interesting observation on sruti proportions

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, July 14, 2001 2:25 PM
> Subject: [tuning] "typical" sruti ratios [was: what a dope (user)]
>
>
> You can see here the three Pythagorean equivalents for
> the sruti names given by Lentz. The three types of
> srutis have the following Pythagorean ratios and
> interval sizes in approximate cents:
>
>
> Ratio Lentz's 5-limit Indian
> 2^x * 3^y ~cents equivalent sruti name
>
> | -19 12 | 23.46001038 81/80 pramana
> | 27 -17 | 66.76498529 25/24 nyuna
> | 8 - 5 | 90.22499567 -- purana

I was playing around with these Pythagorean srutis on an
Excel graph, trying to see what kind of proportions they
exhibit among themselves, and found something very interesting.

First, I noticed just by changing the scale of the y-axis to
a major unit of 22.5 cents that the approximate proportions
of the cents-values of these three srutis is 1:3:4.

I vaguely remember reading something about this somewhere,
in that the intervals between pitches of the Indian scale
were given in that book as 1, 3, or 4 srutis. The specific
type of sruti would thus have been the purana, tho not so
indicated in that book.

Then I went on and did the math, and look what I found:

90.22499567 / 23.46001038 = 3.845906042 = 3 & 3151/3725
66.76498529 / 23.46001038 = 2.845906042 = 2 & 3151/3725

Isn't it odd that these proportions both have *exactly*
the same fractional part? Or is it a property of Pythagorean
tuning that makes this so?

The denominator, 3725, is prime-factored into (5^2) * 149,
in case there's any significance in that.

Very curious,

-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

monz wrote:

> First, I noticed just by changing the scale of the y-axis to
> a major unit of 22.5 cents that the approximate proportions
> of the cents-values of these three srutis is 1:3:4.

I think you've discovered 53 tone equal temperament.

> Then I went on and did the math, and look what I found:
>
> 90.22499567 / 23.46001038 = 3.845906042 = 3 & 3151/3725
> 66.76498529 / 23.46001038 = 2.845906042 = 2 & 3151/3725
>
> Isn't it odd that these proportions both have *exactly*
> the same fractional part? Or is it a property of Pythagorean
> tuning that makes this so?

The largest sruti is the sum of the other two.

90.22499567 / 23.46001038

= (66.76498529 + 23.46001038) / 23.46001038

= 66.76498529 / 23.46001038 + 1

Graham

> From: <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, July 15, 2001 1:56 AM
> Subject: [tuning-math] Re: interesting observation on sruti proportions
>
>
> monz wrote:
>
> > First, I noticed just by changing the scale of the y-axis to
> > a major unit of 22.5 cents that the approximate proportions
> > of the cents-values of these three srutis is 1:3:4.
>
> I think you've discovered 53 tone equal temperament.

Yup, I noticed that too! Thanks.

>
>
> > Then I went on and did the math, and look what I found:
> >
> > 90.22499567 / 23.46001038 = 3.845906042 = 3 & 3151/3725
> > 66.76498529 / 23.46001038 = 2.845906042 = 2 & 3151/3725
> >
> > Isn't it odd that these proportions both have *exactly*
> > the same fractional part? Or is it a property of Pythagorean
> > tuning that makes this so?
>
> The largest sruti is the sum of the other two.
>
> 90.22499567 / 23.46001038
>
> = (66.76498529 + 23.46001038) / 23.46001038
>
> = 66.76498529 / 23.46001038 + 1

Hmmm... thanks for explaining this, Graham. Could you go
into a little more detail as to why the fractions work
out the way they do? It's interesting to me, with my
numerology fetish.

-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

"The" "Monz" wrote:

> > The largest sruti is the sum of the other two.
> >
> > 90.22499567 / 23.46001038
> >
> > = (66.76498529 + 23.46001038) / 23.46001038
> >
> > = 66.76498529 / 23.46001038 + 1
>
>
> Hmmm... thanks for explaining this, Graham. Could you go
> into a little more detail as to why the fractions work
> out the way they do? It's interesting to me, with my
> numerology fetish.

Pythagorean tuning can be defined using two intervals. So you can never
have more than two Pythagorean intervals that are all independent. That's
the only mathematically interesting thing I see here.

Graham