Thursday, September 28, 2006

Out of Town

I'm going to San Antonio for a while to do a photo shoot, see my mom, and hang out with some old musician friends.

Not sure when I'll return, but it won't be for at least a week or so.

In the mean time, all six installments of The Harmonic Implications of the Overtone Series are now on this front page, so if you start at the bottom and read the posts one-at-a-time to the top, you'll be the only other theorist in the world who understands music as well as I do. ;^)

Wednesday, September 27, 2006

Harmonic Implications of the Overtone Series, Part VI

After four installments dealing with the harmonic implications of the overtone series, it only took one to outline the elementally simple contrapuntal implications the series prescribes. I could have spent a lot more time on the subject of counterpoint, but I am leaving town this weekend and I wish to bring this series to a conclusion before I depart. Besides, the elemental underlying laws of contrapuntal movement are enough to know, as learning to actually compose counterpont will almost always lead the student toward more traditional texts. Knowing the underlying fundamental laws governing contrapuntal motion will provide one with a good "BS Detector" so to speak.

There is one last musical element that the series has implications concerning, and that is rhythm (I have not worked through to any conclusions concerning melodic implications of the series as of yet, so that will have to await further inquisitions and inspirations).

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 which exists between the lowest perception of pitch and the fastest perception of rhythm. Further lowering and the former pitch will reemerge into the audio realm as a regular pulse, or a simple rhythmic continuity.

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. Each interval of the overtone series 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.

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 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 sapce. 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.

The perfect fifth's resultant is evidenced in Waltzes, and has been used in popular music as far back as the twelfth 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.


Well, that ends this series for the time being. I think that with a couple more revisions to the presentation order and the eye of a good editor, this will soon be in a formally publishable format. Atleast I have it web archived for copyright now.



Tuesday, September 26, 2006

Harmonic Implications of the Overtone Series, Part V

With harmonic canon explained, the natural next step is to explain the contrapuntal implications of the overtone series.

My presentation is the other way around from the historical development of Western Art Music - modal counterpoint having preceeded the harmonically driven variety - but there are reasons for this: 1) The harmonic implications of the series are more important to tonal/modal musicians today (Most of whom are involved with the jazz and popular generas), and 2) The art of counterpoint was really not perfected until the harmonic implications of the series were incorporated into it.

Whereas a harmonic continuity can be thought of as a stratum in which the chord tones transform in a manner prescribed by the root motion type - there is only one "most natural transformation" for any given root motion - counterpoint is just exactly the other way around: It is the interplay between the expected and unexpected transformations of the chord tones that lends vigor and freedom to the process so that two or more melodic trajectories can be combined to good effect.

In the first example above I have presented an extended version of the harmonic overtone series with the intervallic ratios and interval names added: I went all the way up to where the minor second appears (I could not fit the entire annotated series onto one system, so it continues on the second set of staves).

Along with contrapuntal developments in early Western modal music, the intervals were classified by theorists, and various rule-sets were developed to teach students how to write in the polyphonic styles. Like any early groping theoretical attempts in virtualy any science, these rule-sets were clumsy and sometimes even contradictory. For a time, thirds were considered consonances, but sixths - their octave displacement inversions - were not, for instance. Also, perfect fifths were not allowed to move in parallel, but perfect fourths - their octave inversions - were.

As with any true solution, the answer to both interval classification and parallel prohibitions is simple, and is explained by the implications of the series.

1) All adjacent intervals contained within the first seven partials of the series - plus their compounds and inversions - are consonances:

A) The Perfect Octave at 2:1

B) The Perfect Fifth at 3:2

C) The Perfect Fourth at 4:3

D) The Major Third at 5:4

E) The Minor Third at 6:5 (And 7:6)

F) The Major Sixth at 5:3

G) The Minor Sixth at 8:5

2) All adjacent intervals after the seventh partial - plus their compounds and inversions - are dissonances:

A) Major and Minor Seconds

B) Major and Minor Sevenths

C) Major and Minor Ninths

Note: I didn't bother with the ratios for the dissonances, as the laws that govern them are so simple that they are not required.

3) Consonances which are superparticular ratios* (Or even particulars) in both octave inversions are Perfect Consonances:

A) Perfect Octave/Perfect Unison

B) Perfect Fifth/Perfect Fourth

* In superparticular ratios, the difference between the two terms is one, as in 2:1 (2- 1= 1).

4) Consonances which are superparticular ratios in only one position are Imperfect Consonances:

A) Major Third/Minor Sixth= 8:5

B) Minor Third/Major Sixth= 5:3

The First Law of Contrapuntal Motion: Parallel Perfect Consonances are Prohibited.

