Dovetailing Harmonic Canons

Over the course of the las few posts, as I have presented all of the secondary dominant function harmonies - secondary dominant triads, secondary dominant sevenths, secondary diminished sevenths, secondary dominant seventh minor ninths, secondary dominant seventh diminished fifths, secondary augmented triads, secondary augmented sevenths, and substitute secondary dominants - I have given a few examples of harmonic canons. As I had hoped, these efforts have coalesced into a greater understanding of this technological approach, and have enabled me to develop a methodology for creating Dovetailing Harmonic Canons, which I am now prepared to present. Since I have not given an example of a Triadic Harmonic Canon to this point, we will start there.



In the above series of examples we return to the progression I initially employed to demonstrate secondary dominant seventh chords, and with which I composed an accompanied Harmonic Double Canon. The top staff shows the diatonic triadic transformations over a constant root bass.

Though the progression does not sound inherantly canonic in this form, the diatonic framework for the canon is nonetheless present, simply awaiting the composer's attentions to bring it out. This diatonic canon is extracted on the second staff, and all that needs to be done to make it a strict canon is to create minor triads over the dominant and tonic degrees in measures six and seven of that example by employing B-flat and E-flat respectively.

The third stave presents the canon with that aforementioned modification in conjunction with chromatically introduced leading tones for a series of secondary dominant triads. At this point, the canonic nature of the upper stratum becomes easily perceivable to the astute listener. I also introduced a secondary diminsihed seventh in the final cadence figuration - the vii d7/V - to smooth out the voice leading and also to show how these sonorities are employed in this two-strata texture: The texture changes from three real parts to four momentarily to introduce and resolve the secondary diminished seventh, and then it goes back to triads over a constant root bass line.

On the fourth staff, secondary augmented triads are introduced. Note that since the target chords are of the minor gender, the augmented fifth is notated enharmonically as a minor sixth and is tied across the barline to become the minor third of that target chord. I also introduced a dominant seventh on the final quarter note to demonstrate that there is the oportunity to change to a tetradic texture in the upper stratum at this point. Introducing the seventh in a new voice while tying the root over to become the fifth of the target chord will start a series of crosswise transformations and is a highly effective technique.

Then, on the fifth staff from the top, I have changed the canon to 3/4 time to get a surface rhythm of constant quarter notes, and have also introduced a modulation to the relative minor with a second secondary diminished seventh at the end just to point out that it is there.

As nice as that example is, a much more artful way to introduce the canonic elements is one-by-one with a Dovetail device, as I have demonstrated in the final example of the progression. Since the canonic voice is three measures long, the first time through is the diatonic version, then the leader introduces the raised leading tone figure at the first repeat of the canonic element and the followers continue with that in strict canon, and finally the leader introduces the enharmonic secondary augmented triad's minor sixth for the third repeat of the canonic figure, and it too is copied in strict canon by the following voices: The strictness of the canon is never compromised. Note that because of the length of the canonic voice, the progression's series of falling perfect fifths/rising perfect fourths must be extended all the way to B-flat to give the third and final following voice the oportunity to fully participate in the canon. This increasing distance from the original tonic coupled with the piling up of increasingly dissonant secondary dominant sonorities creates a highly effective and highly charged canon, despite the simplicity of it on it's face. It also presented me with the oportunity to use a very Medieval-sounding cadence to return to the tonic that is quite tasty in this context (The mediant degree is lowered and minor, the subdominant degree is a dominant seventh with no fifth, and the dominant seventh is missing it's leading tone, which combine to create a hollow and primitive sounding cadential figure).



The same proceedure can be followed with a tetradic upper stratum of seventh chords, as I have demonstrated before, and have recapitulated above. In this instance, since all the transformations are crosswise or delayed/interrupted crosswise transformations, all four of the voices do not share all og the canonic elements, rather a Harmonic Double Canon is created with two voice pairs.

After the top stave's transformation blueprint, I have extracted the diatonic canonic framework beneath. Again, just change the seventh chords on the dominant and tonic degrees to minor sevenths with B-flat and E-flat respectively (And in combination for the tonic degree), and the canon is strict.

Secondary dominant sevenths are introduced on the third staff, and the French Sixth-derived V7(d5)'s dress up the lower canonic voices in the penultimate example.

In the final Dovetailing example, I used the delayed or interrupted crosswise transformation pattern associated with the secondary dominant example to begin the canon, but without the chromatically introduced leading tones, because the original diatonic canonic figure was so bland. The canonic elements are only two measures long since this is a double canon, so I didn't have to lengthen the progression. In fact, it could be shortened by a measure and all four of the voices would still have had the oportunity to fully participate in the canon, and that could have been even more effective. However, I wanted to demonstrate how a secondary V7(m9)/V could take the place of the vii(d7)/V I used in the triadic examples to achieve a similar effect in this virtual five-voice texture.

What these harmonic canon sketches remind me of when I listen to them (And, they are just sketches at this point, and require more levels of ornamentation to be employable in actual musical compositions), are some of the sketches of M.C. Escher that smoothly transform one element into another in a seamlessly logical way. In fact, this dovetailing harmonic canon technique is the exact musical equivalent of that visual art technique.



In the following posts I plan to present harmonic canons with two, three, and four different root progression types in their repeating patterns to further explore this idiom.

The so-called "German Sixth" Chord

Here's another sonority unique to Western music - but not limited strictly to art music proper - that comes down to us with a useless nationalistic appelation: The so-called German Sixth chord. This chord - or rather these chords - are known to us with a jazz background as Substitute Secondary Dominants, or simply as "Sub-Fives". Because the jazz theory explanation for these chords is so much more convienient and descriptive than even Dr. Cho's traditional-based rationale for them, this is the terminology that I will employ.



The theoretical derivation of these chords is based on the fact that they share an enharmonic tritone with the regular secondary dominant that targets the same chord. I have presented this in the first measure of the example above where you can see that the V7/V and the SubV7/V share enharmonically the tritone between C and F-sharp/G-flat.

In their employment in traditional music, the Substitute Secondary Dominants are not notated as dominant seventh chords, however: The minor seventh of that chord is notated as an Augmented Sixth in the first version, which I have presented in the second measure of the example, and the fifth is also notated as a Doubly Augmented Fourth in the second version, which is presented in the third measure. These enharmonic modifications of the sub-fives have to do with their respective traditional resolutions.

On the middle staff I have given these respective traditional resolutions where the first version of the now Enharmonic Substitute Secondary Dominant resolves directly to it's target, in this case the V chord. Note that a parallel perfect fifth results from this resolution, and that there is no transformation: The chord tones maintain their respective functions with the enharmonic minor seventh just doubling the root of the target chord, presumably momentarily before it moves down to become the seventh again. In early employments of this chord, composers tended to hide this parallelism by delaying the resolution of the fifth with a flat-6 to 5 suspension. Later composers who favored this chord, such as Chopin, had no qualms about allowing for the parallel fifth to occur directly. In fact, there are so many parallel fifths in Chopin's music that they really do constitute an aspect of his style, which is quite "jazzy" with respect to the voice leading.

The second two measures of the middle staff present the traditional resolution of the second version of the enharmonic sub-five that has the notated doubly augmented fifth. This chord progresses to a I(6/4) sonority - which admittedly does not exist independently, but rather is only a suspended version of the V chord - before finally resolving to the V7. Note here that the final dominant seventh has a doubled root, but no fifth. Obviously this is going to be problematic if harmonic canon writing is the goal. Not only that, but the resolution figure already has a constant root, so in the virtual five voice method for writing harmonic canons, this resolution will create parallel octaves with the constant root bass.

Now, if a triadic texture is being composed in the upper stratum, no such problem would arise so long as the texture is momentarily changed to four real parts. The upper stratum simply recieves a diminished triad and the bass creates the SubV by arranging itself a major third below the lowest note in that diminished triad. In this sense, substitute secondary dominants are really more appropriate for the essentially triadic transformational environment: The upper triad then has a proper clockwise transformation and the bass maintains it's constant root motion with no problems with parallelisms. I suppose I should mention at this point that accompanied harmonic canons can obviously be composed with a triadic upper stratum and a repeating root progression pattern over a constant root bass, but they would not be double canons, which has been my particular interest to this point. I suppose I should compose one of those to demonstrate some of these sonorities in that context.

In order to employ substitute secondary dominants in a virtual five voice texture over a constant root bass - as I have been doing thus far in my examples since the Secondary Dominants and Harmonic Canons post - the regular clockwise transformation that strong decending root motions normally get would have to be employed. This creates problems if the target chord is of the major gender because an augmented second would result if the resolution were made directly. In order to maintain the glass-smooth voice leading I prefer, the target chord must either be of the minor gender, or it must be changed to the minor gender momentarily. I have demonstrated this on the bottom staff of the example. This resolution sounds very unusual: The minor seventh or enharmonic augmented sixth resolves outward as per usual, but the tritone does not "do it's thing". This, combined with the unexpected minor gender of the target in this particular resolution, can only be described as a peculiar effect. Nothing could better explain why I use these chords sparingly: They are simply highly problematic in their various methods of employment.

