1. Mr. Big Noodles

    Mr. Big Noodles Theory God

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    There's not much about fugal writing or analysis on this forum. Let's change that, shall we? I'll start with a quick overview of what a fugue is, talk about form in 18th century fugue, then examine a fugal exposition.

    We will cover the following:
    1. Definition of fugue
    2. A very quick rundown of the differences between Renaissance, Baroque, and 20th century fugue
    3. Structural elements of a fugal exposition
    • Subject
    • (Real) Answer
    • Countersubject
    • Link
    • Bridge
    "Fugue," from Latin "fuga," means "to flee." We get the word "fugitive" from the same root. Fugue involves multiple voices chasing each other by using imitative counterpoint. The conception and use of fugue has changed several times throughout history. When most folks say "fugue," they are typically referring to the high Baroque style of fugue typified by J.S. Bach.

    Fugue began in the 16th century as the ricercar (Latin for "to seek," where we get the word "research" from), which involved many themes, had non-imitative sections, and stuck to a small number of tonal areas (entrances only on the tonic and dominant). During the 17th century, the number of themes in fugal writing decreased dramatically until it was winnowed down to a single theme, which we call the subject. Composers preferred to make the entire thing imitative, and gradually worked into other keys. By Bach's time, fugue was the most highly developed process of imitative counterpoint. In the 20th century, fugue has a number of treatments but tends to have a more expansive harmonic scope and may involve any number of techniques from different idioms.

    Fugues is not a form, but rather a technique or a process. That said, 18th century fugues tend to based on the pattern of an exposition section followed by a body consisting of several episodes and entries. For now, let's focus on an exposition. Fugues have a fixed number of voices. The most common numbers are 3 and 4 voices. Here is what the beginning of a three-voice fugue looks like:

    [​IMG]

    I have the subject beginning in voice 2, but the voices can enter in any order. Focus on the subject and answer right now. The answer is the same thing as the subject, just transposed. Every voice will either enter with a subject or an answer. As soon as the last subject/answer is finished, the exposition is over and the body begins.

    Let's look at the second fugue from Bach's Well-Tempered Clavier, Book 1. The fugue starts at 2:08 in this video:



    [​IMG]

    This is the exposition section with only the subject and answer. In general, subjects must either start on scale degree 1 or 5. If the subject starts on 1, the answer will start on 5. If the subject starts on 5, the answer will start on 1. This fugue is in C minor, so you can see that the subject starts on 1. Once the subject is done, the answer comes in in the upper voice. The answer is literally the same exact thing as the subject, but transposed up a perfect fifth. Because all the intervals are the same, this makes it a real answer. (We will cover tonal answers at another time.) There is a bit of space between the first two entries and the third voice in measure 7. Voice 3 has the subject, which is again on scale degree 1. As soon as voice 3's subject is done, that's the end of the exposition.

    If this were a four-voice fugue, then we would expect another voice to come in with the answer. As soon as the answer finishes, it would be the end of the exposition.

    Now that we have the basic architecture of the exposition, let's fill in the blanks. The first bit I want to talk about is the countersubject. Technically there are rules to writing a countersubject, such as it has to be invertible counterpoint and be used for motivic material elsewhere in the fugue, but I see "countersubject" being used to mean the thing that happens in the first voice when the second voice comes in with the answer. If you write free counterpoint to the answer, you'll be alright. This stuff can get much more technical, but I like to show that fugue is not some big, insurmountable thing that you need to be a genius to write.

    [​IMG]

    In the table I provided, I say that there is something called a link between the subject and answer. A link is an optional bit of music that helps to connect the subject to the answer, because they don't always fit flush. There are two interpretations you can make with this fugue in particular. The first is that this fugue does not have a link and that as soon as the subject is done, it moves on to the countersubject. The other interpretation is that the C in the alto voice right before the soprano's entrance is the link. Yes, a single note. That's how links work.

    The bridge is similar to the link, but it tends to be more extensive. Why? When you compose the answer, you typically move to another key. The next entry is going to be in the old key though, so you have to modulate back. Bridges are usually a couple measures long.

    When the third voice enters with the subject, the other two voices should keep going with free counterpoint or something that is motivically related to the subject or countersubject.
     
