Re: AW: Custom Format

[Kevin Barry]

On Thu, Apr 01, 2021 at 08:24:08PM +0200, David Kastrup wrote:
> 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.
I didn't mean to imply that theorists used frequency counters. The
medium for devising tuning systems was the (mainly theoretical rather
than practical) monochord (lengths of a string basically).

> 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.
Equal temperament is closer to Pythagorean tuning than meantone
temperament, and it divides the comma among fifths - like other well
temperaments - not thirds.

> 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.
I think you are overestimating the effectiveness of the heuristic
methods available before root extraction was developed - either that or
the ability of music theorists to apply their Babylonian lessons. There
are a few attempts in the early 17th century by theorists to calculate
the 12th root of 2, but they contain errors, so I think it was not as
easy as you imagine.

Kevin

P.S. I think we have reached the limits of the usefulness of this
discussion, and in any case we've gotten quite far from the original
topic, so I'll stop feeding it now.