Musical Implications of the Harmonic Overtone Series: Chapter VI

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The Secondary Subdominant System

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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.

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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.

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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.

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Ah, the problems associated with blustery fall days.

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Musical Implications of the Harmonic Overtone Series: Chapter V

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The Series' Prediction of Canon

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As I mentioned in chapter four, the series, quite by itself when left to its own devices, will create canons. There are two circumstances required for this to take place: The root progression pattern has to be all of a single type or it must be a repeating pattern of two or more types, and the upper texture must follow proper transformational logic or at least a logic of all the same kind. If this is done casually within the diatonic system, non-strict diatonic canonic voices will result in the upper transformational stratum, but if the original established intervallic relationships are scrupulously followed in subsequent iterations, strict canons will result.

There are a couple of exceptions to note: In a three voice texture, if two root motion types are selected which result in alternating clockwise and counterclockwise transformations, then the chord tones will not get to play all of the roles within the triad (Root, third, and fifth). This will result in an incomplete canon in which two voices follow properly, but the third voice is "free" so to speak. Likewise, in a four voice texture, if the single or multiple root motion types don't allow for the voices to play all of the tetradic roles, then a four-voice canon will not result: But a double canon will. The double harmonic canon is what is actually implied in the series, but I will display a true four voice canon with two root motion types as well.

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In Example I I have presented the old reliable initial proof in the triadic version. In Example II I extracted the three voice diatonic canon out of it. Obviously, this canon does not draw any attention to itself, but as we embelish it, it will.

By Example III, where I introduce secondary dominant triads into the mix, the canonic nature of the transformational stratum begins to become noticeable. Then, in Example IV, I introduce a nice altered dominant sonority available in triadic textures: The augmented triad. This second of the symmetrical harmonic structures - being a loop of major thirds - also generates a hexa-phonic whole tone scale just as the French and Italian-derived secondary dominants do.

Finally, in Example V, I dovetail the diatonic, secondary dominant, and augmented triad versions together to produce a three voice canon which increases in harmonic and melodic interest over the course of the phrase.

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On this page of examples, I have presented the same continuity example with tetradic upper strata. Since all of the transformations are crosswise, one pair if voices alternates between the roots and fifths, while the other pair alternates between the thirds and sevenths (With the intermediate triads with doubled root). This results in a double canon instead of a four voice canon. So, the series-implied falling fifth root motions create a double canon, as you can see.

The process is the same as with the triadic examples: Example VI is the basic continuity, Example VII is the extracted diatonic double canon, Example VIII adds secondary dominant sevenths, Example IX adds French-derived V(d5m7) chords, and finally, Example X combines aspects of all of the previous versions into a dovetailing strict double canon.

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In Example XI we have a root progression pattern of two types: A half-progression followed by a progression. This is the same pattern I used in the second integrated modality proof, but this version is diatonic and has a triadic upper stratum. Example XII adds secondary dominant triads, and Example XIII is only a two voice canon with a third "free" voice because the alternating clockwise and counterclockwise transformations do not allow all of the voices to take all of the chordal degrees.

Example XIV is the same root motion pattern with a tetradic upper stratum, and as you can see, it does not suffer from the same flaw. In fact, due to the clockwise/crosswise alternations, this will result in a true four voice canon since all four of the voices get to play all four of the chordal functions.

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Here in Example XV we have another version of the second integrated modality proof which displays the fact that it is a true four voice harmonic canon (I labelled it a double harmonic canon, which it is, but is also a true four voice canon).

There are harmonic canons present - but well hidden - in some of Bach's preludes and chorale harmonizations, so the technology is not new by any means. I'm virtually certain that Bach was aware of these since he was such a transcendental contrapuntal master, but since these canons can be incidental to producing repeating root motion patterns, the very remote possibility does exist that some of them were simply artifacts of his harmonic compositional process. So far as I know, neither Bach or any of his students ever mentioned them pedagogically. Probably the most famous harmonic canon in all of music history is the Pachelbel Canon in D, which uses a I, V, vi, iii, IV, I, ii(6/5), V progression, which has alternating regressive and super-progressive root motions right until the end, and that was written a generation before Bach's time (Five years before Bach's birth, if memory serves), so I'm fairly certain this technique was well known in the north German school. I find it rather odd that I have never come across any mention of it within that period's context, but then I don't read German.

We need not stop here with melodic developments either, as we could create a long, drawn out Pachelbelian variation set from these progressions, but the proofs which I wanted to present are complete with this example: The point is that the series works out canon on its own if it is allowed to realize its resolutional desires in a cyclically repeating manner.

