Fugal Science, Volume 1, Number 2: Three-Voice Fugue v1.1

I've had a major breakthrough, and now this piece and the two-voice solo guitar fugue are finished! The breakthrough came in the form of a rhythmic subtlety that actually lead me to modify the subject, which is proof that constantly working with thematic materials one develops over the course of years leads to ever greater perfection. After all, I devised this fugue subject all the way back in 1993, so it was in the same form for over twenty-two years!

Here is the audio for today's Fugue for String Trio

It's an AIFF file, so you'll need QuickTime activated, and opening it in another tab will allow you to listen and follow the score.

The modification to the subject appears in measure four, where the half-note is now tied to an eighth note, followed by another eighth note, instead of the former dotted-eighth-note and sixteenth. This provides another level of resistance, and it's just super tight. I hit upon this minor masterstroke when a cross-rhythmic clash occurred, which left me with the choice of modifying the accompaniment, allowing the cross-rhythms, or... modifying the subject. It was a deeply satisfying moment, and I'll point out how this rhythm first appeared, and how it became viral, infecting the entire piece, when we get to those points.

This rhythm mod is also applied to the tonal answer, obviously, and otherwise, all the music on this page is the same as before, so no additional commentary is necessary.

Our new rhythm is also in the bass in measures nineteen and twenty, but it was there already in the previous version. This is not, however the point of origin. That comes in the next episode. Aside from the modified thematic statements, all of the music on this page is as it was before too.

We also get the new rhythm in the bass of thirty-four and thirty-five, but it was this way before. Measure thirty-five is the point of origin, though: In the two-part fugue for solo guitar, the dotted-eighth/sixteenth versus the sixteenth-note sextuplet was way too hard to play, so I made the eighth notes straight. The eight-note triplet against the straight sixteenth note will be played as a descending flam, which sounds cool and is not hard to execute.

At measure forty-seven, we get the new rhythm in the subject in parallel with the same rhythm in the bass, but this is just an infection point I went back and changed, not the point where I decided to modify the subject. Again, the music is otherwise the same.

Measure fifty-eight is also another infection point I changed after the fact, and it's amazing how profoundly this seemingly minor modification changes things. The effect is much better this way. Other than our new rhythmic elements, The Song Remains the Same.

It was at measure seventy-two that I hit upon the conundrum that lead me to modify the subject. In all previous iterations of the episode/interlude, the mod appeared, but it rhythmically clashed with the subject if I did it here, so I avoided it at first. Not satisfied with that, I tried the cross-rhythmic clash, but that was no good either. Finally, I decided to modify the subject, and Eureka! The music on this page is otherwise the same, but wow, what a difference this seemingly insignificant modification makes. This is the kind of thing that makes being a mature composer so fun and rewarding.

I did add a dotted-eighth-rest in the lead at measure seventy-six, but that was to get rid of the unison, which due to phase amplification, was way too loud with the soundfonts.

The final infection points are in measures eighty-six, where I used the modification as the pickup into the coda/hyper-stretto, and then in ninety-four, where I used the mod to lead into the conclusion. This creates a very interesting rhythmic canon in diminution with the modification of the augmented subject in the bass, which is like the cherry on top of this piece. Finally, I've obviously decided to let the viola play only accompaniment for most of the piece, as it gets the first thematic statement in the exposition, and also the first one in the recapitulation/three-part canon. The piece is perfectly balanced by this.

Now I'm going to finally, after twenty-one years, start working on the rewrite of the string quartet, v1.0 of which I handed in for a graduate level Invertible Counterpoint and Fugue class all the way back in 1994, when I was doing doctoral work at UNT.

Happy New Year!

Fugal Science, Volume 1, Number 2: Three-Voice Fugue v1.0

As I continue to rush through these pieces, I'm finding some minor errors, which is typical for me. But there is a goal in my barging into these two collections, as I'm anxious to form a more perfect big-picture of the series, by getting all of the initial versions finished. At that point, I'll be able to more properly edit the constituent pieces to their perfect versions, and then the collections will be completed.

Today's piece has an earlier date range in the copyright notice, as you can see. The first unsuccessful version of this one was clear back in January of 2012, basically four years ago. I had no idea then, but I was very close to the correct solution, which was to have three-voices continuously. The form and content were so close, all I had to do was rewrite the two-part sections in three voices, rearrange which instruments got what in the thematic entrances, and I was done. Every measure still has the same content, just rearranged.

