Saturday, October 22, 2016

Fugal Science: Vol. 2, No. 2 - Three-Part Fugue for String Trio

Both the guitar fugue and this one adhere to the continuously-voiced method I used in volume one, but hereafter it is not like that. Also like the previous guitar fugue, this one does not modulate, except for the real answer on the dominant in the exposition; that will change for the next two fugues as well.

Here's today's audio file, which is again just the sound fonts I compose with.

Three-Part Fugue for String Trio

An AIFF file, so you'll need to use QuickTime.

We're a whole step lower than before at g-minor now, and the tempo has broadened from ninety to eighty-one BPM. With the real answer at measure six, we get the main counter-answer, which still includes the head of the subject in augmentation, but unlike the guitar fugue, this time we'll get to see some of what that portends. I use an alto, soprano, bass entry scheme, which is necessary to set up for not just the rest of this fugue, but the rest of the fugues in this volume.

The first appearance of the four measure episode happens at measure sixteen, and again, it has to be in this configuration.

For the first middle entries we get a remarkable three-voice mensural canon that leads into a perpetual canon. First, the subject in canon at two measures of delay/three measures of overlap does an Escher morph into itself in augmentation, which would be a single measure of delay in the original note values. Then, the augmented subject canon does an Escher morph into a doubly-augmented version of the subject, which has a diatonicised body, and no tail figure. This would be a half-measure of delay in the original note values.

During the doubly-augmented segment, we get some super-hot dissonances in which the bVI(M7) appears in voicings in which the root is above the major-seventh, yielding a sizzling minor ninth. The order of entry back at the beginning of this section - soprano, alto, bass - had to be that way to yield this wonderful chain of effects. Dissonance flow, as I call it. Again, any musical effect can be a resource, and yield a beautiful effect, if used in the right context with a compelling logic.

As the doubly-augmented statements leave the stage, they re-transition back to the augmented statements, creating a perpetual canon. The augmented statements do not work out to go back to the original note values, so the doubly-augmented version reenters to prove the perpetual canon, and then we get the second episode, in the same configuration as before, but now in augmentation. This is a unique feature of this particular fugue.

After the second episode, it's time for the one and only interlude, which is seven measures this time, as it comes to a full stop to introduce the recapitulation canon at one measure of delay. This is the same perpetual canon as before, but now starting out in the original note values.

After the recapitulation/perpetual canon has proven itself, the piece winds down over an ostinato of the tail figure. This is bad-to-the-bone cool, but it gets ever more impressive during the next two fugues. The ending is to a pluperfect resolution in which all of the voices - ti-do, sol-do, and re-do - converge on the tonic. Only the guitar fugue uses a minor tonic resolution, which is a logical idiomatic consideration. By the Ricercare, all five voices will resolve to the tonic. It is a major philosophical element of this volume.

I have to make a digital fair copy of the four-part fugue, so I'll be back in a few days with that.

Friday, October 21, 2016

Fugal Science: Vol. 2, No. 1 - Two-Part Fugue for Solo Guitar

Whereas the first volume had a diatonic subject that took a tonal answer, and could appear in either the major or minor mode, this second volume has a chromatic subject that requires a real answer, and it only works in minor. Both volume's subjects are five measures in length, and make five-part canons at the octave and one measure of delay, so that in the fifth measure all five measures of the subjects are heard simultaneously, but this one lends itself to some truly astonishing contrapuntal devices.

Since this subject is more restrictive, the two-part fugue here does not need to be very long to get everything said that can be said in two voices: a mere fifty-four measures. Also, a wide ranging modulation scheme is neither required nor desirable here, so tonic and dominant are the only regions used in all four of the fugues. There is not, however, any way to combine this subject with an all-purpose episode, as in volume one, so here I required a third element: an interlude. When both an episode and an interlude are present, the difference is a matter of length as much as character; our episode here is four measures, while the interlude is six. The episode is built around a rising chromatic line, while the interlude's device is a descending chromatic line. After every two appearances of the episode, an interlude is required to define the form and avoid monotony. This fugue is so short, the interlude only appears once.

