Topic: A useful generalization of epimorphicity

2 scales

Thread (99 messages)

From: Mike Battaglia (2013-01-23)
Subject: A useful generalization of epimorphicity

I was just about to write a thing about "epimorphic notation systems,"
when I realized that the definition of "epimorphicity" that's on the
wiki is much more specific than the one that Keenan and I have been
using in our discussions for about a year or two. I'll call it
"abstract epimorphicity" to avoid confusion. The definition is:

--
A "regular tuning system" is a set of musical intervals, expressed
logarithmically as real numbers, which has the additional structure of
being a free abelian group under addition. If we impose an equivalence
relation on the set of regular tuning systems such that two elements
are equivalent iff a group isomorphism exists between them, the
resulting equivalence classes are called "abstract regular tuning
systems." A subset S of an abstract regular tuning system G is called
"abstractly epimorphic" if there exists an element h in Hom(G,Z) such
that the restriction of h to S is a bijection.
--

Abstract epimorphicity is particularly useful in regular temperament
theory when studying scales for abstract regular temperaments that we
haven't assigned a tuning to yet, such as when considering the
periodicity blocks of the Ares lattice. It's also a useful property in
scale theory in general, quite apart from any consideration of
intonation; it comes up a good amount in the study of MODMOS's, and is
particularly useful when looking at higher-dimensional generalizations
of MODMOS's. It's also the thing that it seemed like everyone else was
just calling "epimorphicity" back when we were talking about product
words and rank-3 scales.


This definition leads to a very simple and elegant generalization of
epimorphicity and weak epimorphicity for periodic scales, which is
defined as follows:

--
For any periodic scale p, there exists a group G which is completely
generated by taking Z-linear sums of the elements of Im(p). We can say
that p is "weakly epimorphic" if there exists an element h in Hom(G,Z)
such that the restriction of h to the domain Im(p) is a bijection. If
p is monotone and weakly epimorphic, then it is also "epimorphic."
--

This definition is completely compatible with the existing one on the
wiki, but additionally allows us to speak of the epimorphicity of
tempered periodic scales, as well as of periodic scales that are
mapping-agnostic. We can't do this under the existing definition,
since it's JI-specific; we can only talk about the epimorphicity of
transversals of tempered scales, or the epimorphicity of MODMOS's with
a specific mapping, etc.

-Mike

PS: If you really want to get abstract, you can see that abstract
epimorphicity is really just a property the subsets of some free
abelian group, without regard as to whether or not that group
represents musical intervals at all. I don't know where else in music
theory groups appear, so I didn't bother with that. If for some reason
you do care about this, you'll note that my elements in Hom(G,Z) are
basically just generalized vals, so it might be possible to go even
further and define the concept in a sensible way for arbitrary
Z-modules if you want, or even for arbitrary modules.

From: Mike Battaglia (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

Oh, one more super important thing:

On Wed, Jan 23, 2013 at 7:35 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
> it comes up a good amount in the study of MODMOS's, and is
> particularly useful when looking at higher-dimensional generalizations
> of MODMOS's.

Once you start looking at abstract epimorphic scales in abstract
regular tuning systems, you quickly realize that parallelograms and
their higher-dimensional equivalents are really fundamentally
important. These are the things we'd usually call "Fokker blocks," but
now without any concept of mapping. They appear naturally when you
start looking at product words, for instance, and in general have lots
of important mathematical properties.

Here's an example of why they're important: for instance, someone,
somewhere - thought I can't find where now - once talked about
constructing a graph of arenas, such that two arenas are connected if
they share a wakalix. Or maybe it was a graph of wakalixes which are
connected if they share an arena. I don't remember what it was, but
both are good ideas. Whatever it was, once you've worked that out, the
thing you just figured out is now a mathematical structure which
applies to -every- JI subgroup, and even to every tempered lattice.
The wakalixes are still wakalixes no matter what ratios you map to the
axes, and no matter what tuning map you use, and no matter whether
things are tempered, etc. The graph that you worked out is still
valid, so once you've worked it out for some rank-n abstract regular
tuning system, the results now apply to every rank-n JI group and
every rank-n temperament. Your graph and all of your wakalixes are
fundamental to the structure of the free abelian group itself.

We've been calling these things Fokker blocks. But, some wiki
articles, like "Weak Fokker Blocks," don't even make sense unless
you're assuming that JI is being built in from the start. So it looks
like Fokker blocks require ratios to be involved. So what should I
call these things?

