Why Music Works: Chapter One

How the Harmonic Series Generates the Three Diatonic Contextual Systems: Alpha, Beta, and Gamma

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter One:

My foremost intention with this monograph is to, at long last, describe in a comprehensive way the basic musical forces that exist, and to present them and their resultants in a way that will be of use to composers. These forces all spring from a tension that is inherent in the harmonic series, which desires a resolution. Allowing or thwarting this desire for resolution is what allows music to express things.

While many twentieth-century theorists attempted to describe this through mathematics, their efforts failed because of one simple oversight: The overwhelming majority of possessors of musical minds are not mathematically inclined. Our minds generally work in terms of shapes and sounds: If we can visualize a thing or hear a thing in our minds, we can freely manipulate it to our heart's content. Numbers don't look or sound like anything to us, so they might as well not exist in our world (With the exception of proportions and draw-out geometry). Eventually, this made me return to the beginnings of scientific thought, which was actually called natural philosophy. Sir Isaac Newton described himself as a natural philosopher. Music theory has never even reached the Newtonian stage of evolution because the fundamental forces have never been defined and demonstrated in a rigorous and logical way.

A natural philosopher looks at the God-given constant in a thing, and extrapolates its implications out to describe the observable phenomena (Or hearable phenomena, in this case). For music, that constant is the harmonic overtone series, and so I have used musical proofs based on the harmonic series instead of mathematical proofs to demonstrate all of the resolution possibilities present, as well as all of the manifold contextual results that it can produce.

*****

CHAPTER ONE:

The overtone series is a harmonic system, so music is a harmonic system. If we look at the evolution of western art music, the harmonic series underpinnings were known from ancient Greek writings, but the implications of those writings were only discovered bit by bit by starting at the beginning, which was the unaccompanied melody of Gregorian plainchant. From plainchant, an admittedly oversimplified synopsis would proceed to organum, fauxbourdon, polyphony, traditional homophony, and then jazz, which was the real end result in the twentieth century: The seventh chords were finally ubiquitously used as tonics, including the overtone sonority itself in blues music.

Where the problem arose with both so-called traditional music theory and the succeeding jazz music theory was that the elements were never separated into their pure states: Pure harmony and pure counterpoint. By the time Joseph Schillinger properly described pure harmony, the "classical" guys had jumped the tracks and gone off into the desert of desiccation and musical nihilism known as - to cite but a single representative monicker - atonality (Though admittedly, some good did come of this for film music, but that's not what they intended). Meanwhile, even the most prominent Schillinger student of all, George Gershwin, never really exhibited in his works the revelation that was pure harmony: In other words, like all jazz cats, he didn't give a rip about the transformational voice leading that Schillinger revealed.

Finally, Schillinger himself never coupled his pure harmony with the overtone sonority, so he missed a crucial link, and that's why many of his ideas stray into the weeds of pure speculation. Admittedly, The System was a hodge-podge put together in haste by some of his students after his sudden and untimely death, so there is the possibility - regardless of how remote - that he did in fact make this connection. In any case, I made the connection independently due to the accidents of fortunate circumstance combined with my natural curiosity, and so here we are.

As mentioned previously, the natural philosopher's approach is to begin at the beginning, which in the case of music is with the harmonic overtone series, that every natural sound carries with it (A computer generated pure sine wave, which is an artificial phenomenon, would be an exception to this). In nature, the recognizable timbre of a sound is defined by the attack transients, formants, amplitudes, envelopes, and the phase relationships between the partials in the overtone series, which are individually pure sinusoidal periodicities.

Unfortunately, the pure waveform soundfont I found does not work in iTunes for some reason - it works fine in Encore though - so we're stuck with an organ sound that has relatively few harmonics present in it. If you play the overtone series in 12√2 equal temperament, this is what you get (I will address the simple logic of twelve-tone equal temperament much later in this book).

EXAMPLE 1:



Listen to Example 1

Though TTET does not render the mathematically perfect ratios of overtones on a vibrating string or in a vibrating column of air - which themselves vary minutely, especially on a string - it is quite close enough: A mathematically perfect overtone series sounds virtually the same, as many years of programming digital synthesizers taught me. To put a finer point on it, this 12√2 rendering of the series sounds like a so-called dominant seventh chord, as did the perfect ratio renderings I've created in years past with instruments such as the Synclavier. So please, spare me the, "TTET destroyed tonality" nonsense: The ratios in TTET may be irrational, but the concept isn't - 1/3 is an irrational number too, but a child can understand the idea of a third part of something. Likewise, 1/12 of an octave is a perfectly rational concept, even if the resulting ratios are expressed by irrational numbers.

Under the system are the various eras of western art music development - Plainchant, Organum, Fauxbourdon, Polyphony, Homophony, and Jazz - which roughly correspond to composers coming to understand ever higher partials in the harmonic series. That this understanding was primarily intuitive and not analytical need not concern us, as in musical composition - and performance - it has usually been intuition that lead the way, with theorists describing practice later.

Next I have labelled the partials 1-8, which I will call P1 through P8. It is useful to draw a distinction between partials and harmonics, as the fundamental generator of the series is not a harmonic: Only the overtones P2 and on are harmonics. By labeling the partials starting with the fundamental as P1, the series works out for us the pure intervallic ratios, which are found on the third, fourth and fifth lines.