The reason for this is twofold: 1) The two sets of overtones for pitches in perfect relationships interweave so well as to cancel out any sense of melodic independence - the two pitches blend so well as to become one in the ear of the listener - and, 2) in the diatonic system all octaves and unsons are perfect, and only the tri-tones are different among the fifths and fourths: There is almost no oportunity for variety with parallel perfect intervals.

In so-called simple counterpoint, it has been taught that parallel perfect fourths are OK in three or more voices in situations where the voices will not be inverted at the octave (Which would produce parallel perfect fifths), but this is really wrong. Nonetheless, it's a convention, and I'm sure I'll never be able to stamp the notion out. I would only point out though, that if you take octave inversion as being a natural feature of all correctly written counterpoint then parallel perfect fourths must be proscribed. I hold to the view that all correctly written counterpoint inverts at the octave, naturally, but I too write in the relaxed "simple" style as well.

Interestingly - to me anyway - the allowance for parallel perfect fourths in traditional simple counterpoint and traditional harmonic voice leading (Which is just a simplified form of counterpoint) comes from the harmonic implications of the series.

Remember those parallel perfect fifths in the transformations between diatonic tetrads involved in super-progressions and super-regressions? There you have it: In another inversion, those would be parallel perfect fourths. Composers just intuitively "borrowed" a feature of harmonic transformation to relax the laws of contrapuntal motion.

The Second Law of Contrapuntal Motion: Parallel Imperfect Consonances are Allowed.

The only thing limiting the number of parallel thirds and sixths you can string together is taste. There are never more than two major thirds/minor sixths in succession, or three minor thirds/major sixths succession in the diatonic system, so there is plenty of variety in the genders one hears, and the overtones of two pitches in imperfect relationships do not blend so well as to allow the tones to completely unify in the ear.

The Third Law of Contrapuntal Motion: Parallel Dissonances are Proscribed.

Intervals in imperfect but consonant relationships are sonorous in quality, and so mediate between the perfect consonances and the dissonances. To modern ears, minor sevenths and major ninths may sound "pretty" in a harmonic context, but in a contrapuntal context, they are not allowed by the implications of the series. Personally, I hate the details of musical acoustics - I find the topic mind-numbingly boring and I can never retain any of it - but I have studied the subject enough to realize how the implications of the series make these three laws sensical from an acoustic standpoint.


From these three simple laws of musical motion everything else about counterpoint can be extrapolated: Any intervallic sequence may be justified by oblique or contrary stepwise motion; Parallel unequal fourths and fifths are allowed; Parallel name-only dissonances are allowed if the sonority is the acoustical equivalent of an imperfect consonance, &c.

The idea that dissonances must be resolved is really a stylistic affectation, as are things like prohibitions against "direct fifths" (But, "horn fifths" are OK (?!)) &c. I have written counterpoint in the jazz swing style, and I can attest to this fact.

What is happening in counterpoint, as I mentioned earlier, is that the composer is playing with the active tone/passive tone dichotomy inherent in the series. In harmony the root progressions mandate which transformation is most logical (But, composers have traditionally used contrapuntal "surprises" in that contex too, obviously), but in counterpoint it is the less certain transformations which are exploited to achieve independence between the melodic trajectories.

By dispensing with the ponderous rule-sets that are usually taught, counterpoint can be presented in a simplified form that allows for the student's own style to develop. After all, the rule-set of so-called modal counterpoint merely explains aspects of the personal style of Palestrina, and the rule-set of so-called tonal-counterpoint only explains the personal style of Bach. If the student wishes to learn these styles (As I did), he is certainly free to do so, but the underlying laws governing musical motion in counterpoint as implied by the overtone series are elementally simple.


It would not be a good idea to allow me to get ahold of that quill pen.

Sunday, September 24, 2006

Harmonic Implications of the Overtone Series, Part IV

Last time I got to the point where I demonstrated that it is harmonic implications of the overtone series which predict canon. In this installment I want to pursue that a bit more.

Anytime you create a harmonic sequence which has either the same root motion type, or a pattern of root motion combinations of the same types, and you are using the proper transformations in your voice leading, you are creating harmonic canons, whether you are aware of it or not. The only exception to this is if the voice transformation types do not create a complete cycle: For instance, if they simply alternate back and forth between clockwise and counterclockwise transformations, as I shall demonstrate.

Here is our first continuity example again, but this time I have extracted the very simple diatonic harmonic canon on the second system. This may not look or sound much like a canon, but it can be dressed up to make the canon more noticable as well as more perfect.

On the third system I have added secondary leading tones and made sure that all of the targets are minor triads to make the canon strict and more apparent. Then, in the fourth system, I added a further series of alterations to get augmented triad sonorities.

