The Subgroup Naming Conundrum

It's my opinion that the best way to resolve the subgroup naming
crisis is to get rid of them, and to get rid of limits, since they
don't have any real auditory relevance. If you treat porcupine as a
2.3.5 temperament, it's not like 7 and 11 are really going away. I
have a few ideas about directions that could lead to this result:

1) One way to do this is to work directly with abelian groups of
countably infinite rank. Then we can just something like a 2.3.5.11.13
these subgroup tunings as being a full-limit tunings, in which 7 is
remarkably complex, and in which 17, 19, and the higher primes are all
there in the tuning, but you can either choose to or not to use them.

2) A perhaps simpler way to do the above would be to develop a notion
of "complexity" that goes DOWN as more primes are added, because the
least-complexity transversal of the entire temperamental lattice now
has simpler intervals in it.

3) Another way to do the above is, rather than working with the
complexity of the transversal, to instead work with the complexity of
each cosets of the JI lattice corresponding to each point of the
lattice, and compute complexity recursively. For example, one can
treat each point as corresponding to some coset of JI. Each coset is
itself a lattice, which can be assigned some kind of complexity. Once
you can assign complexity to a particular lattice, you can then
compute it recursively for temperamental lattices in which every point
is isomorphic to a particular coset with a complexity, by just giving
each point in the temperamental lattice a "weight" corresponding to
the complexity of the corresponding coset, and then computing the
complexity again for this new, weighted lattice. (I've been having
trouble working this out in a way that's basis-invariant.)

4) Another way to do the above is, rather than working with the
complexity of entire lattices, to just work with complexity for the
primes. One way to do this is to draw a convex hull around the set of
primes in the temperamental lattice, and then to compute something
like "weighted area" of this shape, where adding a point that has a
low weight increases the size of the convex hull, but has very little
impact on the area.

5) Some solution that mixes #4 with #2, so that adding more points to
this shape makes everything better.

The only one I have a concrete approach to doing is #4, where my
tentative algorithm is something like
1) Let P be a set of n-tuples representing the location of each prime
on the temperamental lattice of a rank-n temperament. So in meantone,
assuming the basis is [2/1, 3/2], P would be {|1 0>, |1 1>, |0 4>}.
(Just transpose the mapping matrix and make each row an entry in the
set, geez.)
2) Compute 2^P, the power set of P.
3) Each entry in 2^P will trace out some shape on the lattice.
Calculate the number of points within the convex hull of this shape.
Multiply it by a complexity factor equal to the Tenney Height (or
Tenney-Euclidean Height) associated with the subset you're working
with.
4) Add em all up.

This still isn't perfect though, because adding another prime still
increases the complexity. So the real challenge is to come up with a
prime-based notion of complexity that satisfies #2, which I think
would be a huge improvement.

If you have any ideas about any of these, please feel free to type the
relevant information here, and I will compress it and send it back
out, and then you can re-compress it and send your newly compressed
version out, and we'll go back and forth a bit until we arrive at a
better mouse trap for complexity.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> It's my opinion that the best way to resolve the subgroup naming
> crisis is to get rid of them, and to get rid of limits, since they
> don't have any real auditory relevance. If you treat porcupine as a
> 2.3.5 temperament, it's not like 7 and 11 are really going away. I
> have a few ideas about directions that could lead to this result:
>
> 1) One way to do this is to work directly with abelian groups of
> countably infinite rank. Then we can just something like a 2.3.5.11.13
> these subgroup tunings as being a full-limit tunings, in which 7 is
> remarkably complex, and in which 17, 19, and the higher primes are all
> there in the tuning, but you can either choose to or not to use them.

But how do you decide how every prime is mapped? We've had long, heated discussions about decisions like how to extend 11-limit marvel to 13-limit. It seems like this would require an infinite sequence of such decisions...

> 2) A perhaps simpler way to do the above would be to develop a notion
> of "complexity" that goes DOWN as more primes are added, because the
> least-complexity transversal of the entire temperamental lattice now
> has simpler intervals in it.

How would this help with naming?

Your other suggestions all seem to describe functions that measure some property of a temperament similar to complexity or badness. How would those help with naming either?

keenan

On Thu, Nov 3, 2011 at 9:06 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> But how do you decide how every prime is mapped? We've had long, heated discussions about decisions like how to extend 11-limit marvel to 13-limit. It seems like this would require an infinite sequence of such decisions...

