Kevin Barry <barr...@gmail.com> writes: > Thu, Apr 01, 2021 at 05:03:58PM +0200, David Kastrup wrote: >> Kevin Barry <barr...@gmail.com> 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.
Square roots can be calculated decimally with a system similar to long division. The system for cube roots is similar though more tiresome. The square root system basically relies on (10a+b)²=100a²+20ab+b² and successively fits the next largest possible digit b. The scheme for cube roots is just the same. > In order to divide a comma equally among a number of fifths you need > to be able to do that. (Equal temperament is just a special case where > you divide the comma over a full circle of twelve fifths.) General solutions for complete cubic _equations_ were derived by Arabic mathematicians before Europe got through comparatively early Medieval stages. -- David Kastrup