I think that Keenan was the first person to take the quantum leap and
look at the properties of these blocks in a completely
mapping-agnostic way and tie the whole thing in with the Zabka paper
and all that, so maybe they should maybe be Pepper blocks, or
Fokker-Pepper blocks or something?

Alternatively, since these things are really general and abstract, and
encompass an entire equivalence class of Fokker blocks, we could call
them Mother Fokker blocks. God, I've been waiting for years to make
that joke.

-Mike

From: Mike Battaglia (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 8:17 AM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> I think that Keenan was the first person to take the quantum leap and
> look at the properties of these blocks in a completely
> mapping-agnostic way and tie the whole thing in with the Zabka paper
> and all that, so maybe they should maybe be Pepper blocks, or
> Fokker-Pepper blocks or something?

Keenan didn't like this much but thought "Abstract Fokker Block" was
OK. So abstract Fokker block it is.

-Mike

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> A "regular tuning system" is a set of musical intervals, expressed
> logarithmically as real numbers, which has the additional structure of
> being a free abelian group under addition. If we impose an equivalence
> relation on the set of regular tuning systems such that two elements
> are equivalent iff a group isomorphism exists between them, the
> resulting equivalence classes are called "abstract regular tuning
> systems." 

Yipe! You are saying all rank N tunings are the same?

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> I think that Keenan was the first person to take the quantum leap and
> look at the properties of these blocks in a completely
> mapping-agnostic way and tie the whole thing in with the Zabka paper
> and all that, so maybe they should maybe be Pepper blocks, or
> Fokker-Pepper blocks or something?

How does a Pepper block relate to my definition of blocks in an abstract regular temperament? As is so often the case, I'm not sure of your definitions.

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> Keenan didn't like this much but thought "Abstract Fokker Block" was
> OK. So abstract Fokker block it is.

Dammit, Mike, that leaves my definition out in the cold! I object.

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 10:51 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> Yipe! You are saying all rank N tunings are the same?

No, of course I'm not saying they're all the same. I'm saying that the
rank-n abstract regular tuning system is a mathematical construction
which is the equivalence class of all of them, and intentionally so.
It's just a roundabout way of saying to look at Z^n, and treat it as a
lattice of intervals that is regularly generated, but for which we
haven't specified a mapping nor a tuning.

This is a useful construction in scale theory; picking the JI group
and the tuning only comes into it specifically when we want to
evaluate how "good" scales are, not when we want to study the
structure of rank-n scales themselves. For instance, Keenan proved a
while ago (I think) that hexagons which tile the lattice always have
three step sizes; this is a theorem about Z^n and its dual module, not
about any particular JI group or tuning. Likewise, Zabka (and Keenan?)
proved that the product word of two MOS's is the thing I called an
"abstract Fokker block" below; this is also a theorem about Z^n and
its dual module. And the same applies to epimorphicity.

The same applies if you want to study the way in which wakalixes
connect arenas, for instance, or understand the structure of the domes
of an abstract Fokker block together: these are very general
scale-theoretic things which immediately apply to scale construction
in ALL JI or tempered groups. It's only when you want to start
separating the "good" scales from the "bad" that you need to worry
about mappings and tunings, but there's still plenty of work to be
done in understanding the structure of general rank-n scales which is
far more abstract than that.

> Dammit, Mike, that leaves my definition out in the cold! I object.

You already have something called an abstract Fokker block?? What is it?

-Mike

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
>
> On Wed, Jan 23, 2013 at 10:51 AM, genewardsmith
>  wrote:
> >
> > Yipe! You are saying all rank N tunings are the same?
> 
> No, of course I'm not saying they're all the same. I'm saying that the
> rank-n abstract regular tuning system is a mathematical construction
> which is the equivalence class of all of them, and intentionally so.

Unless you can define "all of them", you have neither an equivalence class nor a definition.

> It's just a roundabout way of saying to look at Z^n, and treat it as a
> lattice of intervals that is regularly generated, but for which we
> haven't specified a mapping nor a tuning.

If you want to say Z^n, why not just say Z^n?

> This is a useful construction in scale theory; picking the JI group
> and the tuning only comes into it specifically when we want to
> evaluate how "good" scales are, not when we want to study the
> structure of rank-n scales themselves. 

Sure, but now you are talking about a JI group, not Z^n.

