tracking the threads

Hi Paul and Gene and everybody else,

I'm trying to read this list, and having trouble. Maybe you
can help. There were several ongoing topics between Gene and
Paul...

-----------------------------------------------------------------
1. Whether stretched TOP = Kees tuning
-----------------------------------------------------------------

>Graham, if he was indeed talking about the same thing (minimax
>"Kees tuning"), claimed that:
>
>If the corresponding TOP tuning has impure octaves, the "Kees"
>tuning is simply a uniform stretch/compression of the TOP
>tuning so as to acheive pure octaves.
>
>If the corresponding TOP tuning has pure octaves, the "Kees"
>tuning will actually be different! This is not too surprising,
>though, since any TOP tuning with pure octaves remains Tenney-
>optimal, striclty speaking, if the octaves are detuned
>slightly, as long as that's not enough to raise the *maximum*
>damage over all the primes (and hence over all ratios). So the
>original TOP tuning would have been somewhat arbitrary in this
>case anyway.
>
>I think Graham is right. And after he sees the correction
>below, I think Gene can prove it.

> <<3 -5 -6 -1 -15 -18 -12 0 15 18||
>
> with TM basis {56/55, 64/63, 77/75} and mapping
>
> [<1 3 0 0 3|, <0 -3 5 6 1|]
>
>The stretched TOP tuning has pure 7s, and the Kees tuning has
>the error of 3, over log(3), equal to the error of 5, over
>log(5).
>
>Stretched TOP:
>
> <1200 1915.578 2807.355 3368.826 4161.472|
>
>Kees:
>
> <1200 1915.929 2806.785 3368.142 4161.357|
>
>While the tunings are pretty close, they clearly are different.

So the answer is "no"?

I gather this was related to Paul's question about the hexagonal
contours on his plot of projective "vals"... Paul, could you
refresh us on what that question was and how it related to the
above thread? Thanks!

-----------------------------------------------------------------
2. Whether Gene's standard vals always have the lowest TOP
damage of any val.
-----------------------------------------------------------------

>>>>I vaguely remember that these are ones which, when TOP-
>>>>tuned, have the same TOP damage as the standard vals. But
>>>>since the standard vals themselves can often be beaten
>>>>in this regard, I'm probably remembering wrong.
>>>
>>>It depends on your definition of "TOP damage", I suppose. By
>>>the definition I was using, a standard val does have minimal
>>>TOP damage, and any val which does also is a semistandard
>>>val.
>>>
>>>For example, in 64-et we have <64 101 149| as the standard
>>>val, and it and <64 101 148| are the two 5-limit semistandard
>>>vals. Taking 64*<1 p3 p5|, where p3=log2(3) and p5=log2(5),
>>>and the difference to the vals, and dividing by 1, p3, p5
>>>respectively gives <0 .2761 .2599| for <64 101 148| and
>>><0 .2761 -.1708| for <0 101 149|. In the sense of maximum
>>>weighted absolute value, the TOP damage is identical. To
>>>choose one out of the list of semistandard vals as best, you
>>>could first get the list, which isn't hard, and then choose
>>>the best according to some other standard.

>>So you're not doing what I was talking about, which was
>>TOP-tuning first and then comparing the TOP damage. The
>>maximum weighed absolute value of these difference vectors
>>(one element of which is necessarily zero) seems rather
>>uninteresting, since it doesn't translate to a statement
>>about all the intervals in the tuning the way the TOP
>>damage of a TOP-tuned temperament does.
>>
>>My idea was that another way of defining the "standard val"
>>would be to choose the one which, when TOP-tuned, had the
>>lowest TOP damage.
//
>>...properties such as the fact that optimality remins even
>>if you restrict your attention to intervals within 1 octave,
>>or whatever bound on span you wish.
//
>>I was hoping that this would also be the one that when Kees
>>tuned, would have the lowest Kees damage, but you seem to be
>>saying (in another thread) that this isn't necessarily so
>>beyond the 5-limit.

>I think they should get less and less alike for higher prime
>limits.

So I take it standard and semistandard vals do not minimize
TOP damage in the resulting TOP tunings?

And is there agreement that the val that does do this is not
that same val that does it for Kees damage in Kees tuning?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >While the tunings are pretty close, they clearly are different.
>
> So the answer is "no"?

The answer is no, but in practice it's usually yes.

> So I take it standard and semistandard vals do not minimize
> TOP damage in the resulting TOP tunings?

Not unless you use my definition, which isn't what Paul means by it.

> And is there agreement that the val that does do this is not
> that same val that does it for Kees damage in Kees tuning?

The thing minimizing TOP damage by Paul's definition isn't really a
val but a tuning. When did Kees damage get into the picture, and how
is that defined?

>> >While the tunings are pretty close, they clearly are different.
>>
>> So the answer is "no"?
>
>The answer is no, but in practice it's usually yes.

Ok.

>> So I take it standard and semistandard vals do not minimize
>> TOP damage in the resulting TOP tunings?
>
>Not unless you use my definition, which isn't what Paul means by it.

See my previous post.

>> And is there agreement that the val that does do this is not
>> that same val that does it for Kees damage in Kees tuning?
>
>The thing minimizing TOP damage by Paul's definition isn't really a
>val but a tuning. When did Kees damage get into the picture, and how
>is that defined?

I'll let Paul answer this.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I gather this was related to Paul's question about the hexagonal
> contours on his plot of projective "vals"... Paul, could you
> refresh us on what that question was and how it related to the
> above thread? Thanks!

The hexagonal contours correspond to the contours of maximum 'Kees'
when the 'vals' are interpreted to have pure octaves, and they also
correspond to the contours of maximum Tenney error when the 'vals'
are interpreted to be TOP-tuned in each case. This observation is
pretty much equivalent to Graham's claim -- minimax 'Kees' tunings
are stretched or compressed TOP tunings. I was wondering if such a
situation remained true in the 7-limit, where the Kees contours would
be rhombic dodecahedra . . .

> So I take it standard and semistandard vals do not minimize
> TOP damage in the resulting TOP tunings?

Right, they don't do so in general.

> And is there agreement that the val that does do this is not
> that same val that does it for Kees damage in Kees tuning?

I don't know for sure yet, but in the 5-limit, it basically *is* the
same val.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > >While the tunings are pretty close, they clearly are different.
> >
> > So the answer is "no"?
>
> The answer is no, but in practice it's usually yes.

Hmm . . .

> > So I take it standard and semistandard vals do not minimize
> > TOP damage in the resulting TOP tunings?
>
> Not unless you use my definition, which isn't what Paul means by it.

Huh?

> > And is there agreement that the val that does do this is not
> > that same val that does it for Kees damage in Kees tuning?
>
> The thing minimizing TOP damage by Paul's definition isn't really a
> val but a tuning. When did Kees damage get into the picture, and how
> is that defined?

I thought you defined it yourself: For a ratio n/d, it's

mistuning(n/d) / log(l),

where l is defined as the largest odd number that is a (not
necessarily prime) factor of either n or the d.

>> I gather this was related to Paul's question about the hexagonal
>> contours on his plot of projective "vals"... Paul, could you
>> refresh us on what that question was and how it related to the
>> above thread? Thanks!
>
>The hexagonal contours correspond to the contours of maximum 'Kees'
>when the 'vals' are interpreted to have pure octaves, and they also
>correspond to the contours of maximum Tenney error when the 'vals'
>are interpreted to be TOP-tuned in each case. This observation is
>pretty much equivalent to Graham's claim -- minimax 'Kees' tunings
>are stretched or compressed TOP tunings.

But didn't Gene show that claim to be false?

>I was wondering if such a situation remained true in the 7-limit,
>where the Kees contours would be rhombic dodecahedra . . .

I thought this was shown to be untrue too (?).

>> So I take it standard and semistandard vals do not minimize
>> TOP damage in the resulting TOP tunings?
>
>Right, they don't do so in general.
>
>> And is there agreement that the val that does do this is not
>> that same val that does it for Kees damage in Kees tuning?
>
>I don't know for sure yet, but in the 5-limit, it basically *is* the
>same val.

Great. Well keep us posted on developments! I think this stuff
is really exciting.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> I gather this was related to Paul's question about the hexagonal
> >> contours on his plot of projective "vals"... Paul, could you
> >> refresh us on what that question was and how it related to the
> >> above thread? Thanks!
> >
> >The hexagonal contours correspond to the contours of
maximum 'Kees'
> >when the 'vals' are interpreted to have pure octaves, and they
also
> >correspond to the contours of maximum Tenney error when the 'vals'
> >are interpreted to be TOP-tuned in each case. This observation is
> >pretty much equivalent to Graham's claim -- minimax 'Kees' tunings
> >are stretched or compressed TOP tunings.
>
> But didn't Gene show that claim to be false?

