How a Sumerian could approximate 12-tET

Hello all,

I've just uploaded to the files section one of the
papers I'll be presenting in El Paso next weekend,
"How a Sumerian could approximate 12-tET":

/tuning/files/monz/12tet.doc

It's a Microsoft Word document.

Enjoy!

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "monz" <joemonz@y...> wrote:

You wrote that the Sumerians could take cube roots. Do we know they
had a general algorithm for this? Obviously, if they could take both
square and cube roots they could construct 12-et, assuming they had a
musical system in place that would make this a sensible development.

> From: <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, October 27, 2001 11:30 AM
> Subject: [tuning] Re: How a Sumerian could approximate 12-tET
>
>
> You wrote that the Sumerians could take cube roots. Do we know they
> had a general algorithm for this? Obviously, if they could take both
> square and cube roots they could construct 12-et, assuming they had a
> musical system in place that would make this a sensible development.

First thing to note: there are thousands upon thousands
of clay tablets with Sumerian and Babylonain writing on
them, still buried in the ground in Iraq. Second thing:
of the thousands of tablets that *have* been excavated
and now are housed in museums around the world, only a
tiny fraction of them have been deciphered and published.

There are many Babylonian tablets with lists of both
square and cube roots. There are far fewer Sumerian
tablets which actually contain sophisticated math.
But taking the caveats above into consideration, it is
still very possible that some day these sophisticated
Sumerian math tablets, if they do indeed exist, will
eventually come to light.

But as I demonstrate on my webpage
"Speculations on Sumerian Tuning"
http://www.ixpres.com/interval/monzo/sumerian/sumeriantuning.htm

1) the Babylonians retained the Sumerian logograms,
along with additional Akkadian (= Babylonian) words,
to represent important components of math calculations;

2) the Sumerian logograms alone give the essence of the
problems, with the added Babylonian words merely supplying
detail or grammatical necessities.

Therefore, I assume in all of my Sumerian speculations
that the Sumerians had already figured out the algorithms
and results, centuries before they were conquered by
the Babylonians and their own culture was subsumed into
the Babylonian.

I have not actually *seen* any tablets that show the
algorithm for finding square or cube roots, which is
precisely why I went into such detail here... to show
that, and how, it *could* be done. The only evidence in
the tablets is lists of the resulting roots, without
specifying the algorithm.

This is the case with much of Sumerian/Babylonian math.
There exist thousands of tablets giving lists of coefficients,
with no real indication where those numbers came from.

On the other hand, there are also thousands of tablets
containing math problems which require the use of those
coefficients, thus showing us several of the algorithms
that they did use and teach.

The famous Babylonian "tuning tablet", CBS 10996, shows
a method of tuning where the intervals are first tuned by
4ths/5ths and then checked by 3rds/6ths. This seems to
me to indicate that some type of tempering was indicated.

Mostly, what I'm really doing with this work is to
simply publish the relevant sexagesimal numbers, in
hopes that they will (or won't) turn up on tablets that
are examined in the future, thus confirming (or denying)
that the Sumerians were actually interested in 12-tET.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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