Tuesday, October 17, 2006

Musical Implications of the Harmonic Overtone Series: Appendix I


Harmonic Musical Examples


Obviously, real actual music combines all five of the musical elements - harmony, counterpoint, melody, rhythm, and form - but the major conception of a piece may be biased significantly toward a certain one of these elements. The two examples in this appendix I composed out of harmonic continuities - some of the same continuities I used in the previous chapters, in fact - and so I am going to present them as harmonic musical examples.

Since I am a guitarist, I write music for the guitar, primarily, and my criteria for guitar studies are simple: They have to be fun to play and fun to listen to. In other words, if they are not good enough to add to my performance set list, they never see the light of day.

I also like to cover several bases with any studies I write, and so addressing a compositional issue and a performance issue simultaneously makes them all the more valuable. In this case, the problem I faced was that the four and five voice continuities I was writing (Three or four voices plus a constant root bass part) were "pure music" and not executable on the guitar. Then I got the idea that I could play them as single lines - just arpeggiate them - and a set of lineal studies was born. I love to perform but hate to practice - and I refuse to waste time playing scales because scales aren't very musical - so I was needing some single-line pieces anyway: These killed two birds with a single stone, so to speak.


This is the first of the lineal studies I wrote, and it uses the second continuity example of chapter two, which demonstrated a series of four half-progressive motions, and turned around with a series of five progressive motions. In this version, there are four half-progressions followed by four progressions, which modulates the first phrase to the dominant region of V after one harmony per measure for eight measures.

The second phrase, which begins on the dominant tonal level, combines the two progression types of the first phrase in alternation: Half-progressive motion followed by progressive motion. Since there are still four of each type of root motion, and there is still one harmony per measure, after this eight bar phrase the piece modulates to the dominant of the dominant tonal region of II.

In this third phrase, the alternating root motion types are retained, but they are reversed: The progressive motion now preceeds the half-progressive motion. These two successive modulations in the dominant direction have wound up the rubberband, and so at the end of the third phrase I am able to let the airplane fly: The phrase not only turns around, but it returns all the way to the tonic with the six progressive motions in a row of the final four measures of that third phrase (The last two of which are implied over missing real roots).

The details of the voice leading are kind of nebulous, and I play around with the idea of implying a four voice upper stratum while using three voice transformational logic. This would not be possible in a static harmonic stratum, but with this compound line it is a great resource.

After the initial polyphony-implying scalar line from the tonic to the dominant degree, almost all of the melodic motion implies harmony in the upper stratum, but the bass line has some contrapuntal effects in it. As you can see at the end of each of the first four measures, I use implied 4/2 inversions to add some linearity and directionality to the bass line, while the upper stratum teeters back and forth between implying seventh chords (For the color they add) and triads (To allow for the (4/2) effects).

At the beginning of the second system, I made the diatonic vii(d5m7) into a fully diminished seventh sonority - again, just for the color of it - and then I had it's root function as a real root by moving in a progressive manner to the third degree. In this second half of the first phrase, I used a series of (4/3) effects to get some more directional motions in the bass line. I put one of the crosswise transformation symbols in parentheses there because while you have heard a seventh chord in the previous measure, it does not move down to the third of the target in the same octave: The target chord's third appears an octave higher. This works out sonically due to the higher octave being the strongest overtone, by the way, but it is really just a bit of artistic license, as I wanted the highest voice to continue rising. Remember, tetradic transformations in progressive motion move the voices down, but triadic transformations over the same progressive motions move the voices up: I wanted the rising strata effect of three voices but the color of four, and so I was able to have my cake and eat it too here. Again, this is only possible due to the multiple and nebulous inferences which can be drawn from a single line.

I put all of the implied crosswise transformations in parentheses in the second section for this same reason: I'm working toward a melodic peak at the beginning of the second half of the third section, so I need those voices to keep rising. Three of the last four progressive motions of that third section do not need to be in parenthesis, by the way: That is an error I just noticed. Those voices transform and descend properly.

If you are a guitarist I appologize for removing the fingering and position indicators, but the result was just too messy with them and the analysis present. This ought to be pretty easy to noodle out on a keyboard, though; just remember to play it an octave lower as the clef indicates.


This second of the Lineal Studies series uses the fifth root motion example which contains all of the possible diatonic root motion types excepting for the regressive tritone motion. This study pretty much sticks to a triadic upper stratum over the bass part, so there are no transformational shenanigans going on in this one. What few esvenths are implied - like the fully diminished seventh appearing on the seventh degree again - are really just momentary sonic effects which quickly vanish without upsetting the triadic transformational logic in the least.

Whereas I used two transformation types first in sequence, then in both foms of alternation in the previous example, here I'm breaking the original phrase up into antecedent and consequent phrases and am employing those seperately.

On the third system I don't modulate, I simply change the gender of the mode. This takes the piece on a fantastic journey through some interesting regions. By employing the (6/4) arrangement at the end of this minor mode antecedent which was previously at the end of the consequent, and by still progressing by progressive tritone as expected, a sort of direct/sort of correct modulation is made down a semitone to g-sharp minor. This sounds really cool, if I do say so myself.

This g-sharp minor section is another minor mode antecedent phrase, and it now finishes off with a similar arrangement to the original antecedent, and so it modulates properly by strong root motion to the first of two consequent phrases, this one on f-sharp minor. This minor mode consequent sets up the most jarring "modulation" of all via regressive tritone root motion (So, I saved this last root motion type for the best possible moment) to the key of G major. So, the modulations go down by semitone, down by tone, and then up by semitone before returning to the tonic. Reinterpreting the V of G major as IV of A major gives the return to the tonic a kind of super-smooth effect after the preceeding weirdness.

The rhythmic embellishment on the first of each measure's triplets is the antecedent of the resultant for the perfect fourth, by the way, with the sixteenth note being the irreducible unit instead of a quarter note, as I had in my examples in the chapter on rhythm. I save the consequent for the penultimate measure.

I like this piece quite a bit as well. There are more of these, but these two examples ought to suffice. We'll look at examples which are primarily contrapuntal in conception next.


Fountain pens are mostly a thing of the past as well. Pity.


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