Though self explanatory, 2:1 is the ratio for the inviolable perfect octave, 3:2 is the ratio for the just perfect fifth, 4:3 is a just perfect fourth, 5:4 is a just major third, 6:5 is the large minor third, 7:6 is the small minor third, and 8:7 is the first of the just major seconds. Line four shows the ratios of 5:3 for the major sixth and 8:5 for the minor sixth, and finally we have 7:4 for the minor seventh and 7:5 for the tritone on the bottom line.

OBSERVATIONS:

1. All adjacent intervals and their inversions from P1 through P7 are consonances.

2. All adjacent intervals and their inversions beyond P7 are dissonances.

Neither of these two observations should require any comment.

3. Consonances that remain super-particular ratios when inverted are considered perfect.

4. Consonances that are not super-particular ratios when inverted are considered imperfect.

A super-particular ratio is a ratio in which the terms differ by 1: All adjacent ratios for the consonances from P1 to P7 are super-particular; 2 - 1= 1, 3 - 2= 1, &c. Only those consonant intervals that remain super-particular when inverted are considered perfect, however: The 2:1 octave is unchanged during an inversion - which technically requires two octaves of displacement to avoid the unison - and the 3:2 perfect fifth inverts to a 4:3 perfect fourth, which are both super-particular: 3 - 2= 1 and 4 - 3= 1.

With the major and minor thirds, however, we get 8:5 for the minor sixth and 5:3 for the major sixth, which are not super-particular: 8 - 5= 3 and 5 - 3= 2.

5. The only non-adjacent dissonance with a single skip within the first seven partials is the tritone.

6. The only other non-adjacent dissonance within the first seven partials is the minor seventh itself.

The ratios of 3:1, 4:2, 5:3, and 6:4 are all consonances: Perfect twelfth, perfect octave, major sixth, and perfect fifth. Only 7:5 is a dissonance, it being the tritone, and then 8:6 is another perfect fourth. The minor seventh requires two harmonics to be skipped from 7:4, and it's the only other dissonance from P1 to P7 (And of course, it converts to the 8:7 dissonant major second).

7. The harmonic series is complete at P7; P8 is only present to yield the minor sixth harmonic ratio.

As I said at the beginning of this chapter, "The overtone series is a harmonic system, so music is a harmonic system." Once the adjacent intervals in the series become dissonances at 8:7, the harmonic part of the series is over and the melodic part of the series begins. Sure, inversions of harmonic structures can yield seconds, but a root position harmony in close position will only contain adjacent thirds. While ninths can replace doubled roots in the basic harmonic musical system - and sixths can replace fifths - elevenths and thirteenths are actually acquired from harmony with two transformational strata, which is a subject for much, much later (Transformational stratum one is Root, 3rd, 5th, and 7th, while transformational stratum two is 9th, 11th, 13th, and Root: A ponderous system outside of the basic harmonic nature of music, but one useful for certain effects within a larger musical or extra-musical context). Theorists who claim that twentieth century music was an exploration of the harmonic series beyond P7 are in error.

8. Since the harmonic series contains a dissonant tritone, it is inherently unstable.

The life forces of music are in the tritone - the leading tone and leaning tone impetuses - and the inherent instability of the overtone sonority - a dominant seventh, remember - is what provides music with its possibility for forward motion. This is one reason that wild-eyed criticisms of 12√2 temperament are misguided: Since the harmonic series is inherently dissonant, why would a stable just tuning be an advantage, or even particularly desirable? Since just ratios are only possible with computers, voices and other variable pitch instruments like strings, it is just silly to criticize TTET, which is a perfectly equitable solution for fixed pitch idioms. Later, we will look at just how small the deviations from just are with TTET.

*****

The first thing that the harmonic series from P1 to P7 does for us is to define the format for pure harmony: Since we are dealing with a harmonic system, all we have to do is eliminate the non-harmonic fundamental generator and the superfluous twelfth that it produces to get this. The resulting structure is Root, Root, third, fifth, and seventh, which is the pattern for pure harmony.

1. Pure harmony consists of five total voices.

2. These five voices are divided into a four-part, close position transformational stratum above a constant-root bass part.

3. Only the root is doubled - or trebled after a resolution - in pure harmony.

4. Though the transformational stratum can be in any close position inversion, all harmonies are root position in pure harmony due to the constant-root bass part.

*****

EXAMPLE 2:



Listen to Example 2

The four different notes of the overtone sonority are divided into two pairs: The root and perfect fifth are passive tones, while the tritone involving the major third and the minor seventh consists of the two active tones.

OBSERVATIONS:

1. The root is the foundation of the overtone sonority.

2. Together with the root, the perfect fifth provides context for the dissonant tritone.

Since there are twelve pitch classes in the chromatic system and the tritone involves two of them, there are only six tritones possible. Since there are twelve possible overtone sonorities, this means that each tritone is shared between two possible roots. In this case, if the tritone between B-natural and F-natural is enharmonically notated as C-flat to F-natural, it can belong to the overtone sonority with the root on D-flat and the perfect fifth of A-flat. That means that the two possible roots for each tritone are also in a tritone relationship with each other, as well as the two possible perfect fifths. When you change the context of a tritone with the other perfect fifth a tritone away, you also reverse the functions of the notes involved in the tritone. In this case, the B-natural leading tone becomes a C-flat leaning tone, and the F-natural leaning tone becomes an F-natural leading tone.