Here is the audio: Three-Part Fugue in G Minor

AIFF file, so you'll need to have QuickTime activated in your browser, and opening in another tab will allow you to listen and follow the score.

This one is in G minor instead of A minor, as the rest of them have been so far, as I had made that decision by 2013, which is the last time I worked on it. Also, it is numbered four, instead of two, because I had a more elaborate scheme in mind back then. As is almost always the case, simpler is better.

The exposition is untouched from the 2012 version, as I knew even back then that I had that much right. The answer is tonal here, as you can see, and the counter-answer is a new element after the two-part solo guitar version. Then, the countersubject in the lead at eleven is the same as the guitar version, only with a diminished-scale lick added. With the faster dotted rhythms in measure six, the exposition flagged with only quarter notes in twelve.

At sixteen is the first episode/interlude - it serves both functions - and this is where I went down to two voices previously. Now it is exactly like the corresponding episode in the upcoming string quartet.

The first version of the episode/interlude is non-modulatory, and at twenty-two the first middle entries start with a four-measure delay/one measure of overlap stretto that is also a perfect dovetail between the subject and countersubject one. Countersubject two just fills in the harmonies, and only a minor adjustment was necessary to make it work with the stretto.

The second episode then starts at thirty-one.

This episode is only five measures, versus the previous six, and this time we modulate to the dominant. The middle entries on this level are a stretto of three measures of delay/two measures of overlap. This is just like the organizational plan of the guitar version, so far. Likewise, the combination of episode and subject at forty-four is from the guitar version, but here with three parts it reveals how closely related the subject and episode are. It still modulates to the relative major at the end.

The major middle entries are the expected closer stretto of two measures of delay/three measures of overlap, but now filled in with a third voice. Note that the outer voices are getting all of the thematic materials, with the viola only performing accompaniment. Not sure if this will stand the test of time, but we'll see.

The episodes have been six measures, five measures, and five measures so far, but the episode at fifty-six is only four measures, and it modulates to the subdominant, where we get a three-part version of the dovetail stretto that remodulates back to the tonic.

After the remodulation we get the final version of the episode at sixty-nine, now with a full tonic subject statement in the bass, and extended by a resistance arrival into a nine measure interlude to set up the final three-voice canonic stretto, which begins at seventy-eight. The combination of episode and subject does not appear in the upcoming string quartet version, by the way, as I didn't discover that possibility until I composed the earlier versions of this piece.

After the canonic stretto concludes, we get the coda at eighty-seven, and it's the expected hyper-stretto that has the subject and augmented subject starting simultaneously, but with the third voice, the effect of the minor ninth in eighty-eight is much less jarring. Something gained and something lost, in my estimation. The canonic stretto and hyper-stretto are also unchanged from 2012.

Now I'm free to do a revision of the string quartet arrangement, which I've been anxious to get to for a while now.

Merry Christmas!

Fugal Science, Volume 2, Number 3: Four-Voice Fugue, v1.0

Very happy to have the first version of Fugal Science, Volume 2 completed! Today's piece is Number 3, which is for string quartet. As an aside, I believe I'll arrange Number 2 for wind trio. The subject has two natural places for rests, and when they're augmented and then doubly-augmented, it should give the players plenty of breathing time. They keys are also coming into focus, as I think the guitar and orchestra pieces will be in A minor (No other possible key for guitar, and the oboe limits the orchestra to A minor as well), and then this one will be in g-minor, as the viola limits the piece to that level with a c-sharp to the dominant (I don't think I want the open c-string, but I'm not positive. F-sharp minor could be good too). Since I need flute, clarinet, and bassoon for the trio due to the wide range of the parts, That one may go down to f-minor. Not positive yet, though. Just thinking in writing here. The tempi are also coming into focus: 81 BPM for the guitar piece, 72 for the trio, and 63 for the quartet so far.

Here's the audio file for Fugue for String Quartet.

As usual, it's an AIFF file, so you'll need QuickTime activated in your browser, and opening it in another tab will allow you to follow the score while you listen.

This will be quick, as everything here is the same as the string trio.

At sixteen, we return with the episode version that has the lead line continuing up to prepare for the fourth thematic entry, just as in the five-part Ricercare.

