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Re: AW: Custom Format
From: |
David Kastrup |
Subject: |
Re: AW: Custom Format |
Date: |
Thu, 01 Apr 2021 20:24:08 +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.
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
Re: AW: Custom Format, antlists, 2021/04/01