Friday, October 13, 2006

Musical Implications of the Harmonic Overtone Series: Chapter VI


The Secondary Subdominant System


Though traditional composers extended the range and types of secondary dominants first, it is worth noting that the harmonic overtone series desires to resolve continually in the subdominant direction. Sure, if you start on the diatonic system's overtone chord - the V(m7)/I - you get the primary dominant resolution to the tonic, but if you continue the series-implied falling fifth resolutions past the tonic degree, you enter the subdominant realm immediately at IV, and the next resolution to bVII is to what is considered a second-tier and fairly remote secondary subdominant chord.

Primary secondary subdominant chords are all major triads, or major seventh chords, and if any upper structures are involved, they recieve a major ninth, an augmented eleventh, and a major thirteenth. The primary secondary subdominant chords are therefore Lydian sonorities (They harmonically generate the Lydian mode) versus the secondary dominant chords being Mixolydian or altered Mixolydian sonorities.

The traditional method by which the secondary subdominants have been rationalized, both primary (Lydian major) and secondary (Non-Lydian major, minor, and diminished); and both first-tier (On the second, fourth, and sixth degrees) and second-tier (On the third and seventh degrees), is via the concept of modal interchange. Later, however, I will demonstrate that the overtone series itself generates the Lydian versions of these chords all on its own.

Modal interchange is simply the process of borrowing a chord from a differing parallel mode, and basically any type of chord is available on any diatonic or chromatic degree through this process. The series predicts this through the integrated modality resolutional paradigm.

The most interesting of the sonorities available through modal interchange - to me anyway - are the primary first-tier and second-tier secondary subdominant Lydian chords which, as I said, are actually implied in the series itself. Many rock and jazz composers have used these, and Pete Townshend of The Who, in particular, managed to intuit virtually all of them over the course of his musical evolution between the Tommy and Quadrophenia albums. One of the reasons rock guitarists like these chords so much, and are intuitively drawn to them, is because major triads sound good with overdriven amplifiers, whereas minor triads and any kind of seventh chords simply turn to mush because of... over-saturation of the harmonic overtone series through distortional replication.

The most famous of these sonorities in traditional music is the so-called Neapolitan Sixth chord, which is a first inversion secondary subdominant chord built on the flatted second degree of the mode of the moment. The terminology is ridiculous on more than one level, of course, because it isn't from Naples, it doesn't always appear in second inversion, and this idiotic nomenclature does nothing to properly describe what it is or how it functions. Traditionally, the so-called Neapolitian sonority is most often introduced by the root position primary subdominant and it resolves to the primary dominant triad, which often continues on to a (4/2) arrangement. However, many variations on this scheme have been employed by composers which compounds the ridiculousness of the traditional description. If we want to understand what these sonorities are, where they come from, and how they function, ditching the arcane terminology is positively required.


Just as there are three functions in harmony - Tonic, Dominant, and Subdominant - there are also three degrees which can take on each of those functions, depending on contextual harmonic factors. The primary tonic is the first degree of the mode of the moment, of course, but in the proper harmonic contexts chords on the sixth and third degrees can function as tonic substitutes. Then, the primary dominant occupies the fifth degree, but with the right contextual implications, chords on the third and seventh degrees can function as dominants too (On the third degree in minor resides the derivation for the augmented triad, remember: In the major mode the minor on the third degree doesn't work so well). Finally, the primary subdominant sonority on the fourth degree can have the chords on the second and sixth degrees substitute for it: This is the series-implied origin of the first-tier secondary subdominants.

In Examples 1, 2, and 3 on the top system I have shown the modal interchange derivation for the first-tier secondary subdominants. The idea is to get Lydian sonorities on the second, fourth, and sixth degrees. The fourth degree is diatonic to the major mode, the lowered sixth degree is borrowed from the parallel minor, and the Neapolitan-derived Lydian sonority comes from the parallel Phrygian mode (Where it is used directly and in root position all the time in Flamenco music).

Derivations for the second-tier secondary subdominants are demonstrated on the second system: The Lydian sonority on the lowered seventh degree comes from the parallel Mixolydian, and the Lydian sonority on the lowered third degree comes from the parallel Dorian.

Together these two tiers of secondary subdominants, when combined with the primary dominant and the tonic, create a harmonized scale of all major triads: I bII bIII IV V bVI and bVII. I use this poly-modal system all the time in harmonic writing - especially in the preludes I compose - and as I said, I started picking this up intuitively from Pete Townshend's music as far back as the 70's.

Though in traditional music the secondary subdominants have been used primarily as dominant preparations (And so their use in traditional music has been rather limited and... well... boring), what I and guitarists such as Pete Townshend like to do (And some of the contemporary jazz guys as well) is to use these chords as a complete system - just like the secondary dominant system - to create harmonic continuities which are primarily subdominant in nature, just as the secondary dominant system allows for primarily dominant-function continuities. Later, I will show how the secondary dominant system and the secondary subdominant system are joined at the hip, so to speak, and are both extrapolated by implication from the series.