  2. Mr. Big Noodles

    Mr. Big Noodles Theory God

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    Let's talk a little bit about writing a subject. The subject is going to be harmonized with other contrapuntal lines, so we can help ourselves by building a harmonic progression into the subject. Using common practice standards, this means that you should ideally have a mixture of tonic, predominant, and dominant. If you don't follow this pattern of progression, then the result might sound deflated. To review, this is the flow of scale degree functions in the major mode:

    [​IMG]

    These numerals correspond to scale degrees (for example, in C major, 1 = C, 2 = D, 3 = E, 4 = F, and so on until 7 = B). Roman numerals indicate chords built on those scale degrees, and the letter case indicates chord quality: capital = major, lowercase = minor, lowercase with ° = diminished. In C major:

    I = C
    ii = Dm
    iii = Em
    IV = F
    V = G
    vi = Am
    vii° = Bdim​

    Tonic functions in this system are basically I (with vi being a prolongation), predominant includes IV and ii, and dominant is V and vii°. iii is rare in this style, though it does pop up in falling fifth progressions like IV vii° iii vi ii V I.

    The minor mode requires a bit of tweaking, but is mostly the same:

    [​IMG]

    (In C: )
    i = Cm
    ii° = Ddim
    III = E♭
    iv = Fm
    V = G (Altered via harmonic minor)
    VI = A♭
    ♭VII = B♭
    vii° = Bdim (Altered via harmonic minor)​

    If anyone needs me to review how this works, I'll be happy to. For the moment, I'd like to talk about fugue.

    Returning to Bach's subject, I hear the following harmonization (which I derive from the pitches and behavior of the melody):

    [​IMG]
    Audio here, slow enough that you can hopefully hear the harmony.

    I perceive a harmonic rhythm of two beats, with a little pickup leading into each downbeat. So at a background level, the subject can be harmonized as i iv i6 ii° V7 i, but it makes sense to slip in a chord on the eighth note before those downbeats, making the entire thing i i6 iv V i6 ii°6 ii° I6/4 V7 i. The way that I harmonized this is very mindful of counterpoint and voice leading (particularly in the second half of measure 2), but all I am demonstrating is that there is a chord progression hiding within this subject.

    The other thing you want to do is have a bit of motivic variety in your subject. As I mentioned in the first post, 18th century fugue deals with a single subject (unless you have a double fugue, but let's get to that later). There is not a lot of excess material. With that in mind, you want your subject to have some variety so the rest of the piece does not sound bland and so that you aren't tempted to introduce new materials willy-nilly. Motives have a pitch component, a contour component, but also a rhythmic component. You can see that the subject we are examining has a pattern of two 16th notes followed by an 8th, and also a pattern of two 8th notes (which I accent in my harmonization example), and that makes up most of the subject. At the very end, it breaks up a little with two 16ths going to a quarter note, and then two 16ths going back down. I have broken the subject up into three distinct motives:

    [​IMG]

    Let's look at the characteristics of these motives and how Bach manipulates them to get the most mileage out of each one.

    Motive 1 is a neighbor tone with the rhythm ♬♪. It stays the same throughout the subject, but the rhythm is later used to form motive 3. The reason one would transform a motive is to make a piece coherent while producing variety. This is an important point, so I want to draw attention to it. In all sorts of composition, but fugal writing especially, you want to have a good balance of coherence and variety. If your variety comes from coherence, then the sky is the limit.

    Motive 2 uses the rhythm ♪♪. The rhythmic declamation overrides the pitch content, which we see is very flexible: the first instance of motive 2 is an ascending half-step from G to A♭. The second instance is a descending perfect fifth from D to G; the contour is inverted (abbreviated "inv." in my example), and the interval is expanded ("int. expansion"), which, of course, is a technique that produces variety. Furthermore, it is transposed to start at a higher pitch level than the first one. The third instance uses the inverted form once again, and expands the interval even further to a descending major sixth from D to F. I have used a dotted box to show that motive 2 overlaps with motive 3 here. One interesting thing about this overlap is that it summarizes the history of motive 2 thus far. These are the pitches of motive 2 in sequence: G A♭, D G, D F. Here's motive 3: F G A♭.

    Motive 3 uses the rhythm followed by a longer note value (or a perceived longer note value). That eighth note at the end is actually a 16th note in the score, but I think we hear it as the end of a line and therefore a place of rest. I mentioned before that it comes out of the rhythm of motive 1, but it introduces something that we haven't had before: scalar motion. It also contains a bit of syncopation, which helps with rhythmic variety. The second instance of motive 3 is inverted; it's the three-note scalar figure going down instead of up. It retraces over the steps of the first instance of motive 3, but goes one further to resolve to a tone from the tonic triad: E♭.