Finally, this volume has fully evolved artistic fugues, not greatly simplified ones like in volume one, though the style is very elemental still. While volume three will be art fugues on the diatonic subject, no fourth volume will be required on this subject: these are the art fugues.

Here is the recording for today's fugue: Two-Voice Fugue for Solo Guitar

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

As for the traits of the subject, it is five measures of 4/4 time, it has 21 attacks, the range is a minor ninth, and it contains eleven of the twelve pitch classes. The missing note is the minor second, or Phrygian, degree.

Note that the countersubject has, after the first two measures, the head of the subject in augmentation as the accompaniment. This is massively portentous vis-à-vis the implications for the rest of this volume.

Our episode appears at measure eleven, and it has the chromatic line in the bass. At fifteen, we get a counter-exposition with the subject and countersubject inverted...

... which leads to the inverted form of the episode. When there aren't a lot of possibilities, as here in this two-voice fugue, presenting a counter-exposition in inversion is a very common solution. Getting an inverted episode is less common, however.

It does lead perfectly to our six-measure interlude, however, which is why the exposition has the entering voice in the bass.

The first middle entries at thirty present a perfect dovetail stretto with three measures of delay, and two measures of overlap. Though a single measure of overlap is possible, it had no features to recommend it in this context.

Now it is time for the episode again, and this time it returns in its original configuration - with the chromatic line in the bass - and that leads to the only logical presentation left, which is the recapitulation canonic stretto at a single measure of delay, and four measures of overlap. Here - and in the rest of the fugues - the top voice enters first, as that is necessary to get the ostinato in the bass.

So there you have it. A super-tight fifty-four measures of perfection.

I already have the urtext for the three-part fugue for string trio done, so that should be ready in a couple of days. It is exactly twice as long as this little gem, at 108 measures.

Tuesday, October 18, 2016

Fugal Science: Vol. 1, No. 4 - Five-Part Grand Fugue for Orchestra

Today we're going to look at the urtext version of the five-part fugue, both because I haven't finished the orchestration, and because it will be many less pages to deal with at this point. In fact, all of these have been urtext versions, as there are no performance indications, other than for the tempo. After finalizing the arrangements for the two-, three-, and four-part fugues, this one required heavy revisions to fit into the scheme of the treatise. The solution turned out to be having the fifth voice - which logically provides the fifth degree of the key of the moment - in the center with the violas, as they were the first to enter with the subject at the beginning. That is another simplification strategy that made the piece perfectly logical for this collection, but I did lose some nice effects with the earlier arrangement. But, that's the nature of these fugues as you progress from two to five voices: At each stage, something is gained, and something is lost. I will develop this theme more in the subsequent articles, which will become more and more detailed. These posts are little more than overviews.

Here is today's audio, which is, again, just a recording of the sound fonts I compose with. It's a CD quality AIFF file, so you'll need QuickTime activated in your browser.

Five-Part Grand Fugue.

Use a new tab to listen and follow the score.

The exposition begins with the same order it always has, with exception of the two-voice fugue for guitar, which has no tonal answer. Even there, though, the lower voice enters first, so the countersubject - or counter-answer in this case - is first heard in the bass.

Voice entries continue just as they did in the four-part version, with the only difference here being key and tempo.

Now we can see what the fifth voice does: It is just a slightly ornamented pedal point on the fifth degree of the key. Since I devised the subject to alternate between tonic and dominant, this is perfectly logical. However, since I use the bVI and vi chords, the pedal is sometimes the major seventh, or minor seventh, respectively. Note also that I am able to get a complete V7(m9) sonority at the very end of the exposition. This happens to create a simultaneous cross-relation between the f-natural in the viola part with the f-sharp of the first violins. This creates a momentary augmented octave, which is very nice in this context. In the former arrangement, there was an augmented unison at this point later in the piece, which is even cooler, but, alas, it had to go for this piece to fit into the schema of the treatise. Volume 3 will be art fugues on this same subject, so perhaps I'll be able to work it in there.

