II. Counterpoint and Galant Schemas
30
Mark Gotham
Key Takeaways
- The fugue is a contrapuntal genre popular since at least the 18th century.
- This chapter sets out some key principles of the opening exposition section.
- We introduce new terms including these (click on the term for the glossary entry):
- Subject
- Answer
- Countersubject
- Exposition
- We also meet some fugue-specific variants of familiars terms, notably:
- Voices
- SATB
We now move on from imitative practices common in the 16th century to the fugue, which may be thought of as the 18th-century equivalent and successor to this tradition. Fugues can vary quite a bit, and thus “fugue” is a surprisingly difficult term to define satisfactorily. Fugues in general are contrapuntal compositions that are defined by the strict use of a certain number of independent voices; each voice enters one by one, stating the main theme (subject) of the fugue. In this chapter, we will treat fugue as a process and show how to construct (the first part of) a fugue from first principles. This chapter deals with the basics of writing a fugal exposition from a given subject according to the “high Baroque” technique of Bach, Handel, and their contemporaries.
Basic Definitions
First, let’s define some of the terms we’ll need.
- Subject: A short tune that forms the melodic basis of a fugue, recurring throughout.
- Answer: A transposition of the subject that is also sometimes slightly altered, as discussed below.
- Countersubject: A melodic line that is sounded with (and complements) the subject/answer.
- Exposition: The first part of a fugue, during which each of the voices enters with the subject or answer. Note that this is distinct from the use of the term “exposition” in sonata form; they both refer to a first section setting out the main material, but they otherwise differ.
- Voices: The separate contrapuntal lines, which may be either instrumental or vocal. The number of voices does not necessarily correspond to the number of parts or musicians involved—a keyboard fugue for one player may have three, four, five, or six voices, and there are even some fugues for solo violin (which is quite a compositional feat!).
- SATB: Although a fugue may not involve any singers, it is customary to refer to the lines in a four-voice texture with S (soprano), A (alto), T (tenor), and B (bass) from highest to lowest. The ranges do not need to match those of the vocal parts exactly.
Structure / Voice Entries
Basically, the fugal exposition works as shown in
: one voice begins with the subject, then the next voices enters with the answer while the first continues with a countersubject, and so on.| Voice 1 | Subject | Countersubject | Free counterpoint |
| Voice 2 | Answer | Countersubject | |
| Voice 3 | Subject |
Note that the voices are numbered according to the order in which they enter. While it is perfectly common for the voices to enter in order from highest to lowest or lowest to highest, this is not strictly necessary. However, when a new part enters, it should be the highest or lowest voice at that stage, so it is uncommon for an inner part (alto or tenor) to enter last. That leaves us plenty of options. For instance, the order of voice entries in a four-voice fugue could be:
- In register order from highest to lowest (SATB), or vice versa (BTAS)
- Starting in the middle with the alto: ASTB, ATSB, ATBS
- Starting in the middle with the tenor: TBAS, TABS, TASB
Any further subjects/answers entering (beyond the total number of voices) are described as “redundant” entries.
Subject
When considering how to handle a subject, look at its structure and character.
The structure of a subject can often be thought of in three parts. Like many tonal phrases, the subject charts a course from a distinctive opening to a generic cadence:
- Kopfmotiv (literally “head motive”): an opening gesture
- Fortspinnung: prolongation that may include sequence and/or motor rhythm
- Cadence
Types of character include:
- Toccata style: this has a motor-rhythm feel, is often chromatic, and may have irregular rhythmic placement.
- Ricercar: deriving from the plainsong/motet tradition, this is slow moving, has an antiquated feel, and makes extensive use of suspensions.
Answer
We distinguish between two types of answer: real for an exact transposition, and tonal for one that has been altered further.
A real answer simply transposes the original subject by perfect fifth/fourth. All generic intervallic relationships remain the same between the notes.
We need a tonal answer if the subject:
- starts on sol [latex](\hat5)[/latex] or otherwise uses it prominently at the outset. In this case, we must adjust the transposed version such that sol [latex](\hat5)[/latex] is not transposed up to re [latex](\hat2)[/latex], but instead to do [latex](\hat1)[/latex].
- includes any other V–I suggestion at start. Prominent dominant note in subject becomes tonic note in answer, so T → D progressions often become D → T and vice versa.
- modulates. In this case, split the subject into two phrases, one in each key, and adapt accordingly. If the subject modulates at its end to the dominant, the answer must modulate back to the tonic.
In all cases, we seek to make minimal adjustment such that the subject and answer are still as alike as possible, and both make melodic sense.
Countersubject
Try working up a skeletal version of your countersubject first before fully fleshing it out. It’s frustrating to dash off a lovely tune and then realize that it doesn’t fit. Additionally, sketch the version that will go with the answer at the same time as the version that goes with the subject. It’s just as frustrating to write a countersubject that works beautifully with the subject but which the answer makes harmonically nonsensical.
Indeed, countersubjects are melodies, but they only work if they make sense harmonically as well. To that effect, remember that your countersubject will first appear in a two-voice passage, so consider how best to outline clear larger harmonies (triads and sevenths) with only two voices.