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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


> > Dammit, Mike, that leaves my definition out in the cold! I object.
> 
> You already have something called an abstract Fokker block?? What is it?

http://xenharmonic.wikispaces.com/Fokker+blocks#Expanding the definition

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 12:20 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
> >
> > No, of course I'm not saying they're all the same. I'm saying that the
> > rank-n abstract regular tuning system is a mathematical construction
> > which is the equivalence class of all of them, and intentionally so.
>
> Unless you can define "all of them", you have neither an equivalence class
> nor a definition.

I wrote:
>  If we impose an equivalence
> relation on the set of regular tuning systems such that two elements
> are equivalent iff a group isomorphism exists between them

> > It's just a roundabout way of saying to look at Z^n, and treat it as a
> > lattice of intervals that is regularly generated, but for which we
> > haven't specified a mapping nor a tuning.
>
> If you want to say Z^n, why not just say Z^n?

I wrote:
> PS: If you really want to get abstract, you can see that abstract
> epimorphicity is really just a property the subsets of some free
> abelian group, without regard as to whether or not that group
> represents musical intervals at all. I don't know where else in music
> theory groups appear, so I didn't bother with that.

> > This is a useful construction in scale theory; picking the JI group
> > and the tuning only comes into it specifically when we want to
> > evaluate how "good" scales are, not when we want to study the
> > structure of rank-n scales themselves.
>
> Sure, but now you are talking about a JI group, not Z^n.

Your response makes no sense; I just said, in the very paragraph that
you're quoting, a situation where picking the JI group is necessary.

-Mike

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> For instance, Keenan proved a
> while ago (I think) that hexagons which tile the lattice always have
> three step sizes; this is a theorem about Z^n and its dual module, not
> about any particular JI group or tuning. 

This has four step sizes:

! fivecrys1.scl
First 5-limit crystal ball
7
!
6/5
5/4
4/3
3/2
8/5
5/3
2

This has five step sizes:

! fivecrys2.scl
Second 5-limit crystal ball
19
!
25/24
16/15
10/9
9/8
6/5
5/4
32/25
4/3
25/18
36/25
3/2
25/16
8/5
5/3
16/9
9/5
15/8
48/25
2

The third crystal ball continues the pattern.

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> Your response makes no sense; I just said, in the very paragraph that
> you're quoting, a situation where picking the JI group is necessary.

Mike, could you please slow down and give actual precise definitions? It's coming across to me as nonsense, though I'm sure you have some meaning in mind.

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 12:41 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> Mike, could you please slow down and give actual precise definitions? It's
> coming across to me as nonsense, though I'm sure you have some meaning in
> mind.

It would be helpful if you could state explicitly what about the
following definition you find unclear:

A "regular tuning system" is a set of musical intervals, expressed
logarithmically as real numbers, which has the additional structure of
being a free abelian group under addition. If we impose an equivalence
relation on the set of regular tuning systems such that two elements
are equivalent iff a group isomorphism exists between them, the
resulting equivalence classes are called "abstract regular tuning
systems." A subset S of an abstract regular tuning system G is called
"abstractly epimorphic" if there exists an element h in Hom(G,Z) such
that the restriction of h to S is a bijection.

The above already seems 100% precise to me, so I don't know what part
of it you want me to clarify.

-Mike

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> The above already seems 100% precise to me, so I don't know what part
> of it you want me to clarify.

The above says all rank N tuning systems are the same, which is what you rejected.

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 12:47 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > The above already seems 100% precise to me, so I don't know what part
> > of it you want me to clarify.
>
> The above says all rank N tuning systems are the same, which is what you
> rejected.

Note that I distinguish between a "regular tuning system" and an
"abstract regular tuning system." You probably missed that I wrote
"abstract" the second time.

All rank N regular tuning systems are not the same. As I defined it, a
regular tuning system is any set of intervals, expressed as real
numbers, that also naturally has the structure of being a free abelian
group under addition.

If you declare an equivalence relation on regular tuning systems such
that two are equivalent if a group isomorphism exists between them,
the resulting set of equivalence classes is the set of "abstract
regular tuning systems."

-Mike

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 12:56 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> Note that I distinguish between a "regular tuning system" and an
> "abstract regular tuning system." You probably missed that I wrote
> "abstract" the second time.

If it makes it any clearer, here's a totally abstract definition in
terms of free abelian groups:

For any free abelian group G, a subset S is called "abstractly
epimorphic" if there exists an element h in Hom(G,Z) such that the
restriction of h to S is a bijection.

-Mike

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> All rank N regular tuning systems are not the same. As I defined it, a
> regular tuning system is any set of intervals, expressed as real
> numbers, that also naturally has the structure of being a free abelian
> group under addition.

