Friday, October 20, 2006

Musical Implications of the Harmonic Overtone Series: Epilog


Philosophical Conclusions


The definition of what music is and is not has been disputed for centuries. Meanwhile, sound itself has been quietly knocking on the door and offering to reveal the answer to us.

Many definitions don't even begin with the proper elements: Wikipedia, for example, says, "Music is organized in time and consists of pitch, rhythm, harmony and timbre." Timbre is clearly an extra-musical quality which sound produced by differing means exhibits, and this definition completely ignores formal elements, except by implication from "organized."

The proper five elements of music are harmony, counterpoint, melody, rhythm, and form. Western art music began with monodic plainchant, evolved through the modal counterpoint era, and then to the harmonic system with tonal counterpoint and homophonic music. Rhythmic and formal elements - being simply local and global aspects of the same phenomenon - were mixed in and present from the beginning, and so they evolved concurrently with the other elements. Thus we see that composers intuited music in practice from the simplest melody/rhythm/form-only systems to the most complex harmono-contrapuntal/rhythm/form combinations.

Before the complete harmonic system was ever perfectly codified, however, Western art music jumped the tracks and went off in an anti-musical direction. Even today, there are still some who actually believe that Wagner forshadowed the atonalist movement with his opening to Tristan. Aside from the fact that this famous sonority is just a diminished minor-seventh in sound - in root position, no less - it is obviously just an ancient so-called French Sixth chord with a chromatic lower neighbor to the minor seventh. Hell, the thing even resolves into actually being a classic "French" V(d5m7)/V on the final eighth note of the measure in question before resolving to the primary dominant sonority as expected. Before the resolution it would have been V(4/#2/b)/V. The voice leading is "wrong" from a trasformational logic perspective... because it's a harmono-contrapuntal effect (!), but the chord itself is a simple altered secondary dominant, and not even a remote one at that. Gut-bustingly, even Wagner was ignorant as to what he was intuiting: What he was intuiting was entirely predictable given the implications of the harmonic overtone series, and that was amplification of the leading-tone/leaning-tone properties naturally present in every sound we hear.

That Arnold Schoenberg, the founder of all things "twentieth-century" in music, cited this feeble example from Wagner - and that others like Alban Berg agreed with him - as some sort of premonition of atonality, while a guitar playing cowboy from Texas could even figure out what was really going on... well, let's just say G-d has a great sense of humor... and justice.

Over the course of the past fourteen posts, I have demonstrated how aspects of every musical element - harmony, counterpoint, melody, rhythm, and form - are predicited by implications present in the natural harmony - the overtone chord - that nature bashes us in the eardrums with in every pitch we hear. This overtone chord, which has become known to music theorists as a dominant seventh chord, has desires present in it which wish to proceed: The leading-tone and leaning-tone tendencies, which are contained in the diminished fifth between the major third and minor seventh of this sonority.

If we simply stand back and let this dominant seventh chord do its thing, it works out for us the entire integrated tonality system - all of the secondary dominants and secondary subdominants - as well as the entire integrated modality system - all twenty-four possible tonics and all twelve possible overtone sonorities.

All we have to do is decide between the two possibilities the series implies - resolution to a major target or resolution to a minor target - and then decide if we wish to end up with a diatonic system (Ionian, or one of the viable modes of Ionian), a nona-tonic system ("Melodic" minor, Blues tonality, Flamenco tonality, and the like), or if we wish to operate within the complete dodecaphonic integrated tonal or modal systems: These outcomes depend solely upon how many resolutional cycles we allow, and whether the targets we chose are major, minor, or a combination of both.

What the problem is with a twelve-tone system in which "One tone relates only to another" - Schoenberg's words - is that what he was actually trying to define is a system in which all twelve tones are neutral: This is simply not possible. At all. Ever. No matter how hard you wish it wasn't so, wishing won't make that fact disappear.

In order to discover the truth about a thing, you have to consider every possibility surrounding it, even those you may personally find distasteful, and especially those in which you have a large degree of interest invested. The fact that every pitch in the twelve tone system carries an overtone chord along with it - like so much baggage from past realtionships - and since the relationships between these pitches are actual ratios from the overtone series (Or close enough approximations, regardless of the temperament scheme you chose), there is exactly a zero percent possibility that you can present these twelve pitches in a way that makes them all neutral: The pitches themselves desire to have relationships with each other, and to establish a hierarchy.

Every pitch desires to have a relationship with the tone a perfect fifth below it through the inherant resolutional desires present in it's overtone structure. But, before it can do that, it must become one of the two possible tonics - either a major or minor triad - by acquiring a fifth and a third. Though it seems counter-intuitive, the pitch in question becomes a tonic by being targeted by the overtone chord a perfect fifth above it: The integrated modal system is really an endless Mobius Loop of minor tonics becoming major tonics before acquiring a minor seventh in order to become the overtone sonority on that pitch, and then they reasove and are absorbed - literally - into the pitch a perfect fifth below, where the process begins anew.

In actual musical practice, however, a hierarchy must be established which nominates and establishes one of the pitches as supreme - whether it be as a major or a minor triad - in immitation of the most perfect diatonic system that the series implies, which is the major key or Ionian mode. This tonic may be any of the pitches in the chromatic system, but it must be one of the pitches in the chromatic system, and it can be either major or minor, but it must be one or the other (Though major may end in minor and vice versa: In fact minor tonics wish to become major tonics by the end of pieces, to which hundreds of years of the tierce de Picardie practice can attest).

Once this tonic is established, the entire secondary dominant and secondary subdominant solar system orbits around that star. Around the respective secondary dominants and secondary subdominants orbit their moons, which are the various major and minor substitutes for them which are on the degrees a third above and a third below those degrees. And so, the old pissed-upon Musica instrumentalis - the lowliest of the ancient musical divisions - finally reveals itself to be the real and true expression of Musica Universalis or The Music of the Spheres.

In order to accept any form of what has been called atonality as music - serialism, stochastic process, &c. - you would have to expand the definition of music beyond what the harmonic overtone series defines as musical. You may do that for yourself, but I simply shall not allow for it.

You can argue that what Arnold Schoenberg, Alban Berg, Anton Webern, Elliot Carter, and John Cage created (A woefully incomplete list of infamous musical miscreants) was some sort of sonic art - I am personally 100% certain that all of them (And many more) were (And are) mere charlatans - but there is absolutely no way that you can claim that it is music.

It is not I who prevents the above named "composers" - the first set of composers in history who began decomposing before they died - from being classified as creators of music, it is the actual nature of sound itself which prevents it. In other words, it is not I who rebuke you, but G-d.

I don't expect that I will change the minds of those who refuse to change or believe the truth, but neither do I feel any pity for those who choose to remain willfully ignorant of the musical implications of the harmonic overtone series: They have chosen for themselves the obscurity to which they will be relegated.

One thing I have no fear of is human intuition: Those not blinded by the petty conciets of their puny egos will always recognize music when they hear it. Plus, those who truly love music will always learn how to create it. And so, the general populace will continue to eschew all forms of atonality - as they have for over fifthy years now - the pseudo-intellectual class will continue to pretend that they "get it" while clutching desperately at the figleaf that hides the obvious fact that they have absolutely zero inate and intuitive musical talent, and the real composers who do have talent will continue to churn out real, actual music. You know: The kind of stuff people applaud.

I am perfectly certain of this.

"Music must never forget itself..." - Wolfgang Amadeus Mozart


And so ends the first complete rough draft of the musical implications of the harmonic overtone series. I have, for the first time in music history, shown the relationship between all five of the musical elements - Harmony, counterpoint, melody, rhythm, and form - and the harmonic overtone series. No biggie.