Without the fifth entry, we are left on the dominant level, so the episode at twenty-five is unique in that respect, and it also has a unique voice arrangement, which is heard nowhere else in the cycle. Plus, it requires a fifth measure so it can modulate back to the tonic, and this is what I spent the most time coming up with. As is common for me, the solution came to me as I was drifting off to sleep. The chromatic triplet figure in the lead is very tasty.

This gets us to where we need to be at the interlude, back on the tonic level. In thirty-six, the middle entries start, and everything goes down to two voices, just as in the five-part Ricercare, only an octave higher (One reason I want this in g-minor is because how very high the parts get without the contrabass octave to work with). However, there is a voice-leading break after the cadence - perfectly acceptable, but not as smooth as all the other versions - so I'm still looking for a smoother transition.

This is the same music as in the five-part piece, but as you can see, it goes up to c-natural above the fifth line. G-minor will help, but f-sharp-minor is tempting. The violas could play the b-double-sharp with the open c-string, of course.

Another reason to lower the key is so that the c-natural in the exposition of the Ricercare is the highest note in the cycle.

Finally, these high, excruciating dissonances sound fantastic, but they would be less intimidating for the players a tad lower. F-sharp minor might be the right key.

Now that the three-part Musical Escher Morph has run its course, it's time for the four-voice section.

Besides the top voice, which descends to take over for violin two in the accompaniment, before its thematic entrance, this is all the same as the five-voice version.

The climax of this section - the clash between the cello and viola in measure 109 - is much more tart an octave higher, which is a feature I like a lot about this version.

So finally, we get the episode and interlude in their most normative forms.

Now comes the four-part version of the concluding perpetual canon/Musical Escher Morph, which is the same, other than not having a fifth entry.

This iteration is quite full sounding, and not much seems to be missing, as opposed to the three-part arrangement.

The conclusion over the ostinato is less satisfying than the five-voice finish, but the uninitiated listener won't know that, of course. I also used the chromatic triplet conclusion here, which is revealed for the first time. All voices in all four pieces end on the tonic, by the way.

There is only a five measure difference in the length of this from the five-part arrangement - 149.5 measures versus 154.5 measures - and I may edit this piece down a bit yet. We will see. Now, however, I want to get on the three-part fugue in Volume 1.

Fugal Science, Volume 2, Number 2: Three-Voice Fugue, v2.1

Unexpectedly, the three-part fugue for Volume 2 of Fugal Science came together over just the past three days. Version 1.0 was just the first page here, plus the last two pages: Exposition, episode one, interlude, and recapitulation; version 2.0 was exactly what we have here - with the three-voice Escher Morph and episode two added - but episode two was not in augmentation, as I now have it. That was the last thing I came up with, and it gives this fugue something unique, beyond the three-part version of the perpetual canon at the recap and coda.

If you've heard the five-voice Ricercare for Orchestra already, this version may not strike you as immediately as satisfying, but keeping in mind that it's meant to be heard after the two-part Fugue for Solo Guitar, it then becomes much more impressive. These pieces are a progressive revelation of the subject's possibilities.

Here is the audio file for the Three-Voice Fugue for String Trio, then.

As per usual, this is an AIFF file, so you'll need to have QuickTime activated in your browser, and I suggest opening the sound file in another tab, so you can listen and follow the score. Also, I used string choir sounds, as I never have found any solo string soundfonts that I can tolerate.

On the first three systems, we get the three-voice exposition, which is exactly like the first three entries in the five-part Ricercare, but coming from the two-voice guitar version, the listener won't know that. The answer is real, of course, and that's a new element revealed here. The c-natural in measure seven is the pitch climax of the piece - all of these fugues with an answer will have the highest note in the exposition - and the exposition also goes into the first episode in the same way as the orchestral arrangement. The episode is changed here, however, as the lead line drops an octave, versus continuing up to prepare for the fourth thematic entry/second answer appearance.

Having already finished the finale for the present volume, one of the decisions I had to make was just how much to reveal in this version. The former goes down to two voices at the beginning of the middle entries, but two factors kept me from doing that here: One, I didn't want to reveal that yet, and two, I wanted to maintain three continuous voices throughout. So what we get here is the three-part Musical Escher Morph that is on the dominant level in the Ricercare, but in the tonic here. The difference in effect due to the lower tessitura is quite pronounced, and that will be another feature unique to this version, as I've already decided to put it in the dominant for the string quartet (Which will also go to two parts at the beginning of the middle entries).

