Feb. 21, 2017

Ten Steps Of Fugue Writing

"Spurred on by Canadian pianist Glenn Gould who spent his life immersed in that vast ocean of study musicians know as Bach, let's begin to wade into this topic by asking ourselves 3 simple questions ...
1. To whom does history give universal credit for being hands down the greatest fugue-writer who ever put pen to paper ? ... Glenn Gould tells us.
2. Would there be anything to gain by studying the fugues he left us and make a beginning for ourselves by trying to follow as best we can along lines patterned after the ones he used ? ... again, Glenn Gould tell us.
3. Would there be anything to gain by having not just a familiarity with his compositions but also how he understood harmony and composition at the keyboard and something about his thinking and attitude toward fugue-writing ? ... once again, Glenn Gould has the answer.
A FUGUE is a complete stand alone work characterized by a specialized process, procedure, or style of writing (not a form) in which a theme called a subject enters successively in each voice and sounds with other moving lines to create a polyphonic composition; typically this procedure results in 3 major sections, or parts: a) an exposition -- which introduces the subject in each voice, one by one, b) a development -- in which several more modulating entries of the subject alternate with episodes of less dense texture, and c) a final reentry, or restatement of the subject in the same key as the home key, usually followed by a coda; beyond that, there is an almost infinite variety of twists and turns, shapes and styles, that a fugue can assume.
When the fugue is short, often consisting of nothing more than an exposition followed by an episode in a related key and a conclusion, often with the contrapuntal process more simplified and the writing not particularly strict, it is called a FUGHETTA.
FUGATO refers to a passage of contrapuntal music where only part of a fugue, usually an exposition, appears within a larger composition and in a context that is not otherwise fugal.
It is a useless exercise to argue that a newly composed stand alone piece is a fugato when all of the attributes of a fugue within it are well defined; what has always mattered is how well a piece of music conveys a sense of symmetry, proportion, and a language of warmth and meaning to an audience -- not what the composer has named it, what it looks like, or how well it may line up congruent with something else.
NOTE: Rewriting a fugue of ours to get it to sound to the liking of a self-appointed expert who has opined publicly on social media that to him the writing is "not professional" is a meaningless exercise ... what exactly "professional" fugue writing is, has yet to be defined in music; being humiliated publicly like this has actually happened and would be calamity enough, but it gets worse ... it also fuels Resistance -- the force tending to hold others back from taking a stab at their own composing -- and, for many, it's all the proof they need to never give it a try, depriving the world of whatever they could have created.
The creation of a well-built organ fugue can be compared to constructing a well-built dwelling, with each entry comparable to a room and its voices (S-A-T-B) comparable to the 4 walls of each; when the various rooms or a new building are completed they also need to be linked with passageways which are comparable to the various episodes between entries which lead from one room into the next adjoining one.
The process is also comparable to working a crossword puzzle, with the quickest and most productive way of filling in all the blanks by having an idea of the big picture, working the entire puzzle at the same time, and filling in the easiest parts first.
Obviously it's far more challenging to start in one corner of the puzzle and work laboriously from there all the way across to the other side one tedious square at a time, disregarding any clues the surrounding parts of the puzzle might lend.
In like manner, using the same approach, i.e. to work the whole score at the same time in smaller "chunks" at a time and them joining them together rather than starting with the 1st measure of music and finishing with the last one [See blog, Getting Started With Writing, Part IX], makes better use of the time and provides insights that otherwise might not even enter the composer's mind.
This metaphor (of comparing fugue writing to working a crossword puzzle) is an entirely valid one; working the squares across is akin to developing horizontal melodic lines within a given harmonic framework; working the squares up and down is akin to the vertical harmonies created by the horizontal moving lines as they intersect each other in time.
Zooming out a little bit from there, the various chunks of the crossword puzzle also are akin to the various entries in the fugue; the squares in the puzzle where these chunks join with each other are akin to the material which connects the fugal entries.
The subject is the short phrase or melodic statement from which the whole fugue is developed, and is therefore its most important feature; for the discerning composer it contains within itself, just like a seed, an invisible instruction on what to become.
Subjects can be as short as 2 bars in length or as long as 8 bars, but usually are around 2-4 bars long; not every interesting melody however makes a good fugue subject.
Every well written fugue will have a good subject; no good fugue with a poor subject was ever written, nor shall be; it's no accident that the glorious organ fugues of J.S. Bach all have well thought out subjects.
A shorter subject in terms of length may present a single figure; a longer one always subdivides into members which are duplicated by sequence [See blog, Sequencing] or by repetition, either simply or with variation.