I have based my work upon the previous efforts of every music theorist in all of Western music history, plus the more ancient Greek works that they reference: Hucbald, Prosdocimus de Beldemandis, Heironymous de Moravia, Guido of Arezzo, Anecius Boethius, Gioseffo Zarlino, Johann Fux, Jean-Philippe Rameau, Frederich Marpurg, Johann Albrechtsberger, and all of the moderns: There is hardly a muisic theorist in history who has not contributed to my understanding, and for that I thank them. My work is their work.

The last thing I ever intended was to write a philosophical music theory book. This work is simply the result of my love of music, and the insatiable hunger for understanding that love has created and fuelled. I have a collection of music theory books that would put many colleges to shame - and they are not just decorations or status symbols: I've read and studied them all - and this present work is simply the result of thirty years of working to understand everything I can about music.

My understanding is far from perfect, but it's pretty "effing" good.

Just a couple of shelves worth: The top shelves.


It's not like she's going to melt, or anything.

Thursday, October 19, 2006

Musical Implications of the Harmonic Overtone Series: Appendix III


Melodic Musical Examples


This appendix example is also harmono-contrapuntal in conception, but I used a combination of large scale strategic and small scale tactical processes to create both the melody and the bass line that lend themselves to clear analysis, which is why I chose it.

Before I studied the Theory of Melody book of The Schillinger System of Musical Composition all of the melodies I wrote were almost completely the product of my intuition. Sure, I thought in terms of outlining the harmonies with melodic elements &c., but I didn't use large or small scale planning to organize them. This has been true for almost all composers of history, so far as the written theoretical record is concerned. If any composers did develop these kinds of tactics and strategies - which I suspect some like Bach and Brahms did - they were never recorded in a pedagogical manner.

Though the inherent genius of a gifted intuition has doubtless lead to some of the most sublime melodies ever written, Schillinger demonstrated that melodic elements could be categorized and analyzed to inform the intuitions and intellects of those of us who are not so trancendentally blessed. I know that my own work improved markedly after my exposure to Schillinger, so I'm sure this will work for others as well.

You'll notice in the analysis line of today's example that I have dispensed with the traditional degree analysis symbols. I believe my previous contrapuntal examples have amply demonstrated the shortcomings of traditional harmonic analysis in a contrapuntal context, so now I present a new alternative for the first time.

There are only five possible functions that harmonies can have: Tonic, subdominant, dominant, secondary dominant, and secondary subdominant. Every harmony on every degree of the intedrated tonal/modal twelve-tone system will fall into one of these categories. Tonic, subdominant, and secondary subdominant harmonies can imply either major or minor base triads, but dominant and secondary dominant harmonies will always imply major base triads because they are overtone chords or altered overtone chords (The real root may simply be absent in some cases).

Therefore, there are eight possible harmono-contrapuntal analysis symbols: T, t, SD, sd, D, SD2, sd2, and D2. These symbols correspond to tonic major, tonic minor, subdominant major, subdominant minor, dominant, secondary subdominant major, secondary subdominant minor, and secondary dominant.

Not only does this make more sense for contrapuntal textures, but it simplifies and streamlines the process as well. There is no substitute for degree analysis in harmonic textures because it is that very degree analysis that allows for root progression patterns to be created (Or detected, if you are analyzing the work of another composer: I use analysis in the compositional process, so that is the perspective I'll usually speak from). Beyond that, however, this functional analysis tells you how any particular harmony is actually working within the system implied by the harmonic overtone series, and so it is even more valuable than degree analysis in some ways.

With this system and in this context root progression and transformation indicators are also not needed (Traditional "common practice" composers did not use consistent transformation technology anyway, since it hadn't been figured out back then), which only makes the process less ponderous and frustrating.

There are instances of ambiguity, of course: No system is perfect, but there are certainly less ambiguities with figuring out functional categories than degree orientation in two part counterpoint.


This piece is the thirteenth of the Eighteen Axial Studies I wrote for solo guitar between 1987 and 1997: It dates from 1993. The idea for these came directly from Joseph Schillinger and his concept of the zero-axes of melodies. He used as one of his examples the subject from the D Minor Organ Fugue, which has a zero-axis that is actually played, versus being only implied (This piece is almost universally attributed to J.S. Bach, but I can tell you with 100% certainty after analyzing it and just a ton of Bach's music that he did not compose it. Beyond that, as a guitarist I can tell you with virtual certainty that the original must have been for the lute There is the remote possibility that it could have been for the violin). Modern scholars are now beginning to realize these things).

Anyway, I realized that the played zero-axis concept could lead to a nice series of idiomatic guitar pieces, and since the zero axis could be the root, third, or fifth of a tonic major or minor triad, there were six possibilities for each axis. For the high E string that worked out to E major, E minor, A major, A minor, C major and C-sharp minor. For the B string that lead to B major, B minor, E major, E minor, G major, and G-sharp minor. Finally, the G string lead to G major, G minor, C major, C minor, E-flat major, and E minor. This one, therefore, is the first of the G-Axis Studies in G major.

For this melody's organizational strategy I used two structures: A directional unit and a rotational unit. The directional unit is a series of four notes which all progress in the same direction, while the rotational unit is a series of four notes which make upper and lower neighbor relationships between a repeated pitch.

For the directional units, there are only two possible forms: They either go up, or they go down. The rotational units, however, can appear in four forms: Original, inverted, retrograde, or inverted retrograde.

Both directional and rotational units can appear in conjunct contrapuntal forms or disjunct harmonic forms, though I do not exhaust all of those possibilities in this particular piece.

The directional units are labelled with the letter "A" in the melodic analysis, and the rotational units are labelled with the letter "B." The two possible orientations of the directional unit are labelled A and A', while the four forms of the rotational unit are labelled B, B', B'', and B'''. I have written which orientation is which in the melodic analysis.

The first section of this piece is a ten measure phrase that repeats. The bass line is interesting in that, after a tonic half note, it progresses stepwise up the entire diatonic scale to the seventh degree in a series of quarter notes, and then proceeds in alternating falling fifths and rising fourths through all of the diatonic degrees again, but now in half notes. The phrases turn-around is then an implied four to five. If you sing this bass line, it is actually quite infectiously catchy (I have it as an "earworm" just thinking about it).

Over this bass line, the melody descends through six degrees of the diatonic system before turning around in the second half of the first rotational unit. So, the first two measures are the decending version of the directional unit, and the second two are the original form of the rotational unit.

Though the nature of the directional unit is clear - they always are, by definition - the merge into the first form of the rotational unit makes its nature unclear. So, even though it is a rotation on the subdominant level - fa, mi, fa, sol - I immediately repeat it on the tonic level above in it's original form as do, ti, do, re. These are also intervallically strict in relation to each other.

Now I can use the other orientations of the rotational unit and the mind's ear will follow, at least intuitively. The first variation I present is the inversion labelled B'. Notice that as mi, fa, mi, re it is an intervallically strict inversion as well. Immediately after the inversion, I use the retrograde of the original as do, ti, la, ti to turn the phrase around.

This entire first section is completely diatonic to the major mode, and so it sounds quite happy. In fact, diatonic major sounds happy because it conforms most perfectly with the implications of the series for a diatonic system! I have saved the first appearance of chromaticism and the final form of the rotational unit for the repeat. The other form of the directional unit appears in the repeat as well.

As you can see, I use a D2 in the form of an augmented sixth interval targeting the fifth degree at the beginning of the third system. Along with the resolution to the D this creates a chromatically altered form of the inverted retrograde of the rotational unit, which is highly effective and satisfying.