Down on the fifth system, I changed the time signature to 3/4 and ended up with a fairly interesting little harmonic canon which ends with a deceptive motion to the relative minor. Finally, on the bottom staves, I combined these versions to get a dovetailing harmonic canon in which the interest mounts as it progresses by beginning diatonically, then adding the secondary leading tones, and finally the augmented fifths. It also overshoots the key and ends with a funky and vaguely medieval-sounding cadence.


Same example in a tetradic texture: First I extracted the simple and un-obvious diatonic double canon, then I added secondary dominants, then the French-derived secondary dominants with diminished fifths, and finally, I combined all of the versions to create a dovetailing double canon which increases in complexity and interest as it goes along.


Here is an example which will demonstrate a couple of different things: First, that a repeating pattern of two different root motion types wil create harmonic canons, and second, if these root motion types only alternate between clockwise and counterclockwise transformations, the canon will be incomplete.

The root progression pattern is made up of a half-progression followed by a progression, and the triadic and tetradic versions are presented on the first two staves.

As you can see, the circular transformations in the triadic texture only cycle back and forth between clockwise and counterclockwise motion. That means the chord tones do not get to play all of the roles (Root, third, and fifth), so what you end up with is a two-part canon with a free voice, as shown on the fourth system, instead of a true three voice canon.

The tetradic version does not suffer from this "defect" however, and if you continue the original root motion pattern in strict intervals, it transcends the diatonic system in a very nice harmonic double canon which goes all the way through the chromatic system. Note that, as with the bass line in the final example of the previous post, this results in a twelve-tone row, but one which has inherent harmonic functionality. The harmonic series implies not only integrated chromatic tonality, but so-called tone rows, or serial systems as well! The caviat is, of course, that these tone rows will be harmonically functional, or rather, viable.

And some people still believe music is possible entirely without a harmonic or modal context. Well, it isn't. I can only demonstrate the truth, I can't comprehend it for you.


Yes, my date is on the way. Ta, ta.

Wednesday, September 20, 2006

Harmonic Implications of the Overtone Series, Part III

In the first two installments of this subject (None of this will make sense to you unless you read and comprehend them first), I demonstrated how the implications of the harmonic overtone series lead to tonality, and how the secondary dominant system (Extended with French-derived secondary dominants, fully-diminished seventh secondary dominants, and German-derived secondary dominants) was simply a result of these implications: Every degree with a target degree a fifth below can carry an overtone chord, or an altered overtone chord. I realize that there are other altered overtone chords, and that some of these require the assumed-root treatment, but the demonstration thus far will equip the student desiring to integrate those sonorities with what is needed. I wish to move on.

Today I want to demonstrate how the integrated tonality found in Arnold Schoenberg's "Structural Functions" and Gene Cho's "Harmonic Convergence" actually has a simpler solution than even Dr. Cho's slim monograph posits. Not only that, but several centuries of experimentation and anguished doubt about temperament schemes were also mostly a waste of time: The series tells what is required through its implications.

The second example in this series is reproduced as the first example here. It demonstrates the technologically correct direct crosswise transformation between two overtone chords a perfect fifth apart. But, as I finally brought up in the second installment, it is the tri-tone that is the nuclear energy in music: The root and fifth of the overtone chord are "free" with respect to where they go, but the major third and minor seventh are not; they have leading-tone and leaning-tone energies, respectively, and desire to resolve.

Diatonic chords of the major tonal system can be directly connected by these crosswise transformations, but once a secondary dominant function sonority is assigned to a degree, the more proper implied resolution is in the second example: There is a momentary triad with a doubled root (in the upper strata), and the crosswise resolution's completion is delayed.


There are various rationalizations for the minor modality, and all have some merit or other, but the traditional "pure minor" derivation with minor triads on all of the cardinal degrees (Aolean mode), is really not very sound. For one thing, the Aolean mode is almost never heard in harmony: The minor mode is almost always simply a minor tonic with overtone chords on the fourth and fifth degrees, and perhaps a diminished minor seventh chord on the second degree. In any case, the overtone series makes the minor triad conspicuous by it's absense. Whether you prefer the arithmetic mean versus geometric mean explaination, or the cosine series explaination (My personal prefference), the one thing that the series itself implies is that a minor triad can be targeted. There really isn't any more to understand.

In the third example on this page, I have interjected minor triads as targets, which first become major triads, and then acquire a minor seventh to become overtone chords. This is it in a nutshell: Every tone desires to acquire a perfect fifth and become a tonic, even if it is a minor tonic, but a major tonic is more perfectly in tune with the series (And so, all of those Picardy Thirds throughout history ending minor mode pieces). Then, after its part on the stage is over, the major tonic desires to acquire a minor seventh and be absorbed into a new tonic a perfect fifth below: The previous root is demoted to become the new root's perfect fifth.