The Escher Morph plays out just as it did in the Ricercare otherwise, but at sixty-seven I had to do a new thing: Put the second episode in augmentation. This is the point where it goes up to four voices in the Ricercare (and as it will in the upcoming string quartet), but that had a direct modulation from the dominant to the tonic, as well as a new subject entry associated with it. The original version of the episode just didn't have the right effect, and the idea to put it in augmentation came to me quite quickly. This allows the quickness of the texture to accelerate more gradually and logically into the upcoming interlude.

Now that we've heard two versions of the episode, the third-time logic of the repeat scheme kicks in, and it's time for the interlude, which is the only time we need it in this piece, as the three-voice variant of the perpetual canon begins immediately afterwards, at measure eighty-two.

Keeping in mind that these perpetual canons/Escher Morphs will be new to the uninitiated listener - all the guitar could present was a dovetail stretto, and the rest of the ever-closer stretti - and you can see how familiarity with the five-part arrangement could be a disadvantage here, but after the guitar version, this piece will be quite nice. Familiarity with all four arrangements, however, will foster a greater appreciation for the thematic possibilities that are progressively revealed. Obviously, this is an amazing subject with myriad potentialities, some of which came as surprises to me after devising it. I just thought I was composing a five-part canonic stretto!

I think the string quartet version will now come together quickly, and I'll have the first version of Volume 2 completed! It's obvious to me now that Volume 1 will be the more problematic, but even that is beginning to come into focus.

Fugal Science, Volume 1: Introduction, Part 1

And now we turn to the five-measure subject that is, chronologically, the earlier one. I came up with this theme back in 1993, and completed a four-voice fugue for string quartet on it while a doctoral candidate at UNT in 1994, initially. This was early on in my quest, and the piece was a big breakthrough for me. The problem is, that piece looms large in my development, and I was trying to master Bach's late style then, not really develop my own. As a result, I wasn't really able to key in on how I would reimagine it as a mature composer until I wrote this solo guitar arrangement back in 2010. Then I attempted a three-part fugue on it, but it was not successful, and I wasn't able to put my finger on why until just recently: The problem was, I was trying to bring the four-voice version elements back, instead of bringing the two-part discoveries forward, so it sounded too Bachian, and just not me.

The elements of style I've developed since '94 include continuous - and primarily contiguous - voices: No resting voices awaiting another thematic statement. Also, I was keen on the traditional practice of introducing a new subject appearance over a suspended fourth back in the '90's, which I'm not interested in anymore. Finally, the cumulative surface rhythm in the string quartet is quite a bit more busy than the way I write today, and so after I do a modest revision to the four-voice fugue - I'm free to do that now that the unmolested original is the Andante in Fuga Electronica - I believe I will be ready to tackle the three-voice fugue with continuous voices and a more restrained cumulative rhythm. Then I may be ready for the final five-part Grand Fugue finale. Still up in the air about doing a ground-up re-composition of the quartet arrangement.

Here then, is the sound file for the two-voice Fugue for Solo Guitar

The usual notice applies: This is an AIFF file, so you'll need to have QuickTime Player activated in your browser, and opening the file in another tab will allow you to listen and follow the score simultaneously.

I composed the five-measure subject on the top system as a four-part canon at the octave, only briefly noting that it would work in five voices as well (I was nowhere near ready for five parts then, as the quartet was actually my first four-part fugue).

As mentioned previously, an answer at the octave is really the most natural arrangement for a two-part fugue, so that's what we get here (Answers on the subdominant are an exception to this, by the way, because that's like starting with the answer. The Finale of Fuga Electronica is a perfect example of that). Whereas the string quartet version had no real countersubject or counter-answer - I was way into fractal thinking at that time, and was using small fragments - here, sixteen years later, I composed a conservative walking-bass type countersubject for the guitar arrangement, with the written-out trill figure at the end.

On the bottom two systems is the interlude, and this is both the interlude and the episode for the piece. I was able to use only one intervening element because of all of the upcoming modulations, as you'll see. The original here does not modulate.

Beginning in measure seventeen, we get the first stretto, which is also a perfect dovetail between the subjects and countersubjects. There are four measures of delay, so only one measure of overlap, but the fact that it's a perfect dovetail makes it compelling enough, and I added a diminished-scale lick in the second measure of the countersubject to spice it up. That lick then appears in the bass at twenty-two, precisely at the joint of the dovetail, and by the end of the stretto, where the trill figure appears, we are at exactly the same place as the end of the exposition.

This sets up the second appearance of the interlude, but this time it's shortened from six measures to five, as I affect the modulation to the dominant level into thirty-one.

The modulation is nice and dramatic sounding, and this is the second stretto at three measures of delay/two measures of overlap. It's not a dovetail configuration - and the diminished-seventh arpeggio only appears once - so I eschew the diminished scale lick this time, but add an ascending chromatic line in measure thirty-six. When this has played out, we get the third appearance of the interlude, but this time I put a statement of the subject over the descending chromatic line in the bass, and it suddenly modulates to the relative major by the end of forty-three. The diminished fifth (twelfth) at forty-two is particularly nice, and the section has a slightly forced and humorous effect, which is what I was going for entering the playful major key segment.

As an aside, the five-measure subject of Volume 2 does not lend itself to a major mode variant, which, along with the fact that it requires a real answer, is what makes the chronologically later pieces simpler in terms of both modulation and architecture. My move toward a simpler style here does not mitigate where the fugue wants to go, so whereas the guitar fugue of Volume 2 is only fifty-four measures long, this one stretches out to ninety.

This major key stretto at two measures of delay/three measures of overlap sounds quite sun-lit, and is very pretty. A nice respite from the darkness of the minor mode. It leads us to an arrival at the fourth version of the interlude, but this time it's truncated to four measures, which is within my definition for an episode (Episodes= 1-4mm, Interludes= 5+mm). It also has an entirely different effect coming out of the major, and it modulates to the subdominant perfectly at the end. So, with the strettos getting ever closer, and the interludes getting ever shorter, the pace of the fugue has been accelerating, which keeps interest piqued in the listener.

The subdominant entry at measure fifty-five is the original dovetail stretto, so the diminished-scale licks return, only this time I use the head figure for what would be the tonal answer in fifty-nine to cleverly re-modulate back to the tonic. This produces a marvelous effect.

At sixty, then, we are right back to where we were at the ending of the initial dovetail stretto, which in turn sets up the final appearance of the interlude in the tonic, here lengthened with a tasteful resistance section before the final resolution into the last traditional stretto of one measure of distance/four measures of overlap at seventy-two. This section is canonic in its entirety.

I extend the canon to six full measures by using the written-out trill - the final measure of the countersubject, remember - after the subject statements. This goes all the way back to my fractal thinking of the 90's, when I composed the string quartet arrangement.

After the canonic stretto, I present a two-part hyper-stretto (simultaneous entries) between the subject, in the lead, over the subject in augmentation in the bass. Note that the f-natural in measure eighty-one is the pitch climax of the piece, and that it makes a naked minor ninth (sixteenth) over the e-natural in the bass. This makes a piquant effect on the mild-mannered guitar, but it would probably sound harsh on the less-elegant piano (Pianos stink for counterpoint, in my opinion). It is perfectly justified contrapuntally and harmonically, of course, because it is outlining the compass of the dominant-seventh/minor-ninth. I have also left this section slightly idealized, as a guitarist with average-sized hands would need to put a dotted-quarter rest in the bass at the end of measure eighty-three to reach the d-natural in the soprano (Well, alto, since this is a guitar). Then I use an articulated rising chromatic line over the middle of the augmented theme, and a rambunctious ending to a different version of the written-out trill that ends on a fifth position barre chord for a big, fat final punch.

Beethoven once said something to the effect that it was easier to start with a new theme and compose a new piece than to revise an existing work - the thinking being it is more difficult to rediscover where you were than to continue from where you are - and I can certainly understand that for the vast, sprawling sonata-process pieces he composed, but it is also true, to a lesser extent, with fugue themes that you've worked with for many years. That's really what these blog posts are all about: Getting me re-familiarized with my previous modes of thinking, so I can bring them into my present.

Not positive where I'll take this next, so the next post will have to await the new year.

Fugal Science, Volume 2: Introduction, Part 2

In Part 1 of this introduction, I presented the first fugue in the collection, a two-voice fugue at the octave for solo guitar. Now we fast forward to the final piece, which is a five-part Ricercare for Orchestra. The three and four part fugues have yet to be composed, though I think the three-voice one should be relatively straight forward. Not sure about the four-voice fugue yet, as I can see how there could be some problems working it through.