No, not in the 5-limit.

> >I was wondering if such a situation remained true in the 7-limit,
> >where the Kees contours would be rhombic dodecahedra . . .
>
> I thought this was shown to be untrue too (?).

Yes, assuming Gene's figures were correct, but I haven't checked them
yet. He's made some errors already in this thread . . .

>>>> 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.
>
>>>> 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?
>
>>>> 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).
>
>>>> 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?
>>>
>>> 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.
>
>Linf TOP complexity and Kees complexity -- When these aren't exactly
>the same, which itself is pretty often, they usually seem to be the
>same to three decimal places. I didn't find an example differing by
>more than 1% in my old faithful list of 45 seven-limit temperaments.

-C.

I don't know if you meant to add anything, Carl. This isn't a
complete collection of what's been posted in this thread, such as my
subsequent questions. But it's by far the most interesting thing here
to me right now -- if appropriately defined, at least in some
circumstances, complexity as assessed by a "generators per
consonance" type measure, like we (Dave Keenan is part of this "we")
always used to use, is exactly equivalent to that assessed by a "size
of the periodic structure in the original JI lattice -- length of
vanishing comma (or area/volume of wedge product of vanishing
commas)" type measure. Fascinating, and what I had hoped on a gut
level for a long time. I'd like to see my questions answered!

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>>> 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.
> >
> >>>> 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?
> >
> >>>> 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).
> >
> >>>> 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?
> >>>
> >>> 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.
> >
> >Linf TOP complexity and Kees complexity -- When these aren't
exactly
> >the same, which itself is pretty often, they usually seem to be the
> >same to three decimal places. I didn't find an example differing by
> >more than 1% in my old faithful list of 45 seven-limit
temperaments.
>
> -C.

>I don't know if you meant to add anything, Carl.

Nope, just a recap. Actually, it kind of got sent by mistake.
Mainly, I'm just trying to keep up with developments. With a
baby on the way I haven't been taking the time to think about
much of anything in-depth.

>But it's by far the most interesting thing here
>to me right now -- if appropriately defined, at least in some
>circumstances, complexity as assessed by a "generators per
>consonance" type measure, like we (Dave Keenan is part of this "we")
>always used to use, is exactly equivalent to that assessed by a
>"size of the periodic structure in the original JI lattice -- length
>of vanishing comma (or area/volume of wedge product of vanishing
>commas)" type measure. Fascinating, and what I had hoped on a gut
>level for a long time. I'd like to see my questions answered!

If I may hazard an interpretation of the weaknesses in discourse
in this thread... Gene's not a robot. He's brilliant enough to
make it seem like he can do anything, but my impression is that
these things are just as hard for him as they are for anyone
else... and his particular way of thinking about them is very
brainstormy and perhaps a bit random. He also seems prone to
silence when he doesn't know something (which is better than the
reverse). For this and maybe other reasons, he seems to drop
threads, and this causes folks to be more adamant with him, which
he then resents mildly. Perhaps he intends to get back to them
when/if the answer percolates to the top. Sorry to talk about
you in the third person here, Gene.

Despite your best intentions, Paul (actually, I think it may be
due to them!), you can come across as pushy/annoying/patronizing.
(God knows how I come across -- please feel free to let it rip.)

You're both among the most talented communicators I've known on
these lists (believe it or not), so don't take this badly. And
I've met you both and think you're both fantastic fellows. And
these are just my impressions... could be totally wrong here.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> If I may hazard an interpretation of the weaknesses in discourse
> in this thread... Gene's not a robot. He's brilliant enough to
> make it seem like he can do anything, but my impression is that
> these things are just as hard for him as they are for anyone
> else... and his particular way of thinking about them is very
> brainstormy and perhaps a bit random.

You noticed. In the case of Kees complexity, I don't have it sorted out as
well as I would like, but I'm also not clear what Paul thinks I should be
sorting out in terms of the relationship between Kees and TOP, and also
Kees generator complexity. I looked at the L1 norm business, and as I said,
it was very close, and sometimes exactly the same, but not always.
I can't see how to work it into a metric definition along the lines of L1 TOP.