Augmented Triads and Augmented Seventh Chords

The Augmented Triad is, along with the Diminished Seventh Chord, one of only two perfect symetric harmonic structures in the twelve tone system, as I mentioned in my previous post. Like the diminished seventh chord, the augmented triad has a dominant function and no real perceivable root except in retrospect after it has resolved. This is because, like the diminished seventh chord, any one of the three tones in an augmented triad can be interpreted as the leading tone, so it also momentarily suspends tonality as the diminished seventh sonority does. Unlike the all minor third diminished seventh which requires that a real root be added a major third below one of it's tones to get a fifth voice for four part harmonic canon composition wth a constant root bass, the augmented triad's all major third construction means that the root is the tone a major third below whichever tone is selected to function as the leading tone. This solves some problems associated with the diminished seventh, wherein all the four tones are active and require resolution, but others are created with the raised fifth degree (sometimes notated enharmonically as the flatted sixth) in association with a seventh, as we shall see.



On the top staff above I have demonstrated the theoretical origin of the augmented triad in the first measure: If a leading tone for the minor key is added to the fifth degree of the major triad on the bIII degree, an augmented triad is created. The root on the flat third degree is purely theoretical in the normal resolution for this chord, as can be seen in the following two measures. Here, there is the usual counterclockwise resolution of a Progressive root motion where the note C is interpreted as the fifth degree of the augmented triad, and not the sixth degree. The augmented triad is marked as enharmonic because it is spelled with a C versus a B-sharp, which makes sense since the note is tied over to become the third of the tonic triad. The augmented triad's normal resolution in the minor key, with it's two common tones, is very smooth; and coming from such a highly charged dissonant sonority, it is also very effective.

The real augmented triad - by that I mean the one in which the theoretical and real roots agree - is found in the major key where the fifth of the V triad is raised to get it. Here, there is only one common tone since the raised fifth resolves up to get to the third of the tonic triad, following parallel to the leading tone's resolution to the tonic degree. Again, there are no problems with a triadic environment in the upper stratum with a constant root bass below.

It is not until we try to employ an Augmented Seventh Chord that things get tricky. Using the enharmonic version from the minor key, the augmented fifth (flat sixth degree) sustains over to become the third of the tonic, while the seventh resolves down to become a third also. This results in a tonic triad with a doubled third. This is an irregular or hybrid transformation type. The same situation arrises in the major key where the raised fifth of the dominant chord resolves up by step to the major third of the tonic, while at the same time the seventh also resolves down to the second third. Obviously, this would ruin any harmonic canons if not dealt with properly. There are two ways to approach this and rectify the situation.

The first approach involves using the enharmonic version of the augmented seventh. Since the fifth is supposed to go to the root in a crosswise transformation, it follows that the raised fifth needs to get there in some fashion, or the harmonic canon will be interrupted (Unless the same irregular resolution is used in each place where the augmented seventh appears, which we'll cover later). Starting on the third staff, I have demonstrated the accented passing tone method combined with a delayed resolution of the seventh, or a 4-3 suspension resolution. This same method works in the major, where it now makes more sense to notate the raised fifth as the enharmonic flatted sixth. This is the chord I used penultimate to the resolution in the Theorem of Pythagoras sketch I presented earlier.

There is another way to handle this though. On the lower staff I have shown the seventh resolving down to the third while in the major the raised fifth of the dominant resolves up to the other third, then the passing tone figure is employed to bring the chord of resolution back into it's normative configuration. In the minor, this happens after the tied third occurs across the barline.

By understanding what all the normal and most natural transformation patterns are, unusual situations can be reasoned out and control can be maintained. There is no other way to write harmonic canons than by maintaining strict control over your transformation types in conjunction with employing repeating root progression patterns.

Since an augmented triad is symmetrical, any tone of it can be not only used as a leading tone with associated root a major third below, but any of the three tones can be lowered by a semitone to get any one of three different major triads. These can be used with or without a seventh as dominant function chords in suprise modulations. Also, augmented triads and augmented sevenths can be used to target secondary degrees of the key just like other forms of dominant function harmony: Secondary Dominants, Secondary Dominants with Diminished Fifths (Derived from so-called French Sixths), and Secondary Diminished Sevenths( Or V7(m9) sonorities in a five voice texture with constant root).

Diminished Sevenths and Secondary Diminished Sevenths

The diminished seventh chord is one of only two symmetric harmonic structures, the other being the augmented triad. By this I mean that all the intervals are all the same and the structure repeats at the octave without interruption, unlike diatonic symmetries like major and minor seventh chords which have whole tones or semitones interrupting the octave repititions of their symmetries. For the diminished seventh it is all minor thirds, and for the augmented triad it is all major thirds.



In the example above on the top stave I have demonstrated the theoretical origin of this sonority and it's traditional analysis. If the root of the dominant seventh chord found on the bVII degree of the minor key is raised to obtain a leading tone, the result is a fully diminished seventh chord. In the traditional analysis, the "root" of the diminished seventh progresses up by a semitone to resolve to the tonic, while the rest of the tones move down by whole step or half step to complete the tonic triad. This would have to be analyzed as a strong ascending root progression in which the tones transform in a counterclockwise direction with 3 becoming 1, 5 becoming 3, 7 becoming 5, and the "root" being constant between the two chords. I have indicated this with a perenthetical (1) to the right of the counterclockwise transformation symbol underneath the "S+" root progression indicator. The first two measures show this traditional explaination in the minor, while the second two measures show the traditional resolution in the major.

For the major key, the vii is already a diminished triad, so the minor seventh of that half-diminished seventh needs to be lowered by a semitone to get the fully diminished sonority. Note that the minor key and major key diminished sevenths are enharmonically the very same chord, since the G-sharp and the A-flat are the same tone notated differently. So, you could easily be cruising along in A minor and modulate to C major by introducing the diminished seventh for C major and then resolving it to that relative major tonic: The ear wouldn't know you weren't going to target the minor tonic until the modulation had been made. This effect has been used since before Bach's time.

Since one version of the diminished seventh was created by raising the root of a dominant seventh chord by a half step, it follows that any degree of a diminished seventh chord can be lowered a semitone to get any one of four different dominant seventh chords since it is a perfectly symmetrical harmonic structure. I have demonstrated this on the second staff of the example. Composers from Mozart on used this with increasing frequency to achieve startling modulations, and Romantic era composers used it constantly. Schubert was probably the best at reinterpreting symmetric structures, and his pieces often sound like clouds have obscured the view until the sun comes back out, and the listener emerges into a whole different world. Quite effective.

More than this though, since two of the tones of the diminished seventh can be interpreted to be the leading tone in order to target the tonic minor or it's relative major (Or, vice versa), it follows that any one of the four degrees of the sonority can be the leading tone. I have demonstrated this on the third staff of the example.

As you can see, there are eight possible modulations for this chord if you consider both major and minor tonics: That includes fully one third of all of the available keys in the twelve tone system. Since there are only three diminished seventh chords in the twelve tone system, and diminished sevenths can be created between any two degrees seperated by a whole tone by raising the root of the chord on that degree along with either also raising the third (for minor seventh chord degrees), or lowering the seventh (for major seventh chord degrees), and diminished sevenths can easily be created on any actual degree of the key through similar processes, modulations to any one of the twenty four regions of the home key are always only a couple of chords away.

Previously I mentioned the traditional analysis method for this sonority in terms that may have been interpreted as disparaging. There is a reason for this. Since the diminished seventh is a symmetric structure that can be interpreted in any one of four ways, in actual point of fact, it has no root out of context. In other words, there is no way to know what the "root" of a diminished seventh chord is until it resolves. In this respect, the diminished seventh momentarily suspends toinality, and it has been used to achieve this effect since the early Baroque era (Or even earlier).

The other problem with the diminished seventh is that all four of it's tones are active, and require resolution no matter how you interpret it. That means that in the method of writing trasformations over a constant root bass, parallel octaves are unaviodable unless a voice is sustained through the diminished seventh that is not really a member of that structure (As I did in the penultimate measure of the Theorem of Pythagoras sketch from the previous post). While this is "doable", and some interesting sonic effects can be achieved in this way, it is not the ideal solution.

Since the diminished seventh chord has a dominant function it also has any one of four dominant roots available a major third below any one of the chord tones. Adding any one of these potential real roots will create the sonority of a dominant seventh with a minor ninth. This method allows for the constant root bass line to be employed to write transformations over, and at the same time for parallel octaves to be avoided with the virtual five voice texture this technique requires in an all seventh chord environment. I have given a couple of examples of this on the bottom staff of the above example.

In this context, the diminished seventh has a real percievable root, and all that "real root versus theoretical root" nonesense can be avioded entirely. Notice that in those examples the resolution is the proper delayed crosswise transformation that dominants and secondary dominants normally participate in, and that the minor ninth takes the place of the root momentarily in the normative seventh chord transformations. This is indicated with the numeral "9" in place of the "up arrow" to "1" in the transformation analysis symbol. This method is technologically far superior to the traditional analysis method, especially if the goal is to employ these sonorities in the composition of harmonic canons, as I have given an example of below.



