Tops and lattices

My vector space analysis was designed to find the optimal tuning in
terms of the Tenney metric, so I thought it would turn out to be the
same as TOP. However, I'm getting that 1/4-comma is optimal for
meantone, and TOP had flattened octaves. If Paul would post again the
TOP values for a few commas, making sure they are what he wants, I'll
try to find out what the story is here. Anyway I'm happy with my
approach, which seems to finally do what I was trying to do in the
canonical tuning thread.

Another issue is the naming of spaces and lattices. We have the
following normed vector spaces and lattices living in them:

Space A

Space A for the p prime limit is the normed vector space with norm

|| |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap|

Living inside space A is the lattice whose coordinates are integers.
Should Space A be the Tenney space and the lattice the Tenney lattice?

Space B

Dual to space A is Space B, of linear functionals on A. It is a normed
vector space with norm

|| <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p)|)

Living inside of Space B is the lattice whose coordinates are
integers, the lattice of vals. What is a good name for Space B and the
lattice of vals?

Also in Space B is a special point:

<1 log2(3) ... log2(p)|

I called this SIZE, but it could be called, for example, JIP for the
JI point. What do people think?

It might also be noted that if v is an equal temperament val, then

||v/v(2) - JIP||

is a Tenney-friendly measure of badness for v.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> My vector space analysis was designed to find the optimal tuning in
> terms of the Tenney metric, so I thought it would turn out to be the
> same as TOP. However, I'm getting that 1/4-comma is optimal for
> meantone, and TOP had flattened octaves. If Paul would post again
the
> TOP values for a few commas, making sure they are what he wants,
I'll
> try to find out what the story is here.

You mean something like this?

/tuning-math/message/8374

> Anyway I'm happy with my
> approach, which seems to finally do what I was trying to do in the
> canonical tuning thread.

But you just finished demonstrating that my method leads to Tenney-
weighted minimax, and there can't be more than one solution for
Tenney-weighted minimax -- right?

> Another issue is the naming of spaces and lattices. We have the
> following normed vector spaces and lattices living in them:
>
> Space A
>
> Space A for the p prime limit is the normed vector space with norm
>
> || |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap|
>
> Living inside space A is the lattice whose coordinates are integers.
> Should Space A be the Tenney space and the lattice the Tenney
lattice?

Sure.

> Space B
>
> Dual to space A is Space B, of linear functionals on A. It is a
normed
> vector space with norm
>
> || <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p)|)
>
> Living inside of Space B is the lattice whose coordinates are
> integers, the lattice of vals. What is a good name for Space B and
the
> lattice of vals?

Error space and the temperament lattice??????

> Also in Space B is a special point:
>
> <1 log2(3) ... log2(p)|
>
> I called this SIZE, but it could be called, for example, JIP for
the
> JI point. What do people think?

Seems to make a lot more sense than "SIZE"!

> It might also be noted that if v is an equal temperament val, then
>
> ||v/v(2) - JIP||
>
> is a Tenney-friendly measure of badness for v.

Examples?

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

> You mean something like this?
>
> /tuning-math/message/8374

Not exactly. What about a short list, giving your tuning for 2, 3,
and 5 for various 5-limit commas?

> > Anyway I'm happy with my
> > approach, which seems to finally do what I was trying to do in the
> > canonical tuning thread.
>
> But you just finished demonstrating that my method leads to Tenney-
> weighted minimax, and there can't be more than one solution for
> Tenney-weighted minimax -- right?

No, I just finished finding out what Tenney optimized temperaments
ought to be, under the assumption that that must be TOP. But it
doesn't seem to be, and in fact is more like my canonical temperament
business.

> > Another issue is the naming of spaces and lattices. We have the
> > following normed vector spaces and lattices living in them:
> >
> > Space A
> >
> > Space A for the p prime limit is the normed vector space with norm
> >
> > || |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap|
> >
> > Living inside space A is the lattice whose coordinates are
integers.
> > Should Space A be the Tenney space and the lattice the Tenney
> lattice?
>
> Sure.

Are you and everyone else willing to stick with that? By that I mean
are you really willing for the Tenney space and Tenney lattice to
have a very specific definition, which will not dissolve into mush?
I'm ready to adopt the names myself.

> > Space B
> >
> > Dual to space A is Space B, of linear functionals on A. It is a
> normed
> > vector space with norm
> >
> > || <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p)|)
> >
> > Living inside of Space B is the lattice whose coordinates are
> > integers, the lattice of vals. What is a good name for Space B
and
> the
> > lattice of vals?
>
> Error space and the temperament lattice??????

Except that only a few vals are equal temperaments.

> > Also in Space B is a special point:
> >
> > <1 log2(3) ... log2(p)|
> >
> > I called this SIZE, but it could be called, for example, JIP for
> the
> > JI point. What do people think?
>
> Seems to make a lot more sense than "SIZE"!

I like it better too. The JIP point it is unless someone screams
loudly at this point.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > You mean something like this?
> >
> > /tuning-math/message/8374
>
> Not exactly. What about a short list, giving your tuning for 2, 3,
> and 5 for various 5-limit commas?

Like this?

/tuning/topicId_50628.html#51193

(scroll down)

> > > Anyway I'm happy with my
> > > approach, which seems to finally do what I was trying to do in
the
> > > canonical tuning thread.
> >
> > But you just finished demonstrating that my method leads to
Tenney-
> > weighted minimax, and there can't be more than one solution for
> > Tenney-weighted minimax -- right?
>
> No, I just finished finding out what Tenney optimized temperaments
> ought to be, under the assumption that that must be TOP. But it
> doesn't seem to be, and in fact is more like my canonical
temperament
> business.

Well, i'd like to see your version of meantone that has a lower
maximum Tenney-weighted error than mine!

> > > Another issue is the naming of spaces and lattices. We have the
> > > following normed vector spaces and lattices living in them:
> > >
> > > Space A
> > >
> > > Space A for the p prime limit is the normed vector space with
norm
> > >
> > > || |a2 a3 ... ap> || = |a2| + log2(3)|a3| + ... + log2(p)|ap|
> > >
> > > Living inside space A is the lattice whose coordinates are
> integers.
> > > Should Space A be the Tenney space and the lattice the Tenney
> > lattice?
> >
> > Sure.
>
> Are you and everyone else willing to stick with that? By that I
mean
> are you really willing for the Tenney space and Tenney lattice to
> have a very specific definition, which will not dissolve into mush?
> I'm ready to adopt the names myself.

I'm afraid no one here is the mathematician you are, so you may be
hoping for a bit too much. But I see nothing wrong with these, right
now . . .

> > > Space B
> > >
> > > Dual to space A is Space B, of linear functionals on A. It is a
> > normed
> > > vector space with norm
> > >
> > > || <b2 b3 ... bp> || = Max(|b2|, |b3/log2(3)|, ..., |bp/log2(p)
|)
> > >
> > > Living inside of Space B is the lattice whose coordinates are
> > > integers, the lattice of vals. What is a good name for Space B
> and
> > the
> > > lattice of vals?
> >
> > Error space and the temperament lattice??????
>
> Except that only a few vals are equal temperaments.

I didn't say "equal", did I?

> > > Also in Space B is a special point:
> > >
> > > <1 log2(3) ... log2(p)|
> > >
> > > I called this SIZE, but it could be called, for example, JIP
for
> > the
> > > JI point. What do people think?
> >
> > Seems to make a lot more sense than "SIZE"!
>
> I like it better too. The JIP point

JI point point?