If we extend this sequence through twelve roots, we get the example at the bottom of the page. If the fifths were absolutely perfect 3:2 ratio Pythagorean fifths, we would end up at a point not exactly where we departed from. The so-called Pythagorean Comma of 23.46+ cents sharp. Well, since the octave is the first interval (And most important) in the series, it can't be adjusted. But the fifth can. If you adjust each fifth by 1/12 of a Pythagorean comma, you get... Twelve Tone Equal Temperament. In other words,


Not meaning to yell, but just so I'm clear on that point: The series implies equal temperament (So don't get your panties in a bunch about discrepancies between TTET and the ratios in the series, because they are not only red herring issues and straw man arguements, but they are a result of the desires of the series!

I don't know what about this is difficult to comprehend, but some people do seem to have issues with it.

Finally, note that the upper four voices of the final example make a harmonic double canon. A simple one, to be sure, but it would still be a perfect canon if all the targets were major (But, all of the targets would have to be major: This does not work in a diatonic system). So, it is not really the physical phenomenon of echo which is responsible for canon, as Schillinger argued, but rather canon is one of the things harmonically implied by the series.


Redhead, red dress, and goldfish. Just shoot me.

Sunday, September 17, 2006

UPDATED - Harmonic Implications of the Overtone Series, Part II


UPDATE 09/18: As with the first post in this series, I found after re-reading it a few times that some clarifications are in order.


In the first post on this topic, I followed the harmonic implications of the overtone series from the primordial falling fifth progressive dominant resolution, through diatonic triadic and tetradic tonality, and all the way to the complete series of secondary overtone chords (dominant seventh chords). I also showed the harmonic origins of the so-called French Augmented Sixth chord as a vii(dm7) chord with a raised third in second inversion (In actuality, the first derivation of this chord was from the ii(dm7) of the minor modality, but it is exactly the same paradigm). By freeing this chord from it's ridiculous nationalistic name, it can be employed on any of the degrees of the diatonic system as a V(d5m7) (And, in any inversion), which is where we'll start off today.

Example IV is again the original continuity study, but now the secondary dominants have had their fifths diminished. Note that this results in alternating secondary dominants containing diminished thirds and augmented sixths. There is no problem with this, and it sounds excellent, with or without the constant root bass part. This concept is so simple I'm going to leave it at that.


Another set of chords which have been widely misunderstood, and therefore poorly taught, are the so-called secondary diminished seventh chords. One of the problems with this lack of understanding is, again, ignorance of the implications of the overtone series. What these chords actually are is a set of dominant seventh chords with minor ninths and missing roots. In the voice leading schemata, the ninths will simply temporarily take the place of the roots, as can be seen in the following examples.


Example V - after an initial progressive root movement from I(M7) to IV(M7) - presents some of these "passing diminished sevenths" and, as you can see, there is no way the momentary leading tones can be considered as roots. To be considered as a root of a chord, the transformation must allow for a constant root bass part without running into parallel octaves. As you can see, the transformation does not allow for that. These momentary leading tones are exactly that, leading tones, and so are functioning as the thirds of modified overtone chords.

Therefore, to explain these sonorities with a logical unified theory, a root must be assumed a major third below these secondary leading tones. As is the case with the French Augmented Sixth-derived secondary V(d5m7) chords, this brings these fully diminished chords within the easily understandable realm of the secondary dominant system: Any fully diminished entity targeting any diatonic degree is simply a secondary dominant seventh with a minor ninth and a missing root.


UPDATE 09/18: During the parenthetical half-progressive root motions, the clockwise transformations in the tetradic continuity go thusly: Root becomes third, third becomes fifth, fifth becomes seventh, and seventh becomes ninth, and that is indicated with the 7-> 9 symbol. That ninth, which momentarily replaces the root in the upper strata, functions as the root would in it's return to the fifth in the crosswise resolution after the momentary target triads, and that is indicated by the <-9/5-> symbol after the parenthetical progressive root motions.


So, in measure three of example V there is the origin of this chord with the V(m7m9/0)/V. The degree targeting vi is presented in measure five, and then this example turns around with some good,old fashioned secondary dominant action.

Example VI introduces the V(m7m9/0)/ii and the V(m7m9/0)/iii before it too, returns to the tonic via "normal" secondary dominants. Obviously, these chords can be introduced chromatically anyplace there is a whole step, or modified versions of iii(m7) and vii(dm7) may be used as well.

Note that in the root motion analysis some of the symbols are parenthetical: This is where the roots are assumed, or are "theoretical." You can also see that in the voice leading continuity, the ninth temporarily replaces the root, and takes the root's function and returns to the fifth of the target chord, thus "saving" the crosswise transformation.

Now it's time to address that whole German Augmented Sixth/Substitute Secondary Dominant "thing."


These are the same examples as before, but where the turn-arounds are, I have now introduced the so-called German Augmented Sixth/Substitute Secondary dominant sonorities.