Previously, the second violin and cello parts were exchanged in the episode, but I had to do some irregular resolutions to make that happen, so this turned out to be the more logical and compelling arrangement. Since the exposition is a big double-canon between the first two subject entries and the two answers, I wanted to make sure I didn't break that inexorable logic until I absolutely had to.

With this arrangement, we get a diminished octave between the f-natural and f-sharp at the end of the episode.

Technically, the soprano and contrabass could be exchanged in the first middle entry stretto/perfect dovetail here, like I did in the three-part version, but the effect is simply less good in more than three parts, so that will remain a unique feature of the string trio fugue.

There's a super-hot dissonance progression during the last two beats of this middle entry, where we get a minor ninth followed by a major seventh between the soprano and tenor, and it's seasoned quite spicily with the diminished octave with the contrabass on the final beat. This is a minor dissonance climax, which leads to an even more gnarly one later. Note also that I use a cambiata figure in the tenor part.

We're then into the second episode, which is as before for the first four measures.

Then the fifth measure modulates us to the dominant, and there's an augmented sixth at the very last second between the bass and contrabass.

In the four-part fugue, we had parallel sixths in the inside voices at this point, but in this opened up version, those are now parallel tenths (Plus an octave).

Sorry for the missing half-note f-sharp in the bass in measure 53, but that's the problem with doing these digital fair copies: transcription errors abound afterwards. I've fixed it since I uploaded these images. Then, the f-natural in the contrabass in 57 is supposed to be an f-sharp, which I've also repaired. Way back in the guitar fugue, that was an f-natural, as we didn't have the ascending voice with f-sharp to worry about.

The third episode then has the episode supporting the subject, which is one of the major revelations of the piece, and it modulates suddenly and smoothly to the relative major at the end, to great effect.

Our relative major middle entries - a canon at two measures of delay/three measures of overlap - sounds magnificent in five voices. There is a complete V7(M9) in first inversion on the last quarter note, which sounds amazing going into the next episode.

This shortest version of the episode modulates from the relative major to the subdominant minor, as before, but I have rearranged the parts - a very wonderful feature of this scientific fugue technique I've developed - to set up the next section, which is the one measure overlap stretto that remodulates us back to the tonic. That will also lead to the main dissonance climax of the piece on the next page.

While the soprano and contrabass are in the same arrangement as previously - and the tenor keeps the fifth degree - I now have the original countersubject 3 in the alto, to get the subject entry over the tied second, which is just way too good not to use (It was like this in the previous two fugues). In fact, the following dissonance climax would not be nearly as effective without it.

So here is the dissonance climax of the piece. The degrees of the dominant harmony, from bottom to top, are, M3, m7, R, R, and m9. It's the minor ninth in the lead, with the root underneath that puts the sizzle in the sonority, but now this is enhanced with a second root in the tenor, which has appeared earlier, and so is tied, making a beautiful effect because of the well wrought context. Nobody from Bach's era would have done this, of course.

Then we get the third and final episode on the tonic, in which the subject proves it is an effective bass under the episode materials, now quite awesome with the subject in the contrabass octave.

Note that the tenor gets some nice licks in near the final half cadence, and that I built this wonderful climax using almost nothing but diatonic material in the interior voices. The subtitle of this volume is, The Diatonic Subject, after all, and the aim is to be as simple and transparent as possible, so students of fugue have elements that are easy to grasp at first. That's why the chromatic lines are used so sparingly.

First things to notice in the five-voice version of the recapitulation canon are that I can't use the raised sixth degree anymore - due to the clash with the natural sixth in the subject - and that the solution I came up with causes parallel perfect fourths alternating with parallel perfect fifths; a clear violation of the laws of contrapuntal motion. However, the parallel movement is not stepwise, but by leap, and the effect is just wonderful, with the final leap in the soprano making the false ending sound awesome. I may or may not keep this, but for now, I've decided to give myself license for the exception.