As you can see, treating secondary diminished sevenths as upper structures of a real, actual dominant seventh chord is the most logical solution for the composer, if not the theorist. They become just another special type of secondary dominant sonority like the so-called French Sixth is (Which I will refer to as a V(4/2/b) from now on).

Since there are transformation types other than crosswise in this canon, all four of the voices get all four of the canonic parts in turn. That makes an eight measure canonic voice, and since the root progression pattern that repeats is two measures long, the voices follow at that distance. I composed the canon in the first twelve bars and extracted it starting in measure 13. Since I have delayed the second voice in the first pair until the second measure, it takes eleven measures before all four of the voices exchange all of the parts and participate fully in the canon. That is why I made this canon twelve measures long. I didn't bother analyzing the whole thing since it is a sequentially modulating two bar phrase, and the analysis simply follows it's due course.

The so-called "French Sixth" Chord

When I was a doctoral candidate at The University of North Texas, I had the honor and distinctly providential good fortune to study with the consumate music theory master, Dr. Gene J. Cho. Sometimes I think he was one of the only reasons I was lead to study there, as along with another wonderful theorist, Dr. Paul Dvorak, Dr. Cho was one of only a couple of professors at UNT who had anything of substantive value to offer me at that point; the composition faculty being, as it was, populated with post-modern know-nothings in my view. Dr. Cho reminded me of a musical version of the Star Wars character "Yoda" (No physical resemblance; it was an "attitude thing" coupled with his ESOL pronunciations and Chinese accent), and so I am fond of saying that it was he who was responsible for completing my training as a musical Jedi. He also kept me quite insulated from the temptations of the "Dark Side" of atonality and post-modern dilletanteism.

In any case, Dr. Cho has the most complete "unified theory of tonality" of any theorist in all of music history in my opinion, Arnold Schoenberg being a relatively close second. This exegesis on tonal theory is presented in his startlingly slim monograph entitled Theories and Practice of Harmonic Convergence, which I think is a must for the library of anyone serious about tonal music theory and composition.

One of Dr. Cho's pet peeves was the "nationalistic names" that most traditional theorists use to describe some of the more unique and idiomatic sonorities found in Western art music. His position is that these names describe nothing and are useless if the idea is to convey to the student an idea of what they really are and how they actually function (He used to tell this lame joke about "Irish Sixths" in every class he taught, and I believe I was the only one who ever laughed at it, and I laughed every time I heard him tell it. I love that man and owe him an enormous debt of grattitude). He is right about that, of course, and so today we are going to consider the wrongly so-called French Sixth sonority.



Above I have presented two examples to show the theoretical origin of the so-called French Sixth chord (I should note that Dr. Cho does not share in my opinion about this subject, he being of the opinion that augmented sixth chords have a purely contrapuntal origin, and we got into some spirited exchanges over this topic. As much as I hate to disagree with my former master, for the purposes of harmonic theory I remain convinced that this is the best way to present the subject for direct ease of understanding. IMO, we were arguing semantics just for the pure joy of it).

In the top example I have shown the most usual situation that you find the French Sixth in: A ii diminished to V progression in the minor key wherein the ii is in the second inversion (Diminished fifth in the bass), and it's third is raised as in a regular secondary dominant chord. That is exactly what a so-called French Sixth is: A secondary dominant with a diminished fifth in the second inversion. So, that is why it get's the analysis symbol V(4/3/b)/V: It is the secondary dominant of the five chord in second inversion with the diminished fifth in the bass, and this arrangement creates an Augmented Sixth interval with the raised third degree. All of the augmented sixth chords share this interval, but this one is the easiest to theoretically understand, so we'll deal with the others later.

Note that the transformation between the V(4/3/b)/V and the V is the usual delayed crosswise transformation most commonly encountered with secondary dominants. Note also that the final tonic chord is incomplete because the bass stops transforming and becomes a constant root. This is one of the main reasons why composing chord progressions over a constant root bass line and maintaining the transformations in the upper voices is such an overwhelmingly superior technological method compared to the standard practice of the Baroque through Romantic eras where the bass transforms or not depending only on whim (Once the progression is completed, more plastic bass lines may be composed to the transforming voices using transformations other than the most natural ones, and vigorous melodies as well, as will be seen).

This native minor progression was also used in the major key by both raising the third and lowering the fifth of the second inversion of the ii minor seventh to get exactly the same structure. This would give a D7(d5) in the key of C. The natural place that most closely follows the diatonic pattern of this chord occurs in major is between the vii diminished minor seventh and the iii, as I have shown in the second example. Here, I have employed the constant root bass line and put the diminished fifth in an interior voice, so it is essentially now a root position chord. This is perfectly fine, and any inversion of this sonority may be used as long as the diminished fifth is below the raised third to keep the augmented sixth interval. The octave displacement inversion of the augmented sixth, the diminished tenth, is useable - and I have employed it in a couple of pieces where it goes by very quickly on a weak beat - but it really is not as effective as the augmented sixth. I encourage you to experiment and use your own judgement, of course.

Because the so-called French Sixth is just a special type of secondary dominant, it can target all of the same diatonic degrees that any regular secondary dominant can. If we return to the progression I used yesterday, a secondary V7(d5) can be introduced on the last eighth note of every measure that has a Progressive root motion. This does some wonderful things for that progression. Since the two lower canonic voices alternate between functioning as the root or fifth through their crosswise transformations, those voices can be alternately decorated with the diminished fifths and the canon can not only be maintained, but the canonic nature of those voices is actually enhanced and it becomes much more obvious that they are canonic in nature. Not only that, but since the first vii chord has a naturally diminished fifth that progresses diatonically down by a semitone, the first measures of both of those vioces can have pickup eighth notes since the tonic also progresses down by semitone from the original first measure. This adds a huge degree of elegance to the voice entries, and it even translates to better entries for the constant root bass notes.

Finally, we can allow this eighth note to propigate throughout the entire lower canon so that those two voices become a basic accompaniment texture of constant eighth notes. The canon is still maintained! The sketch for my Theorem of Pythagoras compositional exercise now looks like the following where the last eighth note of the lower canonic voices alternately introduce the diminished fifth. I'll cover the secondary diminished seventh and the enharmonic augmented seventh chords found in the penultimate measure at a later date.



Note that the final C major chord has it's tones arranged in the exact same arrangement and position as they appear in the Natural Harmonic Overtone Series after all the suspensions are resolved. ((To polite applause) "Thank you (Stage bow), Thank you very much." (Stage bow)).

Secondary Dominants and Harmonic Canons

I have decided to leave counterpoint behind for the time being (Sort of), as I am currently involved with several harmonic-approach compositions but no purely contrapuntal ones. As with all obsessive/compulsive types, once I get involved in a particular project, I'm in it to stay until it's taken it's full course.

As mentioned earlier, the Natural Harmonic Overtone Series defines for us the primordial falling fifth V7 to I harmonic progression through it's first seven partials, which spell out a dominant seventh chord. Throughout the late Renaissance and early Baroque eras, this progression was extended to secondary degrees of the key until the entire pantheon of Secondary Dominant Sevenths was completed. The way this works is that in any root progression in a key that is Progressive in nature, and that involves degrees other than V and I, the first chord in the progression can be changed from it's natural state - either a minor seventh or a major seventh - to a dominant seventh chord so that it targets the secondary degree in the same manner as the V7 to I progression does.



In the top example above, I have written a progression that allows for all of the secondary dominants to be displayed. By progressing directly from the tonic to the seventh degree, I have set up for a series of falling fifth Progressive Root Motions that will involve each of the seven degrees of the key.

The top example shows the triadic Circular Transformations. After the Strong Decending root motion from the I to the vii diminished triad that kicks off the progression with it's associated clockwise transformation, all subsequent root motions are of the falling fifth (Or, rising fourth) Progressive type. Note that in the Progressive root motions all of the Circular Transformations are counterclockwise (1 becomes 5, 5 becomes 3, and 3 becomes the root), the triads share a single common tone, and that the voices rise in overall pitch from chord to chord. In the final bar there is a Strong Ascending root motion from the IV to the V to turn the progression around on itself, and it's transformation is counterclockwise opposite of the Strong Decending progression from the first measure to the second. This is an example of a very unbalanced chord progression: Since there are not a variety of root motions that alternate between clockwise and counterclockwise transformations, subsequent repititions of the progression will soon lead to problems of range if an all-triad environment is maintained.

In the second example of this progression, I have presented it in an all-seventh chord diatonic environment. Note that in this example the Progressive root motions transform in a crosswise manner (1 becomes 5, 5 becomes 1; 3 becomes 7, and 7 becomes 3), there are two common tones between adjacent chords, and that the voices fall in overall pitch throughout the progression. In an unbalanced progression involving a majority of Progressive root motions (Which includes the overwhelming majority of progressions one encounters in Western music), potential range problems can be avioded by alternating between periods of triadic textures and seventh chord textures. The importance of having this understanding should be obvious because, after all, true freedom is total control (Besides tending toward the obsessive/compulsive, I am also a "control freak", as my ex-wife will freely attest to).

