A Maximally Varied Stretto Fugue

Two years ago, I introduced an isochronous melody that 1) has no internally recurring patterns and 2) can be consonantly combined with itself at any transpositional level or at any time delay. This is the only such melody of eight notes -- allowing for individual octave transfers, or the melody's wholesale transposition, inversion, or rotation (moving some of the notes from one end to the other) -- that has these two properties.



I doubled the durations, moved some notes up an octave, and embellished this eight-note melody to make the following subject.

I then wrote a stretto fugue based upon this subject that demonstrates its special properties. After a standard exposition, and before a final entry in the subdominant, the subject overlaps with itself at seven different time intervals to create seven different strettos, each time interval one unit smaller than the last. Since each time interval is linked to a certain pitch interval, the choice to use an incremental acceleration of the frequency of subject entries thus stipulates the fugue's succession of key areas. However, the linear pattern of the acceleration translates into a convenient arch pattern of transpositions. After a presentation in the main key (A), this arch pattern takes the fugue along a relatively standard tour through the relative major (C), then its dominant (G) and then its subdominant (F, of sorts) before reversing course through these keys, ending with two subject entries in A minor only a single unit apart. The close proximity of the acceleration's final entries yields three more secondary strettos and requires the use of four voices, up from the three with which I decided to begin.

All of this is shown in the graphic below, and can be watched and heard in the video below.

This fugue has 10 different stretto intervals among 11 subject entries, which earns a ratio of .91 different stretto intervals per entry, and all of the stretto intervals are different. This can be compared to two stretto fugues of J.S. Bach. Contrapunctus VII of The Art of Fugue has 22 different stretto intervals that are all different, as I have shown here. However, with 26 subject entries, this fugue earns a ratio of .85 different stretto intervals per entry, a little lower than mine. The C-major fugue from Book 1 of The Well-Tempered Clavier has 8 different stretto intervals among 24 subject entries (ratio of .33), and duplicates some stretto intervals.