Kevin Barry <[email protected]> writes: > Thu, Apr 01, 2021 at 05:03:58PM +0200, David Kastrup wrote: >> Kevin Barry <[email protected]> writes: >> >> > That's why, as soon as the mathematics (root extractions) required for >> > tempered tuning were discovered, it rapidly became the standard. >> >> I think your history of mathematics is a bit off. Seriously. And I >> have no idea how you think mean-tone tunings work. > I was referring to the family of concentric tunings that include equal > temperament and other "well" temperaments. Without the 17th century > discovery of logarithms and the wide availability of log tables they > would not have been possible - there was no method before then to > calculate the nth root of a number.
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. 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. > In order to divide a comma equally among a number of fifths you need > to be able to do that. If you are working with digital frequency generators, sure. Turns out that they are a pretty recent invention. > (Equal temperament is just a special case where you divide the comma > over a full circle of twelve fifths.) 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. -- David Kastrup