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She could probably get me to eat more fruits and vegatables.

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Musical Implications of the Harmonic Overtone Series: Chapter IV

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The Series' Prediction of Integrated Modality and Equal Temperament

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These two chapters are quite short and only require a single example page, plus they are closely related to one another, so I decided to combine them into a single post.



In Example I I have arranged a twelve bar phrase containing twelve progressive root motions. Each target triad is minor, which then becomes major, and finally, it acquires a minor seventh to become an overtone chord before proceeding to the next target. Not only does this progress through all twelve chromatic degrees, but all twenty-four major and minor tonics and all twelve overtone sonorities as well. This is the series working out for itself the concept of integrated modality in the most direct way possible.

This example also represents, as a musical example, the desire of each and every one of the twelve tones: A tone first desires to rise to the rank of a tonic, regardless of whether or not it is major or minor, by acquiring for itself a perfect fifth. Then, if it is minor, it desires to become a more perfect tonic with a major third. Finally, after the tone has had its time on the stage of the piece, it wishes to acquire a minor seventh, become an overtone sonority, and then it is absorbed into a new tonic a perfect fifth below, where the process starts anew.

Note that during this chromatic cycle, the voices in the upper stratum descend an octave. In Schenker-speak this would be an 8, 7, 6, 5, 4, 3, 2, 1 line, and that points out what a joke Schenker's theories actually are. The lines Schenker describes, in breathless terms which suggest that they are some big, important discovery, are nothing more than natural artifacts which result from any functional harmonic continuity which has a preponderance of progressive root movements and a primarily four-voice texture. And, since progressive root movements are statistically the most common type in all forms of traditional music, and since overtone chords resolving to targets will always imply four voices, that means almost all functional harmonic continuities create, as artifacts, decending lines, whether they be 3,2,1's; 3, 2, 3, 2, 1's; 5, 4, 3, 2, 1's; 5, 4, 3, 2, 5, 4, 3, 2, 1's; or 8, 7, 6, 5, 4, 3, 2, 1's.

I spent a lot of time on Schenker when I was working on my Master of Music degree, and I "Schenked" my share of pieces during that time, so when I finally realized that these descending lines are present with or without the composer's being aware of them, and are in fact a natural side effect of the implications of the overtone series in action - simple musical gravity - I about laughed myself silly. I can't think of any music theory that is a bigger waste of time than Schenker's. Well, there is Paul Hindemith I guess. To be valuable as a music theory, that theory must be of use to those who wish to compose music. Schenkerian analysis does not help the prospective composer one whit.

Finally, note that from a technical standpoint the bass part of Example I is a single-interval twelve tone row (Counting octave displacement inversions as equivalent). Not particularly impressive as a tone row, but it does not have to consist of only a single interval, as you can see in Example II. Here, there are two intervals in the bass part: a half-progressive descending minor third followed by a fully progressive rising fourth. The chords which are exited via half-progression start out as major triads and become overtone sonorities (Which are not functional in nature), and their targets start out as minor sevenths, become overtone chords, and then aquire a diminished fifth to become French-derived secondary dominants. When you create intervallically precise root progression patterns such as these, and make all of the transformations according to hoyle, an interesting entity is created: The harmonic canon, which we will look more closely at in chapter six.

For now I want to point out the brief fifth chapter example. It's quite an irony that the example is so simple, because it took an entire millennium of Western music history to work out the solution to it, which I find positively flabbergasting, but I guess cultural hindsight is 20:20.

The problem here is what is called the Pythagorean Comma. If you take twelve absolutely pure perfect fifths with the overtone series ratio of 3:2, you get the derivation of the chromatic scale as implied by the series (But the series actually implies a series of descending fifths, which we just saw harmonically in Example I above). If you compare that to the same span in perfectly pure 2:1 octaves, it takes seven octaves to match the span of the twelve pure perfect fifths. The problem arises between the e-sharp and the f-natural enharmonics at the top of each series: They are not the same pitch. The e-sharp that the pure perfect fifths generated is circa 23.46 cents sharp compared to the f-natural. This caused ten centuries worth of headaches.

