# About the rules of inversion in BWV 1087 (14 canons)

Basically, I'm a bit confused about how to interpret the idea of "inversion" in a mirror canon.

For example, consider the sixth canon of BWV 1087.

So I simply took the inverted image of this part, which resulted in the following.

This means that the inverted part starts with "A-B-C". However, Gerubach suggests the following inversion in his video, which means that the inverted part will start with "D-E-F#" instead.

He says "other part mirrored at concert A", but I don't see any notation on the manuscript that indicates the type of inversion. Then I tried to play a little bit on my piano, and the inversion in Gerubach's video somehow works better.

Is it related to "B-A-C-H"? Or is it a convention that I missed? Or is it just because the inversion with respect to "A" perfectly matches the harmony of the canon?

Thanks for your time and efforts.

• I'd question an inversion that "works better" that forces E-D-C-B-Bb-A against D-E-F#-G-Ab-A when your proposed inversion merely pits E-D-C-B-Bb-A against A-B-C-D-Eb-E. Jan 2, 2022 at 15:17

Any time we invert something we can choose the axis around which everything will be inverted. This way, we can best fit the inversion with whatever harmonic environment may be desired.

When you inverted the image, you were de facto inverting the music around C, the middle of the staff, because that C stays constant in both the original and the inverted versions.

But we can also invert things around another pitch axis. Let's say we want to invert around A. Our motive begins with E–D–C, so let's invert this: E is a fifth above A, so it inverts to a fifth below A, which is D. D is a fourth above A, so it will invert to an E. And C is a third above A, so it will invert to a third below A, which here will be an F♯. In short, inversion around A results in D–E–F♯.

Thus, by choosing our axis of inversion, our E–D–C can invert either to your A–B–C or to Gerubach's D–E–F♯. And these were only two different pitch axes; choosing others will of course result in new inversions.

Lastly, there are "real" and "tonal" versions of inversion. In other words, you can be really specific with interval sizes to determine exact accidentals, or you can use only generic intervals to get the letter names, determining the accidentals based on a requested harmonic environment. Hence the F♯ in Gerubach's inversion.

By the way, something I just learned today: why are there 14 of these canons? Because Bach, ever obsessed with numbers as he was, was showing us that B A C H = 2 + 1 + 3 + 8 = 14.

• It does seem that the key is that the canon's offset is determined to be some amount. I naively assumed all voices were (effectively) introduced at the same time. Jan 2, 2022 at 15:52
• Thanks for your answer! It seems that we may choose any axis of inversion, and it just a problem of whether the resulting music is harmonic, is that correct? Jan 3, 2022 at 4:00