Why allow non-finite ranks?

> If you declare an equivalence relation on regular tuning systems such
> that two are equivalent if a group isomorphism exists between them,
> the resulting set of equivalence classes is the set of "abstract
> regular tuning systems."

This makes no sense: the only invariant of a free abelian group is the cardinality of its basis--for finitely generated groups, that's its rank. In any case you are putting them all in the same equivalence class--ie, are saying they are the same. But when I point this out, you object. What is it you are trying to say?

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> For any free abelian group G, a subset S is called "abstractly
> epimorphic" if there exists an element h in Hom(G,Z) such that the
> restriction of h to S is a bijection.

In other words, S is a transversal for h.

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 2:29 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > All rank N regular tuning systems are not the same. As I defined it, a
> > regular tuning system is any set of intervals, expressed as real
> > numbers, that also naturally has the structure of being a free abelian
> > group under addition.
>
> Why allow non-finite ranks?

You can either expressly disallow them or treat them as uninteresting;
I don't care. I want every JI group, including Q, to also be a regular
tuning system, and Q is of rank 1, right? So I have no problem with
that; we can limit it to finite ranks if you want.

> > If you declare an equivalence relation on regular tuning systems such
> > that two are equivalent if a group isomorphism exists between them,
> > the resulting set of equivalence classes is the set of "abstract
> > regular tuning systems."
>
> This makes no sense: the only invariant of a free abelian group is the
> cardinality of its basis--for finitely generated groups, that's its rank. In
> any case you are putting them all in the same equivalence class--ie, are
> saying they are the same. But when I point this out, you object. What is it
> you are trying to say?

You said "all rank-N tuning systems are the same," but it wasn't clear
what you meant by that. If you're asking whether or not the
equivalence relation that I defined makes all rank-N regular tuning
systems equivalent, then the answer is yes, and and deliberately so;
the resulting equivalence class is the unique rank-N *abstract*
regular tuning system. The different rank-N non-abstract regular
tuning systems can be thought of as different tunings for the same
unique rank-N abstract regular tuning system.

-Mike

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 2:38 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > For any free abelian group G, a subset S is called "abstractly
> > epimorphic" if there exists an element h in Hom(G,Z) such that the
> > restriction of h to S is a bijection.
>
> In other words, S is a transversal for h.

I didn't think that you were allowed to word things that way because h
isn't a set, but OK.

-Mike

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> You can either expressly disallow them or treat them as uninteresting;
> I don't care. I want every JI group, including Q, to also be a regular
> tuning system, and Q is of rank 1, right? So I have no problem with
> that; we can limit it to finite ranks if you want.

Q+ is of countable rank; the primes are a basis.

> You said "all rank-N tuning systems are the same," but it wasn't clear
> what you meant by that. If you're asking whether or not the
> equivalence relation that I defined makes all rank-N regular tuning
> systems equivalent, then the answer is yes, and and deliberately so;
> the resulting equivalence class is the unique rank-N *abstract*
> regular tuning system.

It's just an abelian group, it seems to me. Sorry, but I think my definition of an abstract temperament is way better.

From: genewardsmith (2013-01-23)
Subject: Re: A useful generalization of epimorphicity

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

> > In other words, S is a transversal for h.
> 
> I didn't think that you were allowed to word things that way because h
> isn't a set, but OK.

Its kernel is a set, and its cosets are sets.

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 3:23 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > You said "all rank-N tuning systems are the same," but it wasn't clear
> > what you meant by that. If you're asking whether or not the
> > equivalence relation that I defined makes all rank-N regular tuning
> > systems equivalent, then the answer is yes, and and deliberately so;
> > the resulting equivalence class is the unique rank-N *abstract*
> > regular tuning system.
>
> It's just an abelian group, it seems to me. Sorry, but I think my
> definition of an abstract temperament is way better.

Yeah, I'm lost. I don't know what page you're on. Somehow, in the
middle of this discussion, you apparently decided that the argument
was whether or not you shouldn't change the definition of an abstract
temperament. From my perspective you might as well have said "my
definition of Hahn distance is way better," which would have been
equally random.

I've defined abstract epimorphicity 3 different ways now. If you still
have questions about what it means, let me know. If you don't think
it's a useful property for a scale to have, you don't have to ever
talk about it. Also, I think I've got enough suggestions about how I
should have defined it the first time now, thanks.

