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Re: AW: Custom Format
From: |
David Kastrup |
Subject: |
Re: AW: Custom Format |
Date: |
Thu, 01 Apr 2021 20:32:38 +0200 |
User-agent: |
Gnus/5.13 (Gnus v5.13) Emacs/28.0.50 (gnu/linux) |
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
Re: AW: Custom Format, antlists, 2021/04/01