This is how jazz theorists justify their concept of substitute secondary dominant harmonies, by the way, though we will see later that this isn't really valid according to the implications of the series.

3. The root and perfect fifth are passive tones, neither desiring to rise or fall.

4. The major third and minor seventh are active tones, desiring to resolve their shared dissonance.

5. The major third is a leading tone, and it desires to rise by a semitone.

6. The minor seventh is a leaning tone, and it desires to fall by either a semitone or a tone.

7. The perfect fifth may rise or fall by a tone as the tritone resolves to a major target.

8. The root may remain stationary or fall a perfect fifth when the tritone resolves.

9. In order for the target sonority to be in root position, the lower root must fall by a perfect fifth.

10. In order for the target sonority to be complete, the upper root must remain stationary.

11. In order to avoid doubling a potential active tone in the target sonority, the perfect fifth must fall by a tone.

*****

When we follow the above observations, the following primordial resolution of the overtone chord is produced.

EXAMPLE 3:



Listen to Example 3

This is how you say, "The End" in music: The overtone sonority with a doubled root resolves to a targeted major triad with a trebled root. If we were to wish a continuation, the unison C-natural in the transformational stratum would have to dissolve with one of the C's going down to a seventh - either B-natural or B-flat.

The falling perfect fifth/rising perfect fourth root motion is called a Progressive root motion, and it will get a capital P in the analysis. When we get to more elaborate examples, they will not be called, "chord progressions" because a progression is this specific type of root motion; rather they will be referred to as harmonic continuities.

Though interrupted by the doubled root in the upper stratum, if one of the C's moved down into a seventh, this would be what is called an interrupted or delayed crosswise transformation, as you can see from the diagram in between the staves: The root and fifth exchange functions, and the seventh and third also exchange functions after the third's resolution into the unison. This would be a P_+ in the analysis, which reads, Progressive _Interrupted +Crosswise transformation. Uninterrupted crosswise transformations, though less than perfectly natural, can also be used when that effect is desired, as I shall demonstrate later.

OBSERVATIONS:

1. This resolution does not yield all seven tones of the diatonic system.

2. This resolution - or similar less perfect versions - does, however, yield all six tones of the ancient hexaphonic Church modes.

The modern Ionian mode implied here was not really a common Church mode, but as we shall see, modal displacements of this formula will produce modes commonly used back then. It is also useful to note that when ending resolutions first appeared, they were primitive versions of this one.

3. In order for the target chord to become a complete seventh chord, the doubled root in the transformational stratum must fall.

4. The doubled root may fall either a semitone or a tone.

5. If the doubled root falls by a semitone, the potential for a diatonic system will be possible.

6. If the doubled root falls by a tone, the potential for a diatonic system will be destroyed.

7. If the doubled root falls by a tone, another overtone sonority will be created.

The rule for creating diatonic systems through this resolutional paradigm is to retain the inflection of the notes present in the preceding chords. Here, for example, one would have the doubled root descend to B-natural, because that note is present in the preceding dominant harmony. Allowing for chromaticism with this paradigm will lead to various integrated modalities; again, we'll see this at a later point.

8. As the diagram shows, 1 becomes 5, 5 becomes 1, 7 becomes 3, and 3 becomes 7 after the interruption of the resolution.

9. Therefore, progressive resolution of the overtone sonority creates an interrupted crosswise transformation.

10. A single additional progressive resolution would complete a diatonic system.

*****

If we allow for that dissolution of the unison C so that the former major third descends to B-natural - becoming a major seventh - and add an additional progressive resolution, the Alpha Contextual System is produced.

EXAMPLE 4:



Listen to Example 4

The Alpha Contextual System has as Alpha Prime the traditional major or Ionian mode. Within this contextual system are the additional sub-contexts known as the displacement modes of Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian.

OBSERVATIONS:

1. With two progressive resolutions from the overtone sonority, the diatonic system is complete.

The only note missing in the single cycle resolution was A-natural, which now appears as the major third of the subdominant harmony.

2. All three possible harmonic functions - dominant, tonic and subdominant - are also now defined.

3. The progressive resolution from the tonic to the subdominant is a less perfect form of progressive resolution.

4. Additional less-than-perfect progressive resolutions will be found in the modal sub-contexts.

5. There exists an additional diatonic Beta Contextual System with resolution to a minor triad.

6. The Alpha Prime tonic scale is 2, 2, 1, 2, 2, 2, 1: The semitones are separated by two tones.

*****

If we start again with the formula and have the initial resolution to a minor tonic, the Beta Contextual System is produced.

EXAMPLE 5:



Listen to Example 5

The Beta Prime mode is like a Dorian mode from the Alpha System with a raised seventh degree. In common practice minor key music, this system was combined with the Aeolian mode to produce the nonatonic so-called melodic minor scale: Roughly speaking, his is the ascending version of that system, and Aeolian was used as the descending form (Though those conventions weren't always adhered to).

OBSERVATIONS:

1. The tonic seventh chord in this system is a highly dissonant minor triad with a major seventh.

This means the third mode here has an augmented triad, of course.

