a question regarding error optimizations

I haven't been able to follow all the miracle generator posts by a
long shot, so this may have already come up before. But I was
wondering if anyone has tried to optimize the generators trough a
given limit by also weighting the cross-set or diamond so that either
the simplest ratios are best approximated or the most complex ratios
are best approximated? (Or perhaps so that the big power of two otonal
identities are optimized and the rest of the cross-set is offset in
relation to that.)

thanks,

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> I haven't been able to follow all the miracle generator posts by a
> long shot, so this may have already come up before. But I was
> wondering if anyone has tried to optimize the generators trough a
> given limit by also weighting the cross-set or diamond so that
either
> the simplest ratios are best approximated or the most complex ratios
> are best approximated? (Or perhaps so that the big power of two
otonal
> identities are optimized and the rest of the cross-set is offset in
> relation to that.)

I'm currently fond of one of these. I posted the miracle generator
that minimises the max absolute error when the errors are weighted
according to the odd limit of the ratio. e.g. I multiply the error in
the 2:3 by 3, the errors in the 4:5 and 5:6 by 5. etc., so I best
approximate the more complex ratios.

That's an interesting idea, favouring the "rooted" (i.e. power of two)
dyads. Tell us what weight you want for each dyad and what generator,
and we'll tell you the optimum size of the generator. You can also
choose between max absolute (MA) or root mean squared (RMS).

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > But I was
> > wondering if anyone has tried to optimize the generators through a
> > given limit by also weighting the cross-set or diamond so that
> either
> > the simplest ratios are best approximated or the most complex ratios
> > are best approximated? (Or perhaps so that the big power of two
> otonal
> > identities are optimized and the rest of the cross-set is offset in
> > relation to that.)
>
> I'm currently fond of one of these. I posted the miracle generator
> that minimises the max absolute error when the errors are weighted
> according to the odd limit of the ratio. e.g. I multiply the error in
> the 2:3 by 3, the errors in the 4:5 and 5:6 by 5. etc., so I best
> approximate the more complex ratios.
>
> That's an interesting idea, favouring the "rooted" (i.e. power of two)
> dyads. Tell us what weight you want for each dyad and what generator,
> and we'll tell you the optimum size of the generator. You can also
> choose between max absolute (MA) or root mean squared (RMS).

This seems like a thread for my new group, practical harmonic entropy.
<Just kidding> In my scale search program, I am currently giving equal
weighting to minimizing overall complexity (sum of complexity of all
dyads, where complexity is just the product of numerator and
denominator), minimizing total error (without any weighting) and
minimizing something like variance, that is that there will be no mode
with a real clam in it.

In the complexity formula I lighly favor rooted otonal dyads by
dividing ratios with powers of two in the denominator by two. This
gave me results that I wanted to see 6/5 more complex than 5/3 more
comlex than 5/4 etc...

But the optimization function is the intelligence of
the whole thing, thats where you say "this is how I think the
numbers relate to my ears" (assuming you are looking for material
to listen to). (Yes Paul, I have downloaded the data you sent
to the HE list and will see if ts any differen than data I'm
using, thanks).

It is only this weekend that I have reassmbled my "music room", so
now I am able to aurally examine whether any of this (my program,
Miracle, white key M-scales, etc...) makes any sense to me as a
musician.

And Jeff, the cream really helped, thanks a lot.

Bob Valentine

>
> -- Dave Keenan

Dave K.,

I hadn't really thought a weighting scheme out for the power of 2
idea, though if something jumps right to mind I'd be interested in the
optimized generators at the 7, 11 and 13 limits.

In what ways do the results differ between max absolute (MA) and root
mean squared (RMS)?

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Dave K.,
>
> I hadn't really thought a weighting scheme out for the power of 2
> idea, though if something jumps right to mind I'd be interested in
the
> optimized generators at the 7, 11 and 13 limits.
>
> In what ways do the results differ between max absolute (MA) and
root
> mean squared (RMS)?

MA takes no account of what the other errors are doing, it only
minimises the errors in the two intervals that have the worst
approximations.

RMS doesn't worry about how big the largest error is, but it considers
all errors as contributing to the pain according to the square of
their size.

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> Or perhaps so that the big power of two otonal
> identities are optimized

Hi Dan -- the terminology is confusing enough already, so I suggest
we try to avoid talking about dyads as otonal or utonal. I understand
you're thinking of an interval like 5:4, but 5:4 is no more otonal
than utonal. From an otonal perspective, it's the fourth and fifth
harmonics, with the fourth harmonic octave equivalent to, and closer
to, the implied fundamental. From a utonal perspective, it's the
fourth and fifth subharmonics, with the fourth subharmonic octave
equivalent to, and closer to, the guide tone. Partch, who introduced
the terms otonal and utonal, makes it very clear that any ratio is
equally otonal and utonal, and it is only with triads and larger
chords where a more otonal or more utonal character can be discerned.

Not that I believe in duality, but the terms otonal and utonal do
relate to a dualistic perspective, and I think it would be best to
avoid clouding their meaning.

--- In tuning@y..., BVAL@I... wrote:

> otonal dyads

Hi Bob, I'm being a complete anal-retentive picky doofus, but see my
last message to Dan on this subject. Partch was nice enough to give
us the language we're speaking; let's honor his memory by trying to
retain clarity when using it. It will only help all of us, especially
since misunderstanding seems to be the rule rather than the exception
in this electronic medium of communication.