The final hyper-stretto is unchanged from the four-part fugue - and almost unchanged from what I composed way back in 1993, the triplet figure in the lead being the only modification - with the bass doubled by the contrabass now. In the orchestration, this will be the only tutti section of the piece, with nothing but the string choir all the way until the recap canon, at which point the winds will enter for the first time. Here, the brass and percussion will enter. In the finale for the second volume, the winds will enter from the beginning.

Next step is to do the analysis versions, but first, I'll do the digital fair copies for the second volume and present overviews of them.

Friday, October 07, 2016

Fugal Science: Vol. 1, No. 3 - Four-Part Fugue for String Choir

Allright, I got the piece transposed. I find it wise to make what I call, "digital fair copies" after the development versions are done, as with all the cutting, copying, pasting, and other modifications, errors can accumulate to the point that the files become corrupted, and will crash the program. I was up to v3.4 with a couple of these, so it was time.

The point of this first volume is to reduce fugue to its irreducible essence, which is the two-part fugue for guitar, and build it up to the pluperfect arrangement, which is the five-part grand fugue. If I had had this information available to me when I first got interested in fugue, my progress would have been greatly accelerated. So I hope it is of benefit to anyone who undertakes this awesome journey of discovery and triumph.

Whereas in the two- and three-part fugues all voices resolved to the tonic at the seams between sections - ti-do, re-do, and sol-do - here in four voices we can now use fa-me (or fa-mi) to get a dyad at the resolution points. To keep this from increasing the possibilities exponentially, I will stick to the simplifying stratagem of having the thematic voices in the soprano and bass, while the accompanying voices will remain in the interior, with the exceptions of the exposition and recapitulation. This means I only have to decide which interior voice is the alto, and which is the tenor.

Here's today's audio file, which is again just a recording of the sound fonts I compose with. It's a CD quality AIFF file, so you'll need to have Quicktime activated in your browser.

Four-Part Fugue for String Choir

I suggest you open the audio in a new tab, so you can follow the score while you listen.

We're now in the key of f#-minor, and the tempo has broadened to 72 BPM, down from 90 BPM for the guitar version, and 81 BPM for the string trio. I'll go back up to 81 BPM for the Grand Fugue so that the tempi loop into 90 BPM for the guitar version of Volume 2.

Other than the key, tempo, and instrumentation, the first three entries of the exposition are just like the version for string trio.

It is not until measure sixteen that we get our new elements, which are counter-answers two and three. No diad here - at least not including the mediant degree - due to the fresh thematic entry, so I use the interior voices to dramatically build up to that revelation. As you can see, the alto leaps up a fifth, and the tenor leaps up a fourth to tee this up. When your texture is primarily stepwise, leaps are dramatic. Likewise, when your texture is mostly leaps, the stepwise motions become dramatic.

At the seam into the first episode, then, we get our first dyad which includes the third degree. The arrangement of voices is quite plastic here, with several possibilities, so I auditioned them, of course. I arrived at this configuration by realizing that the bass is a hard part, the tenor adds profundity to it, the alto interacts with the tenor to get the major second in measure twenty-six, and the slow, soaring soprano is the only remaining choice here.

Note that the bass-tenor have a minor tenth - e#-g - moving into a major ninth - e-f# - in parallel motion, and that it is resolved first back to a minor tenth, and finally a major tenth. This is very beautiful.

For the first middle entries, we have the stretto of one measure of overlap that is also a perfect dovetail. But this time, I go back to the arrangement that I used for the guitar version, which has the subject entering first in the bass. While I liked the other arrangement in three voices, this works better in four parts because of the interaction between the tenor's e# and the bass. The first time the e# appears in measure thirty, it is over the g#-f# of the subject in the bass, giving a nice, dissonant effect for the introduction of the subject in the soprano. While the second time in measure thirty-four, the e# makes a consonant major tenth with the c# in the bass, which sets up the final resolution into the second episode better.

