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Introduction to fugues

ABC's of fugues

A ゛fugue゛ is an imitational style with some rules and does not have a fixed
form. On the contrary, its style can be brought into various musical forms
and styles, that is, fugal sonata or fugal dance music could be possibly
composed. You might find the fusion of a fugal style and a canonic style
even in the art of fugue. Many voices more than two are included in the
fugue and the theme is imitated by each voice throughout.

Theme
A fugue consists of melodic imitations called ゛theme゛ in which neither
length nor register of the theme is limited, and the theme generally starts
at tonic or dominant to help introduce the next coming response smoother.
Most of themes are built in the combination of some distinct melodies.
Theme of Cp.1 from the art of fugue. It starts at tonic(d).


A theme is stated by solo melody at the beginning of a piece, however, in
an ensemble fugue, a free counter theme comes along exceptionally almost
right after the introduction of the theme.
Theme of Fuga a 2 Clav. from the art of fugue. The underlined is a free melody.
The theme in the upper voice starts with dominant(a).


Response
Following an exposed theme, a ゛response゛appears as the thematic
imitation. It should be appeared in a different voice from the one initially
used. Basically, tonic-dominant position in the thematic melody is switched
to dominant-tonic in the response. Now, let's move on to the next various
types of responses.

Real response
It refers to a response that the theme exposed at the beginning of a piece
is totally transposed to the dominant key and imitated. In a real response,
only tonic in the theme is reflected to dominant in the response.
Beginning of Cp.11 from the art of fugue. Highlighted in blue is the response
to the previously stated theme, imitating in dominant key after transposing.


Tonal Response
It refers to a response that tonic in a theme is replaced by dominant and
vice versa. In most cases, therefore, the interval motion of a melody is
changed somewhat. Many fugues in the art of fugue include tonal responses.
Beginning of Cp.1 from the art of fugue.
The 2nd note starting from bar 5 in the response would be ゛e゛ if a real response,
but is actually ゛d゛ here because dominant ゛a゛ on the 2nd note of the theme
is reflected to tonic ゛d゛ in the response.


Inverted Response
It refers to a response that a theme is inversed upside down, pivoting the
3rd note. As a result of inversion, tonic in the theme is interchanged with
dominant in the response and vice versa.
Beginning of Cp.5 from the art of fugue. The first two notes ゛a→d゛ of the theme
exposed in the upper voice is varied to゛d→a゛ in the inverted response in the lower voice.
Tonic and dominant are exchanged.


Plagal Response
It refers to a response stated in subdominant key, which was lead from the
result that when the response to the theme was started at dominant,
dominant was reflected to tonic and the interval motion was faithfully
imitated with. In the art of fugue, plagal response can only be seen Cp.10.
Beginning of Cp.10 from the art of fugue.
The third note ゛a゛ in the theme is reflected to the third note゛d゛ in the response.
This piece features that the response is exceptionally built in two beat behind.


Counter Theme
It is called ゛counter theme゛ that when the counter melody accompanied to
the response at the beginning of the piece is employed several times,
together with later coming themes and responses. There is no clear
definition in terms of melody. The counter theme seldom starts with down
beat, and often makes up for rhythm and beat, insufficient to the theme.
From bar 9 of Cp.3 from the art of fugue. Rhythmic complements are found here and there.

Even thought the melody along with a theme appears in the middle of a
piece, it is generally called counter theme, as long as it reappears in
accompaniment to later coming themes.

Exposition
Typically, but not always, once appeared theme in a certain voice is not
exposed in the same voice again until it is thoroughly used among all
voices. Besides, one theme does not appear in the several voices at the
simultaneously timing, in short, it comes out one by one in accordance with
the repetition of theme-response alternate exposition. This process that
the theme is going through all voices, is called ゛exposition゛

Development
After the exposition is made, the theme free from a fixed rule is flexibly
imitated, which is called ゛development゛. It has no regulations in tonality and
exposes the theme in remote key in some cases. Various factors can be
seen in the exposition other than thematic repetitions.