It goes without saying - or, at least it should - that the pureness of the octave is inviolable, so stretching it should not be an option (But, this is done by piano tuners all the time). The fifth, on the other hand, is clearly subservient to the octave in rank within the series. So, the only logical solution would be to pinch the fifths each by 1/12th of the Pythagorean comma, which comes out to a measly 1.95 cents. Why this obviously implied solution took a thousand years to be implimented is simply beyond me, but it does make for fascinating reading. Sure, I like to hear Baroque music in well tempered harpsichord tunings and at philosophical pitch (And, I tune my guitars to a philosophical pitch based on an A0 of 27.0 Hz versus the typical 27.5 Hz), but fixed pitch instruments, especially fretted string instruments, really have no other viable option than equal temperament. To be fair, at least one ancient Greek musician-philosopher suggested it, so the idea was around, but even as late as Beethoven's day the matter wasn't completely settled. What finally settled the matter was the realization - in the actual practice of composers - of the series' implied integrated modality: After the Romantics, it was all over but the shouting of the period instrument and ancient music crowds.

The series itself is unstable, so the idea that the ideal tuning system should be stable is not only an impossible goal, but the series simply does not imply that it is required (Or, that it is even particualrly desirable). What the series does imply is that any one of the twenty-four possible tonics can be the locus of a composition, and for them to all sound the same relative to each other, twelve tone equal temperament is required.

In the electronic realm, octave divisions up to over 170 divisions can be used, and they sound to most people the same as just intonation (Since at circa 7 cents, the divisions border on the smallest differences that the average person can detect), but such ponderous systems will never offer any potential solution for the human performer with a fixed-pitch instrument in hand. As with every ultimate theoretical solution, twelve tone equal temperament is elegantly simple, and since the series specifies twelve pitches in an octave, it is the only real solution the series implies.

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I've had a lot of very good days fishing, but never one that good.

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Musical Implications of the Harmonic Overtone Series: Chapter III

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The Secondary Dominant System

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Today's music theory hat tip goes out to Dr. Gene Cho, who I studied under while I was a Doctoral Candidate at The University of North Texas in the early to mid 1990's. Dr. Cho's slim monograph, entitled Theories and Practice of Harmonic Convergence is, in my opinion, one of the greatest music theory books of all time. Under Dr. Cho, I finalized my conception of integrated modality, though it took about ten more years for me to work out my present terminology, and to relate it all back to implications present in the inherent desire of the musical force in the harmonic series.

Dr. Cho also gave me the spark of insight which lead directly to my understanding of the secondary dominant galaxy of sonorities, which I will present in this chapter. Though I had begun to gain a grasp on the standard secondary dominants, the so-called substitute secondary dominants, and the so-called secondary diminished seventh chords as far back as the late 70's while at The Guitar Institute SW under Jackie King and Herb Ellis, it was Dr. Cho who made me realize that the more exotic so-called French and German Augmented Sixth chords were part of the secondary dominant system as well. Though I could never get Dr. Cho to agree with me that these sonorities originated with implications present in the harmonic overtone series - he stuck with his notion that they were purely contrapuntal in origin (Which is correct in the historically specific sense, but incorrect in the theoretical sense) - it was nonetheless Dr. Cho who is directly responsible for my ability to present these sonorities and their origin clearly.

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Example I is the original continuity proof again, only this time it contains all of the standard secondary dominant seventh chords. In fact, I wrote that phrase with this and some of the following demonstrations in mind.

Just as harmony itself was intuited backwards from cadential points, so were the secondary dominants. Since the primary dominant is an overtone chord resolving to a tonic target, it was a simple step for composers to decide to target that primary dominant with an overtone chord of its own, and so the secondary dominant concept was born out of intuited implications present in the harmonic overtone series.

Over time, overtone chords appeared on all of the diatonic degrees which could move in a progressive manner. So, from the original V(m7)/I the series progressed to the V(m7)/V, V(m7)/IV, V(m7)/ii, V(m7)/vi, and finally, the V(m7)/iii. Note that for the V(m7)/iii I have left the f-natural so that the sonority is an overtone chord with a diminished fifth. This is the actual point of origin for the so-called French Augmented Sixth sonority, though it actually first appeared in the guise of a V(+4/3/b)/V in the minor mode (Where the vii(d5m7) is the ii(d5m7) chord). As I said, I could never get Dr. Cho to recognize this, but this is what it is: The traditional "French" sonority simply being the second inversion which yeilds the augmented sixth interval.

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By understanding these sonorities as simply being overtone chords with diminished fifths (In a particular inversion per traditional use), they become available as sonic resources on any degree which can have a secondary dominant. I have demonstrated this in Example II.

Now, by starting out in closed root position, these modifications yeild diminished thirds versus augmented sixths: This is not an issue. Both inversions are available, and though they have decidedly different sonic effects, both are usable depending on the sonority and result the composer desires. This gives much more freedom and many more resources than the stilted and clumsy traditional way of looking at these chords.

