The Art of Fugue is based on a single subject, which each canon and fugue employs in some variation:
The work divides into seven groups, according to each piece's prevailing contrapuntal device; in both editions, these groups and their respective components are generally ordered to increase in complexity. In the order in which they occur in the printed edition of 1751 (without the aforementioned works of spurious inclusion), the groups, and their components are as follows.
Simple fugues:
Stretto-fugues (counter-fugues), in which the subject is used simultaneously in regular, inverted, augmented, and diminished forms:
Double and triple fugues, employing two and three subjects respectively:
Mirror fugues, in which a piece is notated once and then with voices and counterpoint completely inverted, without violating contrapuntal rules or musicality:
Canons, labeled by interval and technique:
Alternate variants and arrangements:
Incomplete fugue:
Significance[edit]
Principles of construction[edit]
Loïc Sylvestre and Marco Costa reported a mathematical architecture of The Art of Fugue, based on bar counts, which shows that the whole work could have been conceived on the basis of the Fibonacci series and the golden ratio.[13]
Dominic Florence proposes that a concept he calls "opposition" governs all the methods that Bach uses in Contrapuncti 1, 2, 3, and 5 to create variety. These include changes in "melody (contrary motion), polyphony (contrapuntal inversion), harmony (dissonance), [rhythmic] density (texture), rhythm (syncopation), and tonality (modulation}".[14] For example, Contrapuncti 1 and 2 both switch repeatedly between the keys of A minor and D minor; Contrapuncti 2 and 3 in addition enter F major and G minor, Contrapunctus 2 also visiting B-flat major once in a further "tonal remove": all three begin and end in D minor. He concludes that "Analyses of fugues should focus on continuous, dynamic, organic processes that evolve over time, rather than on the static and discontinuous dismemberment into strictly delineated sections."[14]
Religious interpretations[edit]
In 1984, the German musicologist Hans Heinrich Eggebrecht suggested a possible religious interpretation of The Art of Fugue, which he agreed was impossible to prove: that the work illustrated in musical terms the Christian doctrine of redemption by God's grace alone, sola gratia, rather than by any action an individual can take. Eggebrecht noted the presence in fugue 3 of the presence of the composer's surname Bach in its theme, the sequence of notes B-A-C-H-C#-D. In Eggebrecht's view, this could mean that the composer is not just signing the work, but is placing himself by his grave, accepting sola gratia as he reaches towards the tonic note, which marks the end of the fugue and symbolically the end of his life. Further, the six-note fragment is chromatic, denoting sinful humanity, whereas the work as a whole is diatonic, symbolising God's perfection.[15]
The Russian musicologist Anatoly Milka suggests that the various numbers embedded in the work have a numerological significance related to the Book of Revelation, the last book of the Bible which describes the apocalypse.[15]