Variant Theme
A theme receives all sorts of variants such as a gap of syncopation,
embellishment, inversion, diminution and augmentation while developing.
Theme of Cp.2 from the art of fugue and the variant theme in the last part of the piece.
Variant themes are performed by syncopation.


Embellishment can be seen only at the head of a theme.
Theme from bar 23 of Cp.2 from the art of fugue.


Furthermore, a counter theme may vary in accordance with thematic
variant.
Theme and the counter theme from bar 9,
and the variant theme and the variant counter theme from bar 55
of Cp.3 from the art of fugue.


Stretto
Stretto is that several themes are overlapped and exposed, in other words,
the way that before one theme is concluded, another theme in a different
voice enters. Cp. 5. Cp.6 and Cp7. contain stretti in expositions. We also
call it stretto that each theme enters at one time in a fugue with many
themes.
Exquisite skill of stretto from Cp.5 from the art of fugue.
All sorts of timing areincorporated.


Thematic Fragment
Characteristic figure in the head of a theme may emerge fragmentally in a
piece. False stretto is formed by the theme and the thematic fragment
closely in each different voice in the case that stretto cannot consist of by
using whole length of the themes or that the themes cannot recognizable
when made them to stretto. Cp.5 includes stretto by thematic fragments,
apart from the purpose of these sorts.
Highlighted in blue is thematic fragment seen in Cp.8 from the art of fugue.
Stretto is formed by the theme in alto and the thematic fragment.


Sequence
Sequence refers to the repetition of a motive by one voice or more. Every
time the motive is repeated, its pitch fluctuates, and a typical ゛Bach
sequence゛, as it commonly said, is that the two voices exchange the
motive in turn while the other voice repeats another motive alone like Cp.9.
Sequence seen in Cp.9 from the art of fugue.
The motive in soprano and alto go back and forth.


Next music shows that all four voices repeat their respective motives.
Sequence seen in Cp.6 from the art of fugue.
Melodies appeared in each voice in the first column are repeated in the second column.


Cadenza
Cadenza is to play a free melody in one voice, suspending the music
harmonically at the end or in the middle of a piece.
Cadenza seen in Cp.2 from the art of fugue.
Soprano makes a splendid melody from bar 21-22.


In the tenth canon, a performer is required improvisational cadenza.
Instruction of cadenza in ゛Canon alla Decima゛ from the art of fugue.


Some performers regard the harmonic suspension before coda as the need
of improvisational cadenza, like just before the coda in Cp.1.
Harmonic suspension seen at the last of Cp.1 from the art of fugue.


Organ point
Prolonging tonic or dominant, mostly in bass at the end of a piece, is called
organ point. Organ point (Orgelpunkt) originates in organ music and thus
the name comes from it.
The last of Cp.1 from the art of fugue is continued by Organ point.
Tonic in bass is prolonged.



Various Fugues

Fugues fall into some categories depending on the number and/or variant
style of the theme, and then it is sometimes named after the characteristic
technique itself. If various techniques are mixed together in one fugue, it
may employ a complex name or one picked up from most remarkable
techniques.

Simple Fugue
It refers to a fugue having one theme, free from thematic inversion,
augmentation ordiminution. Cp.1-4 of the art of fugue are the simple fugue.
Beginning of Cp.1 from the art of fugue. No distinct variance in the theme is found in this single theme.


Inverted Fugue
Mainly a fugue with a single theme and a inversed response is said inverted
fugue. It is also called inverted fugue when the inverted theme is exposed
in the middle of the piece after exposition. Cp.5-7 in the art of fugue are
inverted fugues.
Beginning of Cp.5 from the art of fugue. Response is stated in inverted form.


Diminished Fugue
It refers to a fugue that the time value of a theme is diminished in a fixed
rate over the entire piece. A diminished theme does not always appears in
the beginning and sometimes have their own exposition. In the art of fugue,
the diminished theme and the original length of theme coexist in the
exposition. Time value is halved in Cp.6.
Beginning of Cp.6 in the art of fugue. Following the theme in bass,
response in soprano is stated in diminished and inverted form.