Here is the audio file for the Ricercare for Orchestra.

Again, this is a CD-quality AIFF file, so you'll need to have QuickTime activated in your browser, and I suggest opening another tab for the audio so you can follow the score while you listen.

Since the soundfonts are just a convenience for composing and orchestrating within Encore for me, I do put up with some obvious limitations: The woodwinds and brass I chose are all solo instrument sounds, so there are no tutti sections, and the muted strings were not available either, so they are un-muted throughout. This is a tad annoying, as I can't wait to hear the combination of soli winds with muted strings that I start out with here. I hear it in my head as being very etherial and subtly menacing, due to the chromatic nature of the subject.

The score is a MIDI score of my own devising, so it is in concert pitch, and the wind and brass choirs are just reflections of the strings. That means there are 15 channels used, with the sixteenth still available for percussion (I have some definite ideas there, but I'm not ready to include them in the score or sound file yet). Another sound font issue is that there is no piccolo trumpet available to me, so I used a very high and pinched-sounding jazz trumpet timbre. It's not awful, but a properly played piccolo trumpet will sound more... "at home" up there in the stratosphere.

This page, then, is just the initial statement of the subject, which is five measures long, has the range of a minor ninth, and contains eleven of the twelve pitch classes. The only one missing is the Phrygian second degree of ra, which I save for the penultimate note at the very end of the piece.

Due to the highly chromatic nature of the subject, the answer must be real, and not tonal. So, here we have that real answer at the usual interval of a fifth above. This gives the exposition the feeling of rocking back and forth between tonic and dominant more than with a tonal answer, and I use this effect for every modulation in the piece, which only alternates between the tonic and dominant regions throughout. Let me point out again that the counter-answer starting in measure eight is the head of the answer in augmentation, which foreshadows all of the Musical Escher Morphs to come. I'll also note that Encore makes it very convenient to compress the MIDI dynamic range by specifying exactly and numerically what those should be (And I haven't put most of the dynamics in yet), but a few of the ">" accented notes stand out a bit more than I'd like, especially when the brass come in.

Back to the tonic for the third thematic entrance, and now you can see how ascetic my style is these days: Lots of slow movement in the accompaniment - nothing shorter than a quarter note - and I don't worry about achieving an even cumulative rhythm, because I find that gets boring. Far superior to devise the rhythms of the parts subtly, so the cumulative effect is interesting, but not busy.

Here we get a version of the episode - and I mean, the episode, as this is the only one, along with different contrapuntal inversions and voice numbers, in the entire piece - within the exposition. This is unusual, but there is no way to execute a dominant region answer at sixteen: The episode is necessary to set up the next entrance of the real answer at twenty. I devised this subject back in 2003, and had the concluding five-voice perpetual canon done by 2006, and then it sat around like that for seven years until I finally came up with this episode to get the exposition done. That was a good day! And then, the middle entries came together within a couple of months in 2013. These delayed gratification payoffs - a genuine jackpot, in this case - make composing the most interesting and rewarding of all human endeavors, in my estimation.

The high note of the entire piece comes in the second real answer here, which is the c-natural in measure twenty-one. At the beginning of the five-part perpetual canon that is the recapitulation of the piece, this statement is back on the tonic, so this piece is basically a 5, 4, 3, 2, 1, in Schenkerian terms (Not that I put much stock in his theories). This is not normative, obviously, as the most natural position for a pitch climax in a piece would be 2/3 of the way through. 2/3 of the exposition is, however, 19.33 measures, so that's another picture frame through which to view it. The bottom line is, the pitch climax can be the first note, or the last note, or any note in between (Or, there can be no explicit pitch climax at all). You just need to have compelling musical logic to support wherever you put it... or don't (The two-voice fugue for guitar has no unique pitch climax, for example).

Here at twenty-five, we get the final subject statement back on the tonic level, and the five-part texture is complete. Note how spare and reserved my accompaniment style is here. This texture owes much more to Palestrina than Bach, and the harmony is very deep with modal effects, just spicily seasoned by the chromatic subject. Just magnificent, if I do say so myself. So, the exposition is a mighty twenty-nine measures long.