You may wish to set yourself the additional constraint of a countersubject which is invertible at the octave—basically, one that can appear above or below the subject and still work. In this case, again work on both versions as you go, and specifically make sure to:
- Avoid fifths, because fifths invert to fourths, which have special rules in two-voice writing. Stick to unisons (/ octaves), thirds (/ sixths), and in the case of seventh chords, also seconds (/ sevenths). The fifth is not a problem in the free part, so “reserve” that note as the completion of the chord and use it there.
- Similarly, avoid 4–3 suspensions, because the first note is not dissonant when inverted (5–6). Instead, use 7–6 and 2–3 suspensions (which invert to each other).
To create your skeletal countersubject, first look for skeletal patterns in the subject. Find the simple, unembellished form of the subject and treat it as a kind of tonal cantus firmus against which to write your counterpoint (skeletal countersubject), which you can then embellish into a more interesting musical line.
When it comes to embellishing, try to:
- be true to the character of the subject when composing the countersubject
- establish complementary rhythmic motion such that one voices moves while the other is stationary
- make changes to the countersubject when moving between subjects and answers only and exactly where those changes appear in the subject/answer
Free Counterpoint
Frankly, the idea of “free” counterpoint is a bit of a misnomer, especially if you’re writing an invertible countersubject, because the subject and countersubject will leave your options highly constrained. This part is “free” in the sense that it doesn’t necessarily recur later on, so you might like to think of it as a freer part, relative to the even less free subject and countersubject! Alternatively, you can embrace the extreme order and replace the free counterpoint with a second regular countersubject that does recur later.
Links
Your fugal exposition may include short interpolations between the subject/answer entries. We call these links. Links frequently make use of a motive from the subject, for instance with sequential repetition of a short fragment. They may appear between any or all voice entries and may be of varying lengths.
There are many musical motivations for including one or more links. Links provide an opportunity to:
- Change harmony: The end of one subject/answer and the beginning of the next may not match up harmonically. In that case, you can use the link to get where you need to be.
- Change register: If you have a continually descending line, for instance, links can be a useful way of resetting the register so you don’t go outside the instrument’s range.
- Vary the phrase lengths: If you have a subject of exactly two measures and just proceed through several voices entering successively, then you may well want to use a link to make the hypermetrical and phrase groupings a little more interesting. That said, the voices should always enter on metrically comparable positions: for example, if your subject comes in on a pickup, all the subject/answer entries should do the same, at least in the exposition.
- Dovetail entries: Apart from varying the phrase lengths, we can also vary the melodic context in which a voice enters. For instance, if a link develops the Kopfmotiv sequentially, then you can bring in the next subject/answer in seamlessly, as it begins with that motive.
General Matters
Apart from the specific considerations of the fugue, naturally much of the general practice of writing (tonal) music applies here. Remember in particular to:
- Control rhythmic flow: If and when continuous use of a given metrical level has built up (e.g., quarter/eighth/sixteenth notes), consider carefully whether and when it is appropriate to discontinue that motion.
- Graduate the relative strength of cadences, for instance by controlling the scale degrees in the top and bottom voices.
- Write idiomatically for the instrument/s. Remember to observe practical limitations of hand span, instrument range, and the like.
Example
For an example of all of this in action, let’s consider
: the C minor fugue from Book I of Bach’s Well-Tempered Clavier (BWV 847).https://musescore.com/user/32728834/scores/8534750/embed
Note that:
- The exposition runs from from measures 1–8 (the extract here includes the start of the following “episode”).
- All the subject and answer entries are highlighted with blue noteheads.
- There are two regular countersubjects (annotated as CS1 and CS2).
- The answer is tonal.
- The link between entries 2 and 3 (mm. 5–6) and the episode following the last voice to enter (starting in m. 9) are built on sequential handling of a motive from the subject, as indicated with brackets.
Head to dezrann.net for a full version of this Bach fugue, complete with on-score annotations and an aligned audio recording of a real performance. You can even choose between using this friendly version in modern notation with the fugal voices on separate staves, or following along with Bach’s manuscript.
- In the Bach C minor fugue shown above, the answer is “tonal.” Given what has been said about tonal answers in this chapter, try explaining how the subject and answer differ, and why.
- Pick another fugue, identify how many “voices” there are, and locate each voice’s entry and the end of the exposition.
- Try writing your own answers and countersubjects. The template file below provides the subjects for all 48 fugues in both books of the Well-Tempered Clavier. An empty second staff is provided for your practice. (Note that the last note in many of the fugue subjects is given without a stem to indicate the pitch without specifying the duration rhythm.) If you know some of these fugues well, you may want to work on ones you’re less familiar with.
- Once you are feeling confident with writing answers and countersubjects, try writing up a full exposition. Again, you may wish to work on fugues you’re less familiar with.