The whole idea of a fugue is crescendo, textural contrast, and tonal contrast; tonic-dominant tension also should prevail in the exposition, giving a sense of harmonic repose; any new keys Bach introduced after that provided the tonal contrast which would be lost if there had already been much modulation, or change of tonality, in the subject itself; he understood that if the available tonal forces for variety and contrast were brought out too soon, i.e. in the exposition, they would lose their effect in later developments; he also attempted to avoid a spiraling Circle of 5ths until all voices in the exposition had a chance to enter either in the tonic or dominant/subdominant key.
NOTE: In a 4-voice fugue exposition not all of the voices have to be doing something, but if all voices stay busy it does help to establish a sense of 4-voice texture from the outset.  When the subject repeats itself in the exposition in a second voice, it is called the answer.  The answer may be a literal reptition of the subject, note for note (real answer) or with one or two of its starting notes switched to an adjacent notes (tonal answer).  One problem that learners encounter is that they try to use a real answer rather than a tonal answer, which is a lot harder to control as they tend to push the music off key, which isn't good in the exposition of a fugue.  Subjects which finish by pointing strongly to the dominant key, for example, if given a real answer in the dominant, end up pointing strongly to the dominant of the dominant when it is too early in the fugue to introduce a key two doors away on the circle of 5ths before all voices have entered.  While there are no hard and fast rules, this is a rule of thumb (or heuristics), and Bach and his contemporaries certainly followed the rules of thumb the vast majority of the time and normally specifically didn't follow the heuristic for good reason.  The time to extend the tonic-dominant tension and move to a different key center is at the end of the voice 4 entry.

The theorists came along after Bach's death and catalogued what he did and did not do and created rules.  For the record, Bach basically adhered to the rules as his humble and loyal servants, but he never let the rules get in his way.  The big take-away with Bach is that within the so-called rules theorists have distilled from what he observed most of the time in his contrapuntal writing there are all kinds of bold moves that can be introduced to bend or break those rules and a moment or two later do a "pirouette" unnoticed to the listener's ear as if nothing unusual had happened.
A subject which begins on scale degree 1 or 5 may or may not modulate to the dominant key, but when the subject starts on scale degree 3 it needs a tail which either outlines a segment of the dominant key scale or which modulates to the dominant; it will also require a tonal answer (perhaps only the chromatic inflection of a single note) in order to steer clear of a secondary dominant and permit a smooth return to the tonic key for the entrance of the 3rd voice.
Those theorists who teach and maintain that fugues do not start on scale degree 3 are promoting something factually inaccurate; fugues can and may start on the mediant note of the home key provided the subject enters smoothly into or modulates to the dominant and is supplied with a tonal answer.
Concerning rhythm (note values), a good fugue subject could have uniform rhythm throughout, slight rhythmic changes, or several different rhythmic groupings, but the combination of outline and rhythm are what give to a melody its identity.
Rising and falling curves are found in endless variety in music, and rhythmic patterns have innumerable possibilities; with the 2 working together, the melodic interest culminates.
Therefore, IF we've got a fugue subject that's good in terms of length, tonality (key), rhythm, and melodic curve -- those qualities which lend it character and make it capable of contrapuntal elaboration -- and IF we know in the exposition where the answer should follow (See blog, Getting Started With Writing, Part XX, there are rules for this] -- and IF we've got a logical plan in mind for the voice combinations in the various entries ... THEN we've got a fugue.
Technically, a fugue doesn't HAVE to have a countersubject, but very many of them do; this is another moving line which, in combination with the subject, complements it tonally and rhythmically and makes a satisfying melody all by itself.
We prepare ourselves for this by studying some of the best examples that have been left to us, beginning with "the 48" (Well-Tempered Clavier, Parts I and II, of J.S. Bach), moving from there to any of his glorious organ fugues; it's also necessary to get fully informed about harmony, counterpoint, the rules of voice leading, and the general rules of fugue writing.  
The process of fugue writing for the organ successfully practiced by this author can be summarized in Ten Steps (photo) which, as they're carried out, may be permitted to overlap to some extent.
We need to get for ourselves a notebook PC and either download some music writing software to it (such as Finale or Sibelius) or sign up with an on line music writing application (such as Noteflight) if we haven't already, make sure we're set up to write with a blank score [See blog, Getting Started With Writing, Part V], click as many measures as we think we'll need, add an upper and lower voice to the upper staff, then we're ready to begin the exposition.
