Thursday, July 14, 2005

How to teach inversions

Marcus Maroney has a blistering reply to Kyle Gann's proposal to do away with figured bass. I completely agree with Marcus, but teaching a brief introduction to Hindemith's harmonic theory today reminded me of another way to teach inversions.

Hindemith divided all possible chords into six categories:

I - chords with no seconds, sevenths, or tritones (only major or minor triads)
II - chords with tritones, but no minor seconds or major sevenths (dominant sevenths and half-diminished sevenths, some secundal chords)
III - chords with no tritones, but does have seconds or sevenths (or both) (lots of options)
IV - chords with tritones and minor seconds or major sevenths (lots of options)
V - chords with no tritones but indetermininate root (augmented triad and quartal chord)
VI - chords with predominant tritones, indeterminate roots (diminished triad, fully diminished seventh chord)

Within each category except V and VI, there are subdivisions for inversion: Root position = 1, any inversion = 2. (Hindemith had a theory of interval roots that he used to identify roots of chords, which is only confusing with very complex chords.)

I think Kyle would like this system, as it has only two categories to remember for inversion, and no pesky figured bass symbols, simply "1" or "2". Unfortunately, it does nothing to show the dissonance of a 4/2 chord (seventh chord in third inversion) as compared to a 6/5 or 4/3 inversion. Kyle gets all worked up that the second inversion triad is significantly different from the first inversion triad, but the second inversion seventh chord is not. I disagree, as a V 4/3 chord can have an upwardly resolving seventh that the V 6/5 never could (in functional tonal music). But more importantly, these inversions are distinctly different from the third inversion, which has very specific rules of resolution, just like the second inversion triad.

As for using figured bass instead of "1", "2" and "3", figured bass allows indications of many things besides chord inversions. It indicates chromatic alterations in a chord, including the ever-important Ti in minor keys that first-years often forget. Figured bass also provides an easy way of indicating all the different types of suspensions -- including double and triple suspensions and retardations -- and can be used to describe types of sequences (see the new Laitz text for this). What other method of chord labelling provides such power and is more memorable (7th inversions are simply counting down from 7: 7, 6/5, 4/3, 2; all figured bass is about describing intervals up from the bass note)?


Marcus Maroney said...

I was hoping the tone would be more "passionate" than "blistering" :) I do find Hindemith's system fascinating and think it's good to use with more advanced chromatic harmony, to indicate density of dissonance, and then you have a nice connection to Persechetti's book....I personally think figured bass is one of music history's most ingenious inventions and am always saddened that almost no music students know how to realize it in real time. It's truly becoming a lost art.

Daybreak said...

I don't know. It strikes me that one difficulty with the American system is that it confuses figured bass and function. Well, not even that, insofar as the Roman numeral system simply masquerades as functional analysis. But the result of this confusion is that peculiar attention to verticality as opposed to voice leading we find in students. That and an understandable (given the realities of mechanical reproduction) de-emphasis of keyboard skills are arguably more responsible for an inability to realize figured bass in real time than is the separation of written theory and aural skills (though personally I also favor keeping them together).

Of course, one might make the case that some of the tradition of figured bass has been absorbed in comping and playing jazz changes...


Scott Spiegelberg said...

Daybreak (I prefer Anbruch, myself), the problem with vertical versus horizontal thinking is not because of figured bass, but because of the orientation of many theory textbooks, from Piston to Benward to Kostka/Payne. Newer books by Laitz, Clendenning/Marvin, and Roig-Francoli use a more Schenkerian (and hence horizontal) approach, improving the first attempts by Aldwell/Schachter. Schenker used figured bass extensively himself, and it didn't prevent him from hearing the horizontal view.

As for real-time realizations, I had to learn that in grad school, and I think organists still learn this skill. But my keyboard skills components for my students are geared towards hearing and feeling the theory. This involves proper voice leading, but not improvised at sight.

Hucbald said...

Hindemith's books gave me headaches. I eventually unloaded them. Sorry, but I never "got it". I don't think Paul's brain and mine have the same circuitry. Surely my shortcoming and not his.

It's probably conditioning, but when I see V(4/2)/ii, or something like that, I can hear it. Speaking of which, I'm glad for my undergrad degree in jazz comp for the same reason: If someone says "Dominant-seventh, minor-ninth, augmented-eleventh, minor-thirteenth" (Or, "seven, flat-nine, sharp-eleven, flat-thirteen" more likely), I can hear that as well. I've always thought that jazz theory had it all over trad theory in that one particular area: Naming vertical sonorities as seperate entities. Trad guys usually explain a lot of those "color chords" away with passing tone schemes. Too bad the terminology in jazz theory is so lax, and as a result, an unholy mess to sort through.


