José Rodríguez Alvira
Bach must work at several levels to be able to create invertible fugues. The first one is the melodic level, a problem he solved when creating the subject.
The subject begins with a D minor arpeggio. The imitation is at the fifth (starts on A, not on D), so the D minor arpeggio transforms into another D minor arpeggio:
The ascending seconds motive that follows leads - in both versions - from dominant to tonic:
As does the descending seconds that end the subject:
Bach must also deal with the harmonic level by inverting the modulations. In the rectus version, the second answer is in the dominant key (A minor). The inversus uses the subdominant key (G minor). In measure 26, the rectus goes to Bb major (submediant) while the inversus goes to F major (mediant):
It is very useful to use a guide to show how each note is transformed by the mirror inversion:
The episode that begins in measure 13 starts in the G minor key in the rectus and in A minor in the inversus and both meet at D minor. We will analyze measures 13 to 17 step by step. Make sure that you are able to see the complete image and text, you may need to play with your browser's window and text size. Use the arrow buttons beside the text to move ahead or backwards:
The blue box indicates the rectus version, the red box the inversus. We will focus on two voices: the soprano from the rectus that will be inverted by the bass in the inversus and the bass from the rectus that will be inverted by the alto in the inversus. A simplified version of these voices is shown in the lower staves. By analyzing these voices we will be able to find out how does Bach does his magic...
We begin with an A minor chord in the rectus (in blue), transformed into a G minor chord in the inversus (red):
|E (soprano)||A minor's fifth||<->||G (bass)||G minor's root|
|A (bass)||A minor's root||<->||D (alto)||G minor's fifth|
|C (alto)||A minor's third||<->||Bb (soprano)||G minor's third|
The C# in the rectus is the leading tone of D minor, G is the seventh of D minor's dominant chord.
|C# (soprano)||D minor's leading tone||<->||Bb (bass)||seventh of F major's dominant|
|G (bass)||seventh of D minor's dominant||<->||E (alto)||F major's leading tone|
|C (soprano)||seventh of G minor's dominant||<->||B (bass)||C major's leading tone|
|F# (bass)||G minor's leading tone||<->||F (alto)||seventh of C major's dominant|
|B (soprano)||C major's leading tone||<->||C (bass)||seventh of G minor's dominant|
|F (bass)||seventh of C major's dominant||<->||F# (alto)||G minor's leading tone|
|Bb (soprano)||seventh of C major's dominant||<->||C# (bass)||D minor's leading tone|
|E (bass)||F major's leading tone||<->||G (alto)||seventh of D minor's dominant|
We arrive at F major in the rectus and D minor in the inversus. The F major in the rectus is followed by D minor. Back to D minor in both versions!
Bach achieves his magic by ingeniously reinterpreting the function of the inverted notes. Summary of the different functions assigned to each note:
|G minor's fifth||A minor's fifth||G's leading tone (#)||G minor's root||A minor's root||Seventh of F major dominant chord (b)||D's leading tone (#)|
|D minor's root||F's leading tone||Seventh of C major dominant chord||Seventh of D minor dominant chord (b)||F major's third||C's leading tone (♮)||Seventh of G minor dominant chord|
|C major's third||D minor's third||G minor's third (b)||C major's root|
|F major's root|
In measure 19, following the first episode, the alto in the rectus and the soprano in the inversus present the subject in D minor: