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"

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> 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"

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