Yahoo Tuning Groups Ultimate Backup tuning-math Subgroups and temperaments, part 2: smonzos, svals

Last time I wrote about tmonzos and tvals, which are elements in
groups that naturally arise from tempering JI. What's really
interesting is to take the same mathematical approach to handling
homomorphisms on the group of -vals-, rather than to linear
transformations on the group of monzos. At first blush, "tempering out
a val" makes no sense, so we'll have to try to make sense of it
mathematically and see where we get.

Following the convention in my last email, I'm going to treat the rows
in matrices as vals and the columns as monzos. So for instance, <22 35
51 62| would be the row matrix [22 35 51 62]. I'll also say that this
val exists in (Z^4)*, which is the group of linear functionals on Z^4.
Following that convention, linear transformations on this group can be
represented by a new type of mapping matrices, this time in which the
columns are monzos and which right-multiply vals. Here's an example
transformation from (Z^4)* -> (Z^3)*:

[1 0 0]
[0 1 0]
[0 0 -1]
[0 0 1]

Note that this is a mapping matrix which has dimensions that look like
the transpose of one of Graham's mapping matrices; it's of full column
rank. To distinguish these from the other sort of mapping, I'll call
them "V-maps," where the V means they're being applied to vals. The
other mappings I'll call "M-maps," where M stands for monzo.

Matrices like this may seem strange at first. This one has the val <0
0 1 1| in its kernel, but what does it mean to temper out <0 0 1 1|?

We can gain more intuition about what these objects are by considering
that, as the rows of M-maps are vals, so are the columns of V-maps
monzos. In the case of the above V-map, the columns represent 2/1,
3/1, and 7/5 respectively, so given a slight abuse of notation we can
write it as a partitioned matrix [2/1 | 3/1 | 7/5]. If we then
multiply this by a val on the left, we get [V] * [2/1 | 3/1 | 7/5],
which is the same as [(V*2/1) (V*3/1) (V*7/5)]. This matrix now
represents a new "val" in (Z^3)*.

The coefficients of this new val are going to be the same as the
values that 2/1, 3/1, and 7/5 map to, respectively, under the old val.
Since these new vals in (Z^3)* would appear to be vals which are the
subgroup restrictions of the vals you're putting into the
transformation, we'll call them "svals", short for "subgroup vals",
which is a name that Gene came up with a while ago. Furthermore, it
now makes sense what it means for <0 0 1 1| to be "tempered out": if
you add <0 0 1 1| to any val, it's not going to change what 2/1, 3/1,
or 7/5 map to under it. <22 35 51 62| and <22 35 1 12| both have the
same mappings on the 2.3.7/5 subgroup.

This is hopefully simple enough, and next I'll talk about how all the
stuff I wrote about dual homomorphisms from the last part applies
here.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Sun, Jul 8, 2012 at 3:59 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> This is hopefully simple enough, and next I'll talk about how all the
> stuff I wrote about dual homomorphisms from the last part applies
> here.

Since the "dual homomorphisms" I talked about last time apply to
linear transformations on free abelian groups in general, they apply
here too. If we're going to call the above V-map S, and say it's a
linear transformation S: X* -> Y*, then a dual homomorphism S*: Y -> X
exists, and this homomorphism will be one-to-one but not onto. B is
the dual group to the group of svals we just created, and as it
represents the elements that svals are linear functionals of, we can
call them "smonzos" or subgroup monzos, also Gene's name for them.

What does this dual homomorphism mean musically? It becomes obvious if
we look at the same example from before. Since the original V-map
right-multiplied vals going from X* -> Y*, the dual transformation
will left-multiply smonzos and go from Y -> X:

[1 0 0]
[0 1 0] [a]
[0 0 -1] [b]
[0 0 1] [c]

Right away, you can see:
- the smonzo |a b c> maps to the monzo |a b -c c>
- the image of this S-map will be the 2.3.7/5 subgroup of the group of
7-limit JI monzos
- there are certain intervals, such as |0 0 0 1>, which have no smonzo
mapping to them at all (and indeed 7/1 is not in the 2.3.7/5
subgroup).

It's hopefully becoming intuitively clear from this that the dual
homomorphism is a transformation turning 2.3.7/5 smonzos back into
2.3.5.7 monzos. The fact that it's one-to-one means that a unique
monzo exists for every smonzo - for instance, the only smonzo mapping
to |0 0 -1 1> is |0 0 1> and nothing else. And also, the fact that it
isn't "onto" means that intervals which aren't in the 2.3.7/5 subgroup
simply do not lie in the image of this transformation. There is no
2.3.7/5 smonzo which maps back to |0 0 0 1> in the full 7-limit.

