origin of 12edo (was: paul/julia 2)

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, October 26, 2002 9:06 PM
> Subject: [tuning] Re: paul/julia 2
>
>
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> /tuning/topicId_38587.html#40221
>
> <big snip>
>
> > i'm sure you know all this, but my point in reviewing it
> > is to express that, as i see it, equal temperaments
> > are quite sophisticated, quite the opposite of "crude" . . .
>
> ***Isn't it true that even with our "overworn" 12-tET,
> that musicians and mathematicians didn't even have the
> tools to mathematically create this ET until comparatively
> *late* in the overall historical development of tuning???
>
> (Well, I guess the Chinese did it early, but they came
> upon it in a different way, yes??)

the idea of 12-tone equal-temperament has surely been
around ever since the first time someone discovered the
so-called "Pythagorean comma". "tuning by concords"
(i.e., by successive "4ths" and "5ths") was known to
the Babylonians and is documented in the famous "tuning
tablet" known as CBS 10996. see my webpage about the
reconstruction of the Hurrian hymn:
http://sonic-arts.org/monzo/babylonian/hurrian/monzh6.htm

anyone who tunes a 13-tone scale according to this method
will notice right away that the 13th note is extremely close
(~23.5 cents higher, the "Pythagorean comma") to the starting
note, and it would be a natural thing for this person to
consider the possibility of tempering all 12 notes of the
scale to eliminate this small "anomaly" and close the system.

while it's true that so far the only positive documentation
we have shows that the Europeans, and the Chinese only
slightly before them, discovered how to calculate absolutely
true 12edo, as 2^(12/12), in the early 1600s, we do know
from some Babylonian tablets (the most famous one is
YBC 7289) that the Babylonians knew how to find arbitrarily
close approximations to 2^(1/2) (i.e., the square-root of 2).

this is the essential problem in calculating 12edo, as
one only needs to be able to calculate the square-root
of the square-root of the cube-root to find the semitone,
because 2*2*3 = 12. and that can indeed all be done
by hand, if one knows the procedure.

so while the Europeans and Chinese only rediscovered
the mathematics of roots and exponents in the late 1500s,
we know that the Babylonians were familiar with them
back around 1600 BC. and as i show in my webpages,
the Babylonians used certain Sumerian words as "logograms",
much as we use the "Arabic" numerals as symbols representing
our words "one", "two", "three", etc., and that if one
strips out all the Babylonian words, one finds that the
essentials of the math problems are still there, entirely
in Sumerian.

the only conclusion that can be reached from this is that
the Sumerians were aware of the mathematical procedures
which exist today in the Babylonian math problems, and
that this knowledge must therefore date from the height
of Sumerian civilization c. 2500 - 2000 BC. see the
section of my "Speculations on Sumerian Tuning" where
i detail how Sumerian base-60 mathematics could give an
extraordinarily close approximation to 12edo with rather
simple (albeit tedious) calculations:
http://sonic-arts.org/monzo/sumerian/sumeriantuning.htm#12edo

-monz
"all roads lead to n^0"

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

/tuning/topicId_40228.html#40228

> hi Joe,
>
> the idea of 12-tone equal-temperament has surely been
> around ever since the first time someone discovered the
> so-called "Pythagorean comma". "tuning by concords"
> (i.e., by successive "4ths" and "5ths") was known to
> the Babylonians and is documented in the famous "tuning
> tablet" known as CBS 10996. see my webpage about the
> reconstruction of the Hurrian hymn:
> http://sonic-arts.org/monzo/babylonian/hurrian/monzh6.htm
>

***Thanks, Monz for the "refresher" and the "fresher..."

best,

Joe P.

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

> true 12edo, as 2^(12/12), in the early 1600s, we do know

monz, 2^(12/12) = 2. don't you really mean 2^(n/12)?

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, October 27, 2002 6:03 AM
> Subject: [tuning] Re: origin of 12edo (was: paul/julia 2)
>
>
> --- In tuning@y..., "monz" <monz@a...> wrote:
> >
>
> /tuning/topicId_40228.html#40228
>
> > hi Joe,
> >
> > the idea of 12-tone equal-temperament has surely been
> > around ever since the first time someone discovered the
> > so-called "Pythagorean comma". "tuning by concords"
> > (i.e., by successive "4ths" and "5ths") was known to
> > the Babylonians and is documented in the famous "tuning
> > tablet" known as CBS 10996. see my webpage about the
> > reconstruction of the Hurrian hymn:
> > http://sonic-arts.org/monzo/babylonian/hurrian/monzh6.htm
> >
>
> ***Thanks, Monz for the "refresher" and the "fresher..."

i was inspired to do some more calculating, and i came
up with an even simpler base-60 approximation to 12edo,
which only uses 2 sexagesimal places, with a maximum
error from true 12edo of less than 1 cent.

http://sonic-arts.org/monzo/sumerian/simplified-sumeriantuning.htm

it took me about 3 hours to do the entire set of base-60
calculations on this page by hand, proving to me that
it was well within the reach of any educated Sumerian.

