Monday, October 09, 2006

Musical Implications of the Harmonic Overtone Series: Chapter I

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The Descent of Tonality and Modality

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Many notions have been advanced to explain the supposed reasons behind some composer's abandonment of tonality and modality in the early twentieth century: The loss of cultural innocence due to the atrocities of WW I (Further exacerbated by the even more horrific carnage of WW II), alleged seeds of destruction supposedly sown into tonality by tonality itself, &c. Frankly, I don't care one whit about any of the supposed extra-musical social factors - though I believe such notions are bunk - but the idea that tonality and modality contained nihilistic factors within themselves are simply brainless, which I shall expose. My purpose here is to demonstrate that the abandonment of tonality and modality was not only a mistake, but that such a thing is in point of fact impossible within the framework of music and what the harmonic overtone series defines as musical.

I used to find it puzzling that the original avant garde atonalist was Arnold Schoenberg (I will eschew the term "serialist" because there is nothing wrong with twelve tone rows per se, so long as they are able to be interpreted according to the implications of the series), as his early works showed such promise, and his theoretical writings - especially Structural Functions of Harmony - demonstrated a superb grasp of tonal theory. No longer. Schoenberg's problem - and the problem at the core of all proponents of atonality in whatever form (Aleatoric, stochastic, &c.) - was that he either lost sight of the implications of the overtone series, or he never really understood them to begin with.

This is beyond irony, not only because Western music theory began with the scientific investigation into the implications of the harmonic overtone series circa the ninth century AD, but also because the initial data was borrowed from the ancient Greek culture of over a millennium earlier. So, for well over two-thousand years music and music theory had been based upon the implications of the harmonic overtone series, and then this one guy decides, in essence, that the series no longer matters to music and can be ignored. That is not nearly as surprising to me as the fact that so many in the so-called academic world followed him into the abyss, when they should have been able to dismiss him out-of-hand. What does not surprise me, however, is that the many forms of atonality never gained any traction with the public. This is the simple result of general intuition understanding what is musical and what is not according to the implications of the series.

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Just as the original bearer of my nom de web did back around the ninth century, I will begin with the simple presentation of the harmonic overtone series you see in Example I (Though, of course, Hucbald did not have modern notation and was describing aspects of the series verbally). Everything that is musical is implied in this series, and everything outside of the series' implications is either extra-musical, or is not musical at all.

What Hucbald could not possibly have understood at his early point in musical evolution was that the harmonic overtone series made a sonority called the overtone chord, which you can see in Example II: He was dealing with monody, after all, and had almost no conception of the blending of different tones. We today, however, recognize the overtone sonority as being a major minor-seventh, and theorists refer to it as a dominant seventh chord (And, for good reason).

Within the overtone chord there are two types of tones: Passive tones and active tones. The root and fifth, which are in a perfect fifth relationship with each other, are passive tones: They have neither the desire to rise, nor the desire to fall - they can go either way, or remain stationary: The root and perfect fifth are the pillars of every harmonic construct. The major third and minor seventh, which are in a tritone relationship with each other (A diminished fifth in closed root position), are active tones: The major third has a leading-tone tendency and desires to rise, while the minor seventh has a leaning-tone tendency and desires to fall. Far from being the diabolus en musica, as some ancient theorists called it, the tritone between the major third and the minor seventh is the very presence of God in music, and is in fact the musical life force: The impulses contained in each and every harmonic, contrapuntal, and melodic continuity arise right here, within the impetus of this tritone.

If we allow this diminished fifth to self-actualize, it most desires to contract into a major third: The leading-tone rises by semitone, and the leaning-tone falls by semitone. In order for the root of the first chord to remain the root of the target chord in the bass part, it must fall a fifth: It is, however, important to note that it could just as well remain stationary and create a second inversion triad under the target as its counterpart in the upper stratum does. Meanwhile, the fifth of the overtone chord can either move up a whole step to the third of the target, or it can move down a whole step to double the root of the target: While this tone has no particular inherent desire for either outcome, transformational logic dictates that doubling the root of the target is preferable to doubling the third, as the root of the target is another passive tone, and doubling active tones will cause trouble (In the form of parallel octaves) if their desires are both to be met.

