ratio presentation

Well, here's a simple one that I need some help with.

I don't understand the process of finding "common denominators" for
some of the examples on the Monz I-IV-V7-I webpage.

For example, we have here:

"The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions,
and the V7 to 36:45:54:64 = 4:5:6|27:32."

Now, *finally* I understand that Monz was simply trying to reduce the
triad and show the size of the 7th with the use of the "slash," not
trying to do any fancy kind of division... :)

But, how do we get the 36:45:54:64??

The first three numbers 4:5:6 are multiplied by *9* and the 7th,
27:32 has each number multiplied by *2*. Whyzzat??

There must be some kind of "common denominator" or some such in
36:45:54:64 that I'm not seeing.

Why is it presented this way (again?)

JP

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
> Well, here's a simple one that I need some help with.
>
> I don't understand the process of finding "common
denominators" for
> some of the examples on the Monz I-IV-V7-I webpage.
>
> For example, we have here:
>
> "The I and IV triads are tuned to the 5-limit JI 4:5:6 proportions,
> and the V7 to 36:45:54:64 = 4:5:6|27:32."
>
> Now, *finally* I understand that Monz was simply trying to
reduce the
> triad and show the size of the 7th with the use of the "slash,"
not
> trying to do any fancy kind of division... :)
>
> But, how do we get the 36:45:54:64??
>
> The first three numbers 4:5:6 are multiplied by *9* and the 7th,
> 27:32 has each number multiplied by *2*. Whyzzat??
>
> There must be some kind of "common denominator" or some
such in
> 36:45:54:64 that I'm not seeing.
>
> Why is it presented this way (again?)

joseph, this presentation shows the intervals between
*adjacent* notes.

those intervals are 4:5, 5:6, and 27:32.

just major third, just minor third, pythagorean minor third.

now to get the entire chord in lowest terms . . .

note that 4:5 and 5:6 can already be combined into 4:5:6.

but 6 is not 27.

the lowest common multiple of 6 and 27 is 54:

6*9 = 54
27*2 = 54

so we multiply the terms in 4:5:6 by 9, and multiply the terms in
27:32 by 2.

we get 36:45:54, and 54:64.

now we can stick them "back together", and get 36:45:54:64

that's all there is to it . . .

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

/tuning-math/message/3343

>
> joseph, this presentation shows the intervals between
> *adjacent* notes.
>
> those intervals are 4:5, 5:6, and 27:32.
>
> just major third, just minor third, pythagorean minor third.
>
> now to get the entire chord in lowest terms . . .
>
> note that 4:5 and 5:6 can already be combined into 4:5:6.
>
> but 6 is not 27.
>
> the lowest common multiple of 6 and 27 is 54:
>
> 6*9 = 54
> 27*2 = 54
>
> so we multiply the terms in 4:5:6 by 9, and multiply the terms in
> 27:32 by 2.
>
> we get 36:45:54, and 54:64.
>
> now we can stick them "back together", and get 36:45:54:64
>
> that's all there is to it . . .

***Well, this makes a lot of sense, and I'm glad that it didn't say
in the introductory page of this forum that this had to be "Advanced"
Tuning Math... :)

However,

> the lowest common multiple of 6 and 27 is 54:
>
> 6*9 = 54
> 27*2 = 54

Is there a "method" for doing this? Surely one doesn't just
go "multiplying around" until one arrives at the same number?? Or is
that how to do it??

??

Thanks!!!!

JP

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:

> Is there a "method" for doing this? Surely one doesn't just
> go "multiplying around" until one arrives at the same number??

You can find the greatest common divisor of two numbers by Euclid's algorithm, which is related to continued fractions. Calling that
gcd(a,b), we have lcm(a,b) = a*b/gcd(a,b).

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning-math/message/3345

> --- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Is there a "method" for doing this? Surely one doesn't just
> > go "multiplying around" until one arrives at the same number??
>
> You can find the greatest common divisor of two numbers by Euclid's
algorithm, which is related to continued fractions. Calling that
> gcd(a,b), we have lcm(a,b) = a*b/gcd(a,b).

***Thanks, Gene!

See, on the other list, I *said* I was supposed to be studying
Euclid!... :)

the lowest common multiple of 6 and 27 is 54:

6*9 = 54
27*2 = 54

So, let's see:

lcm(6,27) = 162/gcd(6,27)

lcm = 162/3

lcm = 54

It rather looks like it works! :)

JP