In Example 6 I have demonstrated a continuity of all major triads which has only tonic and subdominant function. If the parallel nature of the bII to I resolution at the repeat is bothersome to any of you, the bII may be morphed into a minor chord on the fourth degree (A secondary first-tier secondary subdominant, if you will) by lowering the d-flat to c-natural in the second half of the fourth measure: Jazz and rock guys couldn't care less, of course.

Though I have demonstrated these over real roots in the bass part, in vernacular usage these types of continuities most often appear over pedal points or ostinatos.

The subdominant modulating triadic continuity of Example 7 is a progression that I used to write a jazzy Concerto Grosso over back in the mid-90's (Which reminds me I ought to get that out and freshen it up). I made it entirely out of this progression, so it is a sort of Passacaglia or Chaconne, and I used a myriad of different ostinatos which give it a very driving rhythmic character.

Secondary subdominants are one of the more under-utilized resources in traditional music, but they are certainly inherent in the implications of the series. It is just that, in their subserviant role to the dominant function sonorities, they do not draw as much attention to themselves.


With a single addition to the secondary subdominant system - one that I don't recall having ever encountered in either a musical or a theoretical work before - it is possible to combine the secondary dominant system with the secondary subdominant system, which results in an integrated chromatic tonality proof that is almost exactly like the original integrated modality proof of Chapter IV.

In this version all of the target triads are not minor, so it is neither a harmonic canon, nor is it an integrated modality proof (Since it does not have all twenty-four possible major and minor tonics), but it is an integrated tonality proof because it has dominant or subdominant function harmonies residing on all twelve degrees: Five to the dominant - or left - of the tonic, and six to the subdominant - or right - of the tonic. The tonic itself has a dual function: It is both the tonic and the dominant of the subdominant degree, so there are six of each - dominants and subdominants.

From a theoretical standpoint, the problem with adding a flatted fifth degree to any proposed tonal system is that it would, in point of fact, destroy that system if it were to be considered as a replacement for the natural dominant degree. But, as with so many hasty and superficial analyses, that is not really the case here, as the flatted fifth degree does not replace the natural fifth, and it is actually an adjunct which allows the loop to close in more than one way. It is not only an enharmonic augmented fourth which can loop back to the natural seventh degree, in other words, but it can also be considered as the Neapolitan of the tonic, or more properly, the Neapolitian of the V(m7)/IV: The Neapolitan of the primary subdominant key, to put an even finer point on it.

This final piece in the puzzle of the integrated tonality which the harmonic overtone series predicts completes the integration of the secondary dominant system with the secondary subdominant system for the first time ever. So, regardless of the original rationale for the descent of secondary subdominant Lydian sonorities via modal interchange, they are actually implied perfectly well by the harmonic overtone series itself progressing in a manner consistent with a major-locus twelve-tone modal system: The dominants are to the regressive side of the tonic so that a series of progressive resolutions from the most remote one - the V(m7)/iii - leads by the secondary dominant system to the tonic, and the subdominants are to the progressive side of the tonic so that a series of progressive motions from the closest one - the IV(M7(A11))/I to the most remote one - the IV(M7(A11))/bII - creates a gigantic dominant preparation for the most remote of the secondary dominants.

This is actually what the series implies for an integrated (Chromatic) major twelve-tone modality: A series of secondary subdominants leading away from the tonic which loops into the most remote of the secondary dominants and then returns to the tonic. That is why I saved this demonstration for the last chapter in the harmony-related series: It is the crowning proof of the implications of the harmonic overtone series. The real implications of the series are that it is a tonic falling into the subdominant system which acts as a grand preparation for the secondary dominant system which returns to the tonic.


Ah, the problems associated with blustery fall days.


Blogger Hucbald said...


I wrote: "The dominants are to the regressive side of the tonic so that a series of progressive resolutions from the most remote one - the V(m7)/iii - leads by the secondary dominant system to the tonic, and the subdominants are to the progressive side of the tonic so that a series of progressive motions from the closest one - the IV(M7(A11))/I to the most remote one - the IV(M7(A11))/bII - creates a gigantic dominant preparation for the most remote of the secondary dominants."

It should read, "- creates a gigantic subdominant preparation for the most remote of the secondary dominants."

11:14 PM  
Blogger Unknown said...

Is there somewhere a circle of substitutions? I mean if you're in C you could substitute

C a e C7 Cma7 (and maybe some 2nd subs and then ii

d D D7 F 2nd dom?

e E E7 G? 2nd dom? See what I'm trying to do? I'm trying to expand Steedman& Piston by including substitutions: Subdom,rel minor, parallel minor and extensions and others I don't know about.

4:34 PM  
Blogger Hucbald said...

Hi Ray,

Between the secondary dominant system, the secondary subdominant system, and the integrated modality system, ANY gender of chord is available on ANY natural or chromatically altered degree of whatever major key, minor key, OR MODE you happen to be in. That's the beauty of it: Everything is available at any time in any place. For major and minor triads, the modeal inflections you use determine the effect you get.

For example, if you use a secondary dominant to target a minor seventh chord, you can make the minor seventh Phrygian by using a flatted second degree of it's scale, Dorian by using a major sixth, or you can mix them up chromatically.

Hiope this helps.

6:21 PM  

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