But, what is it that is actually happening here? By that question I mean, what feature of the overtone chord is it which allows and even encourages this modification of the diminished fifth? The answer to that goes back to the active tone/passive tone dichotomy within the overtone structure: By diminishing the fifth, you are changing it from a passive tone which neither desires to rise, nor desires to fall, into an active leaning-tone which desires to resolve down by semitone to the root of the target chord. This second diminished fifth only magnifies the energy present in the natural version of the overtone chord (But, it does not double the energy, because the root remains a passive tone, and therefore a real root).

Finally, it should be noted that the French-derived sonority consists of two tritones a whole step apart, and therefore, harmonically speaking, it generates a hexa-tonic whole tone scale.

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The so-called secondary diminished seventh chords present a new problem: Here, the lowest note in the chord spelling is an active tone - a leading-tone - and therefore it cannot be the real root of the chord: Roots must always be passive tones. The lowest tone in the spelling of a fully diminished seventh chord is what is called a theoretical root. The real root of the chord is a major third below the leading tone, and its presence will create an major overtone sonority with a minor ninth added to it.

It should be noted that, because the fully diminished seventh sonority is a perfectly symmetrical structure (It is a circle of minor thirds), without the presence of the real root, any one of its tones can function as the leading tone, or theoretical root. In context, it is easy to tell which tone ought to be the theoretical root, but in actual point of fact, it is not until the resolution that the suspicion is confirmed or denied. This gives the fully diminished seventh two admirable properties: 1) It momentarily suspends the tonal/modal system's functionality and sense of direction (Or, it at least puts it in some doubt, depending on the prolongation of the sonority), and 2) it allows for some unexpected modulations. Composers have been taking advantage of these properties since just before Bach's time.

In the two examples above I have presented all of the so-called passing diminished sevenths. Example III presents those from the tonic to the mediant, and Example IV presents those from the subdominant to the leading tone: Anywhere there is a whole step in any mode of the moment, one of these chords can be inserted. The root motion indicators are in parentheses because the real roots would not be present except for the analysis.

Note what happens in the transformations: The seventh of the first chord becomes the ninth of the V(m7m9/0), thereby taking the place of the root momentarily. In keeping with its functional nature as a root-substitute, the ninth then resolves back to the fifth, and so the expected delayed crosswise transformation still takes place.

Also take note of the fact that - as with the French-derived sonority - the fully diminished sonority contains two tritones. However, with the fully diminished seventh the tritones are a minor third appart, and they are both fully active. Again, the harmonic series allows for these sonorities because it is only magnifying or doubling (In this instance) the natural leading-tone/leaning-tone force of the original tritone of the overtone chord.

This sonority also harmonically generates a non-diatonic scale: In this case the octa-tonic diminished scale, also called a 1 + 2 scale, because it consists of alternating semitones and tones.

The final little nugget in Example IV is that I have targeted the vii(d5m7) with a fully diminished seventh chord: Although the Locrian mode cannot function as an independent mode, there is nothing keeping the diminished minor-seventh sonority from being targeted by a secondary dominant function chord within a tonal or modal context. I don't recall ever seeing this in any Classical or Romantic works, though I'm sure I'm not the first composer to realize this possibility.

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Once I had the theoretical root versus real root problem worked out, I decided to apply it to another set of secondary dominant function chords that have been historically problematic: The so-called German Augmented Sixth chords. Us jazzers were introduced to these sonorities as substitute secondary dominants, and their theoretical justification was that they shared the same tritone with the primary dominant, but their root was a tritone above. In jazz this chord does not transform in the manner predicted by the implications of the series - in fact, it does not transform at all since all nominal chord functions remain the same between the two sonorities, the first root simply sliding down by semitone to the target root - but in traditional music this sonority is problematic. First of all, the sonority in and of itself is an overtone chord, but it is usually spelled with some enharmonics, and it does not appear to resolve in the functional manner that you would expect an overtone chord to do. This appearance is an illusion: It actually does resolve perfectly in keeping with the implications of the series.

Key to understanding this chord is, again, understanding that the root of its non-enharmonic spelling is an active tone, and so it simply cannot be the real root: It must be handled as a theoretical root. The real root is again a major third below the leading tone of the non-enharmonic spelling of the sonority, which in this case is actually the minor seventh. Adding the real root reveals the true nature of the so-called German chord as a V(d5m7m9) secondary dominant (Or primary dominant).