Aside from abstract epimorphicity, the thing I really did care about
was the generalized definition of epimorphicity/weak epimorphicity of
a scale, which I'll use your terminology about transversals to
simplify as follows:

For any periodic scale p, there exists a group G which is completely
generated by taking Z-linear sums of the elements of Im(p). We can say
that p is "weakly epimorphic" if there exists an element h in Hom(G,Z)
such that Im(p) is a transversal of h. If p is monotone and weakly
epimorphic, then it is also "epimorphic."

This is better than your definition because it allows us to talk about
the epimorphicity of tempered scales and scales which have no mapping.
I'm going to put it on the wiki as Epimorphicity (Mike's definition).
If you think that it generalizes yours so that there's no need to have
the two definitions, feel free to merge them. If not, then we'll keep
them separate.

-Mike

From: Mike Battaglia (2013-01-23)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Wed, Jan 23, 2013 at 3:23 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > You can either expressly disallow them or treat them as uninteresting;
> > I don't care. I want every JI group, including Q, to also be a regular
> > tuning system, and Q is of rank 1, right? So I have no problem with
> > that; we can limit it to finite ranks if you want.
>
> Q+ is of countable rank; the primes are a basis.

This is obviously right, btw; I'm making silly mistakes now. Also,
apologies if I've been cranky in this thread today; I'm having some
issues with sleeping and have slept for about five hours total in the
past three days. I basically woke up Monday, took a 5-hour nap at one
point, and as far as I'm concerned it's still Monday.

If you want, you can limit the definition of a regular tuning system
to be capped at countably infinite rank. The definition is already
automatically capped at groups of rank equal to |R|, since no group
can exceed the cardinality of R itself. It all basically depends on
whether you want R itself to be a "regular tuning system" or not; I
decided to not expressly prohibit it.

-Mike

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

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

> I'm going to put it on the wiki as Epimorphicity (Mike's definition).
> If you think that it generalizes yours so that there's no need to have
> the two definitions, feel free to merge them. If not, then we'll keep
> them separate.

I've revised the article, and also added convexity as a property. I've tested repeated use of the PSLQ algorithm to find a basis for the group of the scale, and it seems to work: apply PSLQ repeatedly, until suddenly the relation you get is very complex, and therefore bogus; that's your stopping point. In theory it should be possible to code a test of whether or not an arbitary scale in Scala format is epimorphic.

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 11:28 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> I've revised the article, and also added convexity as a property. I've
> tested repeated use of the PSLQ algorithm to find a basis for the group of
> the scale, and it seems to work: apply PSLQ repeatedly, until suddenly the
> relation you get is very complex, and therefore bogus; that's your stopping
> point. In theory it should be possible to code a test of whether or not an
> arbitary scale in Scala format is epimorphic.

I made some tweaks. I have a lot more stuff to add to this page,
though, which I'll be doing later. It's likely we'll arrive at
something which is large enough to split off into a few separate
pages, though it's not worth it now.

I'm also going to put up an article on what I was calling "regular
tuning systems," which Zabka calls "unpitched generated tone systems."
As Zabka's name takes precedence and is clearer than mine anyway, I'm
going to go with that.

Long term, we're probably looking at splitting the Mathematical Theory
section into Tuning Theory and Scale Theory subsections. Exciting!

-Mike

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

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

> I'm also going to put up an article on what I was calling "regular
> tuning systems," which Zabka calls "unpitched generated tone systems."
> As Zabka's name takes precedence and is clearer than mine anyway, I'm
> going to go with that.

How does it relate to abstract regular temperaments?

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 12:12 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> I'm also going to put up an article on what I was calling "regular
> tuning systems," which Zabka calls "unpitched generated tone systems."
> As Zabka's name takes precedence and is clearer than mine anyway, I'm
> going to go with that.

Actually, this is all just far too complicated. The term "generated
tone system" is silly; it should just be called a "tone group," which
I've heard Paul use anyway. I do like the term "unpitched tone group,"
but since you've already used the term "abstract" to mean "unpitched,"
I guess I'll just call it an "abstract tone group."

Is every abstract regular temperament also a regular temperament under
your naming scheme?

What do you call regular temperaments that have been given a tuning?

-Mike

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 12:42 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > I'm also going to put up an article on what I was calling "regular
> > tuning systems," which Zabka calls "unpitched generated tone systems."
> > As Zabka's name takes precedence and is clearer than mine anyway, I'm
> > going to go with that.
>
> How does it relate to abstract regular temperaments?

