interval arithmetic at Oracle

Finally getting some press, and perhaps a modicum of support
from a company that could actually do something about it:

http://labs.oracle.com/minds/2004-0527/

-Carl

Interesting article Carl.

How would this be applied to tuning?

Thanks,

chris

On Thu, Jul 22, 2010 at 3:41 PM, Carl Lumma <carl@lumma.org> wrote:

>
>
> Finally getting some press, and perhaps a modicum of support
> from a company that could actually do something about it:
>
> http://labs.oracle.com/minds/2004-0527/
>
> -Carl
>
>
>

I posted it here because it'd come up before, when Gene, Graham,
and I got slightly different values for something we were computing.

-Carl

At 01:31 PM 7/22/2010, you wrote:
>Interesting article Carl.
>
>How would this be applied to tuning?
>
>Thanks,
>
>chris
>
>On Thu, Jul 22, 2010 at 3:41 PM, Carl Lumma <<mailto:carl@lumma.org>carl@lumma.org> wrote:
>
>Finally getting some press, and perhaps a modicum of support
>from a company that could actually do something about it:
>
><http://labs.oracle.com/minds/2004-0527/>http://labs.oracle.com/minds/2004-0527/
>
>-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> I posted it here because it'd come up before, when Gene, Graham,
> and I got slightly different values for something we were computing.
>
> -Carl
>
> At 01:31 PM 7/22/2010, you wrote:
> >Interesting article Carl.
> >
> >How would this be applied to tuning?
> >
> >Thanks,
> >
> >chris
> >
> >On Thu, Jul 22, 2010 at 3:41 PM, Carl Lumma <<mailto:carl@...>carl@...> wrote:
> >
> >Finally getting some press, and perhaps a modicum of support
> >from a company that could actually do something about it:
> >
> ><http://labs.oracle.com/minds/2004-0527/>http://labs.oracle.com/minds/2004-0527/
> >
> >-Carl

Well, I've always believed in interval vectors. Interesting article,
Thanks.

PGH

At 02:46 PM 7/22/2010, you wrote:

>Well, I've always believed in interval vectors.

Very funny! :)

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 02:46 PM 7/22/2010, you wrote:
>
> >Well, I've always believed in interval vectors.
>
> Very funny! :)
>
> -C.

hehe. I guess I wasn't very discrete.

- PGH

On 22 July 2010 22:08, Carl Lumma <carl@lumma.org> wrote:
> I posted it here because it'd come up before, when Gene, Graham,
> and I got slightly different values for something we were computing.

I see that I used the phrase "interval arithmetic" once but I meant
the arithmetic of musical intervals. The calculations we're doing are
linear so extending the floating point precision should be enough.
And it's usually possible to do an algebraic proof that we're
measuring the same thing.

Graham

Graham wrote:
>> I posted it here because it'd come up before, when Gene, Graham,
>> and I got slightly different values for something we were computing.
>
>I see that I used the phrase "interval arithmetic" once but I meant
>the arithmetic of musical intervals.

Gene used it in the sense of the article.

>The calculations we're doing are
>linear so extending the floating point precision should be enough.
>And it's usually possible to do an algebraic proof that we're
>measuring the same thing.

It was more of an aside really.

-Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 22 July 2010 22:08, Carl Lumma <carl@...> wrote:
> > I posted it here because it'd come up before, when Gene, Graham,
> > and I got slightly different values for something we were computing.
>
> I see that I used the phrase "interval arithmetic" once but I meant
> the arithmetic of musical intervals. The calculations we're doing are
> linear so extending the floating point precision should be enough.
> And it's usually possible to do an algebraic proof that we're
> measuring the same thing.
>
>
> Graham

So the two things are really quite different, this form is really
analysis not algebra, right? (The first is a slaughterhouse and
the second a beautiful bullfight...ed.) When you say "interval
arithmetic" in what sense do you mean musical intervals? (There
is counting intervals and measuring them...)

PGH

On 27 July 2010 16:04, paulhjelmstad <phjelmstad@msn.com> wrote:
> So the two things are really quite different, this form is really
> analysis not algebra, right? (The first is a slaughterhouse and
> the second a beautiful bullfight...ed.) When you say "interval
> arithmetic" in what sense do you mean musical intervals? (There
> is counting intervals and measuring them...)

The two things are certainly different. I don't know which "this
form" is, but interval arithmetic proper can be analysis, according to
the link Gene gave before:

http://en.wikipedia.org/wiki/Interval_arithmetic

The article Carl found is about implementing it on computers, so that
you automatically get and upper and lower bound on a calculation.
That's probably a good idea, especially for beginners, because
floating point arithmetic can lead to errors if you don't understand
it -- and most people don't. Then again, I suspect the devil's in the
details, but smart people are working on it and I don't think it's
something I'll have to worry about in this lifetime.