Hence, lately I've been thinking about other things, such as consonance
circles and the planar lattice of breed major/minor tetrad pairs.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > If I may hazard an interpretation of the weaknesses in discourse
> > in this thread... Gene's not a robot. He's brilliant enough to
> > make it seem like he can do anything, but my impression is that
> > these things are just as hard for him as they are for anyone
> > else... and his particular way of thinking about them is very
> > brainstormy and perhaps a bit random.
>
> You noticed. In the case of Kees complexity, I don't have it sorted
out as
> well as I would like, but I'm also not clear what Paul thinks I
should be
> sorting out in terms of the relationship between Kees and TOP, and
also
> Kees generator complexity.

It's the relationship between Kees generator complexity and some sort
of Kees wedgie complexity, *or* the relationship between TOP
generator complexity (if this can be defined) and the TOP wedgie
complexities we've used, that I'd like to see pinned down. Generator
complexity was so important to Dave Keenan and others . . . showing
that the wedgie point of view subsumes or is equivalent to this would
be nice, to say the least.

> I looked at the L1 norm business, and as I said,
> it was very close, and sometimes exactly the same, but not always.
> I can't see how to work it into a metric definition along the lines
>of L1 TOP.

You lost me. What do you mean by this last statement?

> Hence, lately I've been thinking about other things, such as
consonance
> circles and the planar lattice of breed major/minor tetrad pairs.

Awesome.

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

> It's the relationship between Kees generator complexity and some
sort
> of Kees wedgie complexity, *or* the relationship between TOP
> generator complexity (if this can be defined) and the TOP wedgie
> complexities we've used, that I'd like to see pinned down.

The most obvious thing from that point of view is that the formulas
for Kees generator complexity and TOP Linf complexity are quite
similar and lead to similar, and often identical, results. TOP Linf
complexity takes the wedgie and divides the p-q coordinate, where p
and q are primes, by log2(p)log2(q), and then takes the maximum of
the absolute values. Kees generator complexity starts the same way,
but only uses the generator coordinates, where one of the primes is
2. Then it adds the absolute values of the differences between two
of these to the list of things to be maximized.

Generator
> complexity was so important to Dave Keenan and others . . .
showing
> that the wedgie point of view subsumes or is equivalent to this
would
> be nice, to say the least.

Well, the above defines Kees generator complexity in terms of the
wedgie in a manner parallel to TOP Linf if that helps.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > It's the relationship between Kees generator complexity and some
> sort
> > of Kees wedgie complexity, *or* the relationship between TOP
> > generator complexity (if this can be defined) and the TOP wedgie
> > complexities we've used, that I'd like to see pinned down.
>
> The most obvious thing from that point of view is that the formulas
> for Kees generator complexity and TOP Linf complexity are quite
> similar and lead to similar, and often identical, results.

OK, maybe you demonstrate this explicitly? But remember that what I'm also interested in
is relating generator complexity to wedgie complexity (probably L1?) *within* either an
fixed-octave, odd-limit (e.g., "Kees") framework *or* a temperable-octave, no-removal-
of-factors-of-2 (e.g.,"TOP") framework, where gnerator complexity and wedgie complexity
are *both* evaluated in the *same* framework. Does that make any sense?

> TOP Linf
> complexity takes the wedgie and divides the p-q coordinate, where p
> and q are primes, by log2(p)log2(q), and then takes the maximum of
> the absolute values. Kees generator complexity starts the same way,
> but only uses the generator coordinates, where one of the primes is
> 2. Then it adds the absolute values of the differences between two
> of these to the list of things to be maximized.

Minimized?

This is interesting and reminds me of one of the open puzzles we found -- how does the
wedgie complexity relate to the number of notes per 2:1 in the TOP tuning for ETs. Neither
L1 or Linf ended up being proportional to the number of notes, but perhaps the right
complexity measure may have been something like this, where absolute values of
differences needed to be included. Any insight?

> Generator
> > complexity was so important to Dave Keenan and others . . .
> showing
> > that the wedgie point of view subsumes or is equivalent to this
> would
> > be nice, to say the least.
>
> Well, the above defines Kees generator complexity in terms of the
> wedgie in a manner parallel to TOP Linf if that helps.

Unfortunately, I'm far from being able to turn this definition into a general comparison.
How, for example, does one transform from the p-q coordinates to the generator
coordinates in the general case? Seems like some heavy mathematical lifting . . . ?