Two C Major Preludes

In an earlier post, I briefly described a computational method that can be used to classify and visualize perceived tonality in a piece of music.  I'm interested in using techniques like this to extract mathematical information from musical compositions.

Here I will look at two piano works:  The C major prelude from Book 1 of  J. S. Bach's Well Tempered Clavier, and the C major prelude from Shostakovich's  24 Preludes and Fugues, opus 87.  The story is often told that Shostakovich composed this work after being inspired by Bach's W.T.C. during a  1950 commemoration of  the 200th anniversary of Bach's death .

Separated by two hundred years, it is easy to discern that these two pieces are harmonically very different from one another.  I am curious to quantify that difference in some way.

Here is a video showing the tonality map along with a performance of the Bach prelude:

And a corresponding video for the Shostakovich prelude:




In these visualizations, what characteristics distinguish the pieces from one another?  The Shostakovich features a wider variety of tonal centers, tonal centers which are sometimes less distinct --  less pronounced dark red  regions on the map, and tonal centers that move by greater distances from measure to measure.  This last point suggests a way to characterize the harmony of the pieces:  the distribution harmonic distance.  Here is a plot that shows the harmonic distance, measured on the tonality map, for the  Bach and Shostakovich pieces.

It is striking that distances 5, 7, and 9 are especially pronounced in the Bach prelude.  These distances on the tonality map correspond, for the most part, to  occurrences of the  ii7 -- V7 -- I  sequence of chords.  This harmonic cadence establishes a tonality in the I key very strongly.   The Shostakovich harmonies avoid this progression.  This comes as no surprise to the listener, who clearly  gets a weaker sense of tonality and systematic modulation between tonal centers.