This most perfect resolution of the overtone chord - the resolution most in accordance with the implications of the overtone series - is presented in Example III. In this type of root motion (And, I will classify all transformations by root motion type over the course of this series: The falling fifth root motion is defined as Progressive, and gets the letter "P" in the analysis) the most logical transformation type is an interrupted crosswise transformation in the upper stratum: The root of the overtone chord becomes the fifth of the target, while the fifth of the overtone chord becomes the root of the target; and the seventh of the overtone chord becomes the third of the target, while the third of the overtone chord becomes the seventh of the target, but not before resolving to the root of the target first; the chord member-tones simply exchange functions during the transformation. So, there is a momentary triad at the point of resolution before the doubled root descends to again create a seventh chord (Which I have indicated by the b-natural in parentheses). You can see the nature of the crosswise transformation in the diagram between the staves.

The parenthetical b-natural of this example can also be a b-flat: In fact, the series implies that it ought to be a b-flat to create another overtone chord at the target. However, if you wish to create a seven-tone diatonic system, you have to retain the b-natural of the original overtone sonority: Introducing the b-flat will create a nona-tonic or completely chromatic system, depending upon how you handle further resolutions. The implied diatonic tonality is realized in Example IV: The b-natural of the original overtone chord is retained to create a major seventh chord at the tonic, and the e-natural of the tonic is retained to also create a major seventh chord at the subdominant. Both of the root progressions are Progressive, and both of the transformations are therefore delayed crosswise transformations, as you can see in the analysis. The root of the diatonic system is therefore the point to which the overtone chord naturally present in the diatonic system resolves.

If we extract the seven tones present in this dual-resolution, we get the Ionian or Major mode, as seen in Example V: This is the most natural diatonic modality as defined by the implications of the series itself. It is important to always keep in mind that it is the harmonic progression of the series which creates the mode.

Historically, it took Western art music many centuries to evolve to the point that this was intuited properly by theorists of the educated classes (Who were predominantly clergymen at the outset). While there are many reasons for this, in a nutshell, Western art music began as monodic chant for liturgical functions, and evolved through singing in parallel consonances (Both perfect and imperfect) to the point at which complimentary melodic trajectories were combined into counterpoint. From the time of Perotinus Magnus at Notre Dame to Palestrina during the Catholic counter-reformation, this was honed into a razor sharp set of polyphonic techniques. At the latter points during this period, harmonic functions began to be intuited starting at the final cadences (Both for pieces and for phrases) and working backwards. Between Palestrina and Bach these harmonic intuitions destroyed the antiquated Church mode system (Which was the most retrograde aspect of early music), and modern harmony evolved to the point where Rameau finally attempted to describe it in 1722.

Ironically, popular musicians as far back as Perotin's day were using the Ionian mode - or modus vulgaris as some of the priestly academics called it - having intuited the correct solution over a quarter of a millennium before their more educated bretheren. This is humorous to me because it was theology which held the Church musicians back: They were enslaved to the modal system and their ideas of the metric supremacy of three, which were peripheral to implications of the series, but not central to them. If it is theo-logical it ought to have God-logic, and the higher order of God's logic is revealed within the implications of the series: Ideas which are not within these implications, or which obviously contradict the implications of the series, are simply not musical. In any event, by Bach's day the entire system was almost perfectly intuited, if not quite properly explained in the theoretical sense.

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At the top of the second page in Example II, the derivitave modes of the diatonic system are presented along with the harmonized scale. Though less perfect than the Ionan/Major system, the series implies that all of these modes can function independently except for the seventh mode Locrian: Without a perfect fifth, no modality is possible at all: The roof caves in without the support of the pillars, so to speak. Even in the most primitive forms of music - in which melodies are played over drones - the fifth degree is virtually always intuited as being perfect. Indeed, most drones are not simple octaves, but are in fact perfect fifths. Again, a combination of factors lead to Western musicians starting out with some of the less perfect modes before homing in on the most perfect form with the advent of functional harmony: The harmonic system is the most complex of the systems implied in the series, after all, and Western composers had to master melody and counterpoint first.

Unfortunately, when I was coming up, this harmonized scale was presented to me first. Without the preceeding perspective that the harmonic overtone series offers, confusion is inevitable: For years I was under the impression that scales generated harmony. This confusion was by no means unique to me, as members of many generations of musicians have suffered from the same defect in understanding due to sloppy and incomplete pedagogy. The truth of the matter is exactly 180 degrees the other way from conventional understanding: The harmonic progression of the series generates the scales.

Though the series implies that the most perfect possible resolution of the overtone chord is to a major triad, it also implies that a less perfect resolution to a minor triad is possible. I have demonstrated this in Example VII. In this less perfect realization of the series' implications, the leaning-tone seventh resolves down a whole step to the minor third of the target. Otherwise, everything else is the same.

