Being Systematic About Subgroups

On Mon, Jun 4, 2012 at 9:05 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> Aside from the historical argument, you need to consider what a vast and
> complicated can of worms you open up in terms of all of the possibilities
> for JI groups. P-limits give us a systematic way to categorize these groups
> as subgroups, for which you've provided no alternative. I simply happily go
> ahead and treat each group separately, but you are wanting to do something
> else, without any tools to do it with I can see.

All of these different JI groups aren't separate, because they lie in
the complete lattice of finitely generated subgroups of the
overarching multiplicative group of the positive rationals. There's no
need to think of 2.3.7 being a subgroup of 2.3.5.7 in any different
way than that 2.3.5.7 is a subgroup of 2.3.5.7.11 or 2.3.5.7.13 or
2.3.5.7.13/11. Using this principle, you can then compare subgroups of
the same rank with one another.

One way to do so is: say that the multiplicative group of rationals is
called J. For any finitely generated subgroup S of J, express the
generators of the subgroup as weighted vectors in J. Then you can take
their wedge product, and take the Lp norm of the resulting
multivector, with L2 being of possible interest at first, because of
the parallels between that and the Tenney Height of the entire
subgroup expressed as a certain chord. Or you could use other Lp norms
to get different results (Linf = integer-limit?).

Once you can compare subgroups, you can compare temperaments across
same-rank subgroups of the same rank (and hence same codimension):
each temperament can be assigned a "superbadness" which factors in the
complexity of its subgroup and the badness of the temperament. There's
got to be at least one existing badness function which can be
generalized in this way.

Temperaments which are then of fundamental importance would be rank-2,
codimension-1 temperaments on good rank-3 subgroups, e.g. rank-2
temperaments low in superbadness on rank-3 subgroups; note that this
generalizes "the 5-limit" as a good starting point for families. These
temperaments will have good extensions to rank-2, codimension-2
temperaments on good rank-4 subgroups; note that this effectively
generalizes the current role of "the 7-limit" as being primary
codimension-2 extensions of the family-producing temperaments.

> Also, of course, you need
> to consider that if you look at something like 2.3.7-64/63; you are in the
> exact same position as 2.3.5-81/80, except now the new prime which forces
> itself on you is not higher limit (7), but lower limit (5). Why not be
> systematic about it to start out with if that is going to happen?

Say that a&b&c&d&... is a temperament called T in some subgroup S.
Then the "subgroup restriction" T' of T on a new subgroup S' is the
mapping given by a'&b'&c'&d'..., where a', b', c', d', ... are vals on
S' which map all rationals represented by S' to the same number of
steps as do a, b, c, d, ... . A subgroup restriction T' of T can be
called a "proper subgroup restriction" if it is neither contorted nor
insane*. Note that proper subgroup restrictions for arbitrary
temperaments may not exist on arbitrary subgroups.

Likewise, a "subgroup extension" of T is a new temperament T' of the
same rank, in a new subgroup S' of rank(S)+1 such that S is a subset
of S' and ker(T) is a subgroup of ker(T'). If T' is noncontorted and
noninsane, then T' is a "proper subgroup extension" of T.

Once you do all of the above, the following two concepts become
well-defined for any temperament T of rank-2:
1) The lowest-superbadness rank-2 temperament of subgroup rank-r which
is a proper subgroup restriction of T.
2) The lowest-superbadness rank-2 temperament of subgroup rank-r which
is a proper subgroup extension of T'.

#1 lets you find the best rank-2 codimension-1 temperaments for
existing 7-limit, 11-limit temperaments, etc. We only have to deal
with this because we didn't start out with 2.3.7 64/63 and have
2.3.5.7 64/63 245/243 as a proper extension of that - so instead we
have to find Little Orphan Superpyth's parent temperament. But this
should let it happen. Any temperament T' which has the property that
T' is the lowest-superbadness rank-2 proper subgroup restriction of T
on a rank-3 subgroup, and for which T is also the lowest-superbadness
rank-2 proper subgroup extension of T', could be said to "confirm" a
parent-child relationship such that if we had decided to start doing
things this way from the getgo, there'd be little to no chance that we
wouldn't have set things up that way anyway.