These chords are highly problematic from a theoretical standpoint because structurally, they are overtone chords, but the roots are active tones, and therfore they cannot be the real roots. Once I figured out the fully diminished seventh deal, I applied the same process to these sonorities, and bingo: They also have a missing theoretical root a major third below the momentary leading tone. The resultant nomenclature gets a little ponderous, but they nonetheless integrate smoothly and inevitably into the secondary dominant galaxy.

The original "German" chord targeted the primary dominant, and you can see it in measure seven as V(d5m7m9/0)/V. In English that is a rather formidable mouthfull: "Five major, diminished-fifth, minor-seventh, minor-ninth, missing-root of five." Whew. But it is actually what the chord is as implied by the series.

Example VIII presents some so-called German chords targeting other degrees.

So as of just recently, I have a completely unified theory of secondary dominant function chords - regular secondary dominants, secondary dominants with diminished fifths (French), secondary dominants with minor ninths without roots (Fully diminished sevenths), and secondary dominants with diminished fifths and minor ninths and missing roots (German).


The atomic energy at the heart of the overtone chord that creates the primary motivational force in music is the tri-tone. Both the leading-tone impetus and leaning-tone impetus are contained within this interval. Far from being the diabolus en musica, the tri-tone is God.

This pantheon of secondary dominant-function sonorities all "work" according to the implications of the overtone series, because what is happening in them is that the natural leading-tone/leaning-tone tendencies of the tri-tone are being amplified or even compounded. The regular secondary dominants intoduce tri-tones onto degrees which are naturally without them, which increases the resultant energy of the resolution over the diatonic versions. The French-derived secondary dominants increase the leaning-tone energy with the second diminished fifth they introduce (Only the tri-tone between the third and seventh is fully active though, as the root is a real root), and the secondary dominant function chords which are fully diminished seventh sonorities actually contain double tri-tones (which are both active), and that really makes the resolution pungent. Finally, the German-derived secondary dominant chords are counterfeit overtone chords in and of themselves, and the wildly "deceptive" nature of their resolution creates surprise as well as drive (It also introduces the oportunity for "normal" modulation a tri-tone away from the intended target).


UPDATE 09/18: I should have pointed out more clearly that the French-derived secondary dominant chords have two tri-tones a whole step apart (And so they harmonically generate a whole-tone scale), and the fully-diminished seventh chords contain two tri-tones a minor third apart (And so they generate a dimished octatonic (1+2) scale).


I hope you've enjoyed my "Grand Unified Theory of the Harmonic Implications of the Overtone Series." I've been working on it on-and-off for thirty years now, and I about have it perfect... but not quite.


Now, about those Neapolitan Sixth chords...

We'll save Secondary Subdominants for another time.

Saturday, September 16, 2006

UPDATED - Harmonic Implications of the Overtone Series, Part I

This is a re-visit to my earlier Musical Philosophy post, which generated some hilarious responses, all from proponents of adding atonality to the definition of music, and all rather infantile. If I think a comment is worth responding to or worth a viewer's read, I'll post it; if not I won't. So, there were none worth responding to, obviously. And here's a hint: If you use dummy e-mails and obviously idiotic names, I won't even read the comment when it comes in. This is primarily directed at a fan of mine named, aptly, U.R.N. Idiot (I did get a laugh out of the name though. I mean, "Dear Mr. Urine Idiot, Sorry about the bad luck with that name of yours, but as they say, if the shoe fits...").

There are strange people on the internet.


Anyway, I've revised my terminology a tad to tighten up the consistency, and I think all lovers of music theory will be able to derive some use out of this post.


UPDATE 09/17: I suppose I ought to take a moment to describe the logic behind my analysis symbols, as they are slightly different than "normal," if there is such a thing in music analysis. I developed these with the idea that they would be 1) Consistent, 2) Unambiguous, and 3) Executable on a QWERTY keyboard.

A bold capital Roman numeral denotes that the fifth has a major third, while a bold small case Roman numeral indicates that the fifth has a minor third. Fifths are perfect by implication. In the case that the fifth is diminished, a small case "d" will appear in parenthesis after the analysis symbol, and if the fifth is augmented, a capital "A" will appear in the parentheses. To be extra-clear on major thirds with diminished and augmented fifths, the numeral "5" will appear after the modified fifth designator.

Senenths (and ninths &c.) will be described with the M, m, A, d symbology as well, with the numeral following the modifier.


The first section will be using the following example:

As always, every discussion of music must begin with the harmonic series, which has been true since the first music theorists investigated it with their monochords. The entire history of Western Art Music has been the ongoing investigation into the implications of this series.