I'm going to answer your other question first, because perhaps then
you'll see the wavelength I'm on, and why I think abolishing limits
entirely would be a good thing (although if we could work out any of
the complexity algorithms I suggested, it might end up being
unnecessary).

> > 2) A perhaps simpler way to do the above would be to develop a notion
> > of "complexity" that goes DOWN as more primes are added, because the
> > least-complexity transversal of the entire temperamental lattice now
> > has simpler intervals in it.
>
> How would this help with naming?
>
> Your other suggestions all seem to describe functions that measure some property of a temperament similar to complexity or badness. How would those help with naming either?

They would help with naming by removing most of the incentive of
dealing with subgroups at all. If this guy had come up as a killer
13-limit temperament in the temperament finder

http://x31eq.com/cgi-bin/rt.cgi?ets=8ccd_9c&limit=13

then we'd have never had to deal with or invent "Bleu" at all. We'd
just talk about Jerome temperament instead, and people who don't like
working with chromatic harmony or with MODMOS's would just not use the
5. The incentive to push into subgroups wouldn't be there as much,
because we wouldn't have had to get creative with the limit to find
these temperaments. So the "subgroup naming problem" would really end
up just being the "temperament naming problem," which is a much easier
problem to solve and much more straightforward.

There's one obvious type of subgroup temperament that this wouldn't
cover: a subgroup temperament that's just a subtemperament of some
other temperament. For example, machine wouldn't ever appear, since
it's just 2-suprapyth, and likewise, A-team wouldn't either, since
it's 2-mothra. But I think this problem would be solved more directly
by looking at the subgroups of the temperamental lattice, rather than
the subgroups of JI. If you do this, you might find that 2-suprapyth
ends up scoring higher than 1-suprapyth, and you're still free to name
that "machine" if you want. This is more complicated than just coming
up with a better mouse trap for complexity, so I'd rather not deal
with it just yet.

If you work these out, the only function a subgroup would ever have is
to optimize tuning error around only the primes you want, but at that
point the subgroup should just be a free parameter in the error
calculation, not actually define a different temperament. For example,
quarter-comma meantone is optimized around only 2 and 5, and ignores
3, but we wouldn't actually call it a contorted 2.5-meantone or
something.

In fact, if you could set this thing up, then really good 7-limit
temperaments would still pop up in the temperament finder even if you
go to the 13-limit, right? They'd be the same sort of situation as
above, except you'd be dealing with primes 11 and 13 being really
complex, but primes 2, 3, 5, and 7 being less complex. And
temperaments that have really simple, 2, 3, 5, 7, 11, -and- 13 would
still win out anyway.

So if you design it right, you could just search for 31-limit
temperaments and all the really good 7-limit, 11-limit, 13-limit, etc
temperaments would still show up. Then we could just name the best
31-limit temperaments. And if you're really clever, you can just
search for ω0-limit temperaments. Which brings me back to your first
question:

> But how do you decide how every prime is mapped? We've had long, heated discussions about decisions like how to extend 11-limit marvel to 13-limit. It seems like this would require an infinite sequence of such decisions...

It would require not thinking in terms of individual primes, but in
terms of functions that cleverly guide a chain of generators to be
optimized for "all primes" in some clever way.

Here's one approach: the above approach suggests some sort of badness
function, the minima of which would be the most aurally useful
temperaments to the ear, which doesn't have any built in prime
limiter. Somewhere in there is going to be a rank-3 temperament that's
analogous to marvel, but which handles all primes instead of just
2.3.5.7, and out of the infinite set of possible ways to do this,
there's only going to be one lowest in badness. If you call -this-
more fundamental mathematical object "marvel temperament," and treat
the specific 2.3.5.7 rank-3 marvel as just a 2.3.5.7-optimized version
of this more fundamental object, then it's easy to specify alterations
to it - just like we have 12p, 12d, 12lmnop, etc, we could have marvel
with a different 11, a different 17, etc. There are an infinite number
of primes, but since it's impossible to care about the mapping for
every single prime, it's also impossible to run into any sort of
problems with this approach. Then we'd simply need to agree that, for
every rank-n temperament that's important enough to have a name, we'd
just give that name to the rank-n w-limit temperament lowest in
badness. (And then we can still come up with cute-sounding names for
variations if we want, like sensor, sensus, sensational, sensamilla,
whatever.)

