Sumerian 12-EDO? (was: [tuning] fractional exponents of prime-factors)

I wrote:

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, July 18, 2001 3:09 AM
> Subject: [tuning] fractional exponents of prime-factors
>
>
> For a very simple example, if we use vector subtraction to
> calculate what note results from tempering the Pythagorean
> "perfect 5th" by 1/12 of a Pythagorean comma, we get:
>
> 2^x 3^y
>
> |-12/12 12/12| = 2^-1 * 3^1 = 3/2 = "perfect 5th"
> - |-19/12 12/12| = ((2^-19)*(3^12))^(1/12) = 1/12 Pythagorean comma
> -----------------
> | 7/12 0 | = 2^(7/12) = 12-EDO "5th"

Upon re-reading this, it just struck me that perhaps this
was the reasoning which may have led the Sumerians to 12-EDO
(if I'm correct that they did "go there").

I'm not suggesting that the Sumerians conceived of numbers
in exponential terms, especially not numbers which describe
ratios of *frequencies* (well, OK, maybe I am), but if they
did, this calculation would look like this in base-60:

|-1 1|
- |-1,35 1|
------------
| 35 0|

So the "cycle of 5ths" in 12-EDO would be:

exponent
in base-60

2^(6/12) 30
2^(11/12) 55
2^(4/12) 20
2^(9/12) 45
2^(2/12) 10
2^(7/12) 35
2^(0/12) 0
2^(5/12) 25
2^(10/12) 50
2^(3/12) 15
2^(8/12) 40
2^(1/12) 5
2^(6/12) 30

which form an exponent-cycle of +35 (or -25) mod 60.

They certainly knew about the so-called "Pythagorean" cycle,
and my research leads me to believe that they knew how to
calculate 12-EDO. If they realized that the 12-EDO "5th"
is tempered by 1/12 of a Pythagorean comma, as is shown by
my vector subtraction, then there ought to be some evidence
of their reasoning.

On firmer ground, let's take a look at string-length calculations.
The string-length which produces 2^(7/12), which is quite
close to 1772/2655, can be described extremely accurately
in base-60 as 40,2,42,42,15.

Do any of you know of any "ancient mathematics" groups
whose subscribers might know about ancient references
to these numbers, if there are any?
(either groups or references, that is)

love / peace / harmony ...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>; <celestial-tuning@yahoogroups.com>
> Sent: Wednesday, August 01, 2001 12:07 AM
> Subject: [tuning-math] Sumerian 12-EDO? (was: [tuning] fractional
exponents of prime-factors)
>

>
>
> I wrote:
>
> > ... if we use vector subtraction to
> > calculate what note results from tempering the Pythagorean
> > "perfect 5th" by 1/12 of a Pythagorean comma, we get:
> >
> > 2^x 3^y
> >
> > |-12/12 12/12| = 2^-1 * 3^1 = 3/2 = "perfect 5th"
> > - |-19/12 12/12| = ((2^-19)*(3^12))^(1/12) = 1/12 Pythagorean comma
> > -----------------
> > | 7/12 0 | = 2^(7/12) = 12-EDO "5th"
>
>
>
> Upon re-reading this, it just struck me that perhaps this
> was the reasoning which may have led the Sumerians to 12-EDO
> (if I'm correct that they did "go there").

A few more quite accurate base-60 string-length ratios
which are important in pursuing this idea:

Pythagorean comma: 59,11,32,43,11

1/12 Pythagorean comma: 59,55,56,13,9

love / peace / harmony ...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com