2. The subdominant chord in this system is another overtone sonority.

3. The sub-contextual subdominant mode is the scale that the overtone series creates to P11.

The overtone scale is best described as a Mixolydian mode with a raised fourth degree, not, "Lydian flat seven" as some jazz theorists describe it: The home harmony is a dominant seventh, not a major seventh.

4. There exists an additional Gamma Contextual System with resolution from a dominant harmony containing a diminished fifth.

The point of origin for the V(d5m7) chord - and the so called French augmented sixth - is from the V/V in minor, where the chord on the second degree is a ii(d5m7) before the third is raised to make it a secondary dominant. In Alpha Prime that sonority appears as the remote V(d5m7)/iii harmony. I will demonstrate this when we get to the secondary dominant harmonies, but here I'm just demonstrating the three possible diatonic contextual systems that contain two semitones and five tones.

EXAMPLE 6:



Listen to Example 6

The Gamma Prime scale created through this resolution process is best described as a Phrygian mode with the sixth and seventh degrees raised, since the minor second degree is the distinguishing characteristic of the Phrygian mode.

OBSERVATIONS:

1. Diminishing the fifth of the dominant chord makes that fifth into an active leaning tone.

As I just mentioned, this V(d5m7) has a natural origin in the Alpha System. When the fifth was D-natural, it was a passive tone that could theoretically rise or fall during the resolution, so long as the composer is prepared to deal with doubling a potential active tone. Now, the D-flat is a third active tone in the dominant harmony that desires to resolve down by semitone to the new root. This additional impetus increases the resolutional desire of the dominant harmony.

2. The tonic seventh chord is again a highly dissonant minor/major seventh.

3. The subdominant chord is again another overtone sonority.

4. The Gamma Prime tonic scale is, 1, 2, 2, 2, 2, 2, 1: The semitones are not separated.

5. All 21 possible diatonic modes consisting of five tones and two semitones have now been generated.

I will present these in detail in chapter two.

*****

What lead me to the idea of pure musical contextual systems was a phenomenon I noticed with so-called atonal works: They were completely unsatisfying - unlistenable, actually - in a purely musical concert context, but when used in a stage play or a film score, they could become quite effective. What I finally realized is that the play or the film provided an extra-musical context in which these pieces could be effective. Likewise, episodes of atonality within a larger purely musical context that is based on any one of the 21 diatonic modes that are independent sub-contexts - those with major or minor triads as tonics, and not diminished or augmented triads - can also be effective. When music provides its own context, it has to be based on an independent musical contextual system or sub-system, regardless of any arguments to the contrary.

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Imitation Study Number 4 in A Minor

This is an interesting piece. It's a fugue with the answer on the subdominant and in inversion, and the initial statement of the subject has an accompaniment line. This was necessary because the subject begins on the third degree of the mode, which may very well be unique. I've never encountered another subject like this, anyway.

Interestingly - to me - this is actually the very first fugue subject I ever came up with back in 1986 or 1987. I got the idea while studying through Joseph Schillinger's System of Musical Composition. Historically, fugue subjects have begun on the tonic or dominant degrees - or with approach notes to those degrees - and I wondered why not the mediant, since it is a degree of the tonic triad? So, I came up with this wedge shaped theme with the idea of writing a two-part invention for guitar with it. That idea fizzled because I just had no idea how to proceed.

Later, when I was working on my masters degree, I wrote a three-part invention for string trio with the subject, which I really, really like. In that piece, I came up with the chromatic countersubjects and the solution of having the answer on the subdominant degree and in melodic inversion.

So, here we are 23 years later and I've come full circle back to the idea of writing a two-voice solo guitar piece on the subject; only now I have the compositional technique to make it work.

Here's today's MIDI to M4A conversion:

Imitation Study Number 4 in A Minor



Despite the subject's unusual feature of starting on the mediant degree of the minor mode and having a diverging pair of melodic trajectories, it's actually fairly Bachian (Since Bach was my inspiration here). However, you'll notice the G-sharp a the beginning of measure two is followed by a G-natural one beat later: This is strange, and it sets up a tension that I really like a lot. Furthermore, on the last eighth note of measure two, there is a diminished octave - equalling a major seventh in sound - between the C-sharp in the bass and the C-natural in the subject, and the final sixteenth is a diminished seventh (Which equals a major sixth in sound, of course). These bold cross-relations give the piece the dark feel of a lament: Something is just not right with the universe, and so there is quite a bit of expressive sadness to the piece.

This is only reenforced by the true counter-answer in measure three, which is an ascending chromatic line at the beginning, and a descending chromatic line at the end. Pathos. On the last eighth of measure four, we have a diminished fourth (plus an octave) and then a diminished tenth (Which is the inversion of an augmented sixth). This is another highly unusual feature of the exposition.

As I mentioned in the previous post, many times two voice fugues work out better with a thrid statement of the thematic unit, so that two subjects appear and you can reveal the true countersubject: That's the case here. Just as the answer is inverted from the subject, the counter-answer is inverted for the countersubject. IMO, this is very, very cool.



Most of the time I use tail figures for the episodes in my fugues, but in this case the head works better. Since there are losts of wide leaps, this also allows for range adjustments that keep the elements within an executable range on the guitar. I made that adjustment in measure eight, which set up the upcoming middle entries, which are in the relative major.