It is not possible for the implied minor system to be diatonic: The delayed crosswise resolution to the minor tonic requires a minor seventh chord on that degree, hense the parenthetical b-flat, as can be seen in Example VIII. Attempting to use the overtone chord's b-natural would produce an augmented triad in the tonic's upper structure, and that is not viable as a functional tonic at all. Therefore, the minor tonic wishes to resolve further to either another minor seventh on the subdominant degree, or alternately, another overtone chord at the subdominant level. If the tonic's target is another overtone chord, then yet another extension of the cycle yeilds the a-flat for the full nona-tonic or so-called melodic minor system (Which is really a harmonic system). There is another way to yeild this derivative however, which I will explain in a moment.

The resultant nona-tonic mode is shown in Example IX, and the harmonized scale with all of the chordal varieties are shown in Example X. Due to the richness of these resources compared to the pure major system, the nona-tonic minor has been a favorite of composers since it first appeared. Additionally, these extended harmonic resourses have also been used with major tonic chords through the process of modal interchange, which lead to the concept of integrated modality. The series predicts integrated modality quite on its own, though, as I will show down the road.

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As I mentioned, there is another route by which the series can imply the nona-tonic minor system, and that is via a combination of the pure minor system and an overtone chord targeting the minor tonic along with a second overtone chord residing on the subdominant degree: The pure minor system is demonstrated in Example XI, and Example XII demonstrates two things (In order not to run onto a third page of examples for this post): In the so-called harmonic minor system, the b-natural of the overtone chord is retained and the subdominant degree harbors a minor seventh chord. As can be seen, this goes against the implications of the series: There is that impossible augmented structure over the tonic, and there is also the non-contrapuntal progression of an augmented second from the minor major-seventh "tonic" to the subdominant. Though the so-called harmonic minor and double harmonic minor systems can yeild some exotic sounding melodic effects (Over a limited vocabulary of harmonies, as fans of Yngwie Malmsteen no doubt have deduced), they are not strictly speaking in accordance with the implications of the series from a harmonic viewpoint. In fact, they are rather primitive, which is no doubt a large part of their appeal. In any event, "harmonic minor" is neither harmonic, nor is it a functional minor without modification. It's a farce, actually.

If we replace the parenthetical b-natural in the second measure of Example XII with a b-flat, and have the now-repaired tonic targeting a major triad on the subdominant degree, we will then have the other implication of the series if we combine Example XII with the pure minor system of Example XI. Both paths work and both lead to the same destination, and I have no firm preference for either.

In Examples XIII through XV I have demonstrated another nona-tonic system with which we are all familiar: Blues tonality. By allowing for direct crosswise transformations between overtone sonorities on the dominant, tonic, and subdominant degrees (i.e. No momentary triads interrupting the transformations), this poly-mode is generated. Though blues musicians (As well as blues-based jazzers like the swing and bebop guys) have never developed to the level of sophistication where they use proper transformations, their ancestors were nonetheless spot-on when they intuited the possibility of overtone chords on the three cardinal degrees. Interestingly, the diatonic mode of the blues is the tonic Dorian, whose Mixolydian belongs to the subdominant degree; not the dominant. Like some other popular forms, the blues is biased in the subdominant direction with dominant episodes almost exclusively at turn-around points only. This totally turns around by the time blues-based Bebop appears though, as secondary dominants are the rule, rather than the exception in that music.

Their are other possible tonality or modality types implied by the series, of course (Including each and every viable mode in the entire diatonic modal system), and one worth a quick mention is the Flamenco mode, which is basically an overtone chord on the tonic (Or, alternately, a major triad), with the diatonic mode of Phrygian. Working out how that was derived would be good exercise, but I'm through for the day.

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3 Comments:

Blogger Mark said...

You write, "The major third and minor seventh, which are in a tritone relationship with each other (A diminished fifth in closed root position), are active tones: The major third has a leading-tone tendency and desires to rise, while the minor seventh has a leaning-tone tendency and desires to fall."

Clearly, you are stating facts that make perfect sense musically. However, you fail to explain why you feel that the tritone "has" to resolve in a certain way. Do you feel it is due to the inherent "friction" between the two overtone series of the notes in the tt? Also, I find it perhaps problematic that you conveniently stop your overtone series to only include the notes of a dominant seventh sonority, when in fact they continue infinitely in theory. How do you choose when they become inaudible, and hence, irrelevant? Finally, why should the tonic be a point of resolution? Every tonic root has an overtone series with its own inherent dom 7 structure demanding resolution (due to the series structure itself). Therefore, shouldn't music always be demanding resolution to the next chord a fifth down? Why are any tonic resolutions musically satisfying?

