Below is a piano-roll depiction of the beginning of the Fugue in A-flat Major from J.S. Bach's Book II of the Well-Tempered Clavier, aligned with the usual two-dimensional orientation of Western music notation. In this grid, the units of the x-axis are notated sixteenth notes, and the units of the y-axis are semitones. Therefore, the first note, shown on the left in gray, is an eighth note long, and descends three semitones to the next note.
This jaunty subject is built upon a frame of ascending perfect fourths -- 5-semitone intervals -- arranged in stepwise descent, as shown with the notes in red and the blue lines. The second and third perfect fourths occupy twice as much time as the first, which partition this part of the subject's timespan into a quarter-half-half division, or, more generally, a 1-2-2 division, as shown with the brackets above the grid. The filling-in of the second and third fourths -- the gray notes in between the red notes -- sustains the eighth-note pulse of the first three notes, compensating for the written-out rallentando of the ascending fourths.
The 1-2-2 division can also be found in semitones, as the intervallic division of this subject's third perfect fourth -- from top to bottom -- as shown with the brackets to the right of the grid.
This match may seem rather trivial; in fact, it should seem rather trivial. The duration succession is fairly short and indistinctive, and perfect fourths can also be divided diatonically into 2-2-1 and 2-1-2 as well; the second fourth in Bach's subject uses the latter division. But this pitch-time match has important implications for the countersubject, the line that appears in counterpoint with the subject.
Part of the enjoyment of listening to a work like a fugue is made available by the textural incrementalism of its opening: it begins with a monophonic subject that immediately appears again in tandem with a countersubject. Imagining what that countersubject might be given a subject can be a exciting, high-speed challenge for a seasoned listener hearing a fugue for the first time. One goes in with certain assumptions about this countersubject's likely features: diatonic, stepwise motion (especially if the subject is dominated by leaps from beat to beat), imperfect harmonic intervals with the subject, metrical conformity, steady pacing. The only three-note beginning of a countersubject that can completely meet the first four of these five assumptions for the subject's three melodic fourths is the following melody shown in green. The numbers in the green cells indicate the intervals between the notes in the countersubject and simultaneous red notes in the subject: they are all either thirds or sixths.
A deficiency of the beginning of this countersubject is that it lacks steady pacing: it slows down according to the same 1:2 proportions outlined earlier. The subject made up for this rallentando by filling in the fourths with passing motion and octave changes. Any stepwise melody can appear to quicken its pace by filling in its major seconds with chromatic passing tones. But in this case, the countersubject is perfectly suited to do so: by virtue of the subject's 1-2-2 temporal proportions and the placement of its fourths within the scale, a complete chromatic descent from scale degree 8 to scale degree 5 is the perfect complement for the subject, as shown below. The first of the five desiderata listed earlier (diatonicism) is traded for the fifth (steady pacing). The 1-2-2 temporal rate in which the melodic fourths of the subject unfold is matched perfectly to the 1-2-2 relationship between diatonic (Ab-G-F-Eb) and chromatic (Ab-G-Gb-F-Fb-Eb) steps in the top fourth of the major scale.
And, indeed, this is what happens, albeit first in the dominant key, which is the convention for this two-voice moment in a fugue's beginning.
Scholars rightfully highlight the disparity between the exuberance of the diatonic subject and the severity of the chromatic countersubject. As part of his wonderful commentary on the music of the Well-Tempered Clavier, the pianist David Korevaar recognizes that "the descending chromatic line from scale degree 1 to scale degree 5 is the standard signal for a lament – not at all the atmosphere promised in the subject!" It seems to me that this countersubject -- however dour -- is one of the few satisfying lines, and perhaps the most satisfying of them, that the subject promised.
