Kees complexity of a linear temperament

Suppose we have a linear temperament in the strict sense, so that the
octave is the period. We then will have generator and period vals;
suppose v is the val giving the number of generator steps to an
interval q. Then

sup_{q != 1} |v(q)|/|| q ||_kees

gives an interesting definition of complexity. Here || q ||_kees is
the Kees norm; the log of the maximum of the numerator or denominator
of the odd part of q. Hence we have number of generator steps to q,
divided by the Kees norm (expressibility) of q, which has a least
upper bound, which is the Kees complexity.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Suppose we have a linear temperament in the strict sense, so that the
> octave is the period.

Presumably the thing to do to generalize to any rank two temperament
is to multiply by the number of periods to the octave. If we do that,
we have both a Kees error and a Kees complexity, and can put the two
together to get Kees logflat badness, which might be worth exploring.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Suppose we have a linear temperament in the strict sense, so that
the
> > octave is the period.
>
> Presumably the thing to do to generalize to any rank two temperament
> is to multiply by the number of periods to the octave. If we do that,
> we have both a Kees error and a Kees complexity, and can put the two
> together to get Kees logflat badness, which might be worth exploring.

Isn't the Kees complexity of a rank two, codimension-1 temperament
simply the expressibility of the vanishing comma? If not, why not?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Isn't the Kees complexity of a rank two, codimension-1 temperament
> simply the expressibility of the vanishing comma? If not, why not?

Because I'm defining a weighted linear complexity which depends on how
many weighted steps it takes to get to an interval. Instead of "Kees
complexity" another name might be better. What?

In the case of 81/80, we don't get log2(81), but 4/log2(5) for this
complexity. This is because if we start from <0 1 4|, and divide by
logs, ignoring the 0, part, we get [1/p3 4/p5] where p3=log2(3) and
p5=log2(5). Then of 1/p3, 4/p5, and 1/p3-4/p5, the largest absolute
value is 4/p5, which is therefore the complexity. Note I defined the
complexity in terms of a supremum, but as usual one can reduce the
problem to a finitistic one.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Isn't the Kees complexity of a rank two, codimension-1 temperament
> > simply the expressibility of the vanishing comma? If not, why not?
>
> Because I'm defining a weighted linear complexity which depends on how
> many weighted steps it takes to get to an interval.

It seems to me the two measures should be proportional somehow. Or is
your measure dependent on which interval you choose, rather than
applying to the temperament as a whole?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> It seems to me the two measures should be proportional somehow.

I thought they wouldn't be, but apparently I'm wrong; for 5-limit
temperaments based on a single comma, they seem to be in a proportion of
log2(3)log2(5).

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > It seems to me the two measures should be proportional somehow.
>
> I thought they wouldn't be, but apparently I'm wrong;

Whew!

> for 5-limit
> temperaments based on a single comma, they seem to be in a proportion
of
> log2(3)log2(5).

Nice. So "comma complexity" and "generation complexity" do amount to
the same thing after all. Are you with me that this is a significant
observation? Is there a similar proportionality one could come up with
in the "original" Tenney case?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Nice. So "comma complexity" and "generation complexity" do amount to
> the same thing after all.

They do in the 5-limit, which turns out to be easy to show. In higher
limits, of course you get more than one comma, which makes the
relationship to comma complexity less direct. However, what happens in
the 5-limit case suggests in higher limits we still have something
significant, which might be detachable from the generator business,
along the lines of L1 and Linf Tenney complexity. I've been puzting
around with circulating scales today, but I expect I'll look at it soon.

>> Nice. So "comma complexity" and "generation complexity" do amount
>> to the same thing after all.
>
>They do in the 5-limit, which turns out to be easy to show. In
>higher limits, of course you get more than one comma, which makes
>the relationship to comma complexity less direct.

What ever happened to the concept of "straightness"?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Nice. So "comma complexity" and "generation complexity" do amount
to
> > the same thing after all.
>
> They do in the 5-limit, which turns out to be easy to show. In
higher
> limits, of course you get more than one comma, which makes the
> relationship to comma complexity less direct.

But we have other complexity measures, such those you mention below.

> However, what happens in
> the 5-limit case suggests in higher limits we still have something
> significant, which might be detachable from the generator business,
> along the lines of L1 and Linf Tenney complexity.

Comma complexity is proportional to L1 Tenney complexity in the 5-
limit case. Therefore, generation complexity as you've defined it is
proportional (possibly equal!) to L1 Tenney complexity at least in
the 5-limit case. What I'm hoping is that this can be extended to the
7-limit case.

> I've been puzting
> around with circulating scales today, but I expect I'll look at it
>soon.

Awesome! I can't wait, because any such result would be worth
discussing in the 'Middle Path' paper when I revise it again.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Nice. So "comma complexity" and "generation complexity" do amount
> >> to the same thing after all.
> >
> >They do in the 5-limit, which turns out to be easy to show. In
> >higher limits, of course you get more than one comma, which makes
> >the relationship to comma complexity less direct.
>
> What ever happened to the concept of "straightness"?

It's taken care of by the wedge product. Apply a suitable metric (such
as L1) to the wedge product of the vectors, and you have a measure
of "area", which is what we were looking for in the first place
when "straightess" came up. If you still want to
measure "straightness", you can divide this area by the product of
the "lengths" of the vectors . . .