The suite for ‘cello began with thoughts about Bach. Not the person, and not the 6 marvelous suites, but the musical letters B, A, C, H. Wondering how a person versed in just intonation and its modern relative, Intonalism, would tune them. Step by step it became clear: it depends! In a sharp key like A major the A must be tonic and the Bb (“B”) must be up a large, pure, diatonic half step. If the cellist were to take out a ruler, measure the string length of the instrument, divide that length by 16 and put down a finger on the A string at that place, 1/16 of the total length, the perfect Bb would sound. In a sharp key like A major the C natural must be up a pure minor third, so also quite high. It turns out that the same ruler work is feasible: the correct C is at 1/6 of the total length. These numbers are from the pure ratios: the diatonic half step is 15/16 and the pure minor third is 5/6 and these give the sounding string length: if your finger is placed at 1/16 then the sounding string length is 15/16; if at 1/6, then the sounding ratio is 5/6. Back to Bach, the B natural (“H”) is in the same relation to the C as the Bb is to A: another diatonic half step. With the magic of numbers, a pure minor third (A to C) reduced by a diatonic half step (C down to B natural) is precisely a large whole step, ratio 8/9. Put the finger at exactly 1/9 of the string length and the perfect B will sound with a string length of 8/9. In summary, every note of the perfectly tuned Bach can be found by a deaf person with a ruler and a cello.
The fourth movement of the suite uses these ideas directly, with the Bb, A, C, and H finding themselves in a Bach-like step sequence: A major, the tonic, and then B major on the same note “H” from BACH, sounding in harmonic terms as dominant of dominant. Then, unusually, taking another step that might be called “real” like many of Bach’s fugues, up another large whole step from B to the sequence on C# up what is often called a Pythagorean major third from the tonic A. This combination of four notes from Bach and two rigorous large steps led to much of the Maestoso movement and by extension to the entire five movement suite.