Let G be a free abelian group representing a collection of musical
intervals, called an "abstract tone group," M a group homomorphism
from any subgroup of Q+* -> G, called a "mapping," and T a
homomorphism from G -> R, called a "tuning map."

Our main objects of study have been 2-tuples (G,M) and 3-tuples
(G,M,T), particularly where M is surjective and non-injective, called
abstract and (non-abstract?) regular temperaments, respectively.

In contrast, in scale theory, 2-tuples (G,T) are extremely useful to
study. These are what Zabka calls "generated tone systems," and what
I'm tentatively calling "tone groups." Given any subset S of G,
there's an associated scale T(S), which can be or not be periodic,
monotonic, epimorphic, or any of the other properties on the periodic
scale page.

Theorems which apply to the subsets of a tone group, and to the tone
group itself, apply regardless of the mapping you give to it; for
example, the 17-EDO diatonic scale, a subset of a particular rank-1
tone group, has Myhill's property regardless of whether or not you
consider it to be 2.3.7 superpyth[7], 2.3.5.7 dominant[7], or 2.3.7.11
suprapyth[7].

Furthermore, subsets of the groups G, which we might call "abstract
scales," can themselves be useful objects of study without any need to
specify T at all. Generalized versions of the epimorphic, Fokker
block, wakalix, etc properties can all apply to abstract scales.

Theorems which apply to abstract scales do so regardless of what
tuning map or harmonic mapping you give to it. For instance, the
rank-3 abstract scale represented by the pattern of steps "abacaba" is
a wakalix, regardless of whether you consider it to be TOP 7-limit
marvel temperament or TE 11-limit marvel temperament, whether you want
to give it a mapping at all, or whether you've tuned it to be
monotone. Additionally, we might want to consider which set of tuning
maps makes "abacaba" out to be a monotone scale; this is a question
about the abstract scale of G itself, and not about any particular (G,
T).

-Mike

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

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

> Is every abstract regular temperament also a regular temperament under
> your naming scheme?

What gets a name is actually the abstract temperament. 

> What do you call regular temperaments that have been given a tuning?

"A rank r regular temperament in a particular tuning may be defined by giving r multiplicatively independent real numbers, which can be multiplied together to produce the intervals attainable in the temperament. A rank r temperament will be defined by r generators, and thus r vals. An abstract regular temperament can be defined in various ways, for instance by giving a set of commas tempered out by the temperament, or a set of r independent vals defining the mapping of the temperament."

Of course, since you we defining things in terms of the abstract temperamnt we should drop the "multiplicatively independent" here, I suppose.

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

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

> Furthermore, subsets of the groups G, which we might call "abstract
> scales," can themselves be useful objects of study without any need to
> specify T at all. Generalized versions of the epimorphic, Fokker
> block, wakalix, etc properties can all apply to abstract scales.

You are saying being a Fokker block is purely a matter of group theory? Why are we not calling G a group, and leaving it at that, since without M or T that is all it is?

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 3:53 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > Is every abstract regular temperament also a regular temperament under
> > your naming scheme?
>
> What gets a name is actually the abstract temperament.

I don't mean that naming scheme. I just want to know if every
"abstract regular temperament" is also a "regular temperament."

> You are saying being a Fokker block is purely a matter of group theory?
> Why are we not calling G a group, and leaving it at that, since without M or
> T that is all it is?

It's a group that specifically represents musical intervals. For
instance, the group of vals is not an abstract tone group.

-Mike

From: Carl Lumma (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

At 01:32 PM 2013/01/24, you wrote:
>I don't mean that naming scheme. I just want to know if every
>"abstract regular temperament" is also a "regular temperament."

Pardon?

Those two terms are used synonymously, by like, everyone.

If you want to talk about scale theory, use "generated tuning"
or something like that.

-Carl

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 5:17 PM, Carl Lumma <carl@lumma.org> wrote:
>
> At 01:32 PM 2013/01/24, you wrote:
> >I don't mean that naming scheme. I just want to know if every
> >"abstract regular temperament" is also a "regular temperament."
>
> Pardon?
>
> Those two terms are used synonymously, by like, everyone.

They're not completely synonymous though. 1/4-comma meantone is a
regular temperament, but not an abstract regular temperament. I'm
specifically trying to confirm that the set of "abstract regular
temperaments" is a subset of the set of "regular temperaments." I hope
it is.

If it is, then I want to know the name that I should give to things
which aren't abstract regular temperaments.

> If you want to talk about scale theory, use "generated tuning"
> or something like that.

Zabