On Thu, Apr 01, 2021 at 08:24:08PM +0200, David Kastrup wrote: > So? The seventeenth century did not have frequency counters. Tunings > were established (and actually still are to this day: just ask any organ > tuner or accordion tuner) by distributing the beatings of non-pure > intervals across several intervals. Assigning some 5 decimals number to > that frequency is irrelevant since the accuracy of the _relative_ > intervals when tuning is much more important than the _absolute_ > frequencies. I didn't mean to imply that theorists used frequency counters. The medium for devising tuning systems was the (mainly theoretical rather than practical) monochord (lengths of a string basically).
> Meantone tuning has a number of pure intervals and distributes the > impurities among a few others. The commonly known quarter-comma > meantone temperament distributes the impurities over 4 major thirds and > keeps the other major thirds pure. Well-tempered tunings focus on > keeping most fifths pure instead of thirds. Equal-tempered tuning is, > in a manner, a special mean-tone case where 0 fifths are kept pure and > the accumulative error is distributed across the 12 remaining fifth > intervals. Equal temperament is closer to Pythagorean tuning than meantone temperament, and it divides the comma among fifths - like other well temperaments - not thirds. > Sure. And even if you wanted to do this with numbers, the 12th root of > 2 can be calculated by doing a cube root and 2 square roots. And cube > roots were already calculated by Babylonian mathematicians close to > 4000 years ago. I think you are overestimating the effectiveness of the heuristic methods available before root extraction was developed - either that or the ability of music theorists to apply their Babylonian lessons. There are a few attempts in the early 17th century by theorists to calculate the 12th root of 2, but they contain errors, so I think it was not as easy as you imagine. Kevin P.S. I think we have reached the limits of the usefulness of this discussion, and in any case we've gotten quite far from the original topic, so I'll stop feeding it now.
