Double subjects and countersubjects

In measures 59 to 62 the subject is presented by the tenor and bass and the countersubject in the soprano and alto:

Here are the inversions used in these measures:

Voices Inversion
Bass and alto 10th
Bass and soprano 12th
Tenor and alto 8ve
Tenor and soprano 10th

Measures 69 to 72 are similar but the voices are organized differently and the key is G minor. The soprano and tenor present the subject (incomplete in the tenor) while the bass and alto present the countersubject:

How should we look at these measures? Like an inversion of measure 59? If so, here are how the voices move:

Voices in measure 59 Voices in measure 69
bass to the tenor
tenor to the soprano
alto to the bass
soprano to the alto

If we apply a second inversion at the octave to the inversions from measure 59 we get some additional examples of inversion at the 13th:

Voices in measure 59 Inversion Move in measure 69 to Second inversion Resulting inversion
bass and alto 10th tenor and bass 8ve 13th
tenor and alto 8ve soprano and bass 8ve 8ve
tenor and soprano 10th soprano and alto 8ve 13th

Also note that the bass and soprano in measure 69 are identical to measure 9 (second subject in the exposition):

And the tenor and alto in measure 69 are the inversion at the 12th of these two voices:

A simpler explanation is that the bass is doubled at the third by the alto while the soprano doubles the tenor. A labyrinth of inversions of inversions seems to take us back to where we started. It is easy to get lost in Bach's labyrinth...

©2011 José Rodríguez Alvira. Published by

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