The dominant targets the tonic as expected, and then another measure of dominant functionality prepares for the next section. The final two measures present the ascending form of the directional unit for the first time, only now it is intervallically expanded to a series of thirds, whic create, en toto a tonic major seventh arpeggio with the fifth and seventh being reinterpreted as the root and fifth of the D-function chord.

After all the sweet happiness of the first section I throw a couple of wrenches into the works. The second section - which has the function of an interlude - metrically modulates to 3/4 time, and the dominant sonority at the end of the previous phrase "resolves deceptively" in traditional parlance to the sixth degree tonic substitute.

The bass part of the first phrase of the interlude allows for a secondary dominant targeting the primary dominant, while the melody above has created a secondary dominant targeting the preceeding subdominant degree. This combination of the broadening out of the pace combined with the more expressive implied harmonies give this interlude a plaintive, yearning quality. That the expected primary major tonic never appears adds significantly to this.

In this repeat as well I have employed some extended harmonic implications of the series: In the first measure of the fifth system I have used a double chromatic approach to the dominant degree, the final of which creates another augmented sixth. It is important to note that the progression from c-sharp to e-flat is a diminished third, and so the ear hears this as a contrapuntal whole step and not a harmonic third. Kind of a nifty effect.

This primary dominant harbors an augmented triad in the melody, which is kind of unusual in a major key piece, and it sets up another measure which makes the repeat a five-measure phrase. You can see from the functional analysis that a momentary tonic inference is made at the beginning of that final measure, and then a dominant series which is simply a degree exchange between the upper and lower voices.

The third section is the "real" second section: The interlude was designed to be a pause in the texture, which would have become oppressive if it had continued unabated. Here, the two directional units are used in a sequential section, with the original stepwise descending version now following the ascending intervallically expanded form.

The bass part has broadened out further into tied half notes. There is a rest in the second measure of this secion because the guitarist must physically release the low G in order to reach the upper notes: It remains implied.

The disjunct motion followed by conjunct motion implies alternating harmonic and contrapuntal effects in the texture, and it is quite sweetly diatonic again for these eight measures. As you can see, the initial ascending tonic triad and the following ascending supertonic tetrad both reach the same G. This is at the fifteenth fret on the guitar, and so is quite high.

Oh yeah: Restrictions on the distances between voices in counterpoint are poppycock for instrumental music. As you were.

The overall harmonic motion is four measures of tonic, two measures of subdominant, and then the dominant for two measures.


The continuation of the second section - now in its third phrase - offers another deceptive motion from the preceeding dominant to the submediant tonic substitute. This is followed by two measures of a secondary dominant function, which leads to the expectation of a resolution to the dominant, but this is spectacularly thwarted.

On the second system down is the climax of the piece. Remember that ascending scale in the bass of the first section? Though it rubbed up against the zero-axis it never actually made it to the tonic degree. That was a setup, and here is the fulfillment. The G in the bass part - as opposed to the G in the zero-axis - is the resolution of that preparation of long ago: It is also the highest note in the bass part, and since the axial G is an open string, this real unison is played on the guitar (Take that, you single-manual keyboard players).

The bass part descends in a chromatically altered form of the original directional unit into a D2-function sonority in the form of a doubly-augmented fourth. This resolves across the barline into the primary dominant, satisfying both the e-flat and the previous c-sharp that was leapt out of. Remember the c-sharp to e-flat bass motion from the first interlude? Another setup for this climactic resolution. The b-natural in the melody, by the way, is at the nineteenth fret on the guitar: That is the highest note on the traditional classical instrument. This climax is only possible because the D in the bass part is an open string.

In the final four measures of this section we have the denouement of the piece, and again, this only works because the E, A, and D in the bass are open strings on the guitar. I remember when I wrote this how amazed I was that it all worked out so perfectly: It's a great combination of pure music and an idiomatic guitar piece.

As I used an augmented triad earlier, I have answered that with a fully diminished descending directional unit to end the section. This gives a whistful end, and the overall effect of the piece is sort of Romantic, even though it's contrapuntal in conception. This diminished tetrad is also the first intervallic expansion of the descending directional unit to appear.

Following this section is a second interlude. This is almost exactly like the earlier one, except for the fact that the melody is an octave lower, and the resolution is as expected to the tonic, versus the sixth degree tonic substitute.

The repeat is quite different, however: Since the peiece will come to an end after the second time through this interlude, I do not want it to set up the leading-tone/leaning-tone dual target to the dominant degree again. For this reason I have allowed the c-sharp to resolve at the end of the first measure of the repeat, but I immediately introduce the sixth degree tonic substute to set up the final element at the ending of the piece.

In the penultimate measure of the second interlude's repeat is the only secondary subdominant in the piece, and it is the traditional so-called Neapolitan Sixth, but instead of preparing the dominant it resolves to the tonic (I could have put a small letter "t" there, but I was factoring in for the G-axis, which I usually ingnore in the functional analysis, naturally).

This sets up the traditional plagel "Amen" in the last three measures of the piece.


This concludes the appendices for the present version. I suppose I ought to have some examples to demonstrate rhythmic and formal aspects of the implications of the series, but to be honest, I don't concieve of pieces from those perspectives. The musical material I develop combined with the local tactics and regional strategies I employ are what determine the rhythmic and formal properties of the pieces I write. I know of composers who say the first decision they make is what the duration of the piece will be, but that is almost impossible for me to imagine. The possibilities of the material I come up with determine the length for me.

I will write a final epilog with some philosophical conclusions, and then I'll be done with this for the time being.

I suppose that is technically possible.

Wednesday, October 18, 2006


Not bad for an old cowboy in his forty-ninth year.

But then, cute young girls in impossibly embarrassing situations are a dime a dozen.

Right? Riiiiiight.

Musical Implications of the Harmonic Overtone Series: Appendix II


Contrapuntal Musical Examples


Again, real living music combines aspects of all five of the musical elements, but today's examples display a conception which is biased in the direction of counterpoint.

Before composers had intuited out the harmonic system, stepwise contrapuntal effects positively dominated contrapuntal melodies and disjunct harmonic effects were rare, primarily because the methods for handling them hadn't been worked out back then. J.S. Bach was the earliest composer to offer both a summation of almost all that had come before him in compositional history, and a perfect blending of counterpoint and harmony.

I have read some theorists who think that much was lost when harmony invaded counterpoint, but my own view is that the older modal style was simply not fully evolved music. Besides, there is nothing preventing composers today from writing music with a very pure contrapuntal conception if they so choose, and the entire chromatic system is now available to them. Some of Penderecki's music displays these features, for example. As for myself, I tend toward preferring the rhythmic drive that well ordered root motion patterns provide, so my contrapuntal style is biased in the direction of harmonic effects, as you'll see.


This example is the Menuetto Sans Trio from my Irreducible Sonatina of 1992. The piece actually dates from 1986 though, and is in fact only the second piece I ever composed in the contrapuntal style. It is the second oldest piece in my set too, behind the Six Variations in A minor that conclude the sonatina. I understand that Scherzo-type menuettos are supposed to be notated in 3/4 time - and that it is a part of the "joke" of the scherzo - but I prefer the ease of reading and understanding, as well as the reduced measure count, that 6/8 time provides.

The simple and streamlined harmono-contrapuntal style here is not Bachian at all. In fact, the direct inspiration for this piece was the Scherzo of Beethoven's Ninth Symphony, which is my single favorite piece in the entire symphonic literature. This piece is not immitative, however, and is in the simple ternary form of A, A, B, A. The piece also does not modulate, remaining in B minor throughout.