In Example V I have presented some of these sonorities in a way which makes their origins as just another altered dominant sonority clear: All you have to do to produce one of these chords is to start with a fully diminished seventh chord, and then flat its theoretical third (Its real fifth if the real root is in the bass). Another way to look at it is as a French-derived sonority with a minor ninth replacing the root.

In the transformational analysis you can again see that the ninth is substituting for the root momentarily in the upper stratum, only this time it is from a crosswise motion versus the previous clockwise. The ninth still resolves properly back to the fifth after the delay.



For my final parlor trick, I have put German-derived sonorities on all of the degrees which can carry standard secondary dominants using the original harmonic continuity. With this root motion pattern - all strong motion between real roots - the root itself moves up to become the ninth. resolutions are still properly crosswise after the delay, of course.

I cannot stress enough how important it is for you to play these. With and without the bass parts.

The so-called doubly-augmented fourth augmented sixths are just a set of secondary dominants in another enharmonic spelling of the German-derived sonorities: They resolve to an intermediary major triad in second inversion (The tonic, traditionally, but it could be a major triad on any degree) before proceeding on to the dominant. I don't feel the need to present those in this series, but I will in the final book.

Also, the so-called Italian Augmented sixth is just a French-derived sonority with a missing root. It is usually found in triadic textures, and so is a bit of a sick man among the augmented sixths. I feel no particular need to present those at this point either.

Finally, Frederick Chopin was fond of an augmented sixth sonority which had a major ninth over the real root, which made for a 6-5 (Or, 13-12) resolution over the target. I've never used those myself, but he (And Wolf as well) got some good mileage out of them. The main point to reaize, of course, is that all of these types of sonorities are simply altered forms of the standard secondary dominants, and so are perfectly well predicted by implication from the overtone series.

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I'm not sure which would be more dangerous: The undertow or those curves.

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Musical Implications of the Harmonic Overtone Series: Chapter II

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Harmonic Motion Dynamics

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When I was attending Berklee back in the early 80's, some of the jazz theorists I studied with used to put brackets under ii to V progressions in order to make them easier to make out in the analysis. This was the first time I was exposed to an incipient system for labelling root progression types - and it was helpful in detecting root progression patterns - so I not only adopted it, but I also expanded it to label each and every kind of root motion.

My first guide in this effort was Arnold Schoenberg, who in his Structural Functions of Harmony classified root motion types using terminology like strong, super-strong, ascending, and decending, and ranked them by how many common tones the adjacent chords shared and whether the targeted root was a new tone, or one present in the preceeding sonority. While this system did enable me to write much better harmonic continuities due to greatly facilitated pattern recognition - and good and bad continuities are almost always separated by whether the root progression patterns are logical or not - Schoenberg's system did not relate in any but the most casual and incidental ways to the implications inherent in the harmonic overtone series.

Over the course of a couple of decades of private teaching, I began to revise his terminology until I had replaced it en toto with my own system which related all root motion types to the primordial resolution of the overtone chord. This root motion, the falling perfect fifth/rising perfect fourth - as I have mentioned previously - is Progressive, and it gets a capital "P" in the analysis. Calling the root motion progressive is not arbitrary, rather it reflects what the series is actually doing in this root motion/transformation type: It is progressing according to the inherent desires of the musical force present in the series.

The contrary root motion to this, the rising perfect fifth/falling perfect fourth, would therefore be Regressive, and it gets a capital "R" on the analysis line. If the progressive root motion is the ticking of the watch, then the regressive motion is the winding of the spring. The regression from the tonic to the dominant, for example, is just the windup for the pitch: The following dominant to tonic progression is the actual delivery of the ball.

It takes two falling thirds to equal one falling fifth, so a decending third root motion is Half-Progressive in nature. It takes two falling thirds to get the voices to the same point as a single falling fifth does as well, as I shall demonstrate shortly. In the analysis a half-progression is labelled as ".5P." Its complimentary opposite, the rising third, is therefore a half-regression, and so is ".5R" in the analysis layer.

Another set of root motions in the diatonic system are the rising and falling seconds. With these root motions we will get parallel perfect fifths or parallel perfect fourths (Depending on the voicing and inversion) in the upper voices during any of these transformations which have perfect fifths (Or, perfect fourths) between the root and fifth and the third and seventh of the adjacent sonorities. This is not a problem: These parallisms are simply the natural result of these root motion/transformation combinations. Parallel perfect fifths are only "wrong" if the voices involved do not transform: In other words, if the root remains the root and the fifth remains the fifth, or alternately, if the third remains the third and the seventh remains the seventh. In these instances the voices move in a clockwise or counterclockwise circular transformation (Depending on the root motion direction), so they are technically correct. These root motions really mimic leading-tone and leaning-tone resolutions, except that the root is not treated as an active tone (Much more on this when we get to the secondary dominant galaxy of sonorities). For this reason I call the rising second Super-Progressive and the falling second Super-Regressive. They get "SR" and "SP" respectively in the analysis.