#2 gives you a good pathway for how temperaments should extend. The
only challenge I can think of is that there ought to be a way to
compare temperaments on DIFFERENT rank subgroups, because if r and c
are the rank and codimension of your existing temperament, it might be
the case that the best subgroup rank-r+2 codimension-c+2 extension of
a certain temperament is not the same as the best rank-r+2
codimension-c+2 extension of the best rank-r+1 codimension-c+1
extension of that temperament.

-Mike

*/tuning-math/message/19938. In
short, an example of an "insane" temperament would be
http://x31eq.com/cgi-bin/rt.cgi?ets=15_22&limit=2.9.5 - note that this
is the restriction of porcupine to the 2.9.5 subgroup. However, while
this temperament isn't contorted, it has a mapping for sqrt(9/1), but
decides to not call it 3/1. That's insane.

On Mon, Jun 4, 2012 at 11:48 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> One way to do so is: say that the multiplicative group of rationals is
> called J. For any finitely generated subgroup S of J, express the
> generators of the subgroup as weighted vectors in J. Then you can take
> their wedge product, and take the Lp norm of the resulting
> multivector, with L2 being of possible interest at first, because of
> the parallels between that and the Tenney Height of the entire
> subgroup expressed as a certain chord.

BTW, Paul suggested this, although I'm not still sure this is the best
way to do it. But credit goes to him, although I'm not sure if he saw
the parallels between this and the Tenney height of the subgroup
expressed as a chord.

-Mike

On Tue, Jun 5, 2012 at 12:00 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> BTW, Paul suggested this, although I'm not still sure this is the best
> way to do it. But credit goes to him, although I'm not sure if he saw
> the parallels between this and the Tenney height of the subgroup
> expressed as a chord.

D'oh, screwed this up. If the subgroup is 2.3.5, you'd get
log(2)*log(3)*log(5), not log(2) + log(3) + log(5). Whatever.

-Mike

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

> BTW, Paul suggested this, although I'm not still sure this is the best
> way to do it.

It's easy to define "superbadness" for the special case where the generators are primes. It's how to do it for general subgroups which has been the hangup. You might ask yourself if just primes is good enough.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > BTW, Paul suggested this, although I'm not still sure this is the best
> > way to do it.
>
> It's easy to define "superbadness" for the special case where the generators are primes. It's how to do it for general subgroups which has been the hangup. You might ask yourself if just primes is good enough.

I might add that I've been using a such subgroup badness function for some time now, since the so-called "subgroup revolution" got rolling. I suppose you could try to convince yourself it can be trusted if the subgroup is not defined by primes; at least it's defined.

On Jun 5, 2012, at 1:05 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

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

> BTW, Paul suggested this, although I'm not still sure this is the best
> way to do it.

It's easy to define "superbadness" for the special case where the
generators are primes. It's how to do it for general subgroups which has
been the hangup. You might ask yourself if just primes is good enough.

The thing I said should work for non-primes - just express the generators
as monzos in the full limit, take their wedge product and take the norm.
You might call that TE subgroup complexity.

-Mike

At 11:53 AM 6/5/2012, you wrote:
>The thing I said should work for non-primes - just express the generators as monzos in the full limit, take their wedge product and take the norm. You might call that TE subgroup complexity.

With this, there'd be no point to subgroups at all, right?

-Carl

On Jun 5, 2012, at 3:06 PM, Carl Lumma <carl@lumma.org> wrote:

At 11:53 AM 6/5/2012, you wrote:
>The thing I said should work for non-primes - just express the generators
as monzos in the full limit, take their wedge product and take the norm.
You might call that TE subgroup complexity.

With this, there'd be no point to subgroups at all, right?

-Carl

What do you mean? Just do everything in infinite limit JI?

-Mike

>>The thing I said should work for non-primes - just express the generators as monzos in the full limit, take their wedge product and take the norm. You might call that TE subgroup complexity.
>>
>>With this, there'd be no point to subgroups at all, right?
>What do you mean? Just do everything in infinite limit JI?

Sorry, you said *generators*. Yeah, that might work. -C.

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

> The thing I said should work for non-primes - just express the generators
> as monzos in the full limit, take their wedge product and take the norm.
> You might call that TE subgroup complexity.