This is how it's done:
STEP ONE: No fugue worthy of the name was ever crafted, nor ever shall be, upon a poor subject; the first move we make in writing a fugue is to carefully craft a subject (a theme in the tonic, or home, key) with an eye to its range, starting note, melodic outline, and workability; its first few notes constitute its head, and its last few notes are its tail; for longer subjects the material between the head and tail is called the body of the subject; in general, the longer the subject is and the more varied it is rhythmically, the more its material can be worked later on, especially in the episodes.
The subject should not be too long nor too short, make a pleasing line when inverted, capable of contrapuntal elaboration and application of learned devices; if it is capable of a stretto (overlapping of entries before the first entry finishes) this is a plus, but not every good subject is capable of a stretto however.
One of the best ways to engineer a stretto into a subject is to shape it so that an implied tonic-dominant harmony alternates within every other bar (or half bar) of its outline; in other words, for longer subjects, in the 1st bar (or half bar) we can make the head of the subject strongly imply the tonic, then in the 2nd bar (or half bar) make dominant harmony prevail, then in the 3rd bar (or half bar) we can make the notes imply tonic harmony again, then in the 4th bar (or half bar) we can make the notes imply dominant harmony again, and so on.
This method lends itself to a possible stretto, which is nothing more than a strict imitation, or series of imitations, at a given distance from the starting note.
By the time we finish this crafting process we could be left with a subject which may on its face seem to be melodious enough but on the bland side; on the other hand, such a subject is usually very workable.
The subject for Bach's Art of Fugue [See blog, Getting Started With Writing, Part XIX] is short, little more than a bare outline of the tonic chord in slow notes, with a little tail in quicker notes attached; it's without color, without light, and anything but striking, but, just like the most common lump of clay, the composer knew that such material could be worked in every possible contrapuntal manner; the result, at his hand, was an hour and 10 minutes of some of the most beautifully intricate music ever written.
Some of the most striking fugue subjects can be formed by making it begin in slower note values or maybe a dotted rhythm, then have it pick up speed with shorter note values in its body and tail; once the shorter note values appear in the subject it's better to keep these shorter note values going so the moving line doesn't slow down and lose energy.
The answer is subject to certain rules [See blog, Getting Started With Writing, Part XX].
Once we've settled on a workable subject we need to examine its tessitura (range) and bear in mind the highest and lowest scale degrees of its melodic curve; we then compare that with the 4 normal voice ranges (soprano, alto, tenor, bass) and imagine placing the subject as close to the middle of these voice ranges as possible during the various entries in the exposition (usually the bass is last to enter, but not always, not 100 per cent of the time); once we determine what the best starting note would be, in order that the entering subject will not exceed any normal voice range as it plays out, then we'll have a good tonic note in view.
All that remains then is to decide major or minor to determine the key signature, we can mark our blank score accordingly, and then get busy filling in the subject in each voice where it is to enter.
We cannot simply pick a key signature out of thin air and begin writing the exposition without giving some thought to the subject's range and how it dovetails with the ordering of entries being planned for the exposition or we risk having to start all over again.
By following this method, since the normal voice ranges are fixed, the melodic curve of the subject and its range will suggest a good, if not THE best, tonic note for the piece, whether major or minor.
STEP TWO: is to pay close attention to the subject's starting note (usually scale degree 1, 5, or, less commonly, 3), repeat the subject in the dominant key a 5th higher or 4th lower, which is called the answer, decide on what type of answer (real or tonal) is needed, and determine what sort of contrapuntal line can be created which complements the answer harmonically and rhythmically; when this same contrapuntal line reappears with most, if not all, subsequent entries of the subject in the course of the fugue, it is called a countersubject.
This countersubject should make a good melody all by itself and generally move in contrary motion to the answer and in shorter note values whenever the notes in the answer are of longer value; it may (but doesn't have to) have slower notes against faster notes in the subject.
Ideally, when the answer and countersubject are put together, the result should be an animated and energized interaction in consonant implied harmony in mostly contrary or oblique motion and perpetual 8th note or 16th note motion (there are exceptions to this, for good and sufficient reason, such as when both subject and countersubject are moving briefly in quicker notes in similar motion, usually to create a sense of excitement, or if they happen to closely approach at a major second interval for a very brief moment).
If either of these 2 moving lines (subject and countersubject) can serve as a bass for the other, i.e. when they can be exchanged to different octaves and still make consonant harmony, then the music is being written in double (invertible) counterpoint.
These 2 lines need to be well thought out, but, we sometimes have to come back later on, after we've worked an entry or two, and "tweak" (change a note or two) something in the subject or countersubject(s).
Why? ... one of the principle rules of harmony forbids consecutive perfect 5ths, octaves, or unisons in similar motion, and sometimes we find them cropping up unintentionally right in the middle of an entry which wasn't noticed earlier; we're human -- things like this can be overlooked; this necessitates fixing that passage to eliminate the mistake and then going back through all preceding entries and making the same change in the subject or countersubject all the way through, for the sake of consistency.
Such a change is made by introducing a rest or moving a note to an adjacent note.
If in the construction of a certain entry we're finding that one of the lines cannot be squeezed into the space planned for it without voice lines crossing, then we put it in a different voice and move the displaced voice somewhere else; it doesn't mean that there's something wrong with our subject or countersubject(s), that they need to be recrafted or, worse, the entire piece abandoned -- it means nothing of the kind -- we just fix it and move on -- easy-peasy, neat 'n squeezy.
If the subject is non-modulating and has no large leaps, then most often it can be answered by an exact reproduction of itself, interval for interval -- this is called a real answer; if however the subject modulates to the dominant or has a large leap, in order to keep the answer from moving into a secondary dominant (the dominant of the dominant key) before the other voices have a chance to enter, something in the answer is changed tonally to either stay in the dominant momentarily or to point back to the tonic key -- and this is called a tonal answer.
The answer reverses the polarity of the subject; scale degree 1 in the subject is answered with 5, and scale degree 5 is answered with 1; tonal alterations in the answer are always minimal however, perhaps involving only a diatonic or chromatic change of a single note or maybe a shift of a note to an adjacent note and nothing more; it's just a tweaking, the whole purpose of which is to either a) remain in the tonic key briefly when the subject is repeated in the 2nd voice before moving fully into the dominant key or b) create a smooth return to the home tonality for the entrance of the 3rd voice -- both without changing the subject's basic shape or character.
When subjects start on scale degree 1, as many do, and are non-modulating, we think initially of supplying a real answer, with the other notes in the subject determining whether a tonal answer may be needed; a large leap -- a 4th or more -- for example, could imply a tonal answer.
When the subject starts on scale degree 5 we have a couple of choices: if the home key is d minor let's say, and we want the answer in the dominant (a minor), we can do a tonal answer by starting on D (scale degree 2) -- remember that 5 is always answered by 1 -- OR go with a real answer in the subdominant (g minor) which starts on D (scale degree 1) -- 5 still being answered by 1.
A famous example of the latter approach appears in Bach's d minor Fugue BWV 565 in which the subject starts on scale degree 5 with the answer starting on 1 and real but in the subdominant key; the same can be found in the b minor Fugue Op. 26, and many other examples can be cited.
This is why working from a notebook PC with online music writing software can be such a time saver; changes like this can be made quickly and easily without crossing things out or maybe trashing the entire copy and starting over.
If the subject starts on scale degree 3 the subject generally needs to conform to the dominant scale or modulate to the dominant at or near its tail before the answer enters; when the tail of the subject points strongly to the dominant key a tonal answer is needed, many times to adjust the tail of the answer to bend it back tonally to the home key to prepare the way for the subject to enter in the 3rd voice; if the answer is real, the music would escape into the dominant of the dominant before the 3rd and 4th voices have a chance to enter -- which is, tonally, not good to travel this far afield this early in the fugue.
The same applies to the (1st) countersubject; if its last note or two points to the dominant key, then at its first appearance with the answer a chromatic inflection will be needed in this note or two to prevent the music from escaping to the dominant of the dominant this early in the fugue.
TIP NUMBER ONE: AGAIN, PAY VERY GOOD HEED TO THE SUBJECT'S RANGE, i.e. the distance between its highest and lowest note; bear in mind that the subject has to appear in every entry, preferably in unaltered form, and this works best when its range is something less than an octave; when it's an octave or more it may be advantageous to provide at least one melodic leap of a 3rd, 4th or 5th in its outline which can be manipulated later on if necessary to break the outline and reposition a few notes to an adjacent octave, this to keep from exceeding normal voice ranges and prevent the moving lines from crossing; it's perfectly all right to do this with any of the moving lines if needed to get things to work.
TIP NUMBER TWO: NEVER INVERT THE ANSWER -- the subject needs time to establish itself right-side up at the outset, in all voices, and by introducing inversions too early this gets delayed; if the listener interprets the answer as something else getting started rather than recognizing it immediately as a repeat of the subject, the function of the answer is disturbed; if inverting the subject in the exposition is desired when there are 4 voices, it would be better to save this for a redundant (5th) entry once the 4th voice had had a chance to enter; the Op. 18 Fugue is an example where the exposition is supplied with a redundant (5th) entry in which all 4 moving lines are inverted [See blog, Getting Started With Writing, Part XIV].
