Re: [tuning] Re Pythagoras, Hello, Siem.

Hi John, Siem here:

Pythagoreans had the reputation for investigating irrational numbers,
(through geometrical constructions), but Thales before Pythagoras is credited
with the first demonstration of the GEOMETRICAL MEAN--the real basis for the
generation of irrationals. Consequently, I would push awareness of
irrationals back at least to the Milesians.

Specifically, the 'Theorem of Thales' states that the angle inscribed within
a semicircle must be a right angle. This construction generates the set of
so-called 'Pythagorean triangles' and lays the basis for the geometrical mean
by dropping the perpendicular and hence constructing the irrational division
of a line segment.

Such pythagorean triangles were also known to be in use by Egyptian
architects (evident in the proportions of temples). This indicates that
irrational numbers preceded Pythagoras' life time.

Siemen Terpstra

Hello Siemen, Joe Monzo here. Nice to make contact with you.

Your article in 1/1 led me the writings of Ernest McClain,
which in turn led me into what has so far been a summer-long
'dig' into ancient Babylonian and Sumerian music and culture.

I'm glad to be able to discuss this with you, but unfortunately
only have a brief second right now; hopefully I can write
more to you later. As yet, I have nothing on the web about
Sumerian/Babylonian stuff except for a few postings to this List.

While it's true that of the Babylonian texts that have so
far been proven to concern music, none say anything about ratios
or numbers, I believe that I may have found another Babylonian
math text that does provide formulae for calculating 'Pythagorean'
string-lengths. I have a hunch that it may alternatively
be a formula for calculating temperament.

It's absolutely certain that the Babylonians (and probably
the Sumerians before them) knew how to calculate arbitrarily
close rational (sexagesimal) approximations to the square-root
of 2, so I'm wondering if they might have actually even
discovered how to calculate 12-tET.

Anyway, that's just a 'teaser'. When I have more time,
I'll write more about it to you.

Stop by and see us in San Diego if you come to California!

-monz
http://www.ixpres.com/interval/monzo/homepage.html