Monday, January 01, 2007

New Years Resolutions

Happy New Year to everybody with best wishes for a healthy, happy, prosperous and hopefully sane year in 2007. It's a time for making New Year's Resolutions, something I don't often do, but this year was different and I thought I'd share some of them with you.

If you've ever studied music, you know that a scale - let's use the white-rat garden-variety key of C Major - is a collection of available pitches out of which you can construct a melody (the linear element of music) or chords (the vertical element of music.) Putting chords together creates harmony, which is mostly about how chords move from one to another. A cadence is a series of chords that act as punctuation for a musical phrase.

As an example, here's a C Major scale and the different chords you can create out of that scale:

Now, if I take a different collection of pitches – a group of six notes – I can create a “set of available pitches” that can act basically the same way. Theorist Allen Forte labels this particular set or “hexachord” 6-30. If you take the other six notes of the 12 available pitches, its mirror, you come up with its complement, a hexachord that is, in this case, the inversion of 6-30 which I write as 6-30'.
Like a C Major scale, I could use these pitches in any order I want to create a melody and I can create different kinds of chords from them, too. If I take the first three notes and turn it into a chord (I don’t want to call it a “triad” because that implies it’s built on thirds rather than is a chord consisting of three notes, so we tend to call it a “trichord”), then the second three notes and so on, I get this progression of chords built on major and minor seconds.
So now I have four chords created out of all 12 pitches of the chromatic scale. It’s also not terribly interesting because everything is moving in parallel directions, sort of like playing those major and minor triads from a C Major scale exactly as I wrote them in my little example above.

If you notice that the first two chords consist of a major 7th with a major 2nd above the lower note and that the second two chords consist of a major 7th with a major 2nd below the top note, you could also switch around some of the pitches to create chords that would have a minor 3rd above or below the outer notes.
The all-parallel motion is still pretty boring, basically just a “succession” of chords. But if you start changing the direction of some of these voices within the chords, you could get something like this:
An “open” chord moving to a chord in a tighter or “close” position now gives you a sense of a chord progression – there’s a feeling of increased tension and, by reversing that order for the end of these four chords, of the release of that tension. This basically creates a kind of cadence, one that may be open-ended or one that may sound more-or-less resolved.
These two are really my first two resolutions. Ex.5 may sound more open-ended, like the tension still needs to resolve, because of the compact nature of the final dissonance with its Major and Minor 2nds. To listeners used to chords built on Major and Minor 3rds – a C Major or D minor triad, say – a chord like any of these will sound “dissonant,” but dissonance only means there’s unresolved tension. In a collection of non-traditional non-tonal chords, dissonance becomes fairly relative.

But looking at the set of notes in 6-30, you can also group them to create chords built on 3rds that have a look of familiarity about them. If I spelled the F-sharp as a G-flat, that would legally be a C Diminished triad, but I wanted to get the idea of the upward half-step motion to the G-natural in the next chord. It resolves to what would function in the key of D Major as Dominant 7th Chord and because of our familiarity with that sound we sense the way it could (or maybe “ought to”) resolve.

In the second hexachord, these could resolve any number of ways, including chords that sound like they could be “augmented 6th” chords (German 6ths and French 6ths, without getting into a whole raft of discussion from Sophomore Theory class) resolving to Dominant 7th chords. The point is, while they “sound” like those chords, they don’t “act” like those chords, resolving in the same expected ways.
That may be interesting but I find it sounding too close to traditional harmony to be too useful in my own style. It could be used at some half-way point before a full-blown, more final resolution, though, so I’ll keep it in mind as a possibility.

But what is the point of all this? For centuries, composers have built their harmonic structure on a series of chords perceived as “consonant.” Yet a Dominant 7th, a diminished triad or an augmented 6th chord are all dissonant because they imply forward and, as yet, incomplete motion needing resolution. The art of creating a satisfying harmonic progression (or forward motion) is the result of blending the right amount of consonance with the right amount of dissonance.

If it’s all consonant, it will sound very bland (“it doesn’t ‘go’ anywhere”). If it’s all dissonant, it will sound too chaotic and unsettled (“it still doesn’t ‘go’ anywhere, just sits there churning”).

So in a more-or-less dissonant style, you still want to mix in aspects of consonance and dissonance. You can do that by resolving to chords that are “less dissonant” and therefor by comparison “consonant.” Or you can use old-fashioned “consonant” harmonies like C Major or D Minor triads.

In Ex.7, you can see the pitches of 6-30 will create two minor triads – C Minor and F-sharp minor. Its complement, 6-30', will create two major triads – E Major and B-flat Major. But play them together, they don’t sound like a “tonal consonant progression” because they’re a tritone apart, that augmented 4th interval that for centuries was called “the devil in music” because it sounded so unsettling. The more chromatic music became – following the progression from Bach and late-Mozart to Wagner to Schoenberg and so on into the 20th Century – the impact of a tritone has lessened until it’s just another interval.

And with the right kind of intervallic motion in the outer voices, it lessens the tritone’s impact if the bass line, for instance, isn’t jumping a tritone. Ex.7 sounds very smooth, actually. Here’s another example with the same triads but I reverse the order of the last two: in some context, the motion from B-flat to E might sound better than E to B-flat.
Now, if I want to use a “consonant” motion resolving to a “dissonant” motion, I could mix the two by using standard triads in an un-standard way resolving to non-tonal, non-traditional trichords creating a sense of tension that leaves this phrase “open-ended.”
It could be more open-ended if I reversed the last two chords to end on the chord in “close” position. Or I could switch the two hexachords, then opening with dissonant motion resolving to consonant motion (and I’m speaking simplistically, here), cadencing on the F-sharp minor triad.
Or I could use the hexachords in their original order and come up with this possibility:
I find this a very satisfying resolution, personally. I also notice that the top-line of each chord scrambles the letters in the name BACH (using the Old German Notation where B=B-flat and H=B-natural). So if I wanted to, I could "partition" these chords in such a way, I could come up with something slightly different. By taking the two middle chords for my first pair and the two outer chords for my second pair, I now have a different progression of chords, harmonizing Bach's Name!

Each pair of chords now consists of one major triad and a trichord built on seconds. These create a different hexachord, which is the inversion of the hexachord Forte labels as 6-14: the complementary hexachord (its mirror) is just another transposition - a tritone away - of the same set, so both of them are inversions.

However, I didn't care as much for the E-Major cadence, so I thought if I switched out the G-sharp for the last chord with the G-natural in the second, I don't really change the structure of the chords but I get a subtle variation that also gives me a less-final-sounding resolution to E Minor. That could come in handy, perhaps. These chords, then, form the hexachord 6-15 and its complement. And it becomes a way of “modulating” between one hexachord and another.
So, basically, with a mix of consonant and dissonant chords, it all comes back to Bach...

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