In the third version of the progression, I have changed the diatonic chords to dominant sevenths in the second half of each measure to demonstrate the most natural transformation type for the resulting chain of Secondary Dominants. The seventh degree requires that the third and the fifth be raised - unless of course you want the sound of a dominant seventh with a diminished fifth (Which is a very cool sonority, not to mention that it is the origin of the so-called French Sixth) in order to get the resulting V7 of iii. The Natural tendency of the raised third degree of a secondary dominant is for it to resolve to the root of the target chord. This results in the target chord being a triad with the root doubled. Now, the other possibility is for the secondary dominant's raised third degree to "resolve" down chromatically to become the minor seventh of the target chord as we saw in the Minuet by J.S. Bach that I analyzed earlier. Here, I want to use the more natural arrangement to demonstrate the transformation type that is most often used or encountered, and also to fulfill a compositional goal that I have for this progression.

Since the target chord becomes a triad with a doubled root momentarily before it continues on to become another secondary dominant seventh chord, the usual crosswise transformation is delayed. I have indicated this with perentheses around the arrow going to the seventh degree in the transformation analysis symbol. If the less-usual decending chromatic "resolution" of the raised third degree of the secondary dominant was used, the transformation would be the same as the diatonic version, and no parenthesis would be needed. Another thing to note in these natural transformations of secondary dominants is that there is only a single common tone between the adjacent chords. This serves in conjunction with the secondary leading tone to give the progressions a stronger effect than the diatonic versions.

When the progression reaches the V and the I, I have changed the triads on these degrees from major to minor to keep the established pattern going. There is also another reason for doing this, which will become apparent in a minute.

Since there are two tones that change in each measure, I could get a constant quarter note surface rhythm if I put the progression in 3/4 time and changed first one note, and then the other; and since the progression is mostly a series of falling fifths, I decided to use the rhythmic resultant of interference of the interval of the perfect fifth to achieve this, which is the resultant of 3:2= 2+1+1+2.

In the fourth example of the progression, I have opened up the chord voicings to avoid unisoni and to make it easier to visualize. Starting with the top voice in the third measure, you can see the rhythmic resultant of interference applied where 2+1+1+2 becomes a half note plus a quarter note, plus another quarter note, plus a final half note. You can also see this figure in the top voice of the left hand (Assuming we're dealing with an organ piece here) starting in the second measure. These rhythms dovetail to give the overall surface rhythm of constant quarter notes. But there is more to what I have done here. Much more.

In the final example, I have removed the preperatory setup voices involving the I to vii diminished minor seventh progression to reveal that what I have composed through this series of exercises is an accompanied harmonic double canon at the fifth below (Or, at the fourth above). I have arranged the voices so that the two most obviously canonic ones are on the top, and they follow at one measure at the fourth above. The two less obviously canonic voices are underneath and at the fifth below. Since these voices enter on the roots of their respective chords and start a series of decending fifths, I have withheld the constant root accompaniment stratum until the third measure and allowed it to continue with the series of descending fifths in this orchestral sketch. And, this is actually the second section of the introduction to a composition of mine called Theorem of Pythagoras for string choir. I chose the title, of course, because the Pythagorean tuning system's stack of fifths tuning scheme is reflected in the falling fifth root motion here, and I chose the string choir because they can more closely approximate the just fifths in that tuning scheme. Add to that my employment of the rhythmic resultant of the interval of a perfect fifth, and all the details of the composition relate to the perfect fifth. All_of_them.

This method of composing harmonic canons is by no means new, though my employment of rhythmic resultants of interference is relatively unexplored. J.S. Bach used this device constantly in his 371 Four-Part Chorales, in his figuration preludes for various instruments (Where the canons are so well hidden that it takes quite a bit of work to extract them), as well as in many of the episodes in his fugues. Progressions constructed using this device are infinitely more compelling than progressions with similar but casual formations, and any time you have a series of repeating root progression patterns you have the opportunity to compose harmonic canons! These progressions elicit in the listener a sense of wonder and sublimnity, whether they are consciously aware of the canons or not. In harmonic composition, there is no more artful or elegant solution. Period. Exclamation point. That this art has been lost, ignored, not utilized, and never taught is a tragedy of epic proportions, in my estimation, because the technological understanding of harmonic progression mechanics has never been greater than it is now, and by employing ninth chords or other sonorities with extended upper structures many new and unused harmonic canons remain never before used in all of music history.

Site Notice One

I will be travelling to San Antonio this week to visit my mother and some friends, and to take some much needed R&R, so there will be no new posts until at least Thursday 06/23 or later. If you discover this blog while I'm away and are addicted to music theory and composition as I am, please read through the previous posts, as there are only a month's worth to catch up on.

This brief hiatus will give me an oportunity to finish off Zarlino and chart the direction in which I wish to proceed. Currently, I am leaning toward addressing Fux as the last of the historical contrapuntal treatises, but I'm not positive about that as my current compositional endeavors are harmony-based in nature and it appears that will be the case for at least a while.

Also, if you like what I'm doing please consider putting a tip in my PayPal "tip jar", since photo hosting, internet service and the like costs real actual money and I'm just a full time gigging guitarist and private teacher, and so have modest means at my disposal. I would like to think that the information I freely share here is also of some value, as it reflects nearly thirty years of my quest for musical knowledge and understanding.

UPDATE: Before I left today I went through my blog and discovered some comments I was not aware of. I thought I would get e-mail notifications, but I failed to enter my e-mail address in the appropriate field. I'm kind of a computer luddite in some ways, so please forgive that unintentional oversight. I will also update my profile to display my e-mail address if you want to contact me off the blog. I welcome questions, but be aware that they could turn into posts. ;^)

Composition Exercise One

You were probably expecting that the first composition exercise demonstrated here would have been a contrapuntal exercise, since I have spent so much time on that subject. So did I. However, I started writing a series of arpeggio studies for the guitar this month, and this morning over coffee I wrote the second in the series using the progression I analyzed in the Mechanics of Harmony post, so I decided to cover creating simple musical continuities out of harmonic progressions first.

It would be a good idea to review that previous post before continuing here.



As you can see, I started with a simple eight bar diatonic progression that uses all seven diatonic chords and that also has every type of diatonic root progression except for the TriTone as Retrogressive arrangement. The TriTone as Progressive root progression from the IV to the vii diminished divides the original progression into two halves that are mirror images of each other with respect to the root progression patterns: Where there is a Retrogression in the first four measures, there is a Progression in the second four; where there is a Strong Ascending progression in the first phrase, there is a Strong Decending in the second; the Mild Decending movement of the first section is answered by a Mild Ascending motion in the second. The only variation is that at the end of the second phrase there is a Strong Ascending progression from the IV to the V to turn the phrase around on itself. This two chord per bar harmonic rhythm will only appear three times and in this exact same form each time during the piece.

For the second eight bars, I isolated the root progression pattern of the first four bars of the original progression and repeated it in sequence still using the TriTone as Progressive root motion to link the to phrases of the progression. This creates an unusual non-dominant direct modulation to the parallel key a semitone below the tonic, but the previously established pattern makes it work seamlessly. I also changed the gender of the first phrase, making it in the tonic minor key, and I kept with that idea and have the second phrase in minor as well. The idea with this phrase, and the subsequent phrases not on the tonic, was to choose between major or minor by selecting the key closest to the original tonic of A major. So by that logic, G-sharp minor would be closer to A major than A-flat major would.

Just as there is a V to I Progressive root progression at the end of the original phrase, there is a similar progression at the end of the second eight bars. However, the one chord per bar harmonic rhythm is maintained throughout.

For the third phrase, I isolated the root progression pattern from the second four bars of the original progression and have it arranged as a mirror image of the second eight bar progression. It too sequences a semitone, but a semitone up versus the previous phrase's semitone down. It is also divided in half by a TriTone root progression, but this time the sole unused TriTone as Retrogressive form is employed, and so the list of possible diatonic root progressions is completed. Far from being a minor detail, this is a major structural point. I continued the first phrase of the third eight bar progression in the minor mode, but changed back to major for the last four bars of the third progression due to G major being closer to A major than G minor with all it's flats.

The modulation back to A major from G major is made exceedingly smooth by the last two bars having chords shared by both keys. I am actually quite delighted with the effect it has: The last two bars of the final eight bar phrase are in fact identical to the last two bars of the original progression, and here we have the second instance of the two chord per measure harmonic rhythm.

The final eight bars are an exact repeat of the original progression with a final two measures of tonic harmony at the end.

I used Schillinger's concept of Circular Transformation with only some minor adornments to arrange the top three voices of the piece: on the diatonic vii diminished triad I added a diminished seventh to give it the more dramatic color I wanted after the TriTone progression, and also to set up the later adornments in the same places in the third eight bar phrase. In measure 17 I use the same figure to get the sound of a minor triad with a major seventh, and in measure 21 I use the sound of a major triad with an added major seventh. These are only adornments and do not affect the purely triadic Circular Transformations of the top three voices in any way.