Then, here at thirty, we get the second appearance of the episode, now fully fleshed out in four voices. I should mention that I avoid unisons in these sound font MIDI scores because of phase and attack problems, or the flutes and first violins would end in unison with the oboes and second violins. The uneasy closeness between the bass and contrabass also plays a big part in the unfolding drama of the piece, reaching a low-pitch climax between them just before the final episode and interlude.

Now we get the first of two appearances of the interlude, and the disquieting rumbles between bass and contrabass continue. By thirty-nine, the middle entries are set up. Note that the strings have been muted and the winds have been soli this entire time. The mutes will come off for when the strings return next time.

The winds, however, remain soli for the beginning of the middle entries, which begin with a duet between the oboe and clarinet. This is the same stretto as was first in the two-part guitar version, so it makes a perfect dovetail between the subject and countersubject statements.

At the end of the stretto, I present the two-voice version of the episode, and we are now set up for the three-part entries.

I modulate to the dominant level for the tree-voice stretti, and transition by having the first two entries play out over a dominant pedal point (On this, the dominant level, so b-natural).

The bassoon then takes over for the clarinet, and we begin the first Musical Escher Morph, where the augmented form of the subject begins in measure fifty-seven. This is where the strings reenter, now un-muted, as they will be taking over during the doubly-augmented section (I'd hate to asphyxiate the wind players). Note that, in the original note values, the stretto is at two measures of distance/three measures of overlap, but the augmented form of the subject equates to a single measure of delay/four measures of overlap back in the original values: Mensural canons like this are very interesting that way.

By sixty-one, the three voices of un-muted strings are complete for this section, and the soli winds have been moved to the background. Notice that in measure sixty-five, there is a beautiful cross-rhythm exposed by the slow, augmented pace, between the quarter note triplet in the top voice, and the straight quarter in the bass. This goes by quite fast in the original note values, so bringing this out is one of my favorite features of the augmented statements.

As the backgrounded winds finish their augmented statements, they drop out for a much needed break as the doubly-augmented transition begins. The strings are the only choir that can sustain this without committing suicide, so it was both a natural and unavoidable orchestration choice to make. Note again that, in this new doubly-augmented environment, the delay between the statements would be a half-measure. This works only for the head of the subject. I had to modify the tail to make it work.

What I did to make the tail work - well, the body of the subject, leaving out the measure with the triplets - was to keep the doubly-augmented rhythms, but make the line diatonic instead of chromatic. The results are astonishing. So much so, that when I analyzed the results, I was not completely certain that it would sound as I imagined. It just breaks some, "rules" for harmonic interaction. For example, the harmony in measure seventy-three is the C(M7) that lives on the bVI degree of our current tonic key of e-minor. But, the root is in the lead, with the major-seventh underneath, creating the interval of a minor ninth. Then, this is supported by the fifth in the bass, and the major third is missing. The result is a dissonance both empty - in terms of interval density - and searingly hot, and yet it is sublimely beautiful. The dorks of the so-called Second Viennese School - Schoenberg, Webern, et al - used structures like this, and they sounded terrible. What's the difference? Context. They rejected true musical logic, and went off on a tangent in which everything was arbitrary, and therefore ugly. And not just ugly, but meaningless. Here, canon - the supreme musical logic of all musical logic - perfectly justifies everything and the results are stunningly beautiful.

This is no fluke, as you can see in measure seventy-five, where even the fifth is gone, and all that is left is a major seventh on top of a minor ninth! This was an earth shattering discovery for me: The real liberation of the dissonance, done naturally via canon, not artificially and arbitrarily by rejecting musical logic. The three-part harmony then reappears in seventy-seven and seventy-eight, but now as a very sweet sounding major seventh, due to the compelling logic of the context. Anyone who says everything that can be done with tonality/modality has been done is a hack utterly lacking imagination. Flee from them as you would a cloud of poison gas, because that's what such utterances are.

What separates my idea of Musical Escher Morphs from simpler mensural canons is the fact that they re-transition back to an earlier version of the theme, thereby creating perpetual canons of three or more voices. Here, that manifests itself as the reappearance of the augmented form of the subject starting at eighty-three. The winds now rejoin the orchestration, and if this was a real orchestra, they would be in tutti to be more of a force versus the un-muted strings (And there could be more than two of each, depending on the orchestral resources available).

As the doubly-augmented statements return to prove the perpetual canon - the morph doesn't work back to the original note values - the winds exit again in preparation for the brass entrances. Note that this entire three-voice section has been on the dominant level.