Today I want to discuss the harmonic implications of the series. Obviously, the series makes a major minor-seventh chord, also called a dominant seventh chord, and it generates the gravitational force in music. The first and most obvious implication of the series is the primordial progressive root motion, as seen on the second system. This decending fifth or rising forth progression is statistically the most common type of progression in Western Art Music, and popular forms as well.

The technologically correct voice leading for this progression is a crosswise transformation in which the root of the first chord becomes the fifth of the target chord, the third of the first chord becomes the seventh of the target chord, and vice versa, in a purely tetradic environment. Traditionally, the target has been a triad momentarily, as I'll demonstrate shortly.

The implications of this progressive root motion lead to major tonality, which is the most perfect in accordance with the implications of the series for a triadic system. The root is the point at which arrivals or departures can be made in a progressive root motion to and from major triads, as you'll see on the third set of staves. Note that where the texture is triadic, there can only be clockwise or counterclockwise circular transformations between chords. In progressive root motions in a triadic texture, the transformations are counterclockwise: Root becomes fifth, fifth becomes third, and third becomes root.

If you simply read the third system backwards, you get the opposite of progressive motion, which is regressive. Regressive root motion types go against the dominant resolution impulse of the harmonic series. This in no way implies that these root motion types are inferior, rather they are just a resource and are necessary for variety and balance, as you'll soon see.

Note also that I not only analyze the degree and gender of the chords, but the root motion and transformation types as well. I have found this to be of great assistance in composing harmonic continuities, because I can visually check the paterns I'm creating.


UPDATE 09/17: Again, I developed these symbols to be easily executable on a QWERTY keyboard, so ---> = clockwise transformation, <--- = counterclockwise transformation, and <-^-> = crosswise transformation. For the root motion types, P = progressive root motion, R = regressive root motion, .5P = a half-progression, .5R = a half-regression, SP = a super-progression, SR= a super-regression, Ptt = a progressive tri-tone root motion, and Rtt = a regressive tri-tone root motion.


If we take the notes of the major triads of the cardinal degrees and put them into a scale, we get the Ionian or Major mode. It is important to understand that it is the harmonic implications of the series that generate the scale, and not the other way around (Technically, the series itself creates a Mixolydian Augmented Fourth scale). Historically, the melodic and contrapuntal implications of the series were worked out first, but this was because the original medium was vocal, I believe. The archaic Church Mode system also held back musical development for centuries, and this lineal thought lead to a poly-lineal musical conception, or counterpoint. As early as the 1300's though, the common musicians were using the Ionian mode in many of their pieces. In any case, what I'm discussing here today was not initially intuited until just before the time of J.S. Bach, and it was Rameau who first attempted (poorly) to explain it theoretically.

Not until Schoenberg's "Structural Functions of Harmony" were root motion types first categorized (Again, poorly: He obviously hadn't a clue about the implications of the harmonic series), and it was Joseph Shillinger who first correctly worked out the transformational nature of harmonic voice leading. Until Schillinger (And still among most traditional tonal composers of today) harmonic voice leading was a watered down form of counterpoint, gleaned primarily from the chorale harmonizations of Bach (Which explains the situation perfectly).

On the bottom staves you see the resulting harmonized scale, and unfortunately, this is what students are often given first. Without the preceeding understanding, confusion is inevitable: For years I thought scales generated harmony. They don't. It's exactly the other way around.


Instead of presenting the root progression types in sterile isolation, I decided to use some phrased continuities. In the first example, after a super-regression from I to vii(d), the phrase has all of the chords in the diatonic system arranged in progressive order.

It is important to note that the constant root bass part is just a checking tool: The triadic continuity exists solely on the upper staff of each system. After the super-regression's clockwise circular transformation, all of the progressive root motions generate counterclockwise transformations. I turned the phrase around in the last measure with a super-progression from IV to V.

The reason for the terminology of super-progression and super-regression is simple: What is actually happening in these cases is that there is an implied root not present a third below the "root" of the first chord, which progresses or regresses to the target chord. By comparing every root progression type to the primordial natural progression, their various effects can be well understood. In part two, I'll clarify this further with secondary diminished seventh chords, which are actually major minor-seventh chords with a minor ninth and a missing root (If you want to prove this to yourself, try to write a four voice continuity employing these chords over a constant root bass: You'll end up with parallel octaves unles you put the REAL root in the bass line).

Note also that in a triadic texture, the voice leading is not totally smooth where super-progressions and super-regressions are used: The series actually implies a four part texture.

In the example on the second system, I have arranged all the chords in the system in half-progressive order. In a half-progression the two chords only go half way to the fifth the series desires to go to. As you can see, If you were to go directly from the I chord to the IV chord, not only would the root fall the required fifth, but the triadic continuity would make the progressive transformation all at once as well. Again, a half-regression would simply be reading that section backwards.