Hopefully it's now obvious that this "dual homomorphism" is a bit less
mystical than when applied to M-maps: for V-maps, it literally just
consists of rewriting the smonzo as a full-limit monzo.

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

On Sun, Jul 8, 2012 at 4:24 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Hopefully it's now obvious that this "dual homomorphism" is a bit less
> mystical than when applied to M-maps: for V-maps, it literally just
> consists of rewriting the smonzo as a full-limit monzo.

There's one more point here, which is probably the most important of
all. In my first post I mentioned this theorem:

For free abelian groups A, B and group homomorphism H*: B* -> A*,
there does not in general exist a -unique- map H*^-1: A* -> B* which
respects the property that, for all b in B*, H*^-1(H*(b)) = b.

This result is much more meaningful when applied to V-maps instead of
M-maps. In musical terms, this means:

FOR ANY proper subgroup of your JI group, THERE IS NO unique map going
from monzos to smonzos on that group.

It's true. Here are the options that you get, keeping in mind I've
defined everything thus far in terms of groups (or Z-modules) and not
Q-vector spaces:

1) If your target subgroup is saturated but of lower rank than the
original group, you get an infinite number of possible maps.
2) If your target subgroup is saturated and of equal rank than the
original group, then it's an automorphism, and you do get a unique
inverse map. However, then the "subgroup" being defined isn't a proper
subgroup.
3) If your target subgroup isn't saturated, you don't get a map at all.

An example of #1 is trying to go from 2.3.5.7 to 2.3.7/5: there are an
infinite number of maps which send the monzo |a b -c c> to the smonzo
|a b c>. However, they all differ with regards to where they map
intervals in 7-limit JI which aren't in the 2.3.7/5 subgroup, such as
|0 0 0 1>.

An example of #2 is trying to go from 2.3.5 to 2.3.5/3. You do then
get a unique inverse map, but this is also not a proper subgroup of
the 5-limit.

An example of #3 trying to go from 2.3.5 to 2.9.5. Since I've defined
everything in terms of groups and not vector spaces, our structure
isn't rich enough for |0 1/2 0> to exist, and so |0 1 0> doesn't map
to anything. If we upgrade to a Q-space and allow for fractional
monzos, then our situation becomes #2 and you do get a unique inverse
map.

This means that, in general, mapping monzos to smonzos, much like
mapping vals to tvals, is non-trivial. You'll typically have an
infinite amount of ways to do it. Most directly, any purported map
from monzos to smonzos is actually a map that's going to define an
equivalence relation on monzos, and hence instead go from monzos to a
new group of tmonzos, where the generators for the subgroup you were
-trying- to map to are going to map to the unit basis for the group of
tmonzos. This means that to go from monzos to smonzos means to "fake
it" with a temperament! Viewed this way, possible ways to go from
2.3.5 monzos to 2.3 monzos include meantone, schismatic, 2.3.5
superpyth, mavila, avila, the rank-2 temperament eliminating 5/1, and
far stupider things.

This may seem simple now, but I've found lots of things I used to
think were sensible, ended up turning out to be incredibly hairy
because they boil down to the above principle. For instance, it may
seem attractive to represent the 2.3.7 subgroup of the 7-limit as a
7-limit temperament that tempers out 5/1 - but note that there are
other maps which also send |a b 0 c> to |a b c> than that one, and for
more complex subgroups it's not always clear which is the best.
Likewise, it may also seem attractive to map from 2.3.5 vals to 2.9.5
vals by sending |a b c> to |a b/2 c>, but you'll find things get
incredibly tricky for more complex subgroups.

One initial way to mess around with is to define the "canonical" map
from monzos to smonzos to be the pseudoinverse of the V-map defined by
the subgroup, which is what Gene's done. It may not be the best thing
to use in all situations, however. Likewise, you can take the
pseudoinverse of a temperament M-map to give you a way to send vals to
tvals of arbitrary temperament.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Sun, Jul 8, 2012 at 12:58 PM, Mike Battaglia <battaglia01@gmail.com>
wrote:
>
> Is the thing you're calling Lp-tuning here the same thing I've been
> calling Lp-TOP?

It looks like it is, but that we have the numbering convention
flipped: what I was calling Linf-TOP, aka just TOP, you're calling L1,
and what you're calling L1 I was calling Linf. I'm pretty sure I'm
losing my mind because I thought I posted something about how
POL1-meantone and POLinf-meantone are the same, and how I thought that
was interesting, but now I can't figure out wtf happened to that post.
(I was calling it Lp-POT though.)

One thing I don't get is this, under "Dual Norm":

"If r1, r2, ... rn are a set of generators for G, which in particular
could be a normal list and so define smonzos for G, then corresponding
generators for the dual space can in particular be the sval
generators. On this standard basis for G-tuning space we can express
the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the
normal G generator list, then <cents(r1) cents(r2) ... cents(rn)| is a
point, in unweighted coordinates, in G-tuning space, and the nearest
point to it under the G-sval norm on the subspace of tunings of some
abstract G-temperament S, meaning svals in the null space of its
commas, is precisely the Lp tuning Lp(S)."