-monz
"all roads lead to n^0"

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

/tuning/topicId_40228.html#40241

>
> hi Joe,
>
>
> > From: "Joseph Pehrson" <jpehrson@r...>
> > To: <tuning@y...>
> > Sent: Sunday, October 27, 2002 6:03 AM
> > Subject: [tuning] Re: origin of 12edo (was: paul/julia 2)
> >
> >
> > --- In tuning@y..., "monz" <monz@a...> wrote:
> > >
> >
> > /tuning/topicId_40228.html#40228
> >
> > > hi Joe,
> > >
> > > the idea of 12-tone equal-temperament has surely been
> > > around ever since the first time someone discovered the
> > > so-called "Pythagorean comma". "tuning by concords"
> > > (i.e., by successive "4ths" and "5ths") was known to
> > > the Babylonians and is documented in the famous "tuning
> > > tablet" known as CBS 10996. see my webpage about the
> > > reconstruction of the Hurrian hymn:
> > > http://sonic-arts.org/monzo/babylonian/hurrian/monzh6.htm
> > >
> >
> > ***Thanks, Monz for the "refresher" and the "fresher..."
>
>
>
>
> i was inspired to do some more calculating, and i came
> up with an even simpler base-60 approximation to 12edo,
> which only uses 2 sexagesimal places, with a maximum
> error from true 12edo of less than 1 cent.
>
> http://sonic-arts.org/monzo/sumerian/simplified-sumeriantuning.htm
>
>
> it took me about 3 hours to do the entire set of base-60
> calculations on this page by hand, proving to me that
> it was well within the reach of any educated Sumerian.
>
>
>
> -monz
> "all roads lead to n^0"

***This was a *great* experiment, Monz... well worth the time, I
think...

JP

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, October 27, 2002 11:53 AM
> Subject: [tuning] Re: origin of 12edo (was: paul/julia 2)
>
>
> --- In tuning@y..., "monz" <monz@a...> wrote:
>
> /tuning/topicId_40228.html#40241
>
>
> > > > the idea of 12-tone equal-temperament has surely been
> > > > around ever since the first time someone discovered the
> > > > so-called "Pythagorean comma". "tuning by concords"
> > > > (i.e., by successive "4ths" and "5ths") was known to
> > > > the Babylonians and is documented in the famous "tuning
> > > > tablet" known as CBS 10996. see my webpage about the
> > > > reconstruction of the Hurrian hymn:
> > > > http://sonic-arts.org/monzo/babylonian/hurrian/monzh6.htm
> > > >
> > >
> > > ***Thanks, Monz for the "refresher" and the "fresher..."
> >
> >
> >
> > i was inspired to do some more calculating, and i came
> > up with an even simpler base-60 approximation to 12edo,
> > which only uses 2 sexagesimal places, with a maximum
> > error from true 12edo of less than 1 cent.
> >
> > http://sonic-arts.org/monzo/sumerian/simplified-sumeriantuning.htm
> >
> >
> > it took me about 3 hours to do the entire set of base-60
> > calculations on this page by hand, proving to me that
> > it was well within the reach of any educated Sumerian.
> >
> >
>
> ***This was a *great* experiment, Monz... well worth the time,
> I think...

thanks, Joe! kind of makes me feel like Thor Hyerdahl,
building his raft and setting out across the Pacific,
just to prove that it *could* have been done with far
more "primitive" technology ...

actually, once you get used to doing base-60 math, you
wonder why it ever went out of favor ...

i'll argue (on at least a few points, one of them being
math) with anyone who says that the Sumerians who lived
5,000 years ago were more primitive than us. i'm convinced
that *they* invented 12edo ... along with the wheel,
irrigation, architecture, law, etc. etc. etc.

... really an amazing civilization. i sure hope the
US offensive in Iraq doesn't destroy too much of what's
still lying there under the sand.

(... "as he veered off-topic" ...)

-monz
"all roads lead to n^0"