Re: [tuning] Some questions

Keenan Pepper wrote,

> does anyone know of a way to find the "simplest" n-limit ratio
between two others? The stern-brocot tree does this on an infinite
limit, but how can you restrict it to say, 7-limit, without resorting
to guess-and-check?

Not sure if this is the sort of thing that your looking for, but one
easy way that comes right to mind to do this to the "simplest"
overtonal or undertonal n-limit (this would simply give ratios where
"n" or a multiple of "n" is in either the numerator or the denominator
of the ratio), would be by roundings of either an overtonal:

(mN / mD) * L
O = ---------------
L

or an undertonal:

L
U = ---------------
(mN / mD) * L

Where:

O = overtonal, i.e., the prime limit is the denominator
U = undertonal, i.e., the prime limit is the numerator
L = limit, i.e., or simply the "n" in your question
m = the mediant (or freshman sum), i.e., the "ratio between two
others" in your question
N = the numerator of any ratio
D = the denominator of any ratio

So say if your two ratios are 13/11 and 11/8, then the mediant would
be a 24/19. And:

(24/19)*7 9 44 221
---------- = --- , ---- , ----- , ...
7 7 35 175

And:

7 7 14 350
---------- = --- , ---- , ----- , ...
(19/24)*7 6 11 277

Of course you will have instances where a rounding is reducible to a
ratio that no longer satisfies whatever you've set "L" to be, and
where simpler roundings will not fall between two given ratios.

Dan

Earlier I wrote,

So say if your two ratios are 13/11 and 11/8, then the mediant would
be a 24/19. And:

(24/19)*7 9 44 221
---------- = --- , ---- , ----- , ...
7 7 35 175

And:

7 7 14 350
---------- = --- , ---- , ----- , ...
(19/24)*7 6 11 277

However, this should have read:

(24/19)*7 9 17 26 34 43
---------- = --- , ---- , ---- , ---- , ---- , ...
7 7 14 21 28 35

And:

7 7 14 21 28 35
---------- = --- , ---- , ---- , ---- , ---- , ...
(19/24)*7 6 11 17 23 29

(etc.)

As what I meant was roundings of successive multiples.

Dan

Keenan Pepper wrote,

>Also, does anyone know of a way to find the "simplest" n-limit ratio
between
>two others? The stern-brocot tree does this on an infinite limit,

Actually, it doesn't, but a similar idea, known as the mediant or "freshman
sum", does.

>but how
>can you restrict it to say, 7-limit, without resorting to guess-and-check?
>Or is this an open problem?...

Well, one could easily write a computer program to do this by brute force .
. .

>2/1, 3/2-45/32-3/2-8/5-3/2... 15/8, 2/1!

I would not only be interested in the simplicity of the ratio itself but
also the simplicity of the ratios it formed with the other notes in the
scale. For that purpose, drawing the lattice is the best approach.

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D. Stearns wrote:
> Not sure if this is the sort of thing that your looking for, but one
> easy way that comes right to mind to do this to the "simplest"
> overtonal or undertonal n-limit (this would simply give ratios where
> "n" or a multiple of "n" is in either the numerator or the denominator
> of the ratio), would be by roundings of either an overtonal:

After a while pondering, I see what you mean. Set one of the two (numerator
or denominator) to what you want and round the other to the nearest whole
number. Wouldn't that work with any numbers, though, not just multiples of
the limit?

> Of course you will have instances where a rounding is reducible to a
> ratio that no longer satisfies whatever you've set "L" to be, and
> where simpler roundings will not fall between two given ratios.

So it's basically a better form of guess-and-check.

Paul H. Erlich wrote:
>Actually, it doesn't, but a similar idea, known as the mediant or "freshman
>sum", does.

That's what I meant. Funny how something so dumb-looking could be so useful.

>Well, one could easily write a computer program to do this by brute force .

That's what I'm doing. It's sort of "brute force strategically applied,"
which is the best kind.

>I would not only be interested in the simplicity of the ratio itself but
>also the simplicity of the ratios it formed with the other notes in the
>scale. For that purpose, drawing the lattice is the best approach.

Good point. In other words, in lopsided scales (otonal scales are lopsided
toward positive exponents, utonal to negative) notes sound more consonant if
they're on the side it leans to.

Stay Tuned,
Keenan P.

P.S. For those of you who didn't figure it out, "2/1,
3/2-45/32-3/2-8/5-3/2... 15/8, 2/1!" is 5-limit JI for "shave and a
haircut."

Keenan!
http://www.anaphoria.com/cd.html has at the bottom some info on Creation of the Worlds
which structure is explained in http://www.anaphoria.com/cps.html sorry have not quite
figured out sound files so far!

Keenan Pepper wrote:

> First of all, I just realized I have never heard any music in a CPS, or even
> heard a CPS scale or cadence or anything. I understand the theory, but how
> does it sound? I need some web addresses!

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Keenan Pepper wrote,

> So it's basically a better form of guess-and-check.

Right, one where successively larger non-repeating N and D multiples
of L that are not reducible to something < L*1 are converging on a
given mediant.

Dan

Keenan Pepper wrote,

> So it's basically a better form of guess-and-check.

Yes, something like that that just came to mind where successively
larger non-repeating N and D multiples of L that are not reducible to
something < L*1 are converging on a given mediant.

Dan