The second episode has the same arrangement as the first, but is a measure shorter due to the modulation to the dominant, as in the previous two versions, but the third voice makes possible the dramatic octave leap up of measure forty. This produces a particularly excellent effect.

Now the dyadic resolution into the dominant region is much more dramatic with the subject so high up in the sopranino register, and that note a in measure forty-two is also the highest note in the piece. We first heard it as the final answer entry in the exposition, and will hear it a third and final time when I next combine the subject with the episode. Note that the interior voices are in parallel sixths throughout this section, until they finally part company in measure forty-seven. This has a nice, suspenseful effect.

Here we have the subject on the dominant level combined with three of the voices of the episode, and you can now see why the soaring voice was there before: The subject replaces it, and we keep the rest of the voices in the previous arrangement. The dominant voice's tyrannical attempt to take over the piece is thwarted by the sudden and almost magical modulation to the relative major, which is enhanced again with the new, fourth voice.

Our relative major middle entries are in the same arrangement that they've always been - with the addition of a fourth voice now - because this is simply the best possible arrangement of the elements.

The fourth part adds greatly to the sonorities in this relative major stretto, as it does to the following shortest four-measure version of the episode that modulates us to the subdominant level.

So, the modulation scheme is, i, v, bIII, iv, i, which is fairly common for a fugue. This allows for the aspirations of the dominant level answers to be fulfilled with their own time in the sun on the dominant level, and then they're brought back down through the relative major and subdominant levels, and resolved back to the tonic minor with the recap canon.

The subdominant entry/remodulation to tonic is a hard part, as I've mentioned before, and the entry of the tonic is over a tied major second, which is super-excellent, and the reason I have the interior voices as I do; best possible arrangement. Note that, in measure seventy, we have a V7(m9) with the minor ninth between the soprano and alto. This is a dissonance climax, right at the point of the return to the tonic. One of the many things I learned from The Schillinger System is that every musical effect is a potential resource. All one has to do is employ the effects in a logical context as tactical devices in an overall stratagem for them to be effective.

Under the final episode we can now reveal that the subject works as the bass for this section. This is a marvelous level of integration of the elements, and even though this fugue is longer than the 1993/94 original, it is much less prolix due to the greatly reduced number of elements. Back then I was trying to cop Bach's late compositional style. Today I have my own style that is simpler, clearer, and more mechanically efficient (From an objectivist viewpoint). As a side note, my upcoming volume three of this series will be on this same subject, but with a stretto exposition and a freer number of elements.

The pedal point section is richer now in four parts, with the half cadence greatly enhanced.

In the original fugue, this recap canon had the bass entering last, with a A, T, S, B arrangement, but here the order of entry is as in the exposition. So both of those dominant level answers, and the three dominant key subjects - five thematic statements in all - are now resolved to the tonic. Note again that the canon is lengthened to six measures by having the trill figure appended. This doesn't work in the five-part version of the canon, but I came up with something even better there.

This order of entry produces a perfect cadence into the dual suspension, like it was for the guitar, whereas it was imperfect in the original and for the string trio.

And now, finally, the full hyper-stretto is revealed, in which three parts have thematic statements starting simultaneously: Rectus for the soprano, inversus for the tenor, and augmentationem for the bass. The alto has the lone free voice. So far as I'm aware, no other composer in all of music history has ever managed to pull this off. And far from sounding dry or academic, it is magnificent, containing many dramatic sonorities that add up to an overwhelming effect. This section is unchanged from when I first discovered it, all the way back in 1993.

We also get a vastly improved final series of cadences. Gone are the alternate trill figures in the lead from the guitar and trio versions. Instead we get a V(4/2)/IV into measure 103, and a V7(d5/m9)/V (Without root) in second inversion (The sonority the ancient theorists called a German Augmented Sixth, but which is clearly just an altered dominant function sonority with the minor ninth replacing the root) in 103, and then the Picardy Third at the end, which echoes the V/IV! I chose to use the so-called German chord because the original piece was a tribute to J.S. Bach, of course.