While two-part counterpoint tends to sound empty and incomplete on keyboard instruments - especially the piano, which is kind of hostile to counterpoint in general - on the guitar it is quite idiomatic and fully satisfying. The problem and the advantage with two-voice contrapuntal textures, of course, is that the harmonies implied tend to be nebulous - often offering more than one possible interpretation - and the freedom that the melodies have is very great. Keep this in mind as we go through the analysis.

Beginning simply on the tonic of B minor with the interval of a minor tenth, the second half of the first measure proceeds to the subdominant. I put a iv(m7) symbol in the analysis, but what is happening in the melody is actually a contrapuntal effect: The third from the tonic harmony is held over as a syncopation - or a suspension if you prefer - and as a dissonant seventh it then resolves down to become a major sixth. As I say, this is a purely contrapuntal effect. This resolution could also be harmonically interpreted as a ii(6/3) chord, which points out the nebulousness of applying harmonic terminology to counterpoint - especially in two voices - but I chose to call it a subdominant chord because root position inferences take precedence, and the sixth proceeds further to the perfect fifth before the end of the measure due to the triple time.

This subdominant also proceeds to the dominant, as expected, at the beginning of the second measure, and the lower neighbor major second over the root of the dominant is another vigorous contrapuntal effect. In the second half of the second measure, however, a purely harmonic effect is introduced: A rising tonic arpeggio. This is over a third in the bass line - and in harmony we would not want to be doubling the thirds of chords - so this is what I call a harmono-contrapuntal effect: It has properties of each element.

Measure three begins with an implied bVI(6/3), and this is a great example of that harmony functioning as a tonic substitute. The folowing implied bIII can also be thought as a tonic substitute, but again, the eleven-ten resolution above the bass note is a purely contrapuntal suspension-resolution effect. The major ninth at the end of the measure is quitted by leap in the bass, and this is not an issue when the top voice continues by step and the bass arrives at a consonant relationship with it.

The bVI in the beginning of measure four is a subdominant substitute, and the five-four-three progression over the bass note is another highly contrapuntal effect. This prepares for the dominant harmony that concludes this antecedent phrase, which I have labelled as a V(4/2), but this is yet another harmono-contrapuntal effect, as the minor sixth there can also imply a harmony on the supertonic - or even an expected six-five over a subdominant harmony - so this again points out some of the limitations inherent in using purely harmonic analysis in a contrapuntal context. The fa sol there does imply to the ear that it is a kind of IV V with a harmono-contrapuntal syncope present in it. It is quite a nice effect.

For the consequent phrase beginning on the second system, the analysis is the same for the first two measures as the antecedent's was. The third measure of the consequent phrase, however, contains a contrapuntal effect I have labelled as a ii(d5) to i(6/3), even though it is quite apparent to the ear that the supertonic to tonic motion is implying some kind of dominant to tonic progression. From this point on the two melodies are moving in a purely contrapuntal contrary fashion, and this provides a powerful conclusion to the phrase.

In the last measure of the phrase is the alternate to the previously implied V(4/2) effect, and with this version's stepwise movement, it really is more like a subdominant to dominant progression, and then the tonic appears to end the phrase with a purely harmonic effect: A descending tonic arpeggio.

The second section begins exactly as the first sections antecedent and consequent phrases did, but any possible enticipation of yet a third repeat is dashed with a very strange - and I mean highly unusual from the historical perspective - effect which I have simply labelled as (4/2). The subtonic rarely appears in this fashion in tonal counterpoint - it is usually a raised leading tone, especially when it proceeds upward as this one does - but the seven-six-five contrapuntal effect over it is easy to understand, and I can tell you that I was thinking of a compound axial arrangement: The subtonic is actually targeting the dominant degree, while the intervening supertonic is actually targeting the third degree. This gives the bass part here a nice driving effect.

The two sequential and sequencing phrases of the second section alternate between contrapuntal and harmonic effects, but the phrases are very clearly biased toward harmonic effects, and the abundance of progressive root motions adds serious propulsion to the music. The fully diminished sonority at the end of the first of these phrases also turns the phrase around on itself in a powerful manner with the diminished seventh proceeding to the perfect fifth as it does (In an added octave separation, of course).

Where there is a turn-around at the end of the first of these two phrases, the second has a half-cadence effect which allows for the third, and final, phrase. This phrase is almost completely harmonic, or harmono-contrapuntal in nature, and has a nice sol fa me re do gravitational return to the top contained in it.

With the repeated eight measure first section and the twelve measure second section, there is a 16:12 proportion to the form. This reduces to the perfect fourth's ratio of 4:3, which is pretty common in music, and yet another example of the series at work in the intuition of musicians: I certainly didn't plan it, but then, I wasn't surprised to discover it either.

A lot of audience members ask me who wrote this piece, and when I tell them it's one of mine, most of them seem surprised. I must admit that it does sound like it has existed forever, and that I simply discovered or rediscovered it. I love simple little gems like this.


"And now for something completely different" as the old Monte Python routine went. This is a jazz swing tune with the swing written out in 12/8 time. I wanted to show this piece off as an example of countrapuntal composition in the jazz idiom. Though I am aware that others have come up with jazz counterpoint styles, I'm not really familiar with most of them, and the pieces I have heard, like Steve Reich's piece that Pat Metheny used to perform, haven't really impressed me in the least, to be blunt.

The problem has been, from my perspective, that the purity of the straight ahead swing style has not been preserved in the application of the contrapuntal concepts, and much was lost in the rhythmic drive department. I solved that problem by writing the piece using the actual technique that jazz composers use. Since I was a jazz guy long before I moved toward traditional techniques, this was totally natural for me.

I think the history of this piece is kind of humorous, as I wrote the melody back in 1979 when I was studying at the Guitar Institute SW under Jackie King and Herb Ellis. This was an actual assignment, in fact. I was given the harmonic continuity from an old standard - I wish I could remember which one it was - and told to write a swing or bebop style melody to it: The melody you see here is exactly what I came up with when I was twenty-one years old.

Jackie loved this tune, as did I, so I kept it around: I'm glad I did. Last year I was doing arrangements of some of my jazz compositions for a guitar duo who are students of mine, and when I dug this piece out of the archive I realized I could make a solo guitar arrangement of it in the jazz contrapuntal style I'd developed, so composing this piece "only" took me twenty-seven years.

This is actually the concluding "menuetto" of the Scherzo of my first guitar sonata now: There is an opening version of this piece followed by a "trio" variation preceeding this. I figured this was enough for the demonstration.

Since I wrote this piece starting with a standard jazz technique - write the harmonic continuity first, and then the melody - I have put the chord symbols for that progression above the staff as you would see in a regular lead sheet. This way, you can compare the original progression the melody was composed to with the bass line, which I composed using good old fashioned cantus firmus or cantus prius factus technique.

For that bass part, I decided to think just like a jazz bassist does, but with the added dimension of creating the bass line in non-real time and concieving of it as real counterpoint against the melody, versus a simpler improvisational approach. So, I'm holding down the primary structure tones there - the root, third, fifth, and seventh - and I'm adding directionality with series-implied directional units such as chromatic passing tones and chromatic approach tones, but I'm also applying the three immutable rules of contrapuntal motion, almost without exception (There are a couple of exceptions of artistic license).

The pickup is actually an augmented sixth interval, which fits with the joking nature of this piece as a Scherzo. I mean, how much bigger of a joke in a "classical" multi-movement sonata can you get than to make the third movement a swing tune? This interval is actually quite common in jazz music in an incidental way, due to all of the chromaticism jazz bassists use, but I'm applying it in the traditional manner here (Except that it is a pickup).