Scott Spiegelberg said...

Hucbald, jazz theory is great for verticalities, but weak when it comes to horizontal motion. The best thing they have is scale theory, like that by George Russell, but that still doesn't have the power to explain many progressions. My best friend just finished his dissertation on jazz theory, which may correct this problem.

Daybreak said...

Call me Anbruch, call be Daybreak. Call me Jimbo for that matter!

I have nothing against figured bass per se, only figured bass used in connection with roman numerals that pretend to be functions when they often aren't even marking scale steps. It's the conflation of roman numeral and "figured bass" that's the problem, I think, because that way of thinking tends to verticalize the figures rather than understanding them as marking moments in unfolding lines.

Realizing a figured bass is also primarily a performance practice. When you are playing, you are thinking temporally and so progressing horizontally through the music. But that's also why I think the era of figured bass is pretty much over, a realm now for the specialists. For better or worse, keyboard skills are just not seen as essential to training musicians as they once were, so few students will ever achieve the proficiency that would be needed to realize the figures properly.

Realizing the figures on paper is not at all the same thing, because without progression in real time, attention tends to be directed to the verticality that the figures seem to signify. This is especially true of college students, most of whom have not yet internalized the harmonic patterns and candential formulae around which the figures are based.


Daybreak said...

Something similar can be said about jazz notation: like figured bass it is a notation for a practice not a theory. If figured bass has nevertheless been turned into the basis for a theory, that's our problem. In other words, jazz "theory" insofar as it exists is a theory for playing it. It facilitates performance, not theoretical understanding in the way that we theorists like to think about it. And like figured bass, jazz notation only seems to be about the verticalities. The changes—a certain sort of progression in real time—are what count, not the series of apparently determined verticalities that they mark. Which is why you can make substitutions, play inside and outside the changes, and so forth.


Daybreak said...

I also think that the original Piston should not be lumped in with the latter- day verticalizers. Piston's (rather than Devoto's revision) was much closer to a functional theory ala Riemann; in that sense it was more a Harmonielehre than an American style all-purpose theory text.


Scott Spiegelberg said...

A/D/J, I admit to only being aware of Piston in its DeVoto incarnations, so I will remove the original from any list of verticalizers.

You're absolutely correct that jazz changes in practice are about progressions, but in theory they only reflect verticalities. That's why true jazz theory does not use standard chord changes as an analytical tool. Instead, Riemannian functions or Schenkerian graphs are used.

I'm not sure I agree about your characterization of figured bass, though. While strict theoretical interpretation does emphasize a vertical approach, historical theoretical practice (heh, there's a term for ya) has given a horizontal connotation to the use of Roman numerals and figured bass, from Campion's Regle d'Octave to Vogler, Sechter to Schenker, and Salzer's Functional Hearing. Riemann's functions have been assimilated into Roman numeral analysis, so current usage is a hybrid of Schenker's scale steps and Riemann's functions, which I think is the proper way to go.

Daybreak said...

I find Riemann useful and revealing and I find scale step useful and revealing. But I don't think current usage of roman numerals is really a hybrid between scale step and Riemann. Or if it is, we've wound up with the worst of both worlds.

All the emphasis on inversion in roman numerals, for instance, not to mention the precious distinction between upper and lower case to mark major and minor triads—neither of those are particularly helpful in a horizontal sense. What students learn from all of this is that their task consists of affixing the correct label to the verticality. Labelling chords asks them neither to trace the fundamental bass of the progression nor discern the harmonic functions guiding it. No, students simply put down the correct roman numeral with the right figures attached and then move onto the next one, without asking whether the progression between the two makes any sense whatsoever.

What I like about Riemann by comparison is that it makes you decide the function of each moment in the progression: you determine the label not by identifying the vertical sonority but by first making a decision on its basic function within the progression. All those wonderfully baroque additions to the functions are means of marking the transformations of the particular sonority into the function that that basic label (S,D,T) marks. When you go through a passage, marking it first with roman numerals and then the full Riemann nomenclature, you find that the assimilation is far from complete, and that what is lost is the strong functional orientation of the latter.

I guess what really bothers me about the American system is that it is at one with the ideology of the correct answer. While this pragmatic ideology makes for easy testing, it doesn't foster the skill of interpretation. Both scale step and Riemann by contrast ask you to hear something other than that which is simply apparent. Consequently, these systems ask you to consider whether this interpretation or that interpretation of a passage better catches its sense (in terms of the particular theory of course). Personally, I think that sort of interpetation is a useful skill.