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:

>

> 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:

> --- 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