The final root motion type is that by tritone. Obviously, the tritone can mimic progressive motion or regressive motion depending upon whether the motion is from the subdominant to the leading tone, or vice versa. In the former instance it mimics a progressive motion and gets "Ptt" (For Progressive tritone) in the analysis; in the latter it mimics regressive root motion and receives an "Rtt" (For Regressive tritone).

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Instead of demonstrating these root motion types in sterile isolation, I have created some simple musical continuities. I will develop these throughout the course of this series until some of them will eventually wind up in the simple composition examples at the conclusion.

Whoever said, "Writing about music is like dancing about architecture." was a fool, by the way. I have found that the better I can explain music, the better I can compose music. The reason is simple: Being able to explain music clearly means you have a better understanding of it. Better understanding of music leads to better compositions.



In the first five examples we will look at triadic continuities, and in the second five tetradic continuities over the same root progression patterns, as the transformations are different depending upon how many voices are present in the upper stratum. For triadic transformations there are only two possibilities: Clockwise and counterclockwise. In a clockwise transformation the root of the first chord becomes the third of the second, the third of the first chord becomes the fifth of the second, and the fifth of the first chord becomes the root of the second. A counterclockwise transformation is simply the opposite of this.

In Example I I have arranged all of the chords in the diatonic system in progressive relationships with each other. After a super-regressive motion from I to vii(d5), all of the remaining root motions are progressive until the super-progressive motion from IV to V in the final measure. Note that progressive root motions result in counterclockwise transformations. This is always the case in a triadic context. Super-regressions transform clockwise, while super regressions transform counterclockwise: Opposite root motion types always result in opposite transformational directions as well.

Note also that the voice leading is not totally smooth in the SR and SP motions: There are skips of thirds present. Pure transformational harmonic voice leading will always be totally smooth and stepwise if the implications of the series are followed, so obviously the series implies a four part transformational texture where either the root of triads will be doubled, or seventh sonorities will be complete.

I have demonstrated the complimentary opposite correlation in Example II, where after two half-progressive motions all the rest of the root movements are regressive. Also note that, in a triadic texture, progressive root motions move the voices higher in pitch through time, while in regressive root motions the voices subside over time. If you have a keyboard handy, you ought to play these examples without the ties (Which are only present to demonstrate the number of common tones between sonorities).

With our admittedly incomplete triadic upper stratum (According to the wishes of the series), Super-progressive and Super-regressive root motions result in zero common tones, Progressive and Regressive root motions result in a single common tone, and Half-progressions and Half-regressions result in two common tones. The percieved abruptness or smoothness of the root motion/transformation combination depends heavily on this, naturally.

Example III and Example IV demonstrate the same yin and yang principle for the half-progressive and half-regressive root motions respectively. As I said, not only does it take two half-progressive motions to move the root down by a progressive fifth, but it takes those two steps to transform the voices as far as a single falling fifth as well. If you'll note, the I vi IV motion in Example III would result in exactly the same voicing if it went directly from the tonic to the subdominant.

The final continuity, Example V, contains every possible root motion type available in the diatonic system with the exception of a Regressive tritone movement. From measure four to five is the Progressive tritone, and that divides the continuity into a pair of four measure antecedent/consequent phrases. Note that the root motion types are mirrored between the antecedent and consequent: Retrogression is answered by Progression; Super-progression is answered by Super-regression; the Half-progression is answered by a Half-regression; and finally, the Super-progression of the final measure turns the phrase around on itself. This is a perfectly balanced phrase, as the voice leading makes an uninterrupted loop. The previous phrases - with their preponderance of the same types of root motion - are very unbalanced. Obviously, with unbalanced phrases you can run into problems of range if you are not careful or do not know how to manage the rising or falling tendencies. The solution is to alternate between three and four voices in the upper stratum (Triadic and tetradic episodes, respectively), as these also yeild a yin and yang correlation: If the triadic transformation raises the voices, the tetradic transformation will lower them, and vice versa.