The bass voice is a slightly adorned Constant Root: Adding a constant root bass line to a properly transforming triadic or seventh chord progression will never result in forbidden parallels. This is a simple example of Schillinger's concept of Strata: The transforming triads in the upper three voices are one stratum, and the constant root bass line is another. You could concievably - and easily - have a third stratum of triads transforming opposite of their natural inclinations and still have no forbidden parallels with the resulting seven voices. Where there are Mild Decending or Mild Ascending root motions, I decorated the bass line with a passing seventh or a ninth in the bass respectively. Since I'm not using seventh or ninth chords, there is no danger of forbidden parallels with this embellishment. This relatively insignificant looking figure adds an unexpectedly large amount of dimension and depth to the resulting progressions in inverse proportion to it's appearance on the page.

Once I had all of these elements constructed in a whole note/half note voice leading sketch in C major, I selected the key in which it would fit most comfortably on the guitar. Since I do not want any open strings in the fingerings (So the piece can be moved to several pitch levels depending on what my performance set needs at any given time), the low note of F would have given A-flat major, but A major is a much more common guitar key, and it's easier to read with only three sharps as well, so that's where I put it.

Lineal Study No. 1 is in G major, 4/4 time, and has the simplest possible texture of straight eighth notes, so I wanted this one to be in a different time signature and with a small added element of rhythm to it since the overall finished set of Lineal Studies is to be roughly progressive in terms of complexity and technical difficulty (I'm planning six to twelve of these little ditties). So, after I decided on 6/8 time I decided to use the rhythmic resultant of the interval of a perfect fourth, or the ratio of 4:3 (If you look back to the Relativity of Pitch to Tempo & Harmony to Rhythm post, you can see how these resultants of interference are worked out). I chose that because there is a compound tripple meter with four bar phrases, so it's a logical aspect of the composition as a whole. That resultant is 3+1+2+2+1+3, so I took the first half and decorated the first three eighth notes of every measure with it (3x1/16ths+1x1/16th+2x1/16ths), keeping the second group of three as the smaller term 3 but diminuted to the middle sized term of the resultant (An overblown way to say I left them straight eighth notes). I didn't use the second half of the resultant until the penultimate measure of tonic harmony, where combined with a slight ritardando it makes for a spectacularly effective close, despite the direct simplicity of the employment of it that I chose.

It is important to realize that there are many other possible permutations of this original progression available if you change the TriTone dividing progression to other forms, and with repeat schemes &c. this could have been made into an art song, or a movement for a chamber ensemble, or even a symphonic movement. I'm a guitarist and have no ensembles available to me, so I'm writing things that I can play and perform: MIDI renditions of larger stuff is only fun for so long for me. I want to play and perform the music that I write. Single line passages are also one of my technical weak spots, and I have neither the time nor the patience to work on scales and arpeggios, so these pieces will also serve a utilitarian purpose.

One final point: Different root progressions make the triadic voices move in different directions, either upward or downward. Strong Ascending progressions move the voices down, while Strong Decending progressions move the vioces up. Mild Ascending progressions also move the voices down while Mild Decending progressions move them up. And Retrogressions move the voices down, and Progressions move them up. Finally, TriTones move the voices up if they are in an as Progressive arrangement, or down if they are in an as Retrogressive relationship. Knowing this allows you to control where you want to have your voices move, and obviously an unbalanced progression can get you into trouble with ranges if you don't know what you're doing.

Now, the direction that voices move with seventh chords is an entirely different matter, and I'll wait to cover that until later, but you can certainly do some experiments and find that out for yourselves.

Even in small pieces with a modest scope - such as these Lineal Studies which are a minute or less in total playing time - it is not only possible but also highly desirable to construct them with a high degree of craftsmanship. By adhering to this philosophy of structural perfectionism and adding the necessary elements of inspired variety and detail, the resulting miniatures have a high degree of musicality. I always encourage my students to write many, many miniatures as this is the most effective way to master and develop the proceedures necessary to tackle continuities with a broader scope.

MM VI: The Mechanics of Melody

This is another concept of Joseph Schillinger's (Who I've linked to before if you look back a few posts) that I find extremely valuable. In fact, in my opinion his "Theory of Melody" is the single best reason for the study of his "System of Musical Composition". The reason for this should be obvious: No other theorist in music history has ever taken a scientific approach to describing this phenomenon, and the art of writing melodies has always remained in the realm of the mysterious or even the supernatural because of this regrettable fact.

Now, it must be admitted that there are many problems one encounters in the study of Schillinger. First of all, the two volume System is a daunting 1,640 pages in total length. Second, Schillinger's prose is very densly abstruse due to the unalloyed mathematical and scientiffic terminology he employs: Terminology that is profoundly alien to the vast majority of musicians. Then there is the fact that Schillinger was a futurist, and so he attempts to present all the possibilities for music of the future along side of explainations of past and present practice. Additionally, the System was compiled by some of his students after his death, if memory serves, and so there is some doubt that he actually would have wanted the materials presented in the form that they are encountered. Finally, there is the fact that the musical examples he presents that are not by other composers are abjectly lame in the extreme. In light of these formidible and compounded difficulties, it is easy to see why the overwhelming majority of traditionally oriented theorists and composers dismiss him out of hand. This is a shame - not to mention that it's usually an intellectually lazy cop out - because, thought flawed in many ways, Schillinger was nevertheless a genius who had many unique and penetrating insights into the nature of music that can be found nowhere else. In lieu of inserting pitch scales into formulas as he suggests in his over-rational methodology, I took the ideas Schillinger presented and made myself aware of them on an intimate enough level that I was able to allow them to function intuitively as a natural - and normal - part of the compositional process. Since the type of personality necessary for a composer is not usually one prone to an overtly scientiffic or mathematical disposition, I believe this is the way that many of his ideas should be employed. Melody is one of the subjects where a looser, more intuitive approach has yeilded better results for me, whereas the harmonic motion mechanics he explains can be used in practice very closely to the way he describes them in his theories.

So, the first step in coming to an understanding of a subject is to define it as precisely as possible.


A melody is a trajectory of pitch through time.


Once this definition is made, an analog can be used to put it into an understanable perspective. I will use the analog of a quarterback on a football team throwing a football to a receiver.


A melody is like the trajectory of a thrown object, such as a football.


Now, in a perfect vacuum with no gravity, if the QB threw the football, it would launch in a straight line and continue off into infinity on that perfectly straight trajectory. If you continue in the vacuum but add gravity to the calculation, the trajectory of the football would peak at the approximate half-way point between the QB and the receiver. Gravity becomes a force of resistance that the football encounters, and so the trajectory has a beginning and an end with the climax of the trajectory at the half-way point. The trajectory under these conditions would describe a part of a circular curve. But, the football game is not played in a vacuum, so there is another form of resistance that the ball encounters: Wind resistance. In an indoor stadium with no additional wind but only still air for the QB and the ball to deal with, the trajectory would now peak approxamately 2/3 from the QB and 1/3 from the receiver. This is in fact the primordial blueprint for a melodic trajectory as presented by the natural order.

Just as the QB imparts rotational spin to the football to stabilize it as it travels along it's trajectory to the receiver, melodies are often adorned with figures that impart rotational spin to them as well. And these rotations often impart increasing degrees of momentum to the melody on the way toward the climax, and then release the momentum after the climax.



Schillinger describes the Axes of Melody as I have labelled them on the top staff in the example above. The Zero Axis can be the pitch axis, or tonic, of the key, or it can be either the third or the fifth of the tonic major or minor triad of that key. The A and D axes go above and below the zero axis respectively, and they are unbalancing axes, meaning they increase tension as they move away from the Zero Axis. The B and C axes travel toward the Zero Axis from above and below respectively, and they are balancing axes, meaning they release tension as they move toward the zero axis.

Since the subjects involved in immitative or fugal composition are often micro-melodies that offer a microcosm of these principles in actual practice, I have taken two of them from Bach to use as examples. The middle example is from the Two Part Invention No. 8. In the first measure Bach establishes a Zero Axis on the tonic degree, and as the A Axis moves away from it and returns to it in successive movements, momentum is increased as an integral part of an increasing scale of tension. This is a broadening rotational structure. After the climax on the upper octave C, this tension is released with a contracted rotational structure in the decending sixteenth note figure that brings the melodic trajectory back into balance at its return to the Zero Axis. In the bottom exampe, which is the fugue subject from the D minor Organ Fugue of Tocatta and Fugue fame, Bach establishes a Zero Axis on the fifth of the tonic triad and spins the subject out below it. Here, the Zero Axis acts as a form of artificial gravity in an inverted musical environment where the B and C axes are employed to give cycles of tension and release through broadening and contracting rotational periods against the established axis.

This subject will come up again and again in various degrees of depth, but an understanding of these basic concepts can be of inestimable value to a composer who is intent on organizing melodies and climaxes to achieve desired effects.

