Tempering Bach's temperament

Abstract

Rather than listing our serious fundamental objections to the two Bradley Lehman Bach articles—which could wait for a later occasion, if required—more importantly, may we briefly suggest an obviously much simpler interpretation of Bach's ‘squiggles’? Let us consider what a tuner of the period, using his ear alone, basically needed to know. Namely, how to tune pure octaves and fifths, and how to flatten other fifths equally. In this way, he would be equipped to construct a chain of four ¼-comma fifths making a pure third, either to complete a mean-tone tuning in its entirety, or as a basis for starting an unequal tuning. So, instead of turning Bach's apparently coded cipher upside down, as Lehman thought necessary, let us begin from the left-hand side, starting the conventional keyboard tuning from C. The majority of all unequal temperaments at this time were based upon the tuning of initial intervals as in mean-tone, leaving the remainder to be tuned unequally. This is much more likely than Lehman's interpretation in which he immediately starts with the extremely complicated ⅙th-comma tempering.