TIP NUMBER THREE: if the plan is to place the answer in the home key, this is part of invention procedure, not fugal procedure, the subject is called a "motive" or "motif," and the 3rd entry also will be in the home key; in the 2nd section of the Op. 11 double fugue the 2nd subject deliberately and boldly enters in the first 3 voices in the home key of c minor this very way simply to do something different and unexpected; in this piece the 2nd subject is given its own exposition completely stripped of tonic-dominant tension before it enters in the 4th voice in the relative key [See blog, Getting Started With Writing, Part VI].
STEP THREE: is to write a 3rd line in the home key and get it to blend in vertical harmony with the subject and countersubject; if this 3rd line can be maintained throughout each successive entry of the subject, then it becomes a 2nd countersubject, the original one becomes the 1st countersubject; if any of these 3 moving lines can serve as a bass for the other 2, then the music is being worked in triple counterpoint.
As the voices of our fugue enter, one by one, it's easy to see that our mind can quickly get boggled trying to coordinate all this complexity unless we work from a predetermined written plan.
We can keep things straight here by working the whole exposition at the same time, like a "chunk" of a crossword puzzle, by first placing the subject in whatever voice we want it to enter in the exposition; we can then come back and enter the countersubject(s), one at a time.
After deciding on the order of voices in the exposition, the expected plan would be for each voice to commence immediately after the conclusion of its statement by its predecessor; but here we very often find that, instead of this arrangement, an additional passage (called a codetta, interlude, link, or bridge) is inserted.
This bridge could be merely one or more measures used as a connection, or it could be extended into a significant statement; this is very commonly encountered after the 2nd voice has given the answer, and its function is to bring about a smooth return to the home key for the 3rd entry which, in some cases, can form a commentary extending past the moment when the home tonality returns.
STEP FOUR: is to write the entrance of the subject again in the remaining 4th line of the exposition; if the fugue is in 4 voices, then this line will enter after the 3rd line while the other 3 lines go one to play the 1st and 2nd countersubjects and a free voice.
If this 4th free voice can be maintained throughout each successive entry, then it becomes a 3rd countersubject, and the music is being written in quadruple counterpoint.
It isn't always necessary to insert another codetta to link the 3rd and 4th entries, although composers sometimes do.
Don't be surprised if, while you're building the 3rd or 4th entries in the exposition, that you encounter something that isn't making good vertical harmony; this situation will crop up at times, and it may require you to go back and change something in the subject or 1st countersubject to get everything to sound in consonant harmony; it may even be necessary to completely rewrite one of the lines.
Whenever making these changes, the end product should be a moving line that makes a good melody all by itself, times four; this is the secret, if "secret" it may be called, to getting the entire exposition to suddenly sound well, as if it was a miracle.
If we're writing in triple or quadruple counterpoint, it will help to keep the range of each voice within the span of a 6th or 7th, if possible; the more narrow the range between highest and lowest note, the better chance we have of avoiding voice collisions and crossings when the lines are brought together in their various combinations.
Expecting (and having) to edit and adjust the exposition when constructing it, where it doesn't come out perfectly on the first try, does not mean the writing is defective; on the contrary, it simply goes with the territory, and ALL who write fugues are subject to it.
This exposition will then typically end with a full cadence in a related key, usually the relative.
One way to construct a smooth pedal line to transition from the 4th entry to the relative key is to have a look at what's going on in the bass a couple of measures before that, invert that figure, and finish it with an octave leap on the dominant note of the relative key using whatever accidentals are necessary to land on the new tonic.
At this point it helps, and is in fact necessary, to print a hard copy of this 4th entry in the exposition, where all 4 lines sound together; we will refer back to it when it comes time to enter the subject and countersubjects in the remaining entries and to figure the inverse of the subject and countersubjects for possible use in the episodes or coda.