In the first measure you can see that the bass part is just like what a jazz cat might play - 1, 5, 1, M7, m7 - but I'm being careful not to imply parallel perfect consonances or parallel dissonances in stepwise motion. Leaping into dissonances is not an issue in this style, however, as the effects are harmono-contrapuntal in nature. The second E in the bass part in the second measure is an example of this: It is just an octave movement of the first E, so the minor seventh is percieved by the mind's ear as a harmonic effect. There is another augmented sixth at the end of the second measure, and employed as a descending chromatic passing tone, it is entirely stylistically consistent with what a jazz bassist is likely to do.

After the three preceeding descending chromatic passing notes, I used an ascending version at the end of the third measure. This creates a minor ninth to minor seventh progression across the barline - something you wouldn't expect to find in traditional counterpoint, but in this idiom is is really quite cool, and is yet another contrapuntal effect, the result of which is interpreted harmonically.

The 3, 1, 3, 5, d5 progression in the bass in measure four is also a typical jazz bassist kind of a deal, and the following long ascending line with chromatic passing embellishments is also quite idiomatic. This results in parallel minor sevenths into the second beat of measure six, but I wanted the line, and so took the license. I'm allowed.

Once the line has risen for two measures in a slick and jazzy way, it turns around at the beginning of measure seven and descends stepwise for two measures. This is a really swinging little episode.

You can see in the analysis that I am analysing key changes before the tonic appears, and this is a stylistic feature of the ii V I mentality of traditional swing era composers: They thought like this.

Note also that the first four measures of the bass line is a compound axial combination: There is a lineal progression going on at the lowest level, but the D's above, and the approaches to them, create another melodic axis for the bass line. Jazz guys do not organize on this level, that I am aware of, but it does add quite a lot to the quality of the bass part. I bring this up now because I flirt with the same concept in measures ten through thirteen: It adds balance with the linear episode in between and then returns to being a single harmono-contrapuntal line leading to the turn around, which is made via the original augmented sixth, of course.

The second ending starts out the same as the first, but then goes it's own way. Notice that I have labelled the E-flat majorminor-seventh chord in the second measures as a tonic, even though it is an overtone (dominant) sonority. This kind of thing goes on in jazz all the time, and relates back to the blues-based origins of the style.

There is a lot of modal interchange going on in the second ending as well. The ii(d5m7) V(m7m9) of minor targets a major tonic at measure twenty, and then this is followed by a minor-origin iv(m7) bVII(m7) progression. I am following the same stylistic and harmono-contrapuntal protocols in the bass part throughout these areas, and the result has a high level of consistency to it. I am still intermittently implying compound axes in the bass as well.

At the end there you can see that the final motion to the tonic is by an augmented sixth, as the first was, but this one isn't syncopated and it is an octave further apart. This provides an effective close, and the stylistically appropriate I(sus6add9) sonority is the Scherzo's final punch-line.


Yet another old appliance-related incident. I'm sure household embarrassments like this are less common today. Shame.

Tuesday, October 17, 2006

Musical Implications of the Harmonic Overtone Series: Appendix I


Harmonic Musical Examples


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

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

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


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

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

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

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

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

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

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

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


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

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

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

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

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

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


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

Monday, October 16, 2006

Musical Implications of the Harmonic Overtone Series: Chapter X


Formal Implications of the Series


NOTE: I have been going back and revising some of the earlier posts to correct misspellings, awkward grammar, and to get rid of some of the more overwrought sentences. I've been trying to clear up some of the more opaque sections with these revisions as well. In doing this, I have renumbered the chapters, so the chapter numbers in the examples won't match up after chapter four now. I'm revising all of this in the original Encore example file - and I'm using consistent Roman numeral/Arabic numeral logic in it as well - so the final revision which will go into Word and end up as a PDF file will be far better sorted out. I think that after I'm done with the current version, I'll only be a couple of revisions away from having this in a publishable form.


One of the nice things about going over this material multiple times is that each go-round I gain new insights I didn't have previously. So, not only did I extrapolate some melodic implications from the series this time around, but I also got a toehold on some formal implications. This chapter, as it stands now, will be little more than a place holder, as I just started making some connections on this topic since the last time I looked at the rhythmic implications of the series.

Form is really a meta-rhythmic aspect of music - rhythm on a broad temporal scale - so it relates back to the resultants of interference that are derived from the intervallic ratios in the series. Since I am unaware of any composers who have applied this type of logic to form in any other way than intuitively, there is not much material to draw on. Nonetheless, just two quick examples will show that formal elements can be related to the rhythmic resultants of the series' ratios in more than one way.


One of the most ancient musical forms - if not the most ancient - is the simple binary form in 4/4 time with four measure phrases which I have illustrated in Example 1: Note that the local time signature is expanded into meta-rhythmic phrase lengths. The first of these phrases is repeated, and the second phrase brings the piece (Or, the verse) to a conclusion. Obviously, the duple-related time relates to the 2:1 ratio of the octave interval, but the 2*A + B proportions of the form itself do as well. Since the octave is the first interval in the series, it is no surprise that this form was intuited over a thousand years ago.

If you take this form and vary the repeated A-section, you start to move up the series: In that instance you are moving toward an eight measure section - not just a literally repeated four bar phrase - followed by a four measure section. This follows the 2 + 1 antecedent part of the resultant of interference that the perfect fifth's 3:2 ratio gives. Now, if you repeat this entire structure - A8 + B4: A8 + B4 - and then add a bridge to it followed by a final A-section, you get two antecedents of the perfect fifths resultant followed by the consequent: Many songs follow this form or something very close to it.




From this you can see that the possibility of applying the series-derived resultants to formal proportions offers a lot of possibilities which have only casually and intuitively been explored by composers of the past. I really find the resultant's potential application to the meta-rhythmic aspects of form to be far more compelling than their potential use as surface rhythm embellishments, and it certainly offers a more series-derived approach to form than the old Golden Mean method (Which is really nothing more than using one possible ratio all of the time and on every level).

In Example 2 I have presented the underlying skeleton of the twelve bar blues form: Though countless embellishments have been made to this form over the years - from busy Bebop arangements to rousing Rockabilly renditions - this is the irreducible essence of it, and it demonstrates some very interesting characteristics.

If we do a simple measure count for the three cardinal harmonies, we find that the tonic overtone chord appears in 6.0 measures total, the subdominant overtone chord appears in 4.5 measures total, and the dominant overtone chord appears in 1.5 measures total. Multiplication by ten gets rid of the decimal points and yeilds a ratio of 60:45:15, and since all of the terms are divisible by 15 this yeilds 4:3:1. The 4:3 ratio is the series' perfect fourth, of course, and not only is the blues' root motion continuity dominated by fourths and biased toward the fourth degree, but the ratio multiplied gives the number of measures at twelve. Intuition is a powerful force.

This points out another possible meta-rhythmic application for ratios in the series: Durational proportions for tonic, dominant, and subdominant harmonies; durational proportions for secondary key-regions within a piece, &c. The possibilities are mind boggling. But then, my mind is easily boggled.

This concludes the musical subjects addressed by implication in the harmonic overtone series for now. Though many more musical details can be extraplated from the series - the countless totality of them in fact - all of the main aspects of music have now been addressed at least in part. The final section will knit all of these subjects together by looking at some simple musical examples.


What would Mr. Spock estimate the odds for this situation to be?

Sunday, October 15, 2006

Musical Implications of the Harmonic Overtone Series: Chapter IX


The Series' Implications for Melody


Schillinger defined melody as a trajectory of pitch through time, and that is about as concise a definition as one can get. Looking at the big picture in such a general way is quite useful, as one of the things that separates student musical exercises from the work of a competent composer is the student's lack of direction over the long term. In other words, the counterpoint and harmony in student exercises might be technically perfect, but the exercise as a whole might suffer because only the local mechanics were addressed, and not the larger scale axial and trajectorial issues. This is also true, I might add, for harmonic continuities written by students: The voice leading might be technically correct, but in many instances the root motion patterns are not well thought out, or they may be even totally lacking in any organizational structure whatsoever over the course of the phrase.