Now I'm ready to tackle the five-part grand fugue for orchestra. This may take a while, as I have to verify that the elements are deployed properly, and also the minor matter of orchestrating it.

Saturday, October 01, 2016

Fugal Science: Vol. 1, No. 2 - Three-Part Fugue for String Trio

Well, I've gotten to the critical mass point on this project, so I have another post already. While the first post was about a two-part fugue for solo guitar, this one is in three parts, and it's for string trio. It is also in g-minor, whereas the first one was in a-minor.

Our subject and countersubject are the same, but now we also have a tonal answer, a counter-answer, a second countersubject, and a second counter-answer. Note also that the two-part fugue was in a continuous two-voice texture, and this one is in a continuous three-voice texture. This is not how fugue writing has ever been taught, but it is the simplest way to exploit our elements, as you'll see.

I continue this for the four- and five-voice fugues as well, so if it says five-part, it means five parts for the entire time that all voices can be active.

There is still only the one episode, and it continues to appear in six, five, four, and nine measure variants.

Without further ado, here is the recording, which is just the sound fonts I compose with recorded into Garageband (I still use Garageband if I only have a single stereo track to capture, and no MIDI tracks). It's a CD quality AIFF file, so you'll need to have Quicktime activated in your browser.

Three-Part Fugue for String Trio

I suggest opening the audio in a new tab so you can listen and follow the score.

The subject remains the same, but now we get the tonal answer beginning at measure nine, and it has a counter-answer that is as much like the countersubject as it can possibly be. Though I composed the initial version of this fugue back in 2012, this was the last element that came to me, just within the past few months. A glorious revelation that came to me on the toilet.

When the second subject statement comes at measure eleven, we have the original countersubject now joined by countersubject two, which is an accompaniment voice, as simple as it could be, yet it has enough character to be a subject in its own right.

Then our first episode begins at measure sixteen, and there is a new voice here too, of course. Again, note that all three voices converge on the tonic: ti-do, re-do, and sol-do. This is pretty much 180˚ contra the way three-part counterpoint is traditionally taught - you are supposedly required to get diads or triads when possible - but this method allows for the freest interchangeability of the materials.

This is not a hard part, per se, but after much experimentation, this ended up being objectively the best arrangement, though the alto and soprano could technically be switched out.

The first middle entries that begin at measure eleven are the same soprano and bass as in the guitar version, but now joined by a third accompaniment voice in the alto. Also, the entry order is reversed, with the soprano entering first this time. I really love the effect of this arrangement in three-voices, but not in two or four (Not sure about five-parts yet). It is the only place that this arrangement can exist, as the modulating version near the end has the tonal answer head in it, which cannot appear in the bass.

Note also that the third accompaniment voice always appears in the middle, and this is the case with the four- and five-part fugues as well. This decision saves a lot of headaches, as you can find out by rearranging the elements yourself, if you want.

Episode two is one measure shorter than before, as it was in the solo guitar fugue, and it also modulates to the dominant level, as before.

These dominant entries are also a hard part, considering the objective decision to have the accompaniment voices in the interior of the texture. Otherwise, the soprano and bass are the same as in the fugue for guitar.

You may have noticed a slight change in the lead voice of the first episode from the guitar version. There, the first three sequences were diatonically identical, but now the third sequence's penultimate note leaps down a perfect fifth. This was necessary to allow that voice to work accompanying the subject. What I call the dissonance flow is much ameliorated by the third voice, as you can hear. The modulation to the relative is also even more sublime.

Mozart did much of his writing with three essential voices, and said that the interior voice should allow the soprano to soar, and add profundity to the bass. Look at the trajectory of the accompaniment voice in this relative major stretto: It accomplishes both feats with great aplomb. Just a joyous respite from the minor key heaviness.

The four measure version of the following episode is also more dynamic with three voices, with the modulation to the subdominant nicely enhanced. It gets even better with four and five parts.

Now we've arrived at our false recapitulation/remodulation to the tonic. As I mentioned previously, this is a hard part that can only be arranged in this manner.

Once returned to the tonic, we get the third and final version of the episode on the tonic, which is again extended by the pedal point section. Now, however, we have the subject in the bass, revealing yet another contrapuntal combination, and this time the tail figure is not modified for a modulation. Very gnarly. The half-cadence is also improved by the third voice, and the wisdom of not having the lead voice hand off to the subject in the lead is again proven by having the order - alto, soprano, bass - echoing the exposition. Again, this is the most perfect possible arrangement of the available elements. Much work and experimentation went into arriving at this form.

The third voice improves the hyper-stretto coda as well, with the piquant minor-ninth ameliorated by having the fifth of the harmony - a V7(m9) - in the middle voice. The harmonic logic perfectly justifies the minor-ninth in two voices, but probably only the guitar could make it sound right. The piano is far too strident and harsh for counterpoint.

And so two posts in two days! I have to transpose the four-part fugue for string choir from g-minor to f#-minor, so no post tomorrow; that will take a few days.

Friday, September 30, 2016

Fugal Science: Vol. 1, No. 1 - Two-Part Fugue for Solo Guitar

I am now finishing up the final versions of the four fugues of Volume 1, which are a two-part fugue for solo guitar, a three-part fugue for string trio, a four-part fugue for string choir (minus contrabass), and a five-part grand fugue for symphony orchestra. The collection is designed to be performable as a concert, and I will also make a written treatise out of it. This post is the beginning of organizing my thoughts for that.

The subject of this set is a five-measure melodic trajectory that works in five-part canon at the octave and at one measure of delay, so that in the fifth measure, all five measures of the subject are heard simultaneously. The volume is set up to progressively reveal that.

This two-part fugue for solo guitar has the answer at the octave, so it is a real answer in the strictest sense. Bach called these octave fugues two-part inventions, but an answer at the octave is the most natural for two voices, just as the most natural two-part canon is at the octave. A tonal or real answer at the fifth implies more voices to come, and is very awkward to pull off. Many examples in the historical literature actually have a third entry on the tonic to rectify this deficiency. Interestingly, an answer at the fourth is perfectly natural for two voices, because it is sort of like starting with the answer, and having the subject enter second.

A note about objective renditions: I am routinely criticized for the mechanically perfect computer realizations of my music, but I don't think that is in any way valid. First of all, I love to hear my music performed perfectly by the computer, because it is precise enough to stand up to that, and if it moves me without any subjectivity, it is a successful piece of music. Of course I will put performance indications in the final versions, but we're not anywhere near that yet (Though they are developing in my head now).

So, without further delay, here is the recording for today's piece. It is just a recording of the sound fonts I compose with, and it is a CD quality AIFF file, so you'll need to have Quicktime activated in your browser.

Two-Part Fugue for Solo Guitar

I suggest opening a second tab so you can listen and follow the score.

One of the scientific features of these fugues is that as many voices resolve to the tonic as possible at every seam between sections. This is not the way fugue writing has ever been taught, but it is the best way to give maximum flexibility in the application and arrangement of the thematic and accompanying material. This will become clear as we progress through the volume.

The five measure subject starts in the bass, but it could just as easily have started in the soprano. I chose the bass first version because of a subjective reason, which is the idiom of the guitar, and also that the final canon is bass first, for reasons we will address when we get there.

The countersubject, then, first appears in the bass, which I subjectively like better anyway.

The six measure episode - and it is the only episode needed for this fugue, as will be revealed - is what I call a hard part: It cannot be inverted because the voices get separated by two octaves, which would be a very poor sounding unison if inverted.

The first middle entries are in the same arrangement as the exposition, with the bass entering first, and this is a hard part, because the voices become separated by two octaves at the end. This is also a stretto at one measure of overlap, which would be trivial except for the fact that this is, structurally, a perfect dovetail, requiring no modifications other than truncation of the countersubject.

That means the thematic section is nine measures this time - versus ten in the exposition - and the episode is a measure shorter this time too, due to the modulation to the dominant. So, as is desirable, the pace is quickening.