The major mode version of the subject and answer worked out better starting off on the tonic degree of the triad, and the countersubject and counter-answer therefore ended up diatonic instead of chromatic. this provides a happy sounding contrast to the dark sounding exposition.

The second episode that starts at fourteen required another range adjustment in fifteen, whereas the string trio version allowed the violin to continue into the stratosphere.



The second middle entries are all about displaying the contrapuntal inversions of the previously revealed material. the original orientation of answer and counter-answer are first stated, only in the dominant level minor of E, and then the countersubject over subject appears starting in nineteen.

Episode three at twenty-one takes the piece to the subdominant minor region, where I present the answer over counter-answer orientation. At the end of this, we get an augmented sixth, of course. Both of these inverted orientations sound very dark. Darker than one might expect just judging from an intervallic analysis. I've never really been able to figure out why this is - even the consonances sound dissonant - but I really like the effect (Even if it was a happy - or sad, I guess - accident).



The final page gets us to the concluding stretto section - also the first stretto I ever composed - and there is no episode needed to get here because the subject to the previous answer is back in A minor. Is that cool, or what?

The first 1.5 beat overlap appears in measure twenty-seven, and then the 2.5 beat dovetail starts in twenty-eight. To get everything back in sync vis-a-vis the bar line, I relax that back to 1.5 beats in the first half of thirty, and then just the head appears in the second half, which sets up the conclusion.


Obviously, it's bedtime.

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Imitation Study Number 3 in F-sharp Minor

Here's the third entry in the series, but I must mention that the sequence of these in the final set of pieces will surely change. That will depend on the keys, time signatures, and types of pieces these end up being. This one happens to be an actual two-voice fugue with a tonal answer at the fifth, as opposed to the previous two, which were two-part inventions, per Bach's definition, with real answers at the octave.

Number three here also ended up in the tres cool key of F-sharp minor, whereas the previous two were in the boring and overworked key of A minor. This being a collection for solo guitar, at this point I'm thinking that A minor pieces will probably end up alternating with those in other keys... if I end up with enough pieces in other keys. lol.

Since these are solo guitar pieces written in open score, they can also function as duets for students too, which is a nice feature.

The subject of today's fugue was created one day while I was driving from San Antonio to San Marcos when I was a masters student at Texas State. I'm not positive anymore, but I think it was 1989 or 1990. It just came to me while I was daydreaming on the drive, and I sang it out in solfeggio.

First it became a rather lengthy and undisciplined string trio piece, but later it became a very tight fugato for chamber orchestra. I'm sure I'll still use that fugato in a larger piece at some point, but it's nice to have a solo - and duet - guitar version of it.

This is not simply a transcription of the fugato, though: I had to do quite a bit of re-composing to get this guitar version... or perhaps rearranging would be more accurate: The orchestral interjections are gone, naturally, but some of the episodes are redone as well.

Here's today's MIDI to AAC conversion:

Imitation Study Number 3 in F-sharp Minor



As you can see if you read the previous posts in this series, this isn't the stately Musical Offering/Art of Fugue type of subject I used for those pieces. This is a sprightly type of subject with lots of leaps and tied notes that nonetheless gradually accelerates to become a head/tail type of deal.

One of the coolest things about the answer is the counter-answer that works with it: There's a descending chromatic tetrachord at the beginning, and the ascending version at the end. I remember being amazed when I discovered this... and, well, it is kind of amazing. There's an eighth rest at the beginning of the last beat of measure 6 to avoid the unison on G-sharp, but it remains implied and a duet performance could actually execute it. Oh, and I have the high notes in the head of the subject and answer notated as harmonics, but it is technically possible to execute them as normally attacked notes.

If you are writing a two-part fugue, many times - but not always - the exposition will work out better with a third statement so you end up with the subject twice: That's the case here.

Note also the even more amazing feature of the countersubject's beginning: It's yet another ascending chromatic tetrachord!



The first episode, such as it is, is just a single measure spin-out of the subject's tail. One of the things I got better at here was register shifting: To get this kind of music to work on the guitar, it's necessary to shift registers to keep the music within the guitar's quite limited contrapuntal range. The descending minor seventh from F-sharp to G-sharp makes the required adjustment smoothly.

By not chromatically inflecting the sixteenths on the final beat as la and ti, they become fa and sol to effect the modulation to the relative key of A major. The low E's in this section are why the piece ended up in F-sharp minor.

I needed a new version of the countersubject at 11 because the descending chromatic tetrachord wouldn't work there, so it's a simple diatonic accompaniment. Same deal at 14, as an ascending chromatic tetrachord wouldn't work there either, so I used another diatonic version of the countersubject. I had to make another range concession at 15, but it sounds very natural; as if it's a natural part of the piece. Bach was a genius at this, but I'm just now getting really good with it.

Note that these are both subject statements and not answers.

Measure 17 is another single measure episode, and I used another octave displacement to set up the final statement in the relative...



... which is finally the answer, complete with descending and ascending chromatic tetrachords in the counter-answer. This sounds really awesome in the major mode, as you usually don't get this sort of thing in that context. I like it a lot. Love it, even.

The episode at 21 is now two measures: 21 modulates back to the tonic minor with the introduction of the raised submediant and leading tone in the final four sixteenths - a new figure for the fugue - and then 22 allows for the required register adjustment. Note that measure 22 is just like the first episode that modulated the piece to the relative, except that the raised sixth and seventh degrees are retained on the last beat, so there is no modulation.