As good as Schillinger's definition is, and as useful as his axial concepts are, without a thourough understanding of the mechanics of melody on the notational scale, writing them is left almost enturely to the intuition (Which has been the case for most of the history of Western music). So, before we start thinking in the grand schema, we need to focus on the micro-mechanics of melodic movement.

There are two kinds of melodic motion: Conjunct and disjunct. Conjunct motion is stepwise (Either diatonic or chromatic) and it implies contrapuntal effects, while disjunct motion proceeds by leap (Again, either diatonic or chromatic) and implies harmonic effects (Now you can begin to se why melody is best addressed last: It has all of the features of harmony, counterpoint, and rhythm combined in it).

An old axiom among jazz improvisers goes something along the lines of, "There are no wrong notes, only awkward phrasing." This is true: Any of all twelve chromatic tones are available over any harmony, so long as they are employed in accordance with the implications of the harmonic overtone series. In order to learn how to do that, we need to take a look at how notes can function over the various harmonic structures, and how the leading-tone/leaning-tone impetus present in the series can be harnessed and even amplified to get highly directional melodic effects.


On the top system, in Example 1, I have presented Schillinger's concept of the zero axis. The zero axis is basically the gravitational locus of a melody, and it can be the root, third, or fifth of a tonic major or minor triad.

In the second system are the unbalancing axes, which lead directionally away from the zero axes above or below (Gravity in music can be real or artificial: Gravity or anti-gravity), increasing tension as they go. Then, the third system has the balancing axes, which lead back to the zero axes and release tension.

All of the axes in these examples, except for the zero axes, of course, proceed by step and therefore they imply contrapuntal effects. In a nutshell, contrapuntal effects in melodies are the alternation of harmonic and extra-harmonic tones in stepwise melodic progressions. The first of the unbalancing axis examples has the melodic progression D, E, F, G leading away from the zero axis on the tonic of C. So, that axis displays the so-called accented passing tone concept with D and F being extra-harmonic tones between the harmonic tones of E and G (With C as the axis underneath).

The second of the unbalancing axis examples has the zero axis on C again, only this time it is functioning as the fifth of the subdominant chord residing on F. So in this instance, the B and G are the extra-harmonic tones as the augmented eleventh and major ninth respectively. The balancing axis examples are just the other way around over the same harmonies, and so they demonstrate the so-called unaccented passing tone paradigm.

On the bottom system we begin to get to the meat of the matter with two demonstrations of the different types of disjunct harmonic motion within melodies: Primary Structure Harmonic Motion and Secondary (Upper) Structure Harmonic Motion.

Primary structure harmonic motion is simply outlining the triad or tetrad of the moment, or making use of two or more of its tones in a disjunct manner. Secondary structure harmonic motion, on the other hand, makes use of the harmonic tones the series implies which are beyond the seventh: The ninths, elevenths, and thirteenths.

In the first measure of Example 4, the melody simply outlines the dominant tetrad over a zero axis at its root. In the next measure, however, the upper structure major triad consisting of the major ninth, augmented eleventh, and major thirteenth is used. These same tones, when surrounded by chord tones and entered and exited in conjunct motion, imply contrapuntal effects. In this context, however - a context in which they are entered and exited by disjunct motion - they imply higher order harmonic effects.

If you play through the dodecaphonic system at a high enough velocity, you can manage to squeeze all of the notes in the chromatic scale in over any kind of harmonic entity - Chopin made a career out of this - but the moment you start leaping into notes and exiting them by leap, the ear will try to interpret them harmonically instead of contrapuntally.

Since contrary stepwise motion can legitimize any melodic sequence in counterpoint, but harmonic requirements are more selective, greater care must be exercised when you start making melodic leaps.


Since the overtone sonority is in and of itself unstable - containing as it does a tri-tone - it is also the most accomodating to upper structure tones. The series itself gives a major ninth, but so the series-generated minor mode provides a minor ninth. The major ninth is not only from the series, but is also out of the diatonic major system, and then the augmented ninth comes from the nona-tonic blues modality. The series also gives us the augmented eleventh, but almost every series-generated modal system gives the perfect eleventh. The perfect eleventh is the single most problematic tone over any major triad because it creates a minor ninth against the major third below it. This undermines the stability of the major triad because it is a simultaneous clash of wills between a leaning-tone and its target. Not to mention that it sounds like crap. In exceptional harmonic instances I have used the perfect eleventh in a semitone relationship with the major third (tenth) to good effect, but in melodies making harmonic motions it is best avoided. The minor thirteenth and major thirteenth come from the minor and major series-generated modalities, respectively, and both have a very pleasant sound which is much used in jazz music.

Due to the problems inherent with the perfect eleventh over major triads, the tonic function major seventh almost never gets any melodic effects on that degree. For that reason, I put the subdominant major seventh in Example 6, which does have a complete upper structure triad.

Minor sevenths kind of split the difference between dominant and subdominant harmonies, as they can have four - or in some cases five - tones available for making upper structure triads. Perfect elevenths make major ninths with the minor thirds of minor seventh chords, so they are not only workable, they sound very good. The minor or major thirteenths are almost equally workable, and they derive from the Aolean and Dorian modes respectively, but the minor thirteenth is quite a dark effect due to the minor ninth it makes with the triad's fifth. This is not nearly as bad as a minor ninth against a major third, however. I saved the ninth for last, because while the major ninth of Dorian and Aolean can be used with impunity, there is also a minor ninth on the Phrygian chord which can sometimes be used to good effect if you are very careful: Using a minor ninth in a harmonic context over a Dorian or Aolean chord, however, will get you an invitation to the clam bake.

Just as the leading-tone and leaning-tone effects of the overtone chord were amplified or doubled in the harmonic system with the development of the various secondary dominant sonorities, they have also been amplified in the melodic system, and quite remote from the host harmonies from a functional standpoint at that. In Examples 8 and 9 I have demonstrated how artificial leading-tones and leaning-tones can be lept into in a harmonic manner, even though some of them are extremely dissonant over the host harmony, and as long as they target either a primary or a secondary harmonic tone, they work fine. In classical music theory you get appogiaturas, and escape tones and all this other crap - frankly I've forgotten most of that ponderous terminology - and never do you get an explaination as to why they work, and why they carry the effects they do: All you have to do to work out what will and will not work, and why, is to relate it back to the harmonic overtone series and all that it implies.

Another couple of melodic resources to mention are the octa-tonic diminished scale and the hexa-tonic whole tone scale. The octatonic works over fully diminished seventh chords, or their dominant with the real root V(m7m9) chords, and the one that you use begins with a semitone between the leading tone of the moment and the target degree. Whole tone scales work over augmented triads or the French-derived V(d5m7) sonorities. Just start on the root.

In order to explain these concepts fully, we'llhave to look at some actual melodies, which we'll do in the appendix.


Praise the Lord for modern appliances, right ladies?

Musical Implications of the Harmonic Overtone Series: Chapter VIII


Rhythmic Implications of the Series


It might seem counter-intuitive to expect that sound would have rhythmic implications, but it isn't: If you take any given pitch and lower it to under about 15-20Hz (Cycles per second, remember), it will dissappear from human hearing and enter the auditory void - our "deaf spot" which in some ways corresponds to the visual blind spot we all have but seldom notice - which exists between the lowest frequency perceptable as pitch and the fastest frequency perceptable as rhythm. Further lowering and the former pitch will reemerge into auditory perception as a regular pulse, or a simple rhythmic continuity. While in the auditory void between pitch an rhythm, the cyclical repetition is perceptable to touch as vibration.