What I called "interval arithmetic" before is the way musical
intervals add up. So a major and minor third add up to give a perfect
fifth, and two intervals of 5:4 and one of 9:7 may add up to give a
2:1 depending on the mapping that's in effect. In the future I'll be
more careful about the term unless I forget it has another meaning.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 27 July 2010 16:04, paulhjelmstad <phjelmstad@...> wrote:
> > So the two things are really quite different, this form is really
> > analysis not algebra, right? (The first is a slaughterhouse and
> > the second a beautiful bullfight...ed.) When you say "interval
> > arithmetic" in what sense do you mean musical intervals? (There
> > is counting intervals and measuring them...)
>
> The two things are certainly different. I don't know which "this
> form" is, but interval arithmetic proper can be analysis, according to
> the link Gene gave before:
>
> http://en.wikipedia.org/wiki/Interval_arithmetic
>
> The article Carl found is about implementing it on computers, so that
> you automatically get and upper and lower bound on a calculation.
> That's probably a good idea, especially for beginners, because
> floating point arithmetic can lead to errors if you don't understand
> it -- and most people don't. Then again, I suspect the devil's in the
> details, but smart people are working on it and I don't think it's
> something I'll have to worry about in this lifetime.

Yes, very useful. It's kind of interesting, even with Excel, you
get those very small numbers instead of zeros in inverse matrices,
determinants, adjugates, etc. GAP fares better....

> What I called "interval arithmetic" before is the way musical
> intervals add up. So a major and minor third add up to give a perfect
> fifth, and two intervals of 5:4 and one of 9:7 may add up to give a
> 2:1 depending on the mapping that's in effect. In the future I'll be
> more careful about the term unless I forget it has another meaning.
>
>
> Graham

Yes, so I guess you measure intervals and add them, etc. (do you ever
use the term "interval vector"? I'm trying to remember). The interval
vectors I use are musical set theory vectors, where I merely count
the different intervals 1 to 6 (or 1 to 11) in the 12t-ET collection.

When I think about it, although its boring, I could distinguish
intervals that are in chains, which add up to give other intervals,
such as a chain of 4 fifths adding up to a major third. But interval
vectors don't normally show the chains...

PGH

On 28 July 2010 16:22, paulhjelmstad <phjelmstad@msn.com> wrote:

> Yes, so I guess you measure intervals and add them, etc. (do you ever
> use the term "interval vector"? I'm trying to remember). The interval
> vectors I use are musical set theory vectors, where I merely count
> the different intervals 1 to 6 (or 1 to 11) in the 12t-ET collection.

I used to call them "interval vectors" before I learned about the
clash with musical set theory. I think I'd call them "ratio-space
vectors" now but I don't need to talk about them much.

> When I think about it, although its boring, I could distinguish
> intervals that are in chains, which add up to give other intervals,
> such as a chain of 4 fifths adding up to a major third. But interval
> vectors don't normally show the chains...

Which kind of interval vectors?

The mapping matrix for a temperament is a linear transformation. You
can use it to transform the ratio-space vector into something that
follows the correct arithmetic for the temperament you're looking at.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 28 July 2010 16:22, paulhjelmstad <phjelmstad@...> wrote:
>
> > Yes, so I guess you measure intervals and add them, etc. (do you ever
> > use the term "interval vector"? I'm trying to remember). The interval
> > vectors I use are musical set theory vectors, where I merely count
> > the different intervals 1 to 6 (or 1 to 11) in the 12t-ET collection.
>
> I used to call them "interval vectors" before I learned about the
> clash with musical set theory. I think I'd call them "ratio-space
> vectors" now but I don't need to talk about them much.
>
> > When I think about it, although its boring, I could distinguish
> > intervals that are in chains, which add up to give other intervals,
> > such as a chain of 4 fifths adding up to a major third. But interval
> > vectors don't normally show the chains...
>
> Which kind of interval vectors?

The set theory kind.

> The mapping matrix for a temperament is a linear transformation. You
> can use it to transform the ratio-space vector into something that
> follows the correct arithmetic for the temperament you're looking at.

Right, what I am trying to do is apply the same matrix math to
musical set theory interval vectors (MSTIV?)

I just thought today - one can have some sort of vector of chains and disjoint ones, maybe with colored numbers. For example, the vector for my set "A" (014589) is <303630>, so the chains would be <(1,1,1)(0)(1,1,1)(3*,3*)(1,1,1)(0)> and the (3,3) are also cyclic (048 and 159) so could be marked with an asterisk.

Any fool should be able to see that 3 thirds map into an octave,
which is the only correspondence in this set, since everything else
is disjoint. But I have no idea how to use this in a matrix, although
the normal vector <303630> still retains much valuable information.

PGH