For 3 and 4, you may wish to compare your solutions to Bach’s. Note that Bach’s solutions are not the only possible ones, so don’t expect to come up with exactly the same music. Note too that not every fugue has a regular invertible countersubject. Here are direct downloads in .mscz and .pdf formats.
Key Takeaways
- This is an anthology of a different kind and at a larger scale than the usual provision. Instead of showcasing a few, select examples (as the textbook chapters proper do), the goal here is to provide long lists of cases that at least one analyst sees in terms of the chord under discussion.
- Harmonic analysis is a reductive and subjective task and you should fully expect to disagree with some of the entries included here.
- The idea is to provide minimal curation, allowing you to roam freely among potentially relevant cases from across a broad repertoire, making up your own mind about what counts as an 'real' example.
Throughout this textbook, we have provided short examples of the musical subjects under discussion such as a specific chord or progression. These examples have included both simple, ‘prototypical’ versions to clarify how the idea works in principle, and also moments from real pieces in the repertoire.
Inevitably, these chapters only have space for a few such examples, so this final, ‘anthology’ section seeks to provide many more instances, enabling users to see a wider range of cases and with full context. No textbook or anthology can hope to capture the full range of ways in which these chords are used. Indeed, it’s not always clear whether a moment constitutes a ‘real’ example of the chord at hand. Hopefully, this at least provides space to roam and explore those 'edge cases'.
This first instalment of that vision focuses primarily on harmonic matters and on a corpus of nineteenth century songs encoded as part of the OpenScore Lieder Corpus which releases its transcriptions under the CC0 licence meaning that they can be used for any purpose whatsoever without restriction.[1] The tables below list moments identified as relevant by human analysts. So while much of the grunt work of collating lists and retrieving examples etc. has been automated, the analysis itself has not.[2]
Each of the tables below gives the:
- song's metadata: composer, collection title, song name,
- measure number and Roman numeral (figure and key) for the moment in question.
- URL link through to check out (play, download, etc) the score online.
This page currently includes lists for:
- Augmented Sixth chords
- Augmented Triads
- Common Tone Diminished Sevenths
- Modal Mixture
- Neapolitan Sixth Chords
Please get in touch if you would like to see other chords or progressions represented here. For those interested in the computational side, all source material (scores, analyses, code, etc.) is available here.
Augmented Sixth chords
Click here for this textbook's chapter on this topic.
[table “32” not found /]
Augmented Triads
Click here for this textbook's chapter on this topic.
[table “34” not found /]
Common Tone Diminished Sevenths
Click here for this textbook's chapter on this topic.
[table “85” not found /]
Modal Mixture
Click here for this textbook's chapter on this topic.
This table provides examples of modal mixture which is much more difficult define robustly than is the case for the Neapolitan sixth chords, for example. The list here provides examples where:
- Clearly mixed tone counts towards mixture and clearly non-mixed tone counts against it.
- E.g., F-Ab-C-E in C major has both a clearly mixed tone (Ab, the minor sixth) and a clearly non-mixed tone (E, the major third).
- Clearly clearly non-mixed tones do not necessarily preclude mixture if there are also strongly mixed tones.
- Mixture does not necessarily need to work in relation to all forms of the minor mode.
- "Chromatic" notes (that are not diatonic to the primary mode, shared between major and minor, nor mixed) are neutral
- Enharmonic spelling of the chord matters (so Ab might indicate mixture from c minor into C major, but G# probably does not);
- Secondary Roman numerals
- can indicate mixture (are not necessarily excluded);
- are considered against the home (primary rather than secondary) key. E.g., viio/V is considered against the I, primary tonic rather the V secondary.
Complex eh?! I'll provide a clear explanation and demonstration of all these options somewhere accessible soon. For now, every entry on this this makes at least a reasonably strong case for mixture.
[table “31” not found /]
Neapolitan Sixth Chords
This table includes root position (bII) in addition to first inversion (bII6) chords, as well as seventh chords based on both.
Click here for this textbook's chapter on this topic.
[table “33” not found /]
... and many more
Here, finally, are external links to corresponding anthologies on the same topics for other repertoire collections. These don't get the same focus here as they are slightly more experimental and the all important links to scores online are not currently possible. All the same, "more is more" for this large, exploratory style of anthology.
This is a placeholder chapter for forthcoming lists of moments that are notable for particular metrical techniques.
In the meantime, here is a list (from Gotham 2019) of passages in mixed meters (5s, 7s, ... ). The list is organised chronologically by year of composition and limited to notated music and cases in which a single mixed metrical pattern is sustained for at least four iterations such that it might realistically be internalised by a listener as a meter.
These (imperfect) criteria are intended to define a meaningful boundary for the collection guided by what meter 'is'.
[table “53” not found /]
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A pattern that is repeated and transposed by some consistent interval. A sequence may occur in the melody, the harmony, or both.
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The number of scale steps between notes of a collection or scale.
A regularly recurring unit of music that's smaller than an idea, and which is typically transformed across a work. The word "motive" usually refers to pitch material, but other kinds of motives such as rhythmic or contour also exist.
Notes on an upbeat that lead into the first downbeat of a phrase.