Analysis I: Menuet by J.S.Bach, No. 15 From Anna Magdelena

I've decided to wait to post my next Miscelaneous Musings about melodic trajectories until tomorrow, mostly because I got distracted with this piece last night, which is a Menuet by J.S. Bach that is No. 15 in the Anna Magdalena Notebook of 1725 (You'll have to do your own search for it as I found no good descriptions to link to. The Urtext is by Henle, but I have the Edition Peters version, which is complete and unabridged, and I like it a lot. I was not able to locate it so it may be out of print since mine is the printing of 1949). I have wanted to make a guitar transcription of this piece for quite a while, and I also wanted to do a detailed contrapuntal and harmonic analysis of it as well, so I decided to kill several birds with one stone last night when I discovered that it would work fabulously well on the guitar in B minor, and I am needing a small piece in that key for my set. Not only that, but it is the single most supreme example of some of the concepts I've been presenting here over the last month in actual musical practice.

The only way to describe this piece when it is considered in the larger context of Bach's corpus of work is to aptly call it strange, weird, or even bizarre. Whenever someone tells me that strict 1:1 ratio counterpoint should be taught as an all consonance environment, or that strict 1:1 counterpoint is just a theoretical exercise that has no application in musical reality, I wave this marvelous little trifle in their faces. On first listen, it is strikingly dissonant and the chromaticism is just plain unnerving, especially when you consider the source and that it is such a tiny little miniature intended as a practice piece. I can't help but think that if a student turned this in as an exercise for a counterpoint course, the instructor would laugh him out of the classroom. There is a picture in my mind of Bach waking up one morning to dash out a few pieces for his bride and geting this mischevious little grin on his face as he noodles this out, giggling as he goes. It's a fantastic conceit of my imagination, of course, but that's the way I envision it. The piece certainly makes me laugh, that's for sure, and knowing that Bach had a robust sense of humor, I'm virtually certain that this "monstrosity" is intended - at least in part - as a joke.

The following conventions are used in the analysis:


The contrapuntal (intervallic) analysis is on the top line.

The home key harmonic analysis is on the second line.

Regions other than the home key are on the third line if present.

The overall regions are indicated at the bottom.

Intervals are notated as P= Perfect, M= Major, m= minor, A= Augmented, and d= diminished.

The harmonies are in bold with the root progressions in between in plain text.

Subsequent inversions of the prevailing harmony are in brackets and in plain text.





The piece starts out very conservatively with an immitative figure at the octave below (If you disregard the opening B in the bass, which I will probably omit when I perform this), and the intimated root motions are simple and conservatively Progressive in nature. Then, into the fourth measure, Bach suddenly has a Tri-Tone as Progressive root motion which is into a "five of five", and it even has the "melodic" progression of a diminished third in the lead! This is quite disconcerting and humorous. I mean "laugh out loud" humorous. It's like the man went instantly insane or into "The Twilight Zone" or something. The effect is to produce the sound of a V(b5)/V, which is just plain goofy in this context.

Then the bass moves down by step to produce the effect of a Vb6/4/2 of V, which just increases the weirdness factor. When the expected V6 appears, does it's leading tone resolve up as expected? Of course not: It progresses down chromatically to become the minor seventh of a V4/2 of iv (Note that this is actually the regular crosswise transformation that you might expect in an all seventh chord diatonic Progressive root motion, but it's not what secondary dominants are expected to do). Then, the root progresses to the fourth degree as expected, but there is another secondary dominant on that degree: It's leading tone progresses down chromatically in sequence to the previous instance, and I'm expecting a little gray creature to say "Take me to your leader!" at any moment now. Note that all the root motions are simple Progressives though? Abnormalcy via normalcy. Very cool. And riotously funny. Bach uses the Tri-Tone as Progressive root motion in the same place in this phrase as well, with an augmented ninth sonority in the lead just for good measure. The two phrases are structurally balanced in a logical way, despite the fact that their overall character is so uncanny. The insult is then repeated, of course.

The second section starts out with a four measure prolongation of a V to i in the subdominant region. One thing to notice here is the minor seventh at the end of the second measure that is leapt into. In a pure and strict 1:1 ratio contrapuntal environment, this would be unacceptable. Period. However, since this is a harmonic contrapuntal environment, subsequent validations of a prevailing harmony allow for this sort of thing, especially on a weak beat, as we have here. After the resolution to the "tonic" on the subdominant degree, the bass moves down to intimate a V4/2 of V in the relative major region, despite the absense of any G-sharp in the phrase (At least that's the way I hear it - comma - man).

This same V to I sequence is then repeated a whole step lower in the relative major region with only a single minor variation: No seventh is lept into and the IV of the prevailing region is implied instead. These are probably the most "normal" phrases in the piece. Good thing too, because Bach is just about to top all the weirdness that has come before with the single most outrageously mad succession of 1:1 counterpoint I have ever heard from a tonal composer. Hold on to your butts.

What can I really say about measures 17 through 22? The intervallic and harmonic analyses tell the tale and sol, le, la, te, ti do and ti, do, re, me, mi, fa and even re, me, mi, fa, fi, sol are not all that unusual for melodic progressions of this era in and of themselves; but all together one after another in an episode of the strictest kind of 1:1 counterpoint written circa 1725?! Note that despite the seeming outlandishness of the progression the root motions are actually quite logical, and measures 17 and 18 are perfectly reflected in measures 21 and 22 as far as the root progression types are concerned. That's what makes this work, and why it's such a brilliant joke: Beneath the apparent lunacy on the surface is a very logical and in many ways quite conservative construction that is well thought out. That Bach ends this phrase with one of the most common i - iv - V - i cliches of his time just puts an excamation point at the end of this "unforgivable" episode of musical jocularity as far as I'm concerned. I absolutely adore this abjectly idiotic little thing (And I mean "abjectly" and "idiotic" in the kindest of all possible ways ;^D).

The Art of Counterpoint, Part Six

Zarlino has shifted gears now, and is going through several subjects with breakneck speed that I'm just not ready to address yet: Immitation, fugue, and canon; counterpoint that is invertable at the tenth and twelveth; and cadential formulas. Since I want to go through Fux as well, I have decided to finish up Zarlino without further comment and allow Fux to address those subjects for me. We'll get into counterpoint with more than two voices with Fux as well.

One last thing about accented and unaccented beats needs to be addressed, and that is the pattern found in strict 3:1 two voice counterpoint: Strong where there are simultaneous attacks followed by two weak beats. Everything else explained in the reduced rule sets for 1:1 and 2:1 counterpoint can be used in this idiom, and besides the potential for two dissonances over a single note in the lower voice, 3:1 is just like 2:1. The main reason for bringing it up is to complete the accent set possibilities: Every possible accent - or strong beat versus weak beat - scenario is going to be some combination of the possibilities found between the 1:1, 2:1, and 3:1 ratios. For instance, 4:1 is normally thought of as strong, weak, strong, weak, which is just two 2:1 accent sets. Similarly, 5:1 would be a 2:1 accent set followed by a 3:1 accent set, or vice versa.

In the next Miscelaneous Musings post that I am preparing, I will introduce some concepts found in Schillinger's "The Theory of Melody" which I have found to be of nearly priceless value in writing countrapuntal music. Since the natural way to begin to learn this subject is by writing a cantus prius factus and then adding counterpoint to it, it makes sense that the quality of the intital melody will determine the potential for the quality of the resultant piece. So, in order to write two complimentary melodies that make a good contrapuntal relationship, it will first be necessary to understand how to write a single melodic trajectory that is effective and has the potential to have counterpoint added to it.

See you tomorrow.

The Art of Counterpoint, Part Five

When Zarlino goes into what he calls "diminished counterpoint", he basically goes from a loosely defined 1:1 ratio counterpoint where repeated notes are admitted (Versus the strict 1:1 counterpoint I use to teach mechanics that does not allow for repeated notes) to free florid two voice counterpoint with a mixture of ratios. To more precisely address the concept of strong beats versus weak beats, I will make this move progressively using strict versions of 2:1 and 3:1 counterpoint in which there are no repeated notes allowed in either voice.

Objections to my concept of dissonance being allowed in strict 1:1 counterpoint I handle thusly: The duration of the note values in 1:1 counterpoint can vary from an entire measure down to divisions theoretically as small as a 128th note or even smaller if triplets, quintuplets &c. are employed. In that sense, there really are no strong or weak beats in strict 1:1 counterpoint: They can be considered as all strong, all weak, an alternation of the two, or a strong followed by two weak beats in a simple or compound triple meter or even other variations in odd meters. A composer may or may not want to use a progression that goes from a perfect fifth, to a seventh, to a ninth, and then to an eleventh at a slow tempo with a texture of whole notes, but that is simply a matter of taste: The quantum mechanics of counterpoint allow for such progressions regardless. Now, in a simple or compound triple meter where the fifth is on the strong beat and the seventh and the ninth appear on the following two weak beats, and the tempo is moderate and the texture is that of quarter notes or smaller, no objection would be raised even by traditional theorists. My point here is, if it's allowable in one instance, it's allowable in any instance; that's the fact of the matter when you consider it logically and exclude subjective evaluations based on taste or the previous established practices of a certain composer or compositional school.