Now, if you go through the same process with two pitches involved in a harmonic relationship, when they reemerge into hearing, it will be in the form of a rhythmic palindrome instead of an even repeating pulse. This rythmic effect is the result of the interference between the two wave periodicities, exactly as harmony itself is the result of periodic interference between waveforms which have a certain ratio relationship. Joseph Schillinger brought these to my attention, and he also provided the notational methodology for displaying them in visual terms that musicians can understand and perform.

It is interesting to me to be able to see harmonic relationships as rhythmic resultants of interference, but I must admit that I have not employed them in composition all that much. Instead, I am developing a rhythmic methodology which allows harmonic and contrapuntal implications present in the music itself to create rhythmic vartiety. Schillinger insisted that these resultants could be applied to virtually every aspect of music, and if you are seriously interested in his thoughts on the matter, I'd suggest you read his Theory of Rhythm from Volume I of The Schillinger System of Musical Composition for yourself. Frankly, I think there is something rather forced and less than organic about his theorietical methodologies, and I believe that is the result of his continually distancing himself from the implications of the harmonic overtone series, while simultaneously drawing a lot of ideas from it. Schillinger was quite a contradictory individual.

Each interval of the overtone series - its ratio - will result in a unique rhythmic palindrome, and I have demonstrated that for the seven consonant intervals below.


The process for figuring this out in musical notation is mechanically simple.

The first step is to multiply the two terms together to get the number of indivisible units: For the octave that is 2 * 1= 2. Then you must decide what base unit to use: I chose the quarter note. So, in the top space of the upper stave in each system you will see the result of the multiplication process in quarter notes.

The second step is to put the minor generator down below the indivisible unit total: For the octave the minor generator is two quarter notes (I define the minor generator in music notation as the smaller rhythmic unit - in this case the quarter note - and not the smaller number of the ratio, which would be one in this instance: If memory serves, Schillinger does it the other way around).

The third step is to put the major generator underneath the minor generator, and for the octave the major generator is a half note.

Finally, you get the rhythmic resultant of interference between the two terms by dropping plumblines at each attack: For the octave that simply results in an accented quarter note followed by an unaccented quarter note. The first attack in each resultant is accented because that is the place - and, the only place - in the resultant where the two periodicities coincide or have simultaneous attacks. The resultants are on the bottom staves of each system.

The ratio of the perfect fifth is 3:2, and 3 * 2= 6, so there are six quarter notes in the top space of the second example. The minor generator is three, so that comes out to three half notes in the second space. Then, the major generator is two, which comes out to a pair of dotted half notes, as you see in the bottom space. Finally, the rhythmic resultant is an accented half note (Where the two terms attack simultaneously), followed by a pair of quarter notes, and finally an unaccented half note, or 2 + 1 + 1 + 2.

As you can see, as you proceed up the series the rhythmic resultants become more complex. If the ratios are superparticular (The terms differing by one), or if the ratio's terms differ by an odd number, the rhythmic pallindrome will be divisible through it's axis of symmetry, as is the case with every example among the consonances except for the major sixth: It has as its axis of symmetry a dotted half note, as its terms differ by an even number (Two). The minor sixth is divisible through its axis of symmetry because its terms differ by an odd number (Three).

The resultant for the octave is the most common in all of music, and is represented all too well by the Rock and Roll "backbeat" that is so mind-numbingly ubiquitous in popular music.

The perfect fifth's resultant is evidenced in Waltzes, and has been used in popular forms as far back as the twelfth or thirteenth century. Using the first half repeated as an antecedant and the second half in the penultimate measure of a phrase as the consequent, you get this for an eight measure phrase:

||:2 + 1|2 + 1|2 + 1|2 + 1|2 + 1|2 + 1|1 + 2|3:||

As I said, this rhythmic resultant of interference as implied by the overtone series was first intuited hundreds of years ago. The rest remain little explored, and almost nothing has been done with the more complex concords. Personally, I do not start out pieces with a rhythmic conception first (At least, that has been the case so far), so I'm probably not the guy who is best equipped to explore these implications of the series. But, they are there, just awaiting a fertile mind to figure out how to effectively apply them.

I should also note that resultants can be calculated for ratios with more than two terms. A major triad would be the resultant for 6:5:4, for example, and the overtone chord itself would be the four terms of 7:6:5:4. I will add a second page of resultants to demonstrate this for all of the triads and tetrads in the final book version of this series.


Oh sure; happens every day.

Saturday, October 14, 2006

Musical Implications of the Harmonic Overtone Series: Chapter VII


Contrapuntal Implications of the Series


I have added a sidebar section for this series now, as the intial post will soon disappear off of the bottom of the front page, and I do not wish to display more posts per page due to bandwidth considerations. In order to get the title to fit on a single line, I had to use "Musical Relativity Theory," which I thought was kind of humorous. But then, I'm easily amused.


Historically, counterpoint evolved before harmony did, but the contrapuntal desires that tone combinations have originate in the passive-tone/active-tone dichotomy found in the series. This is where the mechanics of melody originate as well, and I am now going to have a tenth chapter which addresses melody, but everything contrapuntal and melodic finds its origin and impetus in harmonic aspects of the series, so it actually makes more sense to present these subjects in the reverse order from their historical order of appearance.

Furthermore, harmony is actually a far more complex system than counterpoint is, despite how counterintuitive that may seem: The problem is that counterpoint, as traditionally taught, is filled with burdensome rule-sets which never explain the underlying logic beneath the concept.

The reason that these rule-sets evolved was because they were based on deduced practices of composers, and not on any underlying scientific logic. So, the rule-set that is traditionally taught as Sixteenth-Century Counterpoint is simply the rule-set which describes the practices of the Giovanni Pierluigi da Palestrina school, and the rule-set that is traditionally taught as Eighteenth-Century Counterpoint is just the rule-set which describes the practices of the Johann Sebastian Bach school.

When I was teaching myself counterpoint, I bought just about every book on counterpoint ever printed in the English language, and studied my way through them all. Books by Benjamin, Fux, Gauldin, Gedalge, Jeppesen, Kennan, Mann, Norden, Piston, Shillinger, Soderlund, Taneiev, and Zarlino, which is not a complete list; only the ones I've kept around. Though virtually all of these authors offered some insight or other that I found useful, none of them ever got behind the descriptioins of the processes to the actual origin of the mechanics.

As I began to compose counterpoint for myself, I began to discard - one by one - all of the rules which I found to be either useless or just plain wrong. Useless rules are - to me - prohibitions which only describe a particular aspect of a composer's style, and not what the series implies is and isn't permissable. Rules that are wrong are simply prohibitions which the series implies are not.

What I was discovering was that, just as traditional harmonic voice leading was a simplified countrapuntal system, traditional tonal counterpoint was only a more intricate one: Traditional tonal counterpoint and traditional harmony are differentiated by the preponderance of contrapuntal effects versus harmonic effects, and that is really all there is to it.

Once the true nature of harmonic transformational logic is revealed and understood, adding contrapuntal effects to the system - and going back and forth between harmonic and contrapuntal effects - becomes quite simple. Writing melodies from this perspective is also easier, as I'll show in chapter ten.


