Fugal Science: Vol. 2, No. 4 - Five-Part Ricercare for Orchestra

EDIT: This is part 8 of 8. Here are the links to the entire series:

Index of Fugal Science, Volumes 1 and 2

Here is today's audio. It's just a string choir arrangement, as I still have to make digital fair copies of this piece, as well as the five-part grand fugue of volume one, in the orchestral arrangements. That will be my next project, and since I have to start by making a new Encore template, it could take a while.

Five-Part Ricercare for Orchestra

I apologize for the left side of the mix being hotter than the right, but I'm chasing some balance gremlins and ran out of patience with it. So no, it's not your imagination - or your system's fault - that the balance is out of whack.

We're back up to a-minor now, but the tempo is a slow 63 BPM to really bring out the super-hot dissonance points. Otherwise, this part of the exposition is exactly like the three- and four-part versions.

This page is exactly like the four-voice fugue as well, with the third entry and the required linking episode.

Whereas the four-part fugue left us on the dominant level, this ricercare finally has the fifth entry on the tonic. Note how the simple means I limited myself to produce a magnificent and moving effect here: The fourth entry of the answer on the dominant has the highest note of the piece at f-natural in measure twenty-one, and then the basses get the lowest note of the piece at e-natural in measure twenty-nine (The lowest note on four string basses without the extension to get c-natural, which is what I'm figuring on for when I hire an orchestra to play it).

With the exposition behind us - a relatively vast twenty-nine measures - we transition back to four voices with the second appearance of the episode, and then the six measure interlude comes along. This sets up for the middle entries.

For the overall plan of this fugue - a ricercare is just a particularly large fugue, or one that goes to distant keys - I chose to go back down to two voices and build back up from there, presenting the extensive canonic possibilities of this amazing subject as I go. Here, we get the two-part canon at three measures of delay/two measures of overlap, that is also a perfect dovetail: The countersubjects require no modification to work, they are just interrupted or joined in progress. This was the same canon I presented all the way back in the guitar version, by the way.

This dovetail sets up the two-voice version of the episode, which leads to the three-voice canonic possibilities, which are amazing.

We directly modulate to the dominant, as in the four-part fugue, and this three-part Escher morph/perpetual canon is exactly as before: The canon at two measures distance morphs into itself in augmentation - this does not work in more than three parts - and then into the doubly-augmented subject with the original head figure and diatonicised tail, and back.

On this page we get the dovetail into the doubly augmented version of the subject.

And now that sublime series of searing dissonances - what I've come to call a dissonance flow - that seem so poignant to me, and then the re-transition back to the augmented subject. I think a good conductor could have a field day with this piece.

Then the head of the doubly-augmented form of the subject reenters, proving the perpetual part of the canon - going back to the original note values does not work - and right at the point where the beautiful dissonance flow would appear again, the contrabass makes a dramatic entrance and a direct modulation back to the tonic. As I mentioned in the previous post, stacking the canon up from the bottom produces some wonderful conflicts that I was able to take advantage of.

The top line breaks the canon by repeating the subject, and then the final conflicts are set in motion in a dissonance flow that is evocative of boiling water to me.

These conflicts come to a head when the cello and bass arrive on a unison f-natural in measure 113. The cello then dips below the bass to play the e-natural, f-sharp, and g-sharp against that f-natural, and that is the climax of the section.This is much more dramatic with the bassoon and trombone doubling the cellos, and the contrabassoon and tuba doubling the basses.

The augmented subject head then reenters, the tension ebbs, and we get our episode again.

That's the second episode since the previous interlude, so now comes the final interlude, lengthened to seven measures this time to get the caesura that prepares for the introduction of the final five-part perpetual canon. The introduction is the anacrusis made with the diminished triad at the top.

Finally, with five voices, we get the full harmonies I intended. At 136 it's a bVI(M7) in first inversion, reading, bottom to top; a-natural, a-natural, c-natural, e-natural, and f-natural. It's the minor ninth between the top voices that makes it so astringent. Then at 137 we get the same bVI(M7) in first inversion, but this time it reads up as; a-natural, c-natural, e-natural, f-natural, and e-natural. This puts the minor ninth between the violas and second violins, so that dissonance is moving down through the voice pairs as the canon unfolds. By 138 the bVI(M7) has become a second inversion sonority reading up as; c-natural, e-natural, f-natural, e-natural, and c-natural, with the minor ninth between the tenor and bass voices. In 139 - bet you guessed it - we get the third inversion of the bVI(M7) reading up as; e-natural, f-natural, e-natural, and a-natural. Finally, we get the root position bVI(M7) in 140, reading up as; f-natural, e-natural, c-natural, a-natural, and c-natural. Now the minor ninths are played out, and we get the root position chord. This logical canonic progression makes this dissonance flow seem profound to me. Also, the note doublings are modal, and not tonal, so that adds to the eerie effect.

Now we re-transition to the original subject, and back, proving the perpetual nature of the canon.

And so we wind down over the ostinato of the tail figure, revisiting the descending minor ninth sequence as we do so, and end up with all five voices on the tonic: the pluperfect resolution.

As I mentioned above, I now need to do digital fair copies of the two orchestrated pieces, which could take a lot of time. In the meanwhile, I'll link these posts together so that anyone who finds one will have the links to all of them.