    The last thing I want to say about the subject at this time is that you should shoot for subjects that outline good intervals. This one begins at scale degree 1 and ends at ♭3. This is a good place to end, because any note of the tonic harmony will sound good against it. You probably wouldn't want to end on scales degrees 4 or 6, because those are lousy notes in terms of harmonic progression: they suggest IV or possibly vi (iv or VI in minor), which does not make sense in terms of phrasing. 1 might be okay but could be boring, 5 is good but probably better if you start your subject on 5 (so the answer comes in on 1, implying tonic harmony). 7 and 2 are possibilities if you start your subject on 1, but you should only do 2 if the answer is coming below the subject so that you don't have an unprepared fourth. Rules, rules, rules. But it sounds good if you follow the rules.
     
    Last edited: Jul 1, 2017
  3. Mr. Big Noodles

    Mr. Big Noodles Theory God

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    I'd like to show you something interesting you can do with the information we have covered so far. When you have fugue-like material in an otherwise non-fugal piece, it's called fugato. Let's look at an example from Cécile Chaminade's piano sonata, Op. 21.

    The first movement is in sonata form, meaning that it has three main sections: Exposition, Development, and Recapitulation. The exposition in sonata form is different from that of a fugue. Basically, it contains two themes and some transitional materials. This is what sonata form looks like:

    [​IMG]

    P = Primary Theme
    TR = Transition
    S = Secondary Theme
    C = Closing Theme​

    The typical major mode sonata form starts in the tonic key (I), modulates to the dominant key (V) for the secondary theme, and recapitulates with the secondary theme in the tonic key (I). Minor mode sonatas are similar, with the primary theme in the tonic key (i), but the secondary theme in the exposition is typically in the relative major key (III). These key relationships are flexible, especially after the mid-19th century, but the keys in that chart should be considered normative. The development section is where material from the exposition is fleshed out and explored, and modulation is used to explore the tonality. The development eventually leads to a dominant chord the home key, preparing for the recapitulation.

    Looking at the exposition section, Chaminade uses a fugal exposition for the transition portion of the exposition. The fugato begins at 1:03 in the following video, but I encourage you to listen through the beginning.



    This is the subject:

    [​IMG]

    The fugato begins in C minor (starting on scale degree 5). The answer (real) at 1:15 is in G minor. There is no bridge, though the last four notes of the subject are arguably a link. The subject comes in again in C minor, but the last measure is cut off to make way for a modulation to E♭ major (the relative major of C minor). When the transition ends, the secondary theme begins (1:43), which you should recognize as being derived from motives from the fugato subject. It's pretty cool, because the secondary theme is developed before it ever begins.

    I feel that fugato sections make a work sound more expansive, because it's a structure within a structure. You can use fugato to make strong and poignant contrasts within a work, calling listeners to attention.
     
  4. Mr. Big Noodles

    Mr. Big Noodles Theory God

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    Imitative counterpoint requires training and is difficult to write without technical knowledge and historical awareness, so most rock and metal musicians never try it. There are some neoclassical metal musicians who run into counterpoint because they are borrowing from Bach and Vivaldi, but whether they pick up imitative counterpoint from those sources is dubious. Yngwie Malmsteen has this track on Unleash The Fury titled "Fughetta". It's a competent two-voice fugue (albeit produced with awful compression), which is kind of embarrassing because he is quoting Bach's "Little Fugue in G Minor," an extremely well-known four-voice fugue.

    (Yngwie also has a tune called "Fugue" from that thing he did with orchestra. It is not a fugue, but rather a mostly homophonic rondo. It has a bit of free counterpoint going on every now and then, but nothing of the order we are talking about in this thread.)

    I have heard throughout my guitar-playing years that metal players stick with power chords because distortion makes any chord that is not a root plus a fifth muddy. (Yet how many black metal bands get their sound from tremolo-picked minor triads?) I have heard the same argument applied to counterpoint. I consider that view bullpucky, because there are many examples to contradict this claim. Here's some YouTube rando's metal rendition of the Little Fugue in G Minor with the voices Yngwie left out:



    He's using some synth, so maybe it's cheating, but the guitar is hardly muddy there.

    Connor Gallagher did some Shostakovich arrangements for metal instrumentation:

    String Quartet No.8, II


    Yeah, that one is pretty muddy, but the mix also leaves a lot to be desired. That's fixable, and it's not any worse off than a lot of metal albums that have gone into press. Dynamics do exist, as Paul Gilbert occasionally proves.

    You could replicate the effects of this rock band arrangement of Fugue 4 from Hindemith's Ludus Tonalis by sticking two ice picks in your ears, but the counterpoint is totally clear:



    Let's move on from appropriations of classical music though. Clearly, some metal players find expressive potential in imitative counterpoint. Emperor's "The Eruption" begins not with a fugue but with a canon at the octave. Canon can be easier to understand and write than fugue, and it has some of the same features. There is also a bit of guitar counterpoint at 3:15, then again toward the end of the song.