Don't get me wrong, I think your ideas have great potential. I have always believed in the overtone series as a harmonic generator. But it seems not quite strong enough for a "musical theory of relativity," at least, not yet.

Mark Feezell, Ph.D.
Lecturer in Music Theory
Meadows School of the Arts
Southern Methodist University
Dallas, Texas
www.drfeezell.com

4:52 PM  
Blogger Hucbald said...

Hi Dr. Feezel,

Tell my old friend Rob Frank, George Pepper says "Hi."

"You write, "The major third and minor seventh, which are in a tritone relationship with each other (A diminished fifth in closed root position), are active tones: The major third has a leading-tone tendency and desires to rise, while the minor seventh has a leaning-tone tendency and desires to fall."

Clearly, you are stating facts that make perfect sense musically. However, you fail to explain why you feel that the tritone "has" to resolve in a certain way."

The tritone does not "have" to resolve at all (See Blues modality). The resolution I gave as an example is simply the MOST NATURAL resolution according to the implications inherent in the overtone series. Many less natural movements are available away from a dominant seventh chord, for example, the traditional deceptive motion to vi or bVI. The resolution of the so-called German Augmented Sixth chord is another. Bottom line is, in free composition, any of the twelve dominant seventh chords can move to any of the twenty-four possible major and minor tonics, or their altered variants, or to any of the eleven other dominant sevenths, for that matter.

"Do you feel it is due to the inherent "friction" between the two overtone series of the notes in the tt? Also, I find it perhaps problematic that you conveniently stop your overtone series to only include the notes of a dominant seventh sonority, when in fact they continue infinitely in theory."

You must be young. ;^) Of course the overtone series continues forever in theory, but the only portion of it that is important to music is the portion that contains the harmonic intervals. That would include the fundamental and the first seven overtones, or, the first eight partials, whichever way you wish to look at it.

The function of the tritone is defined by the passive tones: A B to F tritone belongs to a G7 chord, but that same tritone as Cb to F belongs to a Db7: Enharmonically spelled and in second inversion, that Db7 IS A SO-CALLED GERMAN SIXTH CHORD. I make this dual function more clear in the book version.

"How do you choose when they become inaudible, and hence, irrelevant?"

See above. It's not an inaudibility "thing" it's a harmonic relevance "thing." The only harmonic interval you need to go above partial seven to get is the 8:5 minor sixth, and its root position interval of origin is present within the first seven partials. That is why I stop the series at partial eight/harmonic seven: Beyond that point, music just doesn't "care" anymore.

"Finally, why should the tonic be a point of resolution? Every tonic root has an overtone series with its own inherent dom 7 structure demanding resolution (due to the series structure itself). Therefore, shouldn't music always be demanding resolution to the next chord a fifth down? Why are any tonic resolutions musically satisfying?"

I covered this in my mention of Blues modality, and in many other places. You need to read every installment, because you have obviously not gotten to the "critical mass point" of understanding yet.

"Don't get me wrong, I think your ideas have great potential. I have always believed in the overtone series as a harmonic generator. But it seems not quite strong enough for a "musical theory of relativity," at least, not yet."

I don't get you wrong at all. You simply do not understand yet. This is not music theory, this is, in fact, beyond theory: This is the physical reality of the origin of musical phenomena. I'm working on the book version now, and the title will end up being something along those lines. "The Harmonic Overtone Series: Origins of Musical Phenomena," or something along those lines.

"Mark Feezell, Ph.D.
Lecturer in Music Theory
Meadows School of the Arts
Southern Methodist University
Dallas, Texas
www.drfeezell.com"

Music theory is dead: I've killed it. What you've discovered here is the roughest of rough drafts of musical reality. I explain not only "how to" write music, but also "why" music works. Not only that, but I also demonstrate "why" music evolved as we see from the historical record.

Hucbald is the Aplha and the Omega of Western music theory. LOL!

5:28 PM  
Anonymous mark said...

Let me know when you publish your book. I'd love to read it.

As to whether music theory is dead, well, we'll see I guess... :^) If your "musical reality" is really true, it would have to explain all world musics equally well.

I'll withhold judgment until I can see it laid out in a book. Thanks for your work...it seems very promising.

Mark Feezell, Ph.D.
Lecturer in Music Theory
Meadows School of the Arts
Southern Methodist University
Dallas, Texas
www.drfeezell.com

9:22 AM  

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