So, there are only seven types of root motions:

1) Progressive root motion (Falling fifth/rising fourth)

2) Regressive root motion (Rising fifth/falling fourth)

3) Half-Progressive root motion (Falling third/rising sixth)

4) Half-Regressive root motion (Rising third/falling sixth)

5) Super-Progressive root motion (Rising second/falling seventh)

6) Super-Regressive root motion (Falling second/rising seventh)

7) Tri-Tone root motion (Which is Ptt if it is a falling diminished fifth, or Rtt if it is a falling augmented fourth).

In example III, I used all of the root motion types in a single phrase with the exception of the Rtt (Regressive tri-tone). Note that the Ptt breaks the phrase up into two four measure sub-phrases, and that these sub-phrases have mirrored root motion types: The initial regression from measure one to measure two is answered by a progression from measure four to measure five; then the super-progression from measure two into measure three is answered by a super-regression from measure five into measure six, &c. Well ordered root motion patterns separate good progressions from bad ones, for the most part, and using the extended analysis symbol sets that I do makes the patterns easier to create and to see.

As I said, the series really implies a four-part texture, so the first tetradic example is the same as the first triadic example with the exception of the number of voices. Note that in a purely tetradic texture even the super-progressions and super-regressions have a common tone. Progressions and regressions have two common tones, and half-progressions and half-regressions have three common tones. The number of common tones also has a huge effect on the nature and effect that the various root motions create, obviously.

Now for the (Insert horn fanfare here) parallel perfect fifths in the super-progressions and super-regressions: They are perfectly natural and are not an issue in harmony at all, unless the chordal functions of the tones do not transform! In other words, parallel perfect fifths only sound "crude" when the root remains the root and the fifth remains the fifth (Or the same relationships involving the thirds and sevenths).

If you get the impression that I'm calling jazz vioce leading crude, well, yeah, on a certain level, but not transforming the chord tone functions is simply an aspect of that style. No doubt in my mind that the "proper" transformations sound "cooler" though, which is why I use them (Some of the "style" listeners percieve in the late music of George Gershwin is due to his being a student of Schillinger and employing some of these techniques).


UPDATE 09/17: But, one thing the jazz musicians did properly intuit is that the overtone series implies a four-part texture. This actually began with blues tonality, which is simply a musical system built upon overtone chords on all of the cardinal degrees (I7, IV7, V7 in traditional parlance, I(m7), IV(m7), V(m7) as I designate them).


Finally, as with the primordial progression implied by the series, progressive root motion creates crosswise transformations (As does regressive tetradic motion).

The second tetradic example at the bottom of the page is just a four voice version of the second triadic example. Note all the common tones: This series of falling thirds and one-at-a-time note movements in the transformations creates a very mild effect.

If you want to make a four voice version of the third triadic continuity, that would be good exercise, but I want to go back to the first example for my last point today.


UPDATE 09/17: Also notice that in triadic textures the series of progressive and half-progressive root motions cause the voices to rise in pitch through time (And, regressive/half-regressive root motions in triadic textures would cause them to fall), but in tetradic environments, the same root motion types have the opposite effect. Imbalanced progressions, like these examples, can get you in "trouble" with ranges if they are not balanced out by alternating three and four voice episodes. Noticing all of these details enables the composer to gain total control over his work, and freedom is control.


The pull of the dominant resolution/progressive root motion force in music - well, in sound, actually - lead to composers intuiting overtone chords on every degree of the diatonic system over time. The original continuity was written with this demonstration in mind.

After the super-regression from measure one, the thirds of the chords are raised and/or the sevenths are lowered to get an overtone chord. I did not raise the fifth of the V(d5m7)/iii to show you the harmonic origin of the so-called French Augmented Sixth Chord: In the second inversion, you get the Augmented Sixth interval, but in reality there is nothing "French" about this chord, and that description is useless insofar as explaining what it is. What it is, is merely a secondary dominant seventh chord with a diminished fifth in a particular inversion. All inversions of it are available, however, and on any degree, I might add.

Note that the secondary leading tones are allowed to resolve to the new roots-of-the-moment in this example. This is the traditional way to deal with these chords, but it is not the only way: The secondary leading tone can come down chromatically to give seventh chords as targets (Another thing that sounds vaguely Gershwinian in abundance, but composers have done this as far back as Bach's time in isolation). What this more traditional example's resolutions do is to demonstrate that inserting triads as target chords only delays the transformation type (crosswise), it does not negate it entirely (Or rather, that's the way that the series implies it ought to go).


UPDATE 09/17: Yet another thing to note is that if the continuity started out with a minor seventh chord instead of a diminished minor seventh chord, and the remaining targets were also all made into minor seventh chords (On the dominant, tonic, and subdominant degrees), a very simple harmonic double canon would be created. I have covered that subject in depth previously, but I guess I'll hit it again in the next post (Because it really is a super-cool musical technology).