Here's another approach: right now, we already have a naming
convention for rank-1 temperaments that lets us handle any prime we
wish: the phrase 12p unambiguously reflects a certain rank-1
temperament no matter what the limit, because if v(p) is the val for
12p, then v(p) = round(12*log_2(p)). The mapping we alter things
relative to is the "patent val" - the easily computable val, not
necessarily the one lowest in badness. Likewise, the temperament
12p&19p also unambiguously specifies a temperament, regardless of the
limit, although it may not work spectacularly as you get to really
high primes and such. And then if you want to change the mapping for a
certain prime, you can be like 12p&19zyxlmnop or whatever. We could do
everything relative to some easily computable analogue for rank > 1
temperaments, although specifying n patent vals might not be the best
way to do it (I suspect something like a "patent wedgie" might be
easier).

I like the first option better, but the second might be more
realistic. But however you do it, there are a lot more challenges in
handling countably-infinite limits, but I think the complexity
algorithm suggested above has a number of benefits even outside of
that.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> They would help with naming by removing most of the incentive of
> dealing with subgroups at all. If this guy had come up as a killer
> 13-limit temperament in the temperament finder
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=8ccd_9c&limit=13
>
> then we'd have never had to deal with or invent "Bleu" at all. We'd
> just talk about Jerome temperament instead, and people who don't like
> working with chromatic harmony or with MODMOS's would just not use the
> 5. The incentive to push into subgroups wouldn't be there as much,
> because we wouldn't have had to get creative with the limit to find
> these temperaments. So the "subgroup naming problem" would really end
> up just being the "temperament naming problem," which is a much easier
> problem to solve and much more straightforward.

I like this idea a lot.

I think we have to come up with a new name other than "complexity" for this new kind of measure you need, though, because this is something different from previous complexity measures. Instead of measuring the amount of lattice you need to get some predefined set of intervals, this is measuring something like the average HE of all the intervals that are easy to get to in your temperament. May I suggest "temperamental harmonic entropy" or "THE"?

("Temperamental entropy" is no good because the abbreviation "TE" is already taken.)

> There's one obvious type of subgroup temperament that this wouldn't
> cover: a subgroup temperament that's just a subtemperament of some
> other temperament. For example, machine wouldn't ever appear, since
> it's just 2-suprapyth, and likewise, A-team wouldn't either, since
> it's 2-mothra. But I think this problem would be solved more directly
> by looking at the subgroups of the temperamental lattice, rather than
> the subgroups of JI. If you do this, you might find that 2-suprapyth
> ends up scoring higher than 1-suprapyth, and you're still free to name
> that "machine" if you want. This is more complicated than just coming
> up with a better mouse trap for complexity, so I'd rather not deal
> with it just yet.

The mathy way to say this is that your basic idea could work great for subgroups S of Q such that Q/S is free (because they can consistently be extended to all of Q), but there are complications when Q/S has torsion.

The sub-temperament thing does nicely eliminate the possibility of insane temperaments.

I don't have time to reply to the rest of your post now, but it is a great idea I can tell I'll be thinking about for a while. Infinity-limit temperaments!

The one-sentence motto is: "Don't look for regular temperaments that represent the p-limit; instead simply look for regular musical systems whose common intervals approximate simple JI ones."

Keenan

Random tantalizing question before I go to sleep:

Is every Riemann zeta function peak somehow associated with a specific infinite val?

Keenan

On Fri, Nov 4, 2011 at 4:01 AM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > then we'd have never had to deal with or invent "Bleu" at all. We'd
> > just talk about Jerome temperament instead, and people who don't like
> > working with chromatic harmony or with MODMOS's would just not use the
> > 5. The incentive to push into subgroups wouldn't be there as much,
> > because we wouldn't have had to get creative with the limit to find
> > these temperaments. So the "subgroup naming problem" would really end
> > up just being the "temperament naming problem," which is a much easier
> > problem to solve and much more straightforward.
>
> I like this idea a lot.
>
> I think we have to come up with a new name other than "complexity" for this new kind of measure you need, though, because this is something different from previous complexity measures. Instead of measuring the amount of lattice you need to get some predefined set of intervals, this is measuring something like the average HE of all the intervals that are easy to get to in your temperament. May I suggest "temperamental harmonic entropy" or "THE"?
>
> ("Temperamental entropy" is no good because the abbreviation "TE" is already taken.)

I don't understand - where do you see harmonic entropy playing into
this? I don't expect that this metric will change relative to the
tuning of the temperament. If we were going to work with the first 100
intervals in the temperament, or something, we could simply just take
a transversal of the temperament, and feed those intervals into an
equation such that simpler set of intervals as input = scalar output
that is closer to 0.