For the second middle entries on the dominant level - third thematic statements - we have a reversal, with the soprano entering first. This is a hard part, so that is unavoidable, but it fits into the plan for the fugue perfectly, as we'll see.

Part of the mystery of the one-and-only episode is revealed here: The bass line can support the subject. The treatment of dissonance is quite modern, and I'll have more to say about that in later elaborations on this piece. The stretto is eight measures, so one shorter than previously, but the episode is the same five measures as before, so still a bit of an increase in pace.

As you can see, the end of the episode very beautifully and dramatically modulates the piece to the relative major.

Here the bass comes in first, which is perfect after the episode/subject combination with the subject in the soprano! But note that this section is not a hard part, and could be inverted. The stretto overlap is now three measures, so the plan of the fugue is obvious: Ever closer stretti until the recap. That makes for seven measures in the thematic section, followed by a four measure version of the episode, so things continue to move more quickly.

The end of the episode modulates us to the subdominant in a very elegant way. This sets up a sort of false recapitulation on the subdominant, which is in the form of the first dovetail middle entries, but here I use the tonal answer version of the head figure to remodulate us back to the tonic! Very beautiful and dramatic.

Once back on the tonic level, the final version of the episode on the tonic appears, here extended with the pedal section. After the half-cadence in measure seventy-two, we get the recapitulation, which is the canon at one measure of delay/four measures of overlap. Note that the bass enters first, and this is the most desirable configuration, as the effect of the soprano handing off to the soprano is an objectively inferior arrangement. This fugue is the most perfect possible arrangement of the material, and I did experiment with all of the possibilities to arrive at it.

Note that I extended the canon to six measures with the addition of the trill figure, and this leads to a perfect cadence for the false ending. After that, we get, with the coda, the crowning contrapuntal combination of the piece: A hyper-stretto in which the soprano has the subject, and the bass has the subject in augmentation. This was a magical discovery, and it came to me way back in 1993! It is even more elaborate in three and four parts.

The ending is on a flourish with a secondary version of the thematic trill figure, and a strummed out six-part final barre chord.

Perhaps I'll have the three-part fugue ready for next month.

Wednesday, August 31, 2016

Fugal Science, Vol. 1, No. 3: Four Voice Fugue

Big month for me, as I have finished both the four- and five-part fugues for volume one of Fugal Science. This has been over twenty years in the making! So much has come into focus.

I have worked out the orchestration for both volumes (Volume 2 was completed last year), which will be; solo guitar for the two-part fugues, string trio for the three-part fugues, string choir minus contrabasses for the four-part fugues, and symphony orchestra for the five-part Grand Fugue of volume one, and also for the Ricercare of volume two.

The keys for both volumes will be A minor, G minor, F-sharp minor, and A minor.

These will be realized on the Synclavier, of course, and I have ordered a 64 voice FM/Additive system just for these projects (Fuga Electronica and Sonatae as well). Turns out that it is not practicable to link two 32 voice systems together to get 64 voices, because the Dynamic Voice Allocation makes it impossible to predict which of the two stereo pairs of outputs a voice will appear in. Because of the Synclavier's programmed-in compression effects, that makes level matching impossible. So I'll now have a single 64 voice FM/Additive system, which was always the ultimate dream setup for me, since the 80's. Plus, I'll be able to make one of the 32 voice systems a slave via MIDI Sync, which will give me a separate system dedicated to nothing but sound effects and percussion. Obviously, this will aid mixing.

Both of these volumes of four fugues each will be made into separate dissertations - or treatises, if you prefer - in addition to being made into albums. In this way, I'll finish up my doctoral studies simply for my own satisfaction. Nobody is qualified to award me a DMA or PhD anyway.

I was planning on presenting the piece today, but I did not get the audio recording done in time, so that will have to wait for next month.

I did get the JPG's done though, so here is the exposition.

The simple counter-answer, which has the came central body as the counter-subject, was the last piece of the puzzle to be revealed to me.

It is still in G minor, so I have to transpose it yet as well.