Why is this true? It's not obvious to me why. For instance, you're
talking about a vector space, which you call Lp, and then G is a
subspace of it. There's a strange norm imposed on G by virtue of its
being a subspace of Lp, but in general this norm isn't going to itself
be an Lp norm. Then, you define the G-JIP, and single out, for some
temperament S, the closest G-tuning map to it. In this case, "closest"
means closest under the dual norm to this strange non-Lp norm that
we've imposed on G by virtue of it being a subspace of the Lp vector
space. You then say that this is precisely the Lp optimal tuning for
S.

Why is this, and in what sense is it going to be Lp-optimal? What does
the dual norm to one of these weird norms on G look like, and how does
it relate back to the Lp norm on the original space?

-Mike

🔗Carl Lumma <carl@lumma.org>

TOP uses L1 on intervals and Linf on tunings (they are dual).
TE uses L2, which is its own dual. -C.

At 11:47 AM 7/8/2012, you wrote:
>On Sun, Jul 8, 2012 at 12:58 PM, Mike Battaglia <battaglia01@gmail.com>
>wrote:
>>
>> Is the thing you're calling Lp-tuning here the same thing I've been
>> calling Lp-TOP?
>
>It looks like it is, but that we have the numbering convention
>flipped: what I was calling Linf-TOP, aka just TOP, you're calling L1,
>and what you're calling L1 I was calling Linf. I'm pretty sure I'm
>losing my mind because I thought I posted something about how
>POL1-meantone and POLinf-meantone are the same, and how I thought that
>was interesting, but now I can't figure out wtf happened to that post.
>(I was calling it Lp-POT though.)
>
>One thing I don't get is this, under "Dual Norm":
>
>"If r1, r2, ... rn are a set of generators for G, which in particular
>could be a normal list and so define smonzos for G, then corresponding
>generators for the dual space can in particular be the sval
>generators. On this standard basis for G-tuning space we can express
>the dual norm canonically as the G-sval norm. If [r1 r2 ... rn] is the
>normal G generator list, then <cents(r1) cents(r2) ... cents(rn)| is a
>point, in unweighted coordinates, in G-tuning space, and the nearest
>point to it under the G-sval norm on the subspace of tunings of some
>abstract G-temperament S, meaning svals in the null space of its
>commas, is precisely the Lp tuning Lp(S)."
>
>Why is this true? It's not obvious to me why. For instance, you're
>talking about a vector space, which you call Lp, and then G is a
>subspace of it. There's a strange norm imposed on G by virtue of its
>being a subspace of Lp, but in general this norm isn't going to itself
>be an Lp norm. Then, you define the G-JIP, and single out, for some
>temperament S, the closest G-tuning map to it. In this case, "closest"
>means closest under the dual norm to this strange non-Lp norm that
>we've imposed on G by virtue of it being a subspace of the Lp vector
>space. You then say that this is precisely the Lp optimal tuning for
>S.
>
>Why is this, and in what sense is it going to be Lp-optimal? What does
>the dual norm to one of these weird norms on G look like, and how does
>it relate back to the Lp norm on the original space?
>
>-Mike
>

🔗Mike Battaglia <battaglia01@gmail.com>

Right. I was referring to the norm on vals, and he's talking about the one
on monzo. Apparently dealing with monzos better properties when dealing
with subgroups of the original group, but I'm not sure why yet.

-Mike

On Jul 8, 2012, at 3:16 PM, Carl Lumma <carl@lumma.org> wrote:

TOP uses L1 on intervals and Linf on tunings (they are dual).
TE uses L2, which is its own dual. -C.

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> It looks like it is, but that we have the numbering convention
> flipped: what I was calling Linf-TOP, aka just TOP, you're calling L1,
> and what you're calling L1 I was calling Linf.

That's because I needed to start with the norm on monzos and not on vals.

> For instance, you're
> talking about a vector space, which you call Lp, and then G is a
> subspace of it. There's a strange norm imposed on G by virtue of its
> being a subspace of Lp, but in general this norm isn't going to itself
> be an Lp norm.

It doesn't need to be an Lp norm, it needs to make sense. The Lp norm on monzos we are assuming to make sense, so this does also. In particular, even though we may not have a diagonal matrix of weights, the reason we wanted a diagonal of weights is the important consideration, and still applies.

> Why is this, and in what sense is it going to be Lp-optimal? What does
> the dual norm to one of these weird norms on G look like, and how does
> it relate back to the Lp norm on the original space?