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In Examples VI through X I have presented the previous root progression continuities with a four voice upper stratum. Whereas with three voices the transformations are always circular - either clockwise or counterclockwise - with four voices crosswise transformations become possible. I have made all of these transformations directly with no interrupting triad with doubled root to demonstrate the numbers of common tones between the adjacent sonorities. With the fourth voice, SR and SP motions now have a single common tone, P and R motions have a pair of common tones, and in .5P and .5R motions only one voice moves in the transformation. This is the actual texture that is implied by the harmonic overtone series itself, and you can use the intermediary triads with doubled root at your discression: It is only really imperitave to use them when the progressive motion is from an overtone sonority, and even then it is not positively mandatory (As you can see from the sixth to the seventh measure of Example VI).

Earlier I alluded to the fact that some aspects of super-progressive and super-regressive root motions might be considered problematic. In three voices, treating the root as a real root (Versus treating it as an active third over an absent real root - more on this concept later) results in the skip of a third in the transformation with either of these root motions. In four voices we get parallel perfect fifths or perfect fourths, depending on the voicing and inversion. In these close position strata, parallel perfect fifths are produced, as you can see in the first and last measures of Example VI. As I said before: Since the voices are transforming, this is not a problem: It would only sound "crude" if the voices did not transform. Please understand that I am not intentionally dissing jazz voice leading, in which non-transformational progressions are an integral aspect of the style, but I can not help but point out that non-transformational progressions are not the most attuned to the implications of the series.

I would be remiss if I did not note here that I discovered the elegant simplicity of this transformational logic in The Schillinger System of Musical Composition, and some of the latter ideas in this series come from Joseph Schillinger as well. Though quite controversial - and for many good reasons - Schillinger was one of those flawed geniuses whose work has never been either properly presented, or properly understood. Since some of his students compiled The System after his death, I have no doubt but that he would have presented the material differently, but the real problem with Schillinger's ideas is that, while a few of them (Like this transformational logic) are sublime, many of them are simply ridiculous: There is a lot of crap one must wade through in The System in order to discover the pearls. The reason for this is simple: Schillinger often lost sight of the implications of the harmonic series and went off, wild eyed, into nonsensical and anti-musical speculation. His musical examples also sucked in the worst way (Unless they were not his, of course).

Again, you ought to play these examples without the ties on a keyboard and compare them with the triadic versions. I can explain most of the key features, but there is no substitute for educating your own intuition through actual interaction with the implications of the series.

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Yes: Give me a girl who likes to fish.

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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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Musical Implications of the Harmonic Overtone Series: Introduction

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"There's this thing called the overtone series..." - Leonard Bernstein

The above quotation is from a videotaped interview with the late Maestro, during which the interviewer asked him why he never joined his contemporaries in the atonal avant garde. Say what you will about the man's pecadillos: He was nonetheless far more attuned to the implications of the overtone series than almost any of his peers. From his autumnal interpretations of Brahms - a man I sometimes think was fathered by the series - to his own compositions, this came through again and again over the course of his long and distinguished career.

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My quest to understand music on the most fundamental level possible is a path that I have been travelling for over thirty years now. During that time I have amassed an extensive music library, liquidated the undesirable texts from it, and I am now left with what I believe to be the indespensible essentials: Riemann's The History of Music Theory, Zarlino's The Art of Counterpoint, Fux' Gradus ad Parnassum, Rameau's Treatise on Harmony, Gedalge's Treatise on Fugue, Schoenberg's The Structural Functions of Harmony, Taneiev's Convertible Counterpoint and The Doctrine of Canon, The Schillinger System of Musical Composition, Schenker's Free Composition and Five Graphic Music Analyses, Seigmeister's Harmony and Melody and - my theory teacher from graduate school days - Dr. Gene Cho's Theories and Practice of Harmonic Convergence. There are many more garden variety text books in my library, of course, but the aforementioned are the ones that I consider to be of note, and that have helped form my current understanding. I have been through them all; some several times.

Lacking in all music theories that I am aware of from Western history is a neat and tidy description of why music works, and why it has evolved as we see from the historical record. There is no Einsteinian General Theory of Musical Relativity... yet.

For such a proposed theory to be compelling, it would have to relate directly - in all of its aspects - to the very nature of sound itself. Harmony, counterpoint, rhythm, melody, and form would all have to be explained as having originated within some feature that God and nature have given to sound, and sound alone. There is only one candidate for the feature I am describing, of course, and that is The Harmonic Overtone Series.

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The harmonic overtone series is present in each and every sound you hear: From birds chirping to sirens blaring, anywhere periodic vibrations are generating sound, the series is present. Musical instrument design is positively dominated by the series and its implications: The nodes on every vibrating string and in every oscillating column of air generate it, and instrument design is impossible without the understanding of it. As a guitarist, I can touch the series: I can touch each and every node up to the eighth partial and make them all ring out. I can hear the series in a direct and immediate way, and I use it in the music I write. It is no minor phenomenon of little import. On the contrary, every aspect of music is implied in the series, and every aspect of music is predicted by the series, as I shall demonstrate.