It's defined--I've had something coded up for a long time now. But if the generators are not primes or prime powers, how is it justified?

On Jun 5, 2012, at 3:38 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

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

> The thing I said should work for non-primes - just express the generators
> as monzos in the full limit, take their wedge product and take the norm.
> You might call that TE subgroup complexity.

It's defined--I've had something coded up for a long time now. But if the
generators are not primes or prime powers, how is it justified?

It's a start. I wouldn't say it's totally unjustified.

This way uses the L2 norm, but it might be totally ideal to It would be
better to figure out some magic by using some kind of Keesian integer-limit
norm with hexagons. But this looks easier to compute and a decent
approximation to that.

Paul also liked it because this lets you compare subgroups without looking
explicitly for the lowest-complexity chord of some specific cardinality N,
but lets you compare entire subgroups this way.

On the other hand, ranking 2.9.5 and 2.9.5/3 the same might be a
dealbreaker. Is it?

-Mike

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

> On the other hand, ranking 2.9.5 and 2.9.5/3 the same might be a
> dealbreaker. Is it?

I don't know what your question means.

On Jun 5, 2012, at 9:45 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

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

> On the other hand, ranking 2.9.5 and 2.9.5/3 the same might be a
> dealbreaker. Is it?

I don't know what your question means.

I was asking if it was bad that 2.9.5 gets assigned the same score as
2.9.5/3.

Ignoring 2 which is common to both, 2.9.5 has 5/1, but then the next ratios
containing 5 are 9/5 and 45/1, both with Tenney height of 45.

2.9.5/3 doesn't have 5/1, but has both 5/3 and 15/1 with Tenney height of
15.

In a certain sense, the question of whether 2.9.5/3 is worse than 2.9.5 is
the same as the question of whether 15/1 is worse than 9/5.

-Mike

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

> I was asking if it was bad that 2.9.5 gets assigned the same score as
> 2.9.5/3.

I thought we were computing the badness of temperaments, not assigning scores to groups. How is this score defined?

I was just talking about superbadness, correct?

-Mike

On Jun 6, 2012, at 4:57 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

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

> I was asking if it was bad that 2.9.5 gets assigned the same score as
> 2.9.5/3.

I thought we were computing the badness of temperaments, not assigning
scores to groups. How is this score defined?

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I was just talking about superbadness, correct?

Yes, but I think I didn't get the gist. I thought you meant a badness function that could compare between differnt groups, not compare the groups themselves, whatever the hell that means.

On Jun 6, 2012, at 7:43 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I was just talking about superbadness, correct?

Yes, but I think I didn't get the gist. I thought you meant a badness
function that could compare between differnt groups, not compare the groups
themselves, whatever the hell that means.

I meant a badness function which can compare 2.3.7 64/63 with 2.3.5 81/80.
This means I also want to be able to compare 2.3.7 JI to 2.3.5 JI. What's
the problem?

-Mike

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

> I meant a badness function which can compare 2.3.7 64/63 with 2.3.5 81/80.
> This means I also want to be able to compare 2.3.7 JI to 2.3.5 JI. What's
> the problem?

The problem is you didn't say that.

On Jun 6, 2012, at 11:10 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:

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

> I meant a badness function which can compare 2.3.7 64/63 with 2.3.5 81/80.
> This means I also want to be able to compare 2.3.7 JI to 2.3.5 JI. What's
> the problem?

The problem is you didn't say that.

Oh please. The very first thing I said was

"All of these different JI groups aren't separate, because they lie in
the complete lattice of finitely generated subgroups of the
overarching multiplicative group of the positive rationals. There's no
need to think of 2.3.7 being a subgroup of 2.3.5.7 in any different
way than that 2.3.5.7 is a subgroup of 2.3.5.7.11 or 2.3.5.7.13 or
2.3.5.7.13/11. Using this principle, you can then compare subgroups of
the same rank with one another."

Then I gave a way to do that, then I said

"Once you can compare subgroups, you can compare temperaments across
same-rank subgroups of the same rank (and hence same codimension):
each temperament can be assigned a "superbadness" which factors in the
complexity of its subgroup and the badness of the temperament."

I'm not sure what you think I've been talking about if not this.

-Mike