NOTE: In a 4-voice (SATB) organ fugue typically the exposition does not conclude (and the first episode does not begin) until the pedal line drops out, at which time the 1st episode begins in 3-voice, or, far more commonly, 2-voice texture; in an organ fugue, after the 4th voice enters in the exposition, as long as the pedal line continues to and through the next entry, THAT additional entry is called a redundant (5th) entry and is considered part of the exposition; once a 4-voice exposition has concluded all subsequent entries typically continue in 4 voices, but there's no rule that says they HAVE to -- in other words, at the discretion of the composer (using Bach as the example), one of the moving lines could disappear at the next entry, creating a 3-voice entry; additionally, there is no rule which specifies the length of episodes between entries, but it's best to keep them at least as long as the entries themselves -- in other words, if the entries occupy 4 bars, the episodes between them should be at least 4 bars long so the listener's ear can more easily distinguish between them.
Whenever a redundant entry is super-added to the fugue's exposition, obviously one of the moving lines must be assigned to the bass voice; this will be the subject (Op 18, Op. 31 are examples), countersubject, or, in the case of multiple countersubjects, the 1st countersubject (Op. 33 is an example); any other countersubjects or a free voice typically are not assigned to the bass although, theoretically, there is no rule against it -- the whole purpose of a redundant entry is to bring out some aspect of the subject or countersubject, which can be spotlighted well when assigned to the bass.
STEP FIVE: is to work out on a separate sheet of paper how many additional entries there will be, in which keys, and the order of voice leading for each (the positions of the subject, countersubjects, and free voice) so that duplications might be strictly limited, if not avoided; this sheet of paper will be consulted frequently as work progresses [See menu bar, Biol/Free Scores, Fugue Writing Blank].
We do this by listing the entries by name (1st entry, 2nd entry, 3rd entry) in a vertical column on the left margin of the page, top to bottom, spaced a little apart; we then move to the right of each and list another vertical column for each with the letters "S, A, T, B" for the 4 voices, with each letter followed by a dash.
If we're writing in triple counterpoint we then abbreviate the subject (theme), countersubjects, and free voice with "Th, 1CS, 2CS, F" and add them after the dashes in the order we'd like to arrange them in the various entries.
If the free voice is maintained throughout, where it could also serve as a bass for the others, then we're writing in quadruple counterpoint with a 3rd countersubject which may be abbreviated "3CS."
This blank serves as a blueprint on paper of where we're wanting to go with this fugue of ours, and it will keep things straight as we proceed.
TIP NUMBER FOUR: AND DO NOT SKIP THIS STEP -- we need this sheet of paper in front of us to assemble the framework for the rest of the fugue, and without it we're almost certain to get bogged down in a sea of details.
As soon as we have this figured out we expand our working screen to accommodate the number of measures it will take, then mark off with double bar lines the various entries and episodes which separate them; this is so we can find them quickly while scrolling down (these double bar lines are changed back to single bar lines during the final editing).
Now then, by labeling these blank segments with the words "1st entry D Major, episode, 2nd entry b minor, episode, 3rd entry A Major, episode" etc., these segments may be easily located when scrolling up or down the screen; these labels are deleted upon final editing.
There's no rule which prohibits interpolating another bridge, more or less prolonged, in the exposition between the entrance of the 3rd and 4th voices (Op. 10 is an example);
There's no rule which prohibits super-adding a redundant entry in the exposition in which all moving lines are inverted (Op. 18 is an example);
There's no rule which prohibits super-adding a redundant entry in the exposition in the relative key (Op. 31 is an example);
There's no rule which prohibits super-adding a redundant entry in the exposition which modulate a whole tritone away from the home key (Op. 33 is an example);
There's no rule which prohibits building an entry on the inversion of the subject and countersubjects, simultaneously (Op. 13 is an example);
There's no rule which prohibits the exposition of a 2nd subject in a double fugue being constructed like an invention for the sake of variety once tonic-dominant tension has already been established in the exposition of the 1st subject (Op. 11 is an example);
There's no rule which prohibits starting the subject on scale degree 3 provided the tail of the subject points strongly to the dominant key (Op. 19, 27, 29, and 30 are examples);
There's no rule which prohibits building an exposition by means of these or any other bold moves, as long as they're made with good reason, the general rules of part writing are observed, and the result sounds smooth to the ear [See menu bar, Free Stuff]; doing what's unexpected like this changes the musical landscape from common to uncommon and is the way the boundaries of art have been extended and music has been permitted to grow and develop for thousands of years [See blog, Do The Unexpected, Poetic License].
From here on the remaining steps may overlap some, depending on what captures our working attention at the moment.
