually means v–I in the main key?

Jason Yust · December 2019

In response to Nicolas: It is very interesting to see the variety of perspectives. Ultimately, you come to exactly the same point that I think is the bedrock for me: the students should understand the meaning of what they are writing. The other hard part is separating the practical pedagogical issues from the purely theoretical ones.

You ask a number of questions that are good questions and also things I think most North American theorists would take for granted (possibly missing, one might claim, the points upon which the theory we teach might be debatable). Someone else can speak better than I to the historical reason why we use the upper-/lower-case RN system, but suffice it to say it is universal enough here that I would deem it neglegent to not teach that system. (In other words, this is one of those practical issues, where if I felt it was important enough to teach some alternate system like all-caps Stufentheorie or the Riemannian system, I would have to supplement it with translations into the one they are likely to encounter at any other North American institution, which would be a lot of time taken away from other potentially interesting topics.) The other thing that I think many of us take for granted is this graded system of "in a key" versus "extended tonicization" versus "applied chord." All cases involve the invoking of some other key, the distinctions having to do with the structural status of that key (roughly, how much stuff needs that key as a reference point and are there structural markers such as cadences or thematic beginnings in that key). I suspect we all draw those distinctions in slightly different ways, and the more different pieces I teach to different classes, the less I believe that any of the lines between these different categories are in any way crisp.

The solution you point out of saying something like "(ii – V)/vi" instead of "ii/vi – V/vi" is essentially what I mean when I refer to the "bracketing" solution. It was my impression (when I started this thread) that most people require students to deal with these situations with some such strategy, in order to avoid the latter type of notation.

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