The recapitulation at 23 is a super-close stretto between the subject in the bass and the tonal answer above: Only a single beat of delay. This is an extraordinarily difficult thing to pull off, and I really don't know how I make these things happen; I just "notice" them for some reason. Remember, I just sang this subject while on a boring drive up the interstate: I had no idea the chromaticism would work, or that this stretto was possible at the time. I didn't even work out the tonal answer until later.

I've never had a student who could do this, unfortunately, and in fact, I've never met or heard of anyone else who can do it either (Nobody living, anyway). It must be an inexplicable and rare gift, I guess. Too bad it's not worth any money in today's world. lol.

Notice that I had to make a register adjustment even in the subject to get this to work on the guitar there in 25: One hardly notices it. I hadn't acquired this skill when I came up with this subject twenty years ago, and that's part of the development of the craft aspect of contrapuntal writing: It's nice to have the gift of natural talent, but only time and effort can give you mastery of the details of the craftwork.

The final episode beginning at 26 has a gnarly contrary motion dovetail between the tail of the subject and the tail of the answer; it moves into major seconds and out to octaves. In the orchestral version, these are ninths and double-octaves, which are smoother sounding, but his is really excellent on solo guitar (Or, between two guitars playing a duet). the straight sixteenths that originally appeared back in 21 no reappear in 27, and the piece resolves out to a six-note F-sharp minor chord at 28. That's it: 28 measures and about a minute and a half. Very tight and efficient.



I'm sure I've used this image before, but she's just so striking and natural.

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Imitation Study Number 2 in A Minor

Well, this series has found a name now: Imitation Studies. I'm not calling them inventions because, 1] They do not all answer at the octave - some of these are fugues, IOW - and 2] Bach already used that. lol. Believe it or not, I've completed five of them so far, and the previous one was "perfected" by simply changing the dotted-eighth/sixteenth rhythm of the episodes to quarter/eighth triplets. This works better because the triplet feel comes right out of the tail of the subject. Since that was such a minor change, I decided to move on to the next in the series.

With this piece, however, the dotted-eighth/sixteenth rhythm is in the subject, while the written-out trill is in the countersubject: Therefore the episodes sound more natural and organic with the dotted-eighth/sixteenth figuration of the subject. Nice discovery.

I composed this subject back when I was a doctoral candidate at UNT in the early 90's, and it was the final project for the graduate level Invertible Counterpoint and Fugue course I took there. Since I composed it as a four-part canonic stretto with one measure delays, it took the form of an epic string quartet piece (You can also see the score in PDF format if you want).

Well, while writing this guitar invention version, I discovered some new facets of the subject, so that got me to thinking about revising the fugue. I think I'm coming up with a new compositional process here: Guitar invention or two-voice fugue first, then execute the final fugue for whatever ensemble turns out to be most appropriate. Bonus is, I get a bunch of guitar pieces out of all my ensemble fugues. Go me!

Number 2 here also ended up in A minor, but two of the next three are in the "cool" keys of B minor and F-sharp minor (Number 6 I'm working on is in E major).

Here's todays MIDI to M4A conversion:

Imitation Study Number 2 in A Minor



This is a five measure head, body, and tail subject, so we get the desirable odd bar length and an accelerating rhythm until the point of resistance in four and the cadential figure at the end. The countersubject is straight quarters until the final measure, and then there is a dotted-quarter/eighth followed by the written-out trill (I've come up with several of these written-out trills now, and they are becoming a thematic sub-language for me).

For the main episode I chose a descending chromatic line... hey, I like those.



... and this first iteration is non-modulatory.

Since the subject works as a four-part canon, the organizational scheme is of ever closer answers. Starting in 17 we hear the countersubject above the subject for the first time, but this is interrupted in 21 by the subject overlapping itself by one measure. Then, the rest of the countersubject seamlessly continues out of the subject's cadential figure due to the way I constructed it to work like this (I rock).



In 26 the former episode reappears, but this time it's foreshortened by a measure and modulates to the dominant minor region. Sorry I crammed it onto a single system, but it was a matter of keeping the page count down for this post version.

The sweeping rising thirds in the bass starting at 31.5 - me, sol, ti, re, fa, le - sound wicked pisser, as we used to say at Berklee, and then we get another mesure of the countersubject before the next interruption with now two measures of overlap.

For my esthetic, the third episode has to be different - with Bach everything is beautifully, maddeningly, and organically different - and here I discovered that the subject works over the descending chromatic figure of the episode. This leads to some unusual contrapuntal motions, the coolest of which is the perfect fifth into a diminished fifth from 41 into 42. I need to put this in the string quartet, because it will be even more effective with three or four voices. At the end I modified the tail of the subject and the bass part to set up the modulation to the relative major region. Hot, or not?



We're down to two measures of delay/up to three measures of overlap now, and I changed the second measure of the countersubject in 45 just because it sounds really beautiful this way (Hey, I do have a heart). Same with the bass part in 49 (Mahler said, "Interesting is easy; beautiful is hard"). I thrive on interesting, but I go for beautiful whenever I can.

By the way, the inversion of this stretto would not work, because parallel fifths would be implied. The implied elevenths sound fine (I tried it the other way for grins... but those turned to frowns).