TIP NUMBER FIVE: Common practice manner distilled from the fugue-writing habits of many major composers over the course of several centuries is to STICK WITH RELATED KEYS, FOR THE MOST PART, IN SUBSEQUENT ENTRIES; this has been proven to create smooth modulations and flowing transitions from one key to the next which sound effortless; there's also no rule prohibiting the inclusion of an entry or two in an unrelated key if this can be done in a facile and uniformly refined way, but forcing every single entry to wander all over the Circle sounding in nothing but unrelated keys is of but limited value even as a technical discipline, as the result tends to sound incohesive and its composition will be found laborious and toilsome.
TIP NUMBER SIX: Whenever a voice enters on a note which has been heard immediately before in another voice, this note is considered as being in "common" and should be tied; so if, for example, when the subject appears in an inner voice (alto or tenor) and at one point leaves a note which is afterwards immediately taken by the other inner voice, a short straight line can be drawn in the score connecting the two note heads to help alert the performer, like a flag, that these notes are to be tied; DON'T shorten the first note to half its value, as this inserts a rest -- an unnecessary "hiccup" -- into the moving line when in organ playing the rule is to tie it.
STEP SIX: is to fill in the subject and countersubject lines in all the remaining entries, leaving the 4th free voice blank for now; this work proceeds quickly.
Before moving to the next step this would be the time to carefully check the score for voice crossings or consecutive 5ths or octaves between the moving lines in their various configurations and make the necessary corrections; this may require the tweaking of a note or two in the subject itself, which will then make it necessary to backtrack and do the same for each entry, to get the outline of the subject consistent.
When we take an entry into the minor key the lines always move with the melodic minor scale, i.e. the 6th scale degree is raised a half step when ascending and follows the key signature (of the passage) when descending; not observing this will result in some ugly harmonies on playback and serve as a reminder.
If it happens, according to the plan on paper, that 2 of the lines in a certain entry which start on the same scale degree will have to start on the very same note, then they cannot be adjacent voices (collisions and voice crossings will result); in this situation they will have to be separated so they start an octave apart, OR (more likely) all the lines in this entry will need to be reshuffled; while this is a relatively common occurrence, it is however easily fixed.
STEP SEVEN: if we're writing in triple counterpoint we now go back and fill in all the free (4th) voices in the various entries; we don't just pick notes out of thin air to make a 4-note chord with each moving note; instead we make this 4th line just as singable as the other 3 lines; this is the "secret," if there is one, to get the music to sound with an easy, flowing beauty.
If there isn't sufficient space at some point to squeeze another note between 2 others, we adopt one of the notes already on the page as the note for the free voice and resume the line a beat or 2 later when there's enough room; 2 different lines may share a unison on the same note momentarily, if necessary.
If we're writing in quadruple counterpoint we've already filled in the 3rd countersubjects in each entry, and this step will be skipped.
STEP EIGHT: is to create a stretto, if there is one, immediately following the last entry by starting the subject in one voice then starting it again in another before the first voice can finish; we start it again at the same distance so that the voices create some excitement as they accumulate and "pile up."
If we find, after we begin what looks like a promising stretto, that after the 3rd or 4th bar the harmony isn't working, we don't have to throw it away; we can still continue the passage in imitation which will lend a certain excitement leading into the coda (Op. 29 is an example).
There should be no falling off of energy level in this stretto section by allowing the polyphony to suddenly dwindle down to one voice; it's better, when the subject first enters in stretto, to have 2 more moving lines moving with it in trio texture; these 2 additional lines can be freely invented or countersubjects can be used, which we already know will blend well in combination; when the moment comes these countersubjects can then cease as they take over the subject.
The stretto, when present, is usually followed by the final entry and a coda employing something original; one way to avoid a mundane or cliche ending is by using the secondary theme or countersubjects as working material, which are unique for each work (this author's Op. 4, 7, and 31 are examples); at other times the inverse of the melodic curve of the head of the subject or simply its rhythm might suffice (Op. 22 and 23 are examples); in any case, the coda should connect in some way with what has come before it.
We have it on authority of the strongest fugue writer in history that a good fugue doesn't have to have a stretto -- some of the finest fugues of J.S. Bach have no stretto at all; if a subject doesn't lend itself to stretto, it doesn't mean that it's weak or defective by any means; we just skip this step and move on.
STEP NINE: is to connect all of these separate "chunks" of entries with short modulating episodes; these can be of any length, but 4 or 5 bars is sufficient; they do not each have to be of the same length as the entries, but it sounds smooth when they are, and episodes of uniform length throughout provides for a rapid and weighty development; this length should be considered minimum.
This step is like assembling a string of beads where the individual "beads" are these big entries; when all the beads are connected finally, the fugue is almost finished.