As I'm sure is the case with many artists, I have a very asymmetrical set of talents and abilities. Though I always test in the top one percent in abstract reasoning, my verbal scores are only slightly above average, and my numerical abilities are actually below the fiftieth percentile. This causes me no end of frustration, as formulaic systems such as those presented by Schillinger and Taneiev - especially Taniev, who I think I could greatly benefit from if I could only wrap my brain around his formulas - are, quite simply, beyond my ability to grasp. I only bring this up because it is with much trepidation that I present anything involving numbers for fear of turning off similarly inclined musician-readers.

Ratios, however, I can visualize internally - and I'm betting most musicians can do this as well - so I am going to go over the series and its ratios at this point, because counterpoint simply cannot be understood properly without doing this. Nither can rhythm, which I'll demonstrate in the next chapter.

On the top system and the first half of the second system, in Example 1, the harmonic overtone series is presented to the twelfth harmonic, which is where the first semitone appears. There are several reasons for this, only one of which is to distill out the three immutable laws of counterpoint. If you simply number the partials starting with the fundamental as "1" (Versus numbering the harmonics starting at "1" with the fundamental being "0"), the series actually works all of the ratios out for you, as can be seen.

Within the first seven partials, all adjacent intervals and their inversions are consonances, and beyond the seventh partial, all adjacent intervals and their inversions are dissonances.

Among the consonances there are two types: Perfect and imperfect. Dissonances are imperfect by definition.

It is important to reiterate at this point that the contrapuntal impetus is contained within the overtone chord and its passive-tone/active-tone dichotomy, as per Example 2.

The series explains for us exactly why the consonances fall into two categories. Example 3 has the consonances and their octave displacement inversions presented along with their ratios to demonstrate this. The perfect octave has the ratio of 2:1 and its octave displacement (A double octave, actually) yeilds the same 2:1 ratio. The ratio of 2:1 is what is called a super-particular ratio, which means that the two terms differ by one: 2 - 1= 1. Even I understand aritmetic that simple.

The perfect fifth at 3:2 inverts to the perfect fourth of 4:3, and both of those ratios are also super-particulars.


The major third at 5:4 is a super-particular ratio (5 - 4= 1), but when inverted it becomes a minor sixth, which at 8:5 differs by three (8 - 5= 3), and so is not a super-particular. Likewise, the minor third at 6:5 is a super-particular (6 - 5= 1), but in inversion it becomes a major sixth, which at 5:3 differs by two (5 - 3= 2), and so is not a super-particular.


Understanding this leads us to the three immutable laws of contrapuntal motion:

1) Perfect consonances may not move together in parallel stepwise motion.

2) Imperfect consonances may move together in parallel stepwise motion.

3) Dissonances may not move together in parallel stepwise motion.

This really is all there is to counterpoint: Everything else can be extrapolated out between these three laws of contrapuntal motion, harmonic progression dynamics, and melodic theory, as I shall demonstrate.

Some may question the prohibition against parallel perfect fourths, since so-called "simple counterpoint" allows for these. There is no problem with simple counterpoint, per se but it is not pure counterpoint. As I mentioned previously, traditional harmony and tonal counterpoint both contain aspects of each - and the dividing line between highly decorated harmony and very plain tonal counterpoint is positively impossible to draw - so what is called simple counterpoint is just using the harmonic transformational logic which produces as artifacts parallel perfect fifths and parallel perfect fourths, though the parallel perfect fifths are quite exceptional (But there is a famous "hidden" one which Bach hides with a rest in Contrapunctus I from Die Kunst der Fuge, so they do appear from time to time).

Obviously, if you want to write counterpoint which inverts at the octave, parallel perfect fourths must be avoided, and therein lies the proof: All technically correct pure counterpoint will invert at the octave with no resulting forbidden parallels.

The series also gives us the logic for why this is so. Obviously, parallel octaves sound almost exactly like a single line, and this is because two overtone series which are generated by tones an octave apart blend so perfectly. In fact, the upper tone in the octave relationship is positively merged into the first and most powerful overtone in the series generated by the lower. If the idea is to combine two melodies and have them each maintain an independent nature, parallel octaves must be scrupulously avoided.

The relationship between the two series generated by tones a perfect fifth apart is quite similar: The first harmonic of the upper tone is absorbed into the second harmonic of the lower, and so they blend all too perfectly. The squishy rules allowing for parallel perfect fourths in so-called simple counterpoint aise because the perfect fifth's inversion, the perfect fourth, does not suffer the blending problems of the octave and fifth relationships. However, since counterpoint's intervallic logic is built upon the octave inversion principle, and fourths invert to fifths, these parallelisms must technically be avioded in pure counterpoint.

There is another factor to consider here, however, and that is the fact that all octaves are by definition perfect in any series-implied system: There is an absolute zero percent chance for variety with parallel octaves. Parallel fifths and fourths are similar in that they are all perfect in the diatonic system except for one: So, there is only an 14.3% chance for variety with parallel fifths and fourths. In that one place, however, parallel fifths and fourths are allowable, which leads us to Example 4.

The first two measures of Example 4 demonstrte the point at which parallel fifths and fourths are OK due to the intermediary A4/d5 interval. The rest of the example demonstrates DINO's: Dissonances In Name Only. The augmented second is as a sonoroty the same as a minor third, so it is no problem to move in and out of it in parallel. The same is true for the diminished seventh, which is in reality a major sixth; the diminished fourth, which is in reality a major third; the augmented fifth, which is actually a minor sixth, and so on. These situations become more common as the chromaticism of the counterpoint increases, naturally.

Finally, I want to return to the annotated series to point out a few things which relate back to the series' implications for equal temperament. There are two differing minor thirds at 6:5 and 7:6, then there are three major seconds at 8:7, 9:8, and 10:9, finally, there are several minor seconds from 12:11 - which borders on being a whole step in size - up to and beyond the 15:14 minor second which is used in 7-limit just tuning and the 16:15 minor second which is used in 5-limit just tuning.

You can say you like or prefer a particular temperament scheme if you want - I enjoy one called Kirnberger III for Baroque harpsichord music, for instance - but you must admit that there is no more logical system implied by the series than equal temperament. It solves so much so simply and allows for unrestricted compositional freedom within the implications of the series.


Another purely contrapuntal factor that the series implies and defines is the delayed resolution principle demonstrated in Example 5. It is a purely contrapuntal factor because in pure harmony all rsolutions are, by definition, immediately concurrent with the root motion. In this contrapuntal effect, the transformational stratum is "suspended" for a time before the resolution is allowed to proceed. Over the primordial resolution, this results in the delayed resolutions of 4-3, 9-8, and 7-8, as you can see.

Where these delayed resolutions imply a sequence of parallel imperfect consonances, they can be chained together, as I have shown in Example 6 and Example 7. Obviously, the second example has much more interest as counterpoint than the simple parallel thirds, which are in fact implying a purely harmonic effect.


Our final examples for today, numbers eight and nine, are a simple proof to expose a hidden shortcoming with traditional counterpoint pedagogy: The fact that 5-6 ascending syncopation chains are actually allowable. Virtually all teachers of counterpoint would forbid such a thing, but at the heart of it is what is being implied: As can be seen from Example 8, it is a chain of parallel sixths which is being implied, not a series of parallel fifths. Generations of contrapuntists have refrained from using this resource as well, and for no reason. It sounds perfectly fine too.

What is incorrect are series of 9-8's, 6-5's, and 5-4's, as they imply parallel perfect consonances. Likewise, 8-7's imply parallel dissonances: When in doubt, work it out... as I have done in these axamples.

It's time for "Highly Unlikely Embarrassments" I guess.

Friday, October 13, 2006

Musical Implications of the Harmonic Overtone Series: Chapter VI


The Secondary Subdominant System


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

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

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

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

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

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


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

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

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

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

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

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

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

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

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


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

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

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

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

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


Ah, the problems associated with blustery fall days.