Lastly concerning two voice 1:1 counterpoint, I have been reminded that I failed to address outright the law of melodic leaps. In strict 1:1 counterpoint if either one or the other voice leaps, the exit and entry intervals must be consonances. So, the irreducible law of leaps in 1:1 counterpoint is that leaps can only occur between two consonant intervals. Keep in mind that I am isolating counterpoint from harmony at this time. Later, when we work with three or more voices and are intentionally putting harmonic considerations back into the mix this law will be subject to variation. But by that time, it will be evident that harmony is influencing the counterpoint and that harmonic laws are being borrowed to make the counterpoint freer in that particular context.

In strict 2:1 ratio counterpoint we must begin to deal with the concept of strong and weak beats. The duration of the notes of the respective voices is irrelevant: Where there are simultaneous attacks by both voices, there is a strong beat, where only the diminished voice moves, the beat is weak. It also does not matter if the diminished voice moves by uneven note values - dotted quarter/eighth in a duple meter or quarter/eighth in a triple meter or whatever variation - the non-simultaneous attack is always considered a weak beat or a weak part of a beat.

Just as the normative concept of strict 1:1 ratio two voice counterpoint is that of an "all consonance" environment with dissonance only allowed when both voices enter and exit the dissonance by stepwise contrary motion or by parallel motion involving unequal fifths, the normative concept of strict 2:1 ratio two voice counterpoint is that of an environment in which strong beats have consonances and weak beats may have dissonances as long as the diminished voice moves into the dissonance by stepwise oblique motion. However, just as in 1:1 counterpoint, if both voices move by stepwise contrary motion this normal state may be reversed and the dissonance may appear on the strong beat and the consonance on the weak beat. For example, in the progressions ||10-9|7-6|| and ||6-7|9-8||, or variations on that theme. Beyond even this though - which is simply using the laws that govern strict 1:1 two voice counterpoint in a strict 2:1 two voice contrapuntal environment - there is a much broader concept of counterpoint available with the strict 2:1 ratio that began with the stylistic affectation known as the nota cambiata or the changing note group.



In the above example at 1 is an example of a nota cambiata figure as employed by Palestrina and others of his era. Here is an example of a changing note group in which the first dissonance is quit by a leap directly into another dissonance before the resolution finally comes. Now, that example is over a single note in the bass, but the instance at 2 has the initial dissonance again quit by leap into another dissonance over a different note in the bass. The delayed resolution principle that the changing note group makes possible remains in effect.

The inverted forms of this cambiata figure were not employed by Palestrina, presumably because his hyper-conservative approach to style rejected them as a matter of principle, or taste (Palestrina is one of my favorite composers in all of music history, and is in some ways superior to Bach in my opinion, so please don't take that as a criticism). Nevertheless, they are obviously available and no law of counterpoint would be broken by employing them. In fact, the full quadrant rotation of this figure could be completed and the retrograde variations could be used as well. What does this tell us in the broader view about 2:1 counterpoint? As long as there is a consonance in one or the other positions over the bass note - and no forbidden perfect parallels are implied - leaps into and out of dissonances are restricted only by taste when preceeded and followed by a stepwise departure from and resolution to a consonance. Obviously, this law presents the composer with an apparently vast ammount of potential freedom, and that's true. However, experimentation in this idiom after attaining facility with strict 1:1 counterpoint will shorten this to a finite set of possibilities governed by your individual taste, which is exactly the point of these explorations.

MM V: Harmonic Progression Mechanics

As I have alluded to before, the quantum mechanics of counterpoint and the quantum mechanics of harmonic progressions are two different things. Unfortunately, very few musicians and music educators understand this. As a result, the teaching of these two subjects is surrounded with nearly cabalistic obfuscation and therefore needless difficulty for the student or the aspiring composer. In reducing the largely taste-based rules of counterpoint to a small set of underlying irreducible laws, I have made it much easier on myself vis-a-vis developing an individual contrapuntal style. By jettisoning all of the old rules that are based on taste - or the style of a particular composer or compositional school - and using just the underlying irreducible laws of contrapuntal motion, I can allow my own taste to operate within a contrapuntal environment that offers the broadest possible range of choices, and thereby develop those principles that are only a matter of taste for myself using my own judgement. This is, in fact, the basic set of operations that are required to develop an individual style. The mechanics of harmonic motion is a much simpler issue, and therefore easier to distill out to irreducible principles, than contrapuntal laws are, and so I've decided to introduce the essential concepts at this time.

First, it must be understood that there are different types of root progressions and that these various root progressions have different kinds of effects that can be described fairly easily. Only one traditional theorist, Arnold Schoenberg, has ever made an attempt to enumerate these root progressions and describe them - so far as I am aware - in his book The Structural Functions of Harmony. I have borrowed his idea but revised some of the terminology to better reflect how I think of these various root motions and their effects. Furthermore, I have developed abbreviations for these progression types that can be added to any harmonic analysis so that the root progressions can be visualized. Being able to visualize the root progressions is very helpful because through visualization patterns can be more readily recognized, and it is precisely these patterns that give a chord progression it's cohesiveness and drive. So far as I know, I am the only one who currently teaches the writing of harmonic progressions through pattern making, and all of my harmony-based compositions are written using this technique. I never encountered this approach in either my undergraduate or my post graduate studies, which included taking all the courses required for a DMA in traditional composition. That the writing of chord progressions isn't taught this way is a shame, because then what makes a good progression remains in the realm of mystery and writing them is a hit-or-miss game that depends on the writer's intuition instead of his logical abilities.

The primordial root progression is the falling perfect fifth (Or, rising perfect fourth). Statistically, it is the most common root progression in all of Western music: Traditional, jazz, folk, and popular. As with all things musical, there is a logical reason for this that nature presents to us. If you look back at the Natural Harmonic Overtone Series, you will readily see that the first seven partials spell out a dominant seventh chord. As we know from the V7 - I progression, this sonority wants to have it's root fall a perfect fifth (Or, rise a perfect fourth) to the tonic at cadence points. This dominant progression is often replicated and extended to other degrees of the key. In total, you could have a chain that involves all seven of the diatonic chords in this arrangement: vii dim - iii - vi - ii - V - I - IV - (V - I). The chords on the degrees of this progression chain were often changed from their diatonic versions to dominant seventh chords to get a stronger effect that more closely resembles the V7 - I that nature defines for us. The resultant chords are called secondary dominant sevenths: V7/iii - V7/vi - V7/ii - V7/V - V7 - V7/IV - IV (V7 - I). This most pervasive of all root progressions I simply call a Progressive Root Motion, and it gets the abbreviation "P" in the analysis. The full list is:


1) Root ascends by step: Strong Ascending root motion= "S+"

2) Root decends by step: Strong Decending root motion= "S-"

3) Root ascends by third: Mild Ascending root motion= "M+"

4) Root decends by third: Mild Decending root motion= "M-"

5) Root ascends by perfect fourth: Progressive root motion= "P"

6) Root decends by perfect fourth: Retrogressive root motion= "R"

7) Root ascends by augmented fourth: Tri-Tone as Progressive root motion= "TT~P"

8) Root decends by augmented fourth: Tri-Tone as Retrogressive root motion= "TT~R"


Obviously, the corresponding intervallic inversions are labelled the same (Seconds= sevenths, thirds= sixths, etc.)

Next, the concept of circular transformation of the chord member tones must be understood, as this is the quantum mechanics of chord progression. For this we will need the following example.



Above I have written two versions of the same chord progression; one with triads and the other with seventh chords. The bass line is just a constant root to make it easier to see the root progressions. This chord progression has all of the different diatonic root progression types except for the decending augmented fourth (TT~R). If you look below the "C:" at the beginning of the top example's analysis, you will see 1, 3, and 5 arranged in a circle. This is the basis of the concept of circular transformations: between two triads in any diatonic root progression, there are only two possibilities; clockwise transformation or counterclockwise transformation. This concept is one of Joseph Schillinger's which I have enthusiastically embraced because it cuts through all the contrapuntal pollution inherant in the traditional method for teaching this subject and gets to the crux of the issue.

Between the I and the V chord we have Retrogressive root motion and the circular transformation is clockwise: I have indicated this with an arrow to the right below the "R". In a clockwise transformation the 1 becomes 3, the 3 becomes 5, and the 5 becomes 1. This is the natural way that this particular root progression transforms it's tones when triads are involved. Note that there is a single common tone between the triads. Both progressive and retrogressive root motions share this single common tone feature, but retrogressions sound like they are going "backwards", which is why they got that name.

From the V to the vi there is a Strong Ascending root progression and the transformation is counterclockwise, as the arrow to the left below the "S+" indicates. In strong root motion there are no common tones between the adjecent triads, which is why the effect of the root motion of a second is so powerful or abrupt.

Then as the vi moves to the IV we get a Mild Descending root progression with a clockwise transformation. There are two common tones between mildly connected chords, which is why the effect of these root motions is so soft.

