Re: AW: Custom Format

[David Kastrup]

Kevin Barry <barrykp@gmail.com> writes:

> Thu, Apr 01, 2021 at 05:03:58PM +0200, David Kastrup wrote:
>> Kevin Barry <barrykp@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