The four-voice section then begins back on the tonic level with the brass making a dramatic entrance with the tuba in the contrabass octave. This is not a Musical Escher Morph section, but rather a cauldron of canonic statements that begin at two measures of delay, stacking up from the bottom.

After the brass have stacked up with the entrance of the trumpet, the canon begins to repeat, now with the winds reentering. In traditional fugal style, voices rest before a new thematic entrance, but I defenestrated that idea, preferring the streamlined smoothness I've achieved here. So I rely on the orchestration to bring out the new thematic entrances instead.

The trumpets then break the canon by directly repeating the theme in measure 110, which makes the entrances one measure of delay/four measures of overlap. This sets up a clash between the bass and contrabass that will be the climax of the section.

As can be seen in measure 113, the bass and contrabass arrive on the unison of f-natural, and then the bass dips below the contrabass to e-natural, which creates a too-low minor second, followed by another too-low augmented unison between the f-natural and f-sharp, before the augmented second from f-natural to g-sharp at the end of the measure. These forbidden rumblings are below the usually recognized low interval limits for scoring, but they are brief and subtly dramatic. Once you notice them, they can't be un-heard. I like this effect a lot. Immediately thereafter, the augmented heads begin, but this is just a setup for the return of the episode and interlude to follow.

The last episode we heard was the two-voice version near the beginning of the middle entries, so this - a reproduction of the previous four-part version we heard after the exposition, but without the transition from five-voices - is the second episode since the last interlude, which means it's time for the interlude to reappear as well.

And so now we get the second and final interlude, here extended from the previous six measures to seven, in order to provide a more final resolution before the five-voice perpetual canon/Musical Escher Morph begins, the pickup for which is with the piccolo trumpet at the end of 127. If the form continued with more episodes and interludes, I would have needed to compose additional elements for that, as the repeat scheme for these two has been exhausted. As it is, all four of the episodes are different - three-voice, four-voice, two-voice, and four voice w/o the transition from five, in order - as are both of the interludes - the first six measures, and the second seven. Bach would have made every episode/interlude different, but this is a more thematically efficient scheme, and I think it is more in keeping with the times too.

This page is what I first composed back in 2003 when I came up with the concept for this fugue subject: A five-voice canon at the octave, which uses all five octaves of the orchestra - descending from the soprano octave - and that, resultantly, has all five measures of the subject appearing simultaneously in the fifth measure.

It wasn't until 2005 that I discovered that the original five-part canon would morph into the augmented version of the head figure like this. Probably needless to say, a very exciting revelation. As those augmented statements appear, the strings reappear, as we are now building to the final five-voice orchestral tutti. Note that in measure 136 the F(M7) harmony on the bVI degree is now complete, reading from bottom to top as A, A, C, E, and F, so we have that searing minor ninth on top, but the sonority is much fuller now with the increased interval densities present, versus the previous three-voice iteration.

The canon has the minor ninths progressively falling in the voicings - A, A, E, F, and E in 137; C, E, F, E, and E in 138 (An incomplete harmony); and E, F, E, C, and A in 139 - and then the root position chord in 140 as F, E, C, A, and A, so that the hot dissonances eventually disappear into a resolution at the major-major-seventh. Then the re-transition to the original begins in measure 141, and the tutti winds reenter as well, in the progressive buildup to the conclusion.

By measure 145 in the re-transition of this Musical Escher Morph, we have the full five-voice orchestral tutti for the first and only time. You can now fully conceptualize how the orchestration, and not voice rests, brings out the new subject entrances.

The Musical Escher Morph then proves that it is a five-part perpetual canon at the re-transition back to the augmented variant of the subject, and then we get to hear the delicious dissonances play out again, but this time subtly altered by the ostinato of the tail figure in the bass.

The wind-down is particularly nice over the ostinato, and as every line lands on the tonic, it becomes a drone until the conclusion. In the final measure - 154 (The half-measure is just a written-out fermata) - the piece appears to resolve to a major tonic, but this is just a setup for the final triplet figure, which has as the penultimate note b-flat, which is the only one of the twelve pitch classes that is missing from the subject in the tonic key. That makes the final resolution the most complete of all possible: Five octaves of pure tonic.

Next, I'll introduce Volume 1 of Fugal Science with its two-voice fugue for solo guitar. It is a much longer and more involved piece.