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In many musical pieces, complex harmonic relations can be identified in the chord progression even when the melody can intuitively be "felt" to progress in a very simple step-wise manner.

For instance, in this music analysis video, a Norah Jones song is presented in which one of the guitar voices (the bass?) descends chromatically step-wise, and yet the Roman-numeral analysis demonstrates how complex (recursive) secondary-dominance relations exist between those transitions.

In other words, the chord progression (V7/IV resolving to IV, then V+/vi resolving to vi7 etc) seems very nontrivial, and that it could have well been different in many ways, while still based on secondary dominants - but the melody progression is not likewise complex, instead it's the simplest most parsimonious option possible: go down a semitone with each chord-functional change. How come?

I'm sure this apparent paradox merely betrays a rudimentary knowledge of music theory, but still, how is it best explicated?

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I think the basic theory idea you are looking for is that each tone of a diatonic scale (exluding TI and MI) can be chromatically altered, raised a half step, to become a temporary leading tone to the diatonic tone above. And importantly, only that one altered tone is needed to create the secondary dominants. These alterations can be inserted into ordinary step-wise voice leading patterns. The result will be lots of simple step wise motion, but a fair amount of chromatic complexity. It's sort of "process driven" - meaning you just follow mechanical procedures - so while it sounds complex, perhaps "chromatically rich" is a better description, it isn't really too difficult to understand.

Take a simple diatonic line...

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...insert chromatic half step as leading tones to the step above and add some harmony above enough to clarify the secondary dominant role of the chromatic leading tones...

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Your example song used a descending line so let's just make the line descend...

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...notice that the leading tones are now in the treble part and the descending line of the bass repeats tones, that's just how it works out when we procedurally insert secondary dominants into an ordinary descending diatonic line.

You can insert secondary dominants in this way for the whole scale. However, when the diatonic line moves through the areas of MI FA and TI DO those are already half steps diatonically so it complicates our procedure of inserting chromatic half steps.

Let's take a descending bass and stack up diatonic seventh chords...

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...take note of where the diatonic half steps in the bass occur, then insert the secondary dominants...

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...notice that things are a bit odd at the points of the diatonic half steps. The chord rooted on B is a diminished chord so we can't really make a secondary dominant for it, and the secondary dominant moving the to E chord - noted with ! breaks the smoothly descending bass, because of the rise to F# necessary for a dominant seventh chord.

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...if we wanted to avoid that F# we could do a number of things, but one that I like is a tritone substitution so we can use the F natural in the bass...

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OK, we've worked out the procedure in a fairly mechanical way, and it should be clear that all we have is a series of diatonic seventh chords preceded by secondary dominants of some kind. In the results we can note that:

  • all the movements from tone to tone are simple steps, whole or half
  • almost all tones are diatonic, there are a total of 56 tones to play all the chords but only 5 are chromatic (I highlighted the in blue)

So, on the one hard the procedure gives us chromatically rich music, on the other hand it's still mostly diatonic steps.

My example isn't the progression of the Norah Jones song, but keep in mind I'm only trying to outline the basic theory idea: you can insert secondary dominants into most diatonic progressions. The Norah Jones progression is basically portions of the circle of fifths with secondary dominants inserted. Same concept applied to two different diatonic progressions.

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  • Excellent answer, many thanks Michael
    – z8080
    Apr 10 at 11:02
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I think it really depends on how you define "simple" in terms of this inner-voice chromatic line.

It sounds as if you're equating simplicity with smoothness—that is, how much a line moves. Since the line consistently moves the smallest interval in common Western music, the half step, the line therefore must be simple.

But I would argue the opposite: these consecutive half steps are actually rather difficult, because they're so actively going against the tonal background of the diatonic scale. And making the underlying chord progressions sound convincing in a manner that fits with that descending chromatic scale in the melody is actually very complex.

But there's another thought process to consider: the answer above assumes something of a top-down compositional process. A composer may write this chromatic line up top, then find chords that fit down on the bottom. But relatively few composers actually work this way these days. (But interestingly, it seems to have been how composers wrote in the pre-tonal era.) Especially in popular music, many composers will create the chords first and then find some melodic lines that fit with it. And when composed in this bottom-up fashion, it's quite a bit easier to find a smooth line that fits into all of these various chords.

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  • Very good points & excellent answer Richard, thank you. It's quite useful to bring top-down vs bottom-up processes in composition in explaining this paradox (I knew it was just an apparent one, haha)
    – z8080
    Apr 8 at 14:55

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