If the episodes are constructed upon fragments of the subject, countersubjects, and their inversions or retrogrades, the notes where the joining occurs on each end of the episode can and should be adjusted so the music sounds seamless.
How this is done: we invert, for example, either the subject or one of the countersubjects and start it on one of the inner voices (alto or bass) when the entry concludes; this will be typically on scale degree 1, 3, or 5; we select the line depending upon where it begins and line it up with the alto or tenor at the conclusion of the entry at hand.
After we have that line outlined in note heads in the alto or tenor, we then adjust the accidentals in the line to take it through a key or two as it modulates to the new key; it sounds seamless when we keep it in the same key for a couple of bars before we start adjusting the accidentals; the idea is to conclude the episode with an implied harmony in 2 voices that "fits" with the new key.
After this line is in place the other inner voice may then be invented to rhythmically complement this line and imply a harmony that's singable; once this is done our 2 voice episode is finished, and we should then listen to it and tweak it, if need be, to get it to sound smooth; by being based upon fragments or inversions of the subject or countersubjects this little episode will have a sense of familiar unfamiliarity and seem to blend right into the next entry.
It's important, during these episodes, to keep the melodic motion going; if one of the countersubjects is in longer note values, and it would cause the energy level to drop if used in a 2 voice texture, then it would be better to use some other line instead.
STEP TEN: now it's time to proofread our work, check it for errors in notation or composition (stems on the top staff up or down, voice lines crossing, notes failing to resolve correctly, consecutive fifths, octaves, or unisons moving in similar motion, awkward leaps in a melodic line, misplaced or missing accidentals or rests, etc.) and make the necessary additions or corrections [See blog, Making Mistakes, A Smooth Sea].
Now is when we add our metronome and tempo marks, dynamic marks, hand divisions, and registration suggestions or changes, if any; here is where we convert our double bar lines to single bar lines and erase any labeling inserted earlier to keep things separated and easier to find on a screen.
This is where we go back and spruce things up, put our work under a microscope, get things as animated as we can and as smooth as we can, so that when we're done we can't imagine a single note following another any other way.
When we can say that, when our score is cleaned up and professional looking, our fugue is finished.
We have a look at it now, and if we see something in it that can be worked as a prelude or other type of free work to be paired with it we create some blank space in the score anterior to it and feel free to have at it again.
Sometimes an idea or two for a prelude occurs to us in the middle of writing our fugue and can/should be notated at that time; it's perfectly okay to postpone work on the fugue today in order to give its matching prelude some attention -- we can always resume work on the fugue tomorrow.
It's better to get the big picture first, to map out the prelude's main sections, theme(s), key changes if any, and creating the framework for the piece, before filling in its smaller details.
By following this process of 10 steps, and without hurrying, a well built fugue can be completed from start to finish in a surprisingly short time; it has been battle tested by this author in the creation of many fugues and is very solid; it makes our time productive over a period of a few days, but it's important to remember not to fly through the work so quickly with such an overambitious timetable and unrealistic expectations that we start flaming out or missing some obvious mistakes.
The best results come by working our projects a little bit at a time, chipping away some of it every day like a sculptor frees the subject slowly from a solid block of marble.
If we aren't careful, we can overdo it; we can get so wrapped up in what we're doing that time gets away on us and our energies get completely spent; we can wear ourselves out -- unless we pace ourselves.
We should stop at the first sign of physical or mental fatigue; as we make progress we learn to develop the ability to recognize that point when it arrives, when the mind starts to get numb, and listen to what our body is telling us, that it's time to call a halt for today.
Composers know to arm themselves with patience; they know, with every day's work, when to say "when," that it isn't a sprint, that it isn't the 60-yard dash.
It's a marathon.

ONE FINAL NOTE:  Every century from the Middle Ages to the 20th century has had different definitions of "good" counterpoint.  As we wade into fugue writing, people may criticize our first few attempts because we're not doing what they do, but it's better for us to keep being ourselves.  We're all born unique, and the world is better off when humanity is allowed to express its natural diversity.  A lot of the rhetoric is more a sign of ego than of knowledge, and we could all use a bit more humility.  While counterpoint is the arterial life's blood of music, and everyone will have an opinion about whether or not our fugue writing "sounds good," just how well the aesthetic standards of 17th and 18th century Europe should judge music written by people who aren't living in 17th and 18th century Europe is also a fair question.  All of the music theory and counterpoint and study of harmony in the world are simply tools in the box, and the true purpose is ultimately our art and our choice for expressing our genuine selves.  The rest is simply refining techniques for effectively conveying our intent to performers and audiences.

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