Augmented Fugue
A fugue that the time value is prolonged in a fixed rate over the entire
piece. Like a diminished fugue, the augmented theme and the original length
of theme coexist in the exposition in the art of fugue. Time value is doubled
in Cp.7.
From bar 2 of Cp.7 in the art of fugue. Following the theme in soprano,
diminished theme in alto and enlarged theme in bass are presented.


Double Fugue
A fugue in which simultaneous presentation and combination of two themes
in individual voices are made possible. Firstly stated theme is a first theme
and the next introduced theme is a second theme. As for the structure,
the exposition of the first theme comes first and then later the
combination of the first theme and the second is made. In some cases, the
exposition of the second theme is placed before the combination. There is
no such case that the two themes are combined at the beginning of a piece
among Bach's fugues. Cp.9 and Cp.10 are the double fugues in the art of
fugue.
From bar 44 of Cp.10 in the art of fugue. Two themes are presented at the same time.


Triple Fugue
A fugue in which simultaneous presentation and combination of three
themes in individual voices are made possible. Thirdly introduced theme is
called third theme. The basic construction is the same as double fugue, but
depending on the way how to expose each theme, various variations can be
created. Cp.8 and Cp.11 are triple fugues.
From bar 147 of Cp.8 in the art of fugue. Three themes are presented at the same time.


Mirror Fugue
Mirror fugue has nothing to do with the treatment of a theme, but is
composed allowing a whole piece to inverse upside down. Making it possible
to be a fugue after inversed by pivoting the third note, harmonic responses
have to be considered. Cp.12 and Cp. a3C come under this.
Beginning of Cp.12 from the art of fugue. Lower column is the゛mirrored image゛,inversed upside down.


Multiple Techniques
The above mentioned multiple fugal techniques are occasionally compound
in one piece like inversion, diminution and augmentation of Cp.７, and also
inverted triple fugue of Cp.11 and so forth. As all these are hard to be
categorized in one title, Cp.7 is simply given a name as inverted fugue and
Cp.11 is triple fugue in conventional way.


Canon

Unlike a fugue,゛canon゛is that a follower imitates a leader exactly without a
break. In principle, this imitation continues until the end of a piece. Many
canonic techniques are fully employed among Bach's cycles like゛the art of
fugue゛or゛musical offering゛.

Unison Canon
Simplest canon of all. A follower imitates a leader without a difference of
one note. ゛Round ゛can be said as one kind of unison canons.
Example of the canon at the unison from ゛musical offering゛BWV1079.


Canon at various degrees
A follower begins on a different pitch from the starting pitch of a leader.
Depending on the intervals, they are called ゛canon at the fifth゛or ゛canon
at the octave゛respectively. So-called ゛Goldberg Variations゛BWV988 by
Bach includes the interval canons at the unison to the ninth by leaders and
followers.
Example of the canon at fifth degree from ゛musical offering゛BWV1079.


Inverted Canon
A follower imitates a leader in its inverted form of the leader. In many
cases, among Bach's inverted canons, the follower is generated from
inversion, pivoting the third note of a theme. The note taken as a pivot will
determine the way of inversion, which brings about various intervals
between the two.
Example of the inverted canon from ゛musical offering゛BWV1079.
The follower(blue notes) is inverted by the fifth note(g).


Crab Canon
It is sometimes referred to as retrograde canon in which one voice mimics
the other from backward, that is, a melody starting from left in music is
equivalent to the melody starting from right.
Example of crab canon from ゛musical offering゛BWV1079.Two melodies make progress oppositely.


Augmented Canon
A follower imitates a leader while prolonging its time value in a regular rate.
There is a case that reverted form is incorporated into this. In contrast to
a fugue, a canon in diminution is unlikely because, by diminishing time value,
the follower gets ahead of the leader.
The forth variation from Choral"Von Himmel Hoch"BWV769
The follower in blue imitates the leader while augmenting its time value twice to the leader.