The original episode then reappears at 51, but yet another measure shorter as the piece returns to the tonic for the final stretto... or so it would appear.



The trill figure takes the final canon from five measures to six, and the end is a bit underwhelming, which sets up the hyper-stretto coda: Subject over subject in rhythmic augmentation. This is admittedly a bit weird in two voices, as there is a leapt-into minor ninth at the beginning of 64. Since the implied harmony is so obviously a V(m7m9) though - and this is immediately strengthened by the appearance of the major third - it actually works, IMO. Bach would never have done this (Well, he never did, to my knowledge), but I think it's kind of cool.

I worked up to a nice flourish in 71 and 72 with an implied V(4/2)/iv and then an augmented sixth into the primary dominant. A final - and new - trill figure finishes things off with a final chromatic flourish.

As it stands now, this is a better composition than the original fugue... which is why I have to re-write that puppy.


No doubt about that being hot.

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Invention in A Minor v5.3

This piece is now "finished." I put the quotation marks there because pieces like this tend to get revised over succeeding years - sometimes radically - but it is complete as it is, and I'm quite happy with it. One of the things I like about composing with dispassionate and lifeless notation based MIDI sequencers is that if I like the way it sounds in MIDI, I know a well performed version will be even better.

Here's the m4a sound file:

Invention in A Minor

This is actually version 5.3, as you can see, so it really didn't take all that many versions to get it done.



There are no changes here, so if you want a rundown, see the previous post.



The changes start at the first stretto section in the subdominant minor region. Previously I had both the 2.5 measure delay and the 1.5 measure delay stretti here, but I decided to save the closer one for the upcoming dominant minor. This was the breakthrough idea that allowed me to complete the piece, because I hadn't saved anything for the dominant previously.

There are no changes to the episode starting at 17 or the relative major statement starting at 20 either.



This episode starting at 23, however, is entirely new. The inversion of the major mode statement does not work due to a leapt-into major second - which is an augmented second in the minor version - so this was a natural place to put the harmonized subject in the bass with a melody in the lead. Sounds cool in MIDI, but it would be a PITA to play. Since this is primarily a compositional exercise, though, that's of little practical consequence.

At 27 we get the dominant minor statement with the closer stretto, and this also gives the piece a melodic climax at the C in 28. By saving this for the peak, we get a dramatic pause under it as the next subject statement begins, which sounds nice tres cool. Since the piece is 48 measures long now, 48/28.5= 1.684, which is just about as close as one can get to the Golden Mean of 1.618. Nice, huh?

Saving this stretto for this point also makes it out of kilter vis-a-vis the bar lines, so at the conclusion of the section in 31 I was able to use another nice contrapuntal/rhythmic acceleration lick into the final episode: quarter-eighth, quarter-eighth triplets, and then dotted quarter-sixteenth, dotted quarter-sixteenth. I like this effect, and in performance I'd probably swing the dotted quarter-sixteenth sections anyway - so it doesn't sound so stiff - so the transition is super-smooth.

From the final episode that begins at 32 until the end is the same as the previous version...



... so if you are new to this series and have any questions, check out the previous post.

Now that this has got me exploring some aspects of rhythmic variation in counterpoint - something I'd like to develop more as it's has heretofore been one of my weaker points - I believe I'll write two-voice inventions with all of my previously composed subjects. Series work like this is a great way to develop compositional technique, and after my eighteen axial studies and twenty-four figuration preludes, a series of inventions would be a logical step.

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Invention in A Minor v4.5

At first I thought I was working on a two-voice fugue with this subject, then it became an invention, and now it's actually almost a piece of music. Previously I had the various statements and stretti linked only by lineal licks, but now I've put some real episodes into it.

Here's today's AAC sound file:

Invention in A Minor v4.5



The exposition hasn't changed any, but beginning in measure seven there is an episode that modulates to the subdominant minor region. This is actually a harmonized version of the subject, which you can see if you look at the first dotted-eighth note of every group: It's the note at the corresponding location in the subject until I prepare to modulate at the end. Since the tonic A minor is the dominant degree of the upcoming D minor region, this was easy to accomplish by using the written-out trill on the proper level and introducing the C-sharp leading tone at the end.

An interesting aspect of working in two voices is that you can't easily just work your rhythmic velocity up to a constant eighth note motoric motion - or alternating dotted-eighths and sixteenths, as the case would be here - so that makes one think about how to make the combined rhythm breathe smoothly. This actually requires a bit of thought. Fortunately, the subject has six different rhythmic values in it and the trill tends to scrub out any buildup and so allows for a new beginning, so to speak. That's why I put the trill figure at the end of the episode: Going back into a constant quarter cumulative rhythm sounds excellent with that trill as a re-transition.

Since I've now established that there will be no perpetual motion cumulative rhythm, the stretti beginning in eleven breathe quite naturally from a rhythmic standpoint.



This culminates in a virtual stop in the rhythm at fifteen, which sounds quite dramatically cool at the new pitch climax of B-flat.

There is another episode based on the harmonized subject beginning at eighteen, but I've modified the counterpoint to facilitate a modulation from the subdominant minor region to the relative major. This took a little more head scratching and chin rubbing than the more obvious and easy previous episode, but the result is more interesting, which is what you want in succeeding versions of a musical idea.

