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Celebrating the pipe organ, the King of Instruments

The Art of Fugue

Types of Fugues, Part 2

 

document Mirror Fugues | document Quadruple Fugue | document Four Canons


MIRROR FUGUES
A mirror fugue is a pair of fugues in which each voice (or line) in the second fugue is a mirror image of the first - where the first goes up, the other goes down. In the previous movements only the theme was inverted; in the mirror fugues the entire piece is inverted. This requires Bach to play even more difficult games with his themes, since everything must be designed with its inversion in mind. Bach actually manages to achieve six different types of mirrors in these two pairs of fugues - a particularly stunning feat in that no matter how much Bach is bound up by the mirror fugue’s strict techniques, he still manages to make the music dance.
(See Tim Smith’s Art of Fugue site for an document in-depth look at these mirrors.)


Contrapunctus XII: Two pieces with varied main theme, first upright, then in mirror image
After the emotional turmoil of Contrapunctus XI, a serene pair of fugues clears the aesthetic palate. This pair of four-part fugues uses the principal Art of Fugue theme, but in 3/4 time - the only fugue in this meter. In this movement the two voices pivot around the same note when turned upside-down.


Contrapunctus XII: New variation of main theme
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Contrapunctus XII: Inversion of new variation
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Contrapunctus XIII: Two pieces with varied main theme and its inversion, then in mirror image
This is one of Bach’s most spectacular achievements. Bach takes an especially ornate variation of the Art of Fugue theme and incorporates it and its inversion into both the original and mirrored versions. He adds a special twist by inverting each fugal voice around a different pivot note.


Contrapunctus XIII: New main theme variation
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Contrapunctus XIII: Inversion of new theme variation
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QUADRUPLE FUGUE (“Unfinished”)

Contrapunctus XIV: Four themes, left incomplete
Bach intended this fugue to crown the Art of Fugue - a four-part fugue integrating the themes in their upright and inverted forms. Only three subjects are introduced in the surviving manuscript, however - did Bach die and leave it unfinished, or did he intentionally leave the ending open as a challenge to other composers?

The absence of the principal Art of Fugue theme initially gave rise to speculation that this movement didn’t belong to the Art of Fugue at all. Scholars have pointed out, however, that Bach intended to develop the principal theme in the missing fourth voice.

Composer and organist document Michael Ferguson, who has composed a completion to Bach’s Contrapunctus XIV, writes about his motivations in document On the Audacity of Completing Bach’s Unfinished Quadruple Fugue.


A page from document Michael Ferguson’s completion of Bach’s Contrapunctus XIV.
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Bach’s son C.P.E. Bach wrote over the last bars of Bach’s unfinished fugue,“The composer died at the point in the fugue where the name BACH is introduced as a countersubject.”
(See the document manuscript copy of the unfinished fuge at the J. S. Bach Page.)

 

FOUR CANONS
The canon is the strictest form of counterpoint in that the second voice imitates all the rhythms and all of the intervals of the first voice. The method of imitation can vary, however - the second voice might imitate the first at the octave, or at some other interval (in Art of Fugue Bach uses the 10th and the 12th intervals). The second voice might also imitate the main voice using different note values (e.g. quarter notes become either eigth or half notes) or it may appear as an inversion or reversal of the main voice. Combinations of all of these might be used.
(See document Anatomy of a Canon at Tim Smith’s Art of Fugue Web site for a comprehensive analysis of canon techniques.)

There has been some controversy over the placement of the canons within Art of Fugue as a whole. Originally meant to come between Contrapuncti XIII and XIV, subsequent editions have either placed one canon at the end of each fugal section, or grouped the four together at the end as if an appendix to the rest of the work. The canons are not considered to be essential to the Art of Fugue, but nevertheless showcase Bach’s ability to create exquisite music within the constraints of the canonic form.


Canon in Hypodiapason (at the octave)
Bach uses one of the most basic forms of imitation - placing the second voice at the octave - in this straightforward, gigue-like piece.


Canon in Hypodiapason (at the octave): main theme
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Canon alla Decima Contrapunto alla Terza (at the tenth)
This more inward-looking piece, the “Canon at the Tenth,” incorporates both simple imitation as well as double counterpoint (where the voices exchange registers). The opening theme is imitated not at the octave but at the tenth (octave plus two), which creates new harmonic effects.


Canon alla Decima (at the tenth): main theme
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Canon alla Duodecima in Contrapunto alla Quinta (at the twelfth)
This canon is similar in structure to the Canon alla Decima except that the secondary voice imitates the leading voice at the interval of the twelfth (octave plus five). Chromatic elements and rhythmic variations give this work a special flowing beauty.


Canon alla Duodecima (at the twelfth)
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Canon in Hypodiatessaron (in contrary motion with augmentation)
Bach’s most elaborate canon has two sections in double counterpoint - but Bach shows his mettle by arranging the answering voice in both inversion (contrary motion) and augmentation (lengthened note values). Like the previous two canons, the voice swap registers, this time an octave apart. Listen for an echo of the Agnus Dei from Bach’s B-Minor Mass.


Canon in Contrary Motion: main theme
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document Types of Fugues, Part I | document Art of Fugue home page.