I'm not sure what "look like" means. For L2 norms, we get a bilinear form on smonzso and an associated on on svals.

🔗Mike Battaglia <battaglia01@gmail.com>

On Jul 8, 2012, at 7:10 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

It doesn't need to be an Lp norm, it needs to make sense. The Lp norm on
monzos we are assuming to make sense, so this does also. In particular,
even though we may not have a diagonal matrix of weights, the reason we
wanted a diagonal of weights is the important consideration, and still
applies.

Ok, I get that. But what does this line mean?

"If [r1 r2 ... rn] is the normal G generator list, then <cents(r1)
cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning
space, and the nearest point to it under the G-sval norm on the subspace of
tunings of some abstract G-temperament S, meaning svals in the null space
of its commas, is precisely the Lp tuning Lp(S)."

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Ok, I get that. But what does this line mean?
>
> "If [r1 r2 ... rn] is the normal G generator list, then <cents(r1)
> cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in G-tuning
> space, and the nearest point to it under the G-sval norm on the subspace of
> tunings of some abstract G-temperament S, meaning svals in the null space
> of its commas, is precisely the Lp tuning Lp(S)."

By the "normal G generator list" I mean the normal list for the subgroup G; for instance 2.3.7/5 or 2.6.11, etc. The G-sval norm is the dual norm on the svals of G of the norm induced on the smonzos of G. The subspace of some G-temperament would be, for instance, the subspace tempering out 50/49 in the 2.3.7/5 subgroup.

🔗Mike Battaglia <battaglia01@gmail.com>

On Jul 8, 2012, at 11:39 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Ok, I get that. But what does this line mean?
>
> "If [r1 r2 ... rn] is the normal G generator list, then <cents(r1)
> cents(r2) ... cents(rn)| is a point, in unweighted coordinates, in
G-tuning
> space, and the nearest point to it under the G-sval norm on the subspace
of
> tunings of some abstract G-temperament S, meaning svals in the null space
> of its commas, is precisely the Lp tuning Lp(S)."

By the "normal G generator list" I mean the normal list for the subgroup G;
for instance 2.3.7/5 or 2.6.11, etc. The G-sval norm is the dual norm on
the svals of G of the norm induced on the smonzos of G. The subspace of
some G-temperament would be, for instance, the subspace tempering out 50/49
in the 2.3.7/5 subgroup.

Yeah I followed all that, but why are you saying the tuning map minimizing
the G-sval distance from the G-JIP "is precisely the Lp tuning" for the
subgroup temperament?

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

When you said "is precisely the Lp tuning," I thought you were saying there
was some deep identity showing how the G-sval norm optimal tuning map is
precisely the same as [something in the space of linear functionals on Lp].

But is it rather that you're saying that you're defining the term "Lp
tuning" for a subgroup temperament just to mean the tuning map minimizing
the G-sval normed distance from the JIP?

-Mike

On Jul 9, 2012, at 2:05 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Yeah I followed all that, but why are you saying the tuning map minimizing
> the G-sval distance from the G-JIP "is precisely the Lp tuning" for the
> subgroup temperament?

Because of how I defined Lp tunings for subgroups.

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Mike Battaglia <battaglia01@gmail.com>

On Wed, Jul 11, 2012 at 12:20 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > OK, I get it now. So why does it make a difference to do it this way,
> > rather than to let the norm be induced from Lp* on the G-svals and
> > then let the norm on G-monzos just be dual to that?
>
> You're the one who keeps pointing out that there isn't a unique map taking
> the p-limit monzos to some subgroups smonzos. When you just drop some of the
> primes, or replace primes with prime powers, that doesn't matter, but in
> general a tuning map which works fine for a subgroup G may be completely
> wonky if considered as a map on the full p-limit.

I might be confused about something, but I don't see how it's the same
reason things go wonky. The fact that there's no unique map from
monzos to smonzos doesn't seem related, but after thinking about it
more, the fact that there's no unique map from svals to vals seems
more related. Moving from svals to vals is a transversal problem, much
like moving from tmonzos to monzos is a transversal problem. But
moving from monzos to smonzos is far more "difficult" in that you're
trying to design the preimage of each smonzo a priori and it's
completely arbitrary how you do that. Right?

On the other hand, there is a single unique map taking you from vals
to svals for any subgroup. So I guess that what I don't get is that
you had no problem working the corresponding situation out with the
OETES, so I don't see why there's a problem doing it here. For
instance, every V-map from Lp* -> G* with rank(G) < rank(Lp) has a
nontrivial kernel. Say you looked at the subspace of Lp* orthogonal to
this kernel and used that to come up with a map from G* -> Lp* in the
obvious way. What happens if you then use the restriction of the Lp*
norm onto that subspace? How does that differ from what your'e doing?

-Mike