    The counterpoint isn't imitative or even strict in Animals As Leaders' "Kascade," but there are three melodic ideas happening at once at 1:51. It sounds crisp, clear, and awesome.



    There is room for good counterpoint in our genre. I hope somebody feels inspired and explores some of the possibilities I have spoken about in this thread.
     
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  5. Mr. Big Noodles

    Mr. Big Noodles Theory God

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    Let's talk about techniques for writing "filler" material. I do not mean to imply that this material is any less important for a fugue. To the contrary, it is often essential to break away from the subject in a fugue. I will be detailing some common devices used in transitional sections.

    Turning back to the Bach fugue we started with (WTC, Book 1, No.2), let's examine the bridge between the answer and the third voice's entrance. We will be looking at the part in the red box, but I also want to talk about how it connects to the next bit.

    [​IMG]

    I have extracted the top voice from measures 5-9 and tabbed it out so youse guys can play around with it and see what's going on.

    [​IMG]

    The harmonic analysis shows that there is some modulation. The passage begins in G minor, modulates to C minor, has a couple chords in B♭, then continues in C minor. Remember that the subject enters in the bass voice in measure 7, so it's necessary to get back to C minor by that point. What may not be so obvious from a harmonic analysis is the design of the melody. I have highlighted a pattern in yellow. If you play those notes, you'll notice that the pattern is a neighbor tone figure that travels up the scale, starting at E♭, then going to F, and finally on G. This is called a sequence. A sequence is generally any melodic pattern that repeats at a different pitch level a few times. Sequences usually come in threes.

    I have included the part that accompanies the subject (mm.7-9) because the melodic resolution happens on the downbeat of bar 9. The bridge is not separate from the other materials, but is intertwined to maintain continuity and create forward motion.

    Let's look at the second voice now. This is from the same measures, 5-9. This voice also has a sequential pattern made from a passing tone figure.

    [​IMG]

    Notice that this one also travels up the scale: it starts on C, then on D, then finally on E♭.

    Taken together, the sequence in the first and second voices form what is called a "linear intervallic pattern" or LIP. This is a fancy way of saying that there is a sequence involving two voices. In this case, it is a pretty straightforward LIP: a rising scalar figure harmonized in tenths. We would call this a 10-10 LIP. (A tenth is the same as a third, just displaced an octave higher.) This is a reduction of both voices:

    [​IMG]

    The passage obviously has more than six notes, so I think it is worth mentioning that the other notes are involved in the LIP too. The formatting is all screwed up from the switch to XenForo, but my thread on Schenkerian analysis might help in understanding some of the analytic notation here. If you find that reading too dense, then hopefully you can hear the relationship between the original music (bars 5-6) and this reduction.

    [​IMG]

    This expanded LIP is a 10-3-3 pattern. Listen to the qualitative difference between the tenths and the thirds. Also check out the pattern that Bach has embedded into the thirds: the top voice goes G-A♭-A-B♭-B-C, a chromatic ascent from dominant (G) to tonic (C).

    LIPs are the bread and butter of writing episodes as well. When I get a moment, I'll show how sequences are used to write episodic material in this fugue.
     
    Last edited: Jul 12, 2017
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  6. chopeth

    chopeth SS.org Regular

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    Excellent, I wish I could understand everything posted first and then apply it to my music. Thanks, Schecterwhore! ;)
     
  7. Mr. Big Noodles

    Mr. Big Noodles Theory God

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    Here's a less technical explanation for writing an exposition:
    1. Guitar 1 - Make up a single-line melody, around 2-4 bars. (Subject)
    2. Guitar 2 - Play that melody in exactly the same way, but a perfect fifth higher (up 7 frets, or up a string and up 2 frets) or a perfect fourth lower (down a string, or down 5 frets). (Answer)
    3. Guitar 1 - Write a melody that sounds good against what you did in step 2. (Countersubject)
    4. Guitar 3 - Play the same thing you played in step 1, but an octave higher or lower. (Subject)
    5. Guitar 1 & 2 - Write something that sounds good against what you did in step 4.
    Just doing steps 1, 2 and 4 will put you well on your way without too much pain (3 and 5 are much more difficult), but there is no magical formula. You have to hammer away at it until you have something satisfactory. I think the results are worth the effort though. Composing a really rigorous piece of music is a good way to discover new things about yourself.
     
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