The relative major statement is the last confirmed bit of the piece from the beginning - I'm sure the first 23 measures won't change any - but I had to resort to a lineal link to get to the final episode based on the harmonized subject, which is back in the tonic minor. I'm just not sure what will replace measure 24 at this point, but I'm thinking of statements of some kind on the dominant minor level, which would provide an overall pitch climax on C above the previous high of B-flat. I must admit to being stuck here, but that's usually a good sign, as what comes after much consideration is always - in my experience, at least - the final element that makes the piece perfect (Or pluperfect, if I'm particularly inspired).

For the final version of the harmonized subject episode, I discovered that two consecutive statements of that harmony, with slight variation, would work with a descending chromatic tetrachord as the counterpoint. this is goosebump-inducing cool if the listener is astute enough to follow what's going on, and it's one of those musical effects that can only be perceived with a bit of retrospection. So, 25-27 have the first statement over do, ti, and te...



... while the second statement is over la, le, and sol. It isn't really possible to perceive the new beginning of the subject until half-way through 28, so the effect is quite surprising. Tres cool, non?

There is no change to the closest stretto that begins at 31 - which functions as a recapitulation here - but the hyper-stretto coda that starts at 35 has been improved with some very nifty chromatic, rhythmic, and contrapuntal action.

The last quarter of 38 now has an augmented sixth on it - the only one of those in the piece at this point - and then there is an ascending chromatic tetrachord in 39. Then, the descending quarter triplet that starts in 40 has some cool contrapuntal relationships over the rhythmically augmented trill: m7-M6, m6-P5, and d5-m6 into the m6-m7 and d5 last two-beat triplet. It sounds a bit awkward in the m4a version because MIDI assumes every note is attacked, but when played the trill will be executed with hammer-ons and pull-offs, so it should be very smooth and cool. It's a nice closing contrapuntal device.

Almost more interesting to me though is the cumulative rhythm produced. If you consider a half note as the basic rhythmic unit, starting in 39 the cumulative attacks are 2, 4, 6, 3, 1, which is a very nice accelerating and decelerating rhythmic ratio.

I'm learning some cool things about the musico-psychological effects of rhythm with this piece.


I haven't posted a redhead in a while.

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Invention in A Minor v3.7

What I thought was going to be a two voice fugue in D minor just ten days ago has morphed into a two-part invention in A minor now. I'm kind of bummed on one hand, as the last thing the universe needs is another guitar piece in A minor. OTOH though, years ago I had the idea to write a series of two-part inventions for guitar, but my development wasn't far enough along, and my compositional technique was therefore not up to the task. Today it's a different story, so no telling where this will lead: I've come up with no less than ten timeless subjects over the years, and I realize now that any one of them would work in two voices on the guitar with imitation at the octave.

That's the difference between a fugue and an an invention, right there: Imitation is at the fifth - or more rarely the fourth - in a fugue, while inventions answer at the octave.

I'm still just using single line lineal links between statements - no episodic material as of yet - but I now have all three of the traditional stretto possibilities in it, as well as the two-part hyper-stretto as a conclusion. There are also not nearly as many possibilities as I initially thought, as the melodic inversions (or mirrors) of the stretti yield some minor ninths and major sevenths on strong beats. I'm not squeamish about those at all in five voices - in fact I seek them out in that context - but in less than four voices they can sound, well, icky. Two voices especially. Yes, I tried strict intervallic inversion, but that yields a Mixolydian minor-sixth mode, which sounded positively bizarre in this context.

With the contrapuntal possibilities cut by exactly 50%, the task of reaching a final configuration for the piece will be much easier, and the smaller implied dimensions of an invention over a fugue are actually more appropriate.

I can again provide AAC conversions, as I realized I hadn't updated iTunes on my old 1.67GHz G4 PowerBook, so you can open another window or tab to follow along.

Invention in A Minor

The work-around is pretty ponderous though: I have to export the Encore file from my PowerMac G5 to the shared HD plugged into my Airport Extreme, then download it to the G4 PowerBook, import it to the old version of iTunes there, create the AAC conversion, and then do the whole thing in reverse with the M4A file. PITA.

Honestly, I really do hate OS X 10.5.8. For me, OS X 10.4.11 is much better, so I'm thinking of wiping the HD on my G5 and downgrading the OS... lot of work though.

But I digress...



No linking measure needed in the expo anymore, and that makes the desirable odd three-measure length of the subject more apparent.

First modulation is to the subdominant, and you can see the two longer stretto possibilities there: 2.5 measures of delay at measure 10.5, and 1.5 measures of delay at measure 12. Since you have two entries at .5 distances, they mesh back up to be at the beginning of the measure for the third entrance.



Next modulation is to the relative major, and only the subject-above-countersubject orientation works properly here: The inversion produces an attacked major second - which was an augmented second= minor third in the minor - and I decided I didn't really like that either. After that statement I plan an episode over the dominant pedal on the low E string, and I have a very cool harmonized version of the subject in mind for that.

Then at 20 we're back to the tonic minor for the closest stretto of .5 measure of delay, and then the hyper-stretto between the subject and its rhythmically augmented form. The pedal episode harmonized version will also be an augmented form to set this up better.

I really like where this is going, and like i said, this could lead to another series of solo guitar pieces. I love composing series... I just hope they are not all in A minor and C. lol

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