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.

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