However, I'm still a little bit baffled by the concept of working out
the complexity for an entire lattice in a basis-invariant way, which
is the real goal. But, I think I've just figured it out for the
primes, which I'll address in a separate post.

"Tenney Height of the average interval" is a perhaps intuitive way to
think of it. If you increase the limit, the Tenney Height of the
average interval must always go down, no matter how complex your new
prime is. (You can use Tenney-Euclidean Height instead if you want).
So I envisioned it as being something quite different from HE.

> > There's one obvious type of subgroup temperament that this wouldn't
> > cover: a subgroup temperament that's just a subtemperament of some
> > other temperament. For example, machine wouldn't ever appear, since
> > it's just 2-suprapyth, and likewise, A-team wouldn't either, since
> > it's 2-mothra. But I think this problem would be solved more directly
> > by looking at the subgroups of the temperamental lattice, rather than
> > the subgroups of JI. If you do this, you might find that 2-suprapyth
> > ends up scoring higher than 1-suprapyth, and you're still free to name
> > that "machine" if you want. This is more complicated than just coming
> > up with a better mouse trap for complexity, so I'd rather not deal
> > with it just yet.
>
> The mathy way to say this is that your basic idea could work great for subgroups S of Q such that Q/S is free (because they can consistently be extended to all of Q), but there are complications when Q/S has torsion.
>
> The sub-temperament thing does nicely eliminate the possibility of insane temperaments.

How can Q/S have torsion if the sub-temperament thing eliminates the
possibility of insane temperaments - I thought you just proved that
only insane S will cause Q/S to have torsion?

> I don't have time to reply to the rest of your post now, but it is a great idea I can tell I'll be thinking about for a while. Infinity-limit temperaments!
>
> The one-sentence motto is: "Don't look for regular temperaments that represent the p-limit; instead simply look for regular musical systems whose common intervals approximate simple JI ones."

I think so! Although what do you mean by "common intervals" here?

-Mike

Mike wrote:

>The only one I have a concrete approach to doing is #4, where my
>tentative algorithm is something like
>1) Let P be a set of n-tuples representing the location of each prime
>on the temperamental lattice of a rank-n temperament. So in meantone,
>assuming the basis is [2/1, 3/2], P would be {|1 0>, |1 1>, |0 4>}.
>(Just transpose the mapping matrix and make each row an entry in the
>set, geez.)
>2) Compute 2^P, the power set of P.
>3) Each entry in 2^P will trace out some shape on the lattice.
>Calculate the number of points within the convex hull of this shape.
>Multiply it by a complexity factor equal to the Tenney Height (or
>Tenney-Euclidean Height) associated with the subset you're working
>with.
>4) Add em all up.
>This still isn't perfect though, because adding another prime still
>increases the complexity. So the real challenge is to come up with a
>prime-based notion of complexity that satisfies #2, which I think
>would be a huge improvement.

I don't know about making the complexity go down, but even
to get it to converge I believe you'll have to use something
stronger than Tenney weighting.

More generally, the zeta function must hold the answer to
the subgroup naming conundrum... somewhere...

-Carl

I wrote:
>More generally, the zeta function must hold the answer to
>the subgroup naming conundrum... somewhere...

I've long pointed out that the conundrum is easily solved for
codimension 1 temperaments: good commas are good regardless of
which factors they do or don't contain. -Carl

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Nov 4, 2011 at 4:01 AM, Keenan Pepper <keenanpepper@...> wrote:
> > I think we have to come up with a new name other than "complexity" for this new kind of measure you need, though, because this is something different from previous complexity measures. Instead of measuring the amount of lattice you need to get some predefined set of intervals, this is measuring something like the average HE of all the intervals that are easy to get to in your temperament. May I suggest "temperamental harmonic entropy" or "THE"?
> >
> > ("Temperamental entropy" is no good because the abbreviation "TE" is already taken.)
>
> I don't understand - where do you see harmonic entropy playing into
> this? I don't expect that this metric will change relative to the
> tuning of the temperament. If we were going to work with the first 100
> intervals in the temperament, or something, we could simply just take
> a transversal of the temperament, and feed those intervals into an
> equation such that simpler set of intervals as input = scalar output
> that is closer to 0.