Double Canon
Double canon is that respective followers imitate simultaneously performed
two melodies in leaders. An inverted canon may be incorporated into it.
Canon No.5 of ゛various canons゛BWV1087. Double canon in reversed form is exemplified.


Triple Canon
Triple canon is that respective followers imitate simultaneously performed
three melodies in leaders. An inverted canon may be incorporated into it.
Canon No.13 of ゛various canons゛BWV1087. Triple canon in reversed form is exemplified.


Enigmatic Canon
It refers to not a canonic style but a special notation to write a canon: In
music, canonic leader is just notated and there appended the starting point
of a follower, intervals of canons and keywords like inverted or retrograde.
Enigmatic canon is exemplified from ゛musical offering゛BWV1079. Two clefs and ※ are found.


Spiral Canon
It refers to a canon which is repeated by repeat mark and in which a
follower keeps following a leader endlessly. An ending point is sometimes
stated or sometimes not. If specific ending is not noted, the decision leaves
to a performer.
Canon No.3 of ゛various canons゛BWV1087.
The entire music is below and you cannot find any specific ending.



Compositional Skills and others

Double Counterpoint
Double counterpoint is that simultaneously stated two melodies are
interchangeable upside down.
Double counterpoint seen in Cp.8 from the art of fugue.
Both melodies in bar 170 and bar 182 are replaced upside down.


And it is also referred as double counterpoint that the interval between two
melodies stated simultaneously can be changed. ゛Tenth double
counterpoint゛is for the interval to be changed at three degree, and ゛
Twelfth double counterpoint゛is for the interval to be changed at fifth
degree. In these cases also, positions of both melodies are interchangeable
in each other.
Double counterpoints at twelfth degree seen in "Canon alla Duodecima゛　from the art of fugue.
Comparing bar 74 and bar 75 with bar 76 and bar 77, you can find that
the black notes in upper and lower column are the same degree, but blue notes
between bar 76 and bar 77 are five degree lower than the blue notes between bar 74 and bar 75


As mentioned before, ゛double counterpoint at eighth degree゛makes it
possible for two melodies to be interchanged exclusively. In゛double
counterpoint at tenth degree゛, both a melody, three degree higher than the
original and the original melody can be stated simultaneously like Cp.10.
Tenth double counterpoint seen in Cp.10 from the art of fugue.
Duplication with three degree difference can be seen in soprano and alto.


Mirror Counterpoint
There is no such a term as "mirror counterpoint" officially but the way to
be called so here is a compositional skill that a whole piece is invertible
upside down like Cp.12. The third note functions as a pivot in the art of
fugue.
Beginning of Cp.12 from the art of fugue. The lower column is　the ゛mirror゛ inversed upside down.


Highest technique is demanded on composing a mirror fugue. ; Things to be
put in mind are that the inverted final chord and the second inverted chord
might be generated, and the use of chromatic motion may be undesirable.

Rhythmic Complement
We tentatively name it rhythmic complement. Speaking of fugues, initially
taken beat or rhythm in the exposition is maintained throughout by any
voices, excluding partial exceptions, regardless time signature at the
beginning of a piece. Discontinuing it may be cadenza.
Rhythmic complement seen at the closing of Cp.3 from the art of fugue.
Blue notes in each voice are taking eight beat, making up each other.


Voice Maintenance
In principle, once exposition has set in a fugue, no voice is added
afterwards. However, a brand-new voice is exceptionally increased at the
last of a piece although the voices performed in exposition never disappear
till the end. Once crossed voices never fail to return their original places
and voice crossing does not last so long either.

Register Maintenance
In four parts music, each voice stays within a regular register.
In three parts music, middle part is allowed quite free register.
In two parts music, both parts admit their freedom and do not intrude upon
each other's resister.
Highest and lowest pitches are observed principally in any voices. One
exception lies in mirror counterpoint, due to the technical reason such as a
violation of highest or lowest register.

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