My fascination with the series and its implications began almost from the first time I chimed a harmonic on the guitar, but it really came to fruition during my years as a Synclavier owner/programmer/guitarist in the 1980's. In those days I was in a band and had a management contract, and I did Synclavier programming on the side. I was paid to practice and create and produce new and interetsting sounds. Some days I'd spend eight hours or more just programming timbres on the Syncalvier. I was positively addicted to the thing.

Since the Sync's voice architecture was a combination of additive synthesis and frequency modulation, I was able to reach into the chest cavity of sound and touch its heart. I could grab each harmonic by the throat and adjust its amplitude. I could nudge each harmonic in relation to its neighbors by shifting its phase. I could crossfade several waveforms in sequence to allow the different series iterations to do the bump and grind against each other. I learned about the nature of sound under the tutlage of sound itself.

Though my intuitive understanding of sound - the harmonic series and its implications - was quite well developed by the time I returned to school for a graduate degree at the end of the 1980's, my ability to articulate that understanding and to convey it to others was nearly totally absent. Like many artists, I'm sure, my ability to understand my art and my ability to describe that understanding are two very different things: My abstract reasoning abilities are way beyond my verbal talent, so please be patient and excuse the linguistic crudities and misspellings that you are bound to encounter.

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Though I had heard a little twentieth century atonal stuff - I can't call it music - when I was attending Berklee, it was easily ignored. I was into jazz and rock music, and there weren't really any raging proponents of twentieth century "serious" stuff there. At least, none that I can recall. In any event, twentieth century music struck me as decidedly unmusical from my first encounters with it, but I didn't really put any thought into why I found it distasteful. That all changed when I returned to school to pursue a Master of Music degree.

I'll never forget the first time some pompous gasbag said to me, "You just don't understand this music." when I dismissed an atonal work as inherently anti-musical. Being the hot-head rock guitarist I was, I was furious, of course. My reply was something along the lines of, "I understand it perfectly well, and it's $#!*!" I had no doubt but that I was correct in that assessment, but I was bothered by the fact that I couldn't explain - in irrefutible terms - exactly why that piece was... er... excrement.

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By this time (Circa 1990) I was well aware that the harmonic overtone series was a big, fat major minor-seventh chord - or dominant seventh - and that this overtone chord's inherent instability predicted harmonic progression dynamics. As I got further into it, I realized that traditional functional harmony was a product of nature, and not artifice. As has always been the case, the struggle I had was not with understanding, but with developing concise terminology: Terminology which would allow for the utmost economy of expression in articulating these discoveries, and which would also allow me to explain the musical proofs I was beginning to develop in as exact and scientific a manner as possible.

The key to reaching a critical mass of understanding as well as to developing the required terminology turned out to be... this weblog. In explaining various aspects of music theory here over the past +/- eighteen months, I have narrowed down my choices for verbiage and have homed in on how to realate eveything back to the musical godhead. Obviously, the tipping-point was the recent Harmonic Implications series.

In this outing, I will relate everything to the series except for perhaps melody and form. I am developing some extrapolated deductions pertaining to melodic and formal implications of the series - and I'll certainly present them at least in passing at appropriate points - but I'm not certain my conclusions yet merit entire chapters in this book. However, they very well may be sufficiently developed by the time I get to that point; I'll just have to wait and see.

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As it stands now, there will be nine more posts in this series:

01) The Descent of Tonality and Modality

02) Root Progression and Transformation Types (Root Progression Dynamics)

03) The Secondary Dominant System

04) The Series' Prediction of Integrated Modality

05) The Series' Prediction of Canon (Harmonic Canon)

06) The Secondary Subdominant System

07) Contrapuntal Implications of the Series

08) Rhythmic Implications of the Series

09) Simple Musical Examples

A chapter on melody would make for the perfect ten, so I'm hopeful that will materialize, but I remain uncertain. The next step will be to enter all of these posts into Word, along with cleaned-up versions of the musical proofs, and publish it as a monograph. Perhaps I'll get that done before I go on tour next year.

In any event, those who have an ear to hear and a mind to understand - and who are able to see behind their musical and societal indoctrinations - will be able to reach no other conclusion than I have at the end of this series: If it's called "atonal music," then it follows logically that it must also be "amusical tone."

En garde!

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