Right, but this idea has a lot in common with HE. You're right, it could be independent of the tuning, but it's some measure that turns out to be strongly correlated to the complexity of the simplest ratio that could possibly represent the interval in question. The only difference is that "could possibly represent" is defined in terms of the temperament rather than in terms of interval size.

> "Tenney Height of the average interval" is a perhaps intuitive way to
> think of it. If you increase the limit, the Tenney Height of the
> average interval must always go down, no matter how complex your new
> prime is. (You can use Tenney-Euclidean Height instead if you want).
> So I envisioned it as being something quite different from HE.

But HE is very strongly correlated with Tenney height. That's all I was saying, really.

Maybe it would be better to have a totally new name for this "average Tenney height"-like measure, one which sounds neither like "complexity" nor "harmonic entropy".

> > The mathy way to say this is that your basic idea could work great for subgroups S of Q such that Q/S is free (because they can consistently be extended to all of Q), but there are complications when Q/S has torsion.
> >
> > The sub-temperament thing does nicely eliminate the possibility of insane temperaments.
>
> How can Q/S have torsion if the sub-temperament thing eliminates the
> possibility of insane temperaments - I thought you just proved that
> only insane S will cause Q/S to have torsion?

You're confusing two different things. In my "sane" vs. "insane" post I was talking about the quotient group of two different subgroups, neither of which was Q.

For example, take 2.9.5 and 2.9.5.7. The quotient group between them, (2.9.5.7)/(2.9.5), is free. So it would be possible to have a 2.9.5 temperament and a 2.9.5.7 with the same rank, same size generator, and same name, with neither of them "inane". But the quotient group Q/(2.9.5) has torsion. The coset containing 3 is a torsion element of order 2.

> > The one-sentence motto is: "Don't look for regular temperaments that represent the p-limit; instead simply look for regular musical systems whose common intervals approximate simple JI ones."
>
> I think so! Although what do you mean by "common intervals" here?

Intervals that have low complexity (using the old definition of "complexity"). That is, intervals that can be reached with a small number of generators.

Of course, there is a problem with this for rank > 2 temperaments, because they have infinitely many generators...

Keenan

On Fri, Nov 4, 2011 at 6:12 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > I don't understand - where do you see harmonic entropy playing into
> > this? I don't expect that this metric will change relative to the
> > tuning of the temperament. If we were going to work with the first 100
> > intervals in the temperament, or something, we could simply just take
> > a transversal of the temperament, and feed those intervals into an
> > equation such that simpler set of intervals as input = scalar output
> > that is closer to 0.
>
> Right, but this idea has a lot in common with HE. You're right, it could be independent of the tuning, but it's some measure that turns out to be strongly correlated to the complexity of the simplest ratio that could possibly represent the interval in question. The only difference is that "could possibly represent" is defined in terms of the temperament rather than in terms of interval size.

Yeah, that's what Gene's been calling a transversal.

> > "Tenney Height of the average interval" is a perhaps intuitive way to
> > think of it. If you increase the limit, the Tenney Height of the
> > average interval must always go down, no matter how complex your new
> > prime is. (You can use Tenney-Euclidean Height instead if you want).
> > So I envisioned it as being something quite different from HE.
>
> But HE is very strongly correlated with Tenney height. That's all I was saying, really.
>
> Maybe it would be better to have a totally new name for this "average Tenney height"-like measure, one which sounds neither like "complexity" nor "harmonic entropy".

I don't see why "complexity" is a bad name for it; all I'm after is a
scalar for some temperament that goes up when the primes are further
out on the lattice, and goes down when they're closer to the origin.
The only difference I propose between this and existing complexity
algorithms is that temperaments that have a bunch of primes close to
the origin, but one or two that are really far out, should get a lower
complexity score than existing algorithms do. Graham's worked out some
composite-based complexity measures (see composite.pdf) anyway,
although I haven't delved as much into them yet, so it's not like
there's no precedent for this sort of thing.

> You're confusing two different things. In my "sane" vs. "insane" post I was talking about the quotient group of two different subgroups, neither of which was Q.
>
> For example, take 2.9.5 and 2.9.5.7. The quotient group between them, (2.9.5.7)/(2.9.5), is free. So it would be possible to have a 2.9.5 temperament and a 2.9.5.7 with the same rank, same size generator, and same name, with neither of them "inane". But the quotient group Q/(2.9.5) has torsion. The coset containing 3 is a torsion element of order 2.

So what is "Q" in this case? I assume 2.3.5.7?

-Mike