Back sometime next month.

Fugal Science: Vol. 2, No. 3 - Four-Part Fugue for String Choir

EDIT: This is part 7 of 8. Here are the links to the entire series:

Index of Fugal Science, Volumes 1 and 2

Here is today's audio file:

Four-Part Fugue for String Choir

As usual, it's an AIFF file of the sound fonts I compose with, so you'll need to use QuickTime.

Aside from being down in the key of f-sharp minor now, the exposition is as it was in three voices.

After the third entry, however, we need to use the episode to set up the final one. That's because it's impossible to do a direct modulation to the dominant after the three-part subject statement. This volume was on ice as just the concluding perpetual canon for years - since 2006 - because I couldn't figure out what to do here. So, this is actually the origin point of the episode, though I completed the five-voice fugue first, back in 2013.

With four entries and real answers, the exposition leaves us on the dominant level. This means the episode is on the dominant, and has to modulate back to the tonic for the interlude (After two episodes, the interlude is required). So, instead of four measures, that makes this one five. All you have to do to get the modulation is to make the dominant resolution to a major chord, and then introduce the minor seventh to make it the dominant seventh of the tonic. I was able to make the dominant a major add-9 chord, and then I used the thematic triplet to make that a minor ninth, and down through the tonic to the minor seventh. This took a while to figure out as well. I had more elaborate ideas, but the most compelling solution turned out to be the simplest.

Now we get the four part version of the interlude, and we're ready for the middle entries. The two- and three-part versions maintained their voice numbers throughout, but here I reduce the texture to two voices to demonstrate the possibility of a two-part stretto at three measures of delay/two measures of overlap that is a perfect dovetail. As I alluded to in the last post, the single measure overlap stretto is possible, but trivial, so I didn't include it in the plan.

After the stretto, it's time for the episode again, now back in its normative form.

With the three-voice Escher morph/perpetual canon comes the first modulation of this volume outside of the expositions. It is the same direct modulation as in the expositions, it just continues on instead of returning to the tonic immediately after the answer statement. So, the real answer becomes the tonic and has it's time in the sun.

The plan of this stunning canon is the same as it was in the three-part fugue, only before it was on the tonic: The subject morphs into the subject in augmentation, and then the modified subject in double-augmentation, and then back to the augmented form, all conforming to the laws of pure counterpoint.

Though I didn't break any laws, the dissonances I employ are not something any of the Bach-era composers would have used. Basically, the deceptive resolution to the bVI(M7) is to an inversion with the root above the seventh, yielding a minor ninth. This creates a kind of creepy sounding dissonance flow, with the chord of resolution more dissonant than the dominant that targets it. This excruciatingly beautiful passage is proof that any musical effect can be employed to achieve something wonderful, if it is presented in a logical musical context. There is no musical context more logical than canon.

The canon then re-transitions back to the augmented form of the subject.

Once the perpetual nature of the canon is proven, the piece modulates directly back to the tonic with the cellos entering with the subject in the bass. The order of entry for this canon is reversed, however, which leads to some glorious conflicts.

The canon is broken by the first violins at measure 196, when they repeat the subject. This turns the rest of the section into a roiling cauldron that uses nothing but the subject.

This topsy-turvy action comes to a head in measure 109, when the cellos and violas land on the unison d-natural. The violas then dip below the bases to play the c-sharp, d-sharp, e-sharp triplet against that sustained d-natural. This is the climax of the section, after which, the augmented head figure reappears, the tension ebbs, and the episode appears again.

That makes two episodes, so it's time for the interlude again, and now it's seven measures so that it can come to a full stop before the concluding perpetual canon begins, which is introduced by the diminished triad triplet figure in the first violins.

The concluding perpetual canon is much fuller and more complete in four parts than has hitherto been the case with the three voiced versions.

For example, in measure 132 we now get a complete bVI(M7) in first inversion. Reading from the bottom, major-third, perfect-fifth, major-seventh, and the root on top. I came up with the concept of beautiful dissonance when I was at Berklee College of Music back in the early 80's, but I was writing in jazz and jazz-fusion idioms then, so I was thinking of colorful jazz-fusion harmonies. I intentionally pursued that concept with counterpoint, and this is the flowering of those efforts. Next path will be to recover more complex harmonies in counterpoint (Which reminds me that I need to analyze the Beethoven Op. 131 string quartet fugue).

So, the canon in original note values reenters, and once the perpetual nature of the canon is proven, the piece winds down over the ostinato of the tail figure in the bass. With more than three parts, we can now get the pluperfect ending in which all of the voices come to rest on the tonic. Note the momentary major chord at the beginning of measure 149: This a feint to get the final triplet figure in, which has the missing Phrygian degree, which is the only one absent from the subject on the tonic level. Ta da!

I have to do a digital fair copy of the five-part ricercare, so I'll be back in a few days.

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

EDIT: This is part 6 of 8. Here are the links to the entire series:

Index of Fugal Science, Volumes 1 and 2

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.

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

EDIT: This is part 5 of 8. Here are the links to the entire series:

Index of Fugal Science, Volumes 1 and 2

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.

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

EDIT: This is part 4 of 8. Here are the links to the entire series:

Index of Fugal Science, Volumes 1 and 2

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.