The Tritone root motion that divides the progression into halves is in an "as progressive" arrangement between the IV and the vii diminished triad, and it has a counterclockwise transformation. If it were in an "as retrogressive" arrangement between the vii diminished and the IV, the transformation would be clockwise. Like the progressive and retrogressive root motions, the adjacent triads share a single common tone.

The second half of the progression is a mirror image of the first half: Progressive versus retrogressive, strong decending versus strong ascending, and mild ascending versus mild decending. Note that the transformations are mirrored as well but the numbers of common tones (Or, the lack of them) remains the same. Note also that the pattern of the root motions is easy to visualize with the abbreviations added to the analysis. This construction of a chord progression into two halves with different root progression patterns allows the individual halves to be isolated and used in subsequent variations of the chord progression and that device lends a greater cohesiveness to the whole of the resultant composition.

The second example is the same progression with seventh chords instead of triads. Since there are four chord tones to work with, there is always at least one common tone regardless of the root progression type. There is also a new transformation type, which is Crosswise Transformation, and it is indicated by the symbol with arrows at each end and an up arrow in the middle. In a crosswise transformation the 1 becomes 5, 5 becomes 1, 3 becomes 7, and 7 becomes 3: The chord member tones simply exchange positions or functions. Finally, note that between the iii minor seventh and the ii minor seventh there is a parallel perfect fourth that results from the transformation. In a different inversion of these chords, a parallel perfect fifth would result: Parallel perfect fifths that result from transformations between seventh chords involved in strong root motions are perfectly natural and are not a problem.

We will return to this subject at a later date and treat it in more depth, but I wanted to put this primer here to emphasize the fundamental difference between the countrapuntal and harmonic approaches. This should give you a grester appreciation for the isolated and non-polluted laws of counterpoint I am distilling out.

The Art of Counterpoint, Part Four

After the long buildup to Zarlino's instructions concerning two part 1:1 counterpoint, he ends up treating the subject very briefly. One reason is that Zarlino simply does not have much to say about the issue. For him, it's all consonances and that's that. Several things struck me as I have progressed through this book, none of them unexpected: The rules as presented are hit and miss since many of them are based simply on taste and not upon the fundamental mechanics of counterpoint; Even at this relatively early stage in the history of counterpoint, many of the rules are misnomers resulting from purely harmonic considerations, or "pollution" as I call it; And finally, the melodies of this era are extraordinarily primitive with respct to the effeciency of their mechanical trajectories.

Of course, as I have previously shown, dissonance is available in strict 1:1 counterpoint (Where repeated notes are excluded, unlike Zarlino's freer interpretation of it) as long as both voices enter and exit the dissonance by stepwise contrary motion. More than one dissonance in succession can appear in this context, in fact. Then, the dissonance of the diminished fifth or augmented fourth can also appear as part of a succession of unequal fifths or fourths in strict 1:1 ratio counterpoint. This is lightyears beyond Zarlino's conception, and no doubt he would object to these ideas; ideas which would seem radical in the extreme to him. But nonetheless, these concepts are based - as objectively as possible - upon the mechanics of counterpoint to the exclusion of any other consideration. Then, of course, there is the special instance of syncopated 1:1 ratio counterpoint where syncopation chains can introduce dissinance in rhythmically embellished series' of parallel thirds or sixths. This Zarlino was familiar with, so it's at least a tiny bit surprising to me that he was apparently unable or unwilling to make the rest of the intellectual leap. But, I'm probably expecting too much.

Some of the rules based on taste that Zarlino presents go beyond forbidding dissonance in 1:1 ratio counterpoint, however. The idea that both voices should not leap in the same direction by different intervals into a perfect consonance, for instance. For a few years at the beginning of my counterpoint studies I took this as a rule carved in granite. Then of course, somewhere along the line, I discovered the so-called "horn fifths" that violated this "rule". This was one of the seminal moments that forced me to question everything I had been taught about counterpoint. If they were OK for horns, then they must simply be OK, I reasoned. That was circa twenty-seven years ago, and here I am today distilling contrapuntal mechanics down to it's bare essence; or trying to in any event.

One of the reasons for Zarlino's nebulous definition of 1:1 counterpoint is the quaint mechanical ineffeciency of the melodies he has to deal with. Based as they are on Church chants, repeated notes abound in these text-based melodies: There would have simply been no way for Zarlino to come up with the pure idea of a strict 1:1 ratio countrapuntal environment where repeated notes would be excluded. And, this pure 1:1 ratio concept is absolutely essential for distilling the principles we are after here.

As previously mentioned, the rules of counterpoint began to be polluted by harmonic considerations as soon as the ancients discovered the 6/4 sonority and objected to it's less than perfectly stable nature based purely upon subjective evaluation related solely to their personal tastes, or the mores of their times. Objectively speaking, there is no reason to discriminate against a 6/4 sonority according to the mechanics of counterpoint. In fact, the relative instability of the 6/4 arrangement is a feature which can be used as a resource to good effect, as a moment's pause to consider should make evident. By Zarlino's time, a nascent harmonic sense had begun to emerge, and many of the taste-based rules he presents are based upon harmonic considerations. In my next asside, I will explain why this is a very large and largely unrecognized problem that both obfuscates the teaching of counterpoint, and prevents the true understanding of the dynamics of harmonic progression.

The Art of Counterpoint, Part Three

My reading has now taken me up to chapter forty, where Zarlino will actually begin explaining how to compose two part simple (1:1) counterpoint (Thirty-nine chapetrs of prologue, preparation and rules!). The chapters have been quite short to this point, but chapter forty is quite long. Therefore, I have decided to make a few more points about the reduced laws for two voiced strict 1:1 ratio countrapuntal writing as I employ them. But first, an asside concerning the unusual experimentation of Zarlino and his contemporary Vicentino in the area of harpsichord design.

Nicola Vicentino, who preceeded Zarlino by only three years with the publication of his magnum opus music theory treatise L'antica musica designed a "super-harpsichord" that he called an archicembalo which had, as far as I can discern from the contradictory descriptions of it that I have found, thirty-one notes per octave and as many as six (!!!) manuals, some or all of which seem to have had split keys. He did this in an attempt to recreate the diatonic, chromatic, and enharmonic genera of the ancient Greeks, but what the archicembalo amounts to in reality - since Greek practice will forever remain in the relm of wild-eyed speculation - is a keyboard where just intervals are obtainable in several - but not all - keys. Vicentino was considered a radical member of the avant garde in his time, and he even advocated the abandonment of contrapuntal rules when so doing would allow the composer to dramatically accentuate the text and arrouse the desired passions in the listener, which was another idea he got from the ancient Greeks. Obviously, the archicembalo was profoundly impractical and it never caught on, but it does provide an example of the lengths to which some theoretically astute musicians were willing to go in order to get workable tunings and temperaments that would allow for the ratios of the Natural Harmonic Overtone Series to be used in actual practice.

Zarlino was in the opposite camp from Vicentino, and represents the conservatives. He was, in fact, openly critical of Vicentino and the archicembalo. It may come as a surprise, then, to learn that Zarlino designed and had built for him his own split-key harpsichord with twenty-four notes per octave. Zarlino's aims were more modest, however, and they had nothing whatsoever to do with trying to revive ancient Greek practice, but they served the same real goal: To make music heard that used intervals closer to the just ratios. His extended harpsichord did not catch on either, and as a result of practical considerations, we have ended up with twelve tone equal temperament all these centuries later. I thought this was interesting, so I thought I'd share. Now, back to strict 1:1 counterpoint.

Since I am not going to issue any edicts prohibiting fourth relations with the bass, which I'm sure you've deduced if you've followed me thus far, it follows that parallel progressions of unequal fifths and fourths should be addressed in strict 1:1 ratio counterpoint at this time.



In strict 1:1 ratio two voice counterpoint, I usually employ parallel unequal fourths and fifths in one of the four manners exhibited in the example above. Now, technically speaking, there is nothing wrong with beginning and ending these progressions with perfect fifths or fourths, but in two voice writing my personal taste is to avoid those variations with this device. Of course, that doesn't mean you must follow my lead, because the last thing I want to do is impose my personal prefferences upon you. In fact, the only way to arrive at the irreducible laws of counterpoint is to dispense with matters of taste entirely and simply concentrate on the logical mechanics involved. In fact, I used this example just to have the oportunity to get that particular point across.

Parallel unequal fourths and fifths as I have just described are a very valuable resource - especially in writing for three vioces and more (Where I am not shy about beginning and ending those sequences with perfect fifths and fourths) - because it allows for much freer melodic trajectories. Not only that, but in writing contrapuntally for more than four voices, this device is sometimes almost a necessity in order to maintain control over your voice leading.

By applying the laws of counterpoint we are distilling here combined with your own personal tastes and prefferences, a very personal contrapuntal style can be developed. By identifying the irreducible laws of counterpoint and not bothering with matters of personal or stylistic taste, the maximum amount of flexibility can be achieved. As basic as these laws are, they can even be applied to popular, jazz, and folk idioms to good effect, which I will give an example of later on. It could be a while, though.