Please understand that this is something I have been pondering recently, and I wrote this post in a stream-of-consciousness kind of way to put my own mind in the driver's seat with these concepts. I will study and re-read this a few times and try to clear up and points that might be confusing with a summary at the beginning of the second part of this subject.

Monday, September 11, 2006

Remembering 09/11/2001

Five years ago today, I was living in Adelphi, Maryland. At the time, I was working at the FEMA National Processing Service Center in Hayattsville, a mere three mile commute. That morning was crystal clear - not a whisp of cloud from horizon to horizon - and a crisp nip of Autumn was in the air. I rode my pride and joy, a BMW K1200LT motorcycle, to work that AM, and it was a pleasant little start to the day.

At the office, I logged in and began my morning by processing cases in the queues, just as I always did when I wasn't deployed to the field. As I was working, I overheard someone say, "An airplane just hit the World Trade Center!" I was thinking in terms of a small private aircraft, and not a comercial jetliner, so while I thought it bizarre, I just kept on processing cases.

A few minutes later, I heard another person say, "Another airplane hit the Trade Center!" And I got a real deep, sick feeling. All of us went downstairs to the lunchroom, where the TV's were tuned to the CNN coverage. I walked in just a few minutes before the first tower came down. Somehow, stunned doesn't quite cover the way I felt.

I couldn't really wrap my brain around what was transpiring, so I numbly walked back upstairs to my desk and sat in silence. Within minutes, the news spread that a third aircraft had hit the Pentagon. I had a south facing window, so all I had to do to confirm that was look over my left shoulder. Sure enough, there it was: A black cloud of smoke streaming up from the Pentagon, which was several miles across town.

I'll never forget that day, nor will I forget the aftermath. For over eighteen months after 09/11, since I was a FEMA employee, I worked cases related to that terror attack. It changed me. Forever.

One of the main reasons I got out of the disaster business and returned to music is because of my work on the 09/11 aftermath. I had no problem dealing with natural disasters, but the unmitigated evil that is terrorist mass murder is another thing entirely. I was totally unprepared for that.

These photos were taken at Ground Zero by a FEMA employee who was one of the first FEMA people allowed on the scene. Unfortunately, I cannot recall the person's name, but I thought they were worth saving when they were forwarded to me.

Never forget who did this, and why: They hate freedom, they hate love, they hate the God of Abraham, Isaac, and Jacob, and therefore they hate all of us. "Allah" is historically a pre-Islamic diety who was god of the moon, and now "Allah" is simply another mask that Satan hides behind. If you have a problem coming to terms with that truth, then you have a problem. God is love; "Allah" is hate, and that list of opposites goes on forever.

When I lived in NYC, I used to take summer jobs as a bicycle messenger. I went up those towers to make deliveries all the time. If the weather was nice, I'd always stop on the observation deck and enjoy the spectacular view. Another thing I enjoyed was standing between the towers - where they came together edge-to-edge - and I'd tilt my head back and laugh at the incomprehensible vanishing-point effect.

Nobody ever has to worry about me forgetting that day, or the miserable aftermath.

Monday, September 04, 2006

It's Official: I Have a Management Contract

The Lord works in mysterious ways. Many years ago I was under contract while in a rock band, and it was a disaster. I learned a lot from the experience, but I swore I'd never have a manager again. You know what they say: Never say never.

As I have been rebuilding my neglected music career over the past two years, one thing has become apparent to me: I can't do it all alone, despite my most fervent desires. Doing all my own booking is not only a distraction that keeps me away from practice and composition, but I also do not enjoy it, and so, consequently, I'm not very good at it.

Realizing I needed a partner in this effort, I decided to ask a friend of mine who is a musician - I don't wish to name drop, but he has been a friend for over twenty years and you would recognize the name of his jazz band instantly - how he managed his booking. He turned me on to his manager, and within a couple of e-mails, I knew that this lady understood artists on a deep and fundamental level. She also has an IQ which could instantly vaporize vast seas.

The first draft of her proposed contract - THE FIRST DRAFT! - was all but flawless, and was sensitive to any outcroppings of differences which might arise, so... I signed it.

The contract runs from February, so I have a few months to collect gear, tools, get the press materials together, etc.

I. Am. Psyched.

I'll be touring the college circuit, with higbrow dinner gigs and piano bars thrown in, and she wants to start me off in... ski lodges. Too cool. Er, cold (I hate cold weather and wouldn't ski on a dare, as one of my childhood friends broke both legs on his first day out, and paid for that with several months of rehab). But, I have a bad@$$ 4x4 pickup, so I think I'll get there OK, and sitting by a fire with ski babes sipping warmed cognac in the evenings sounds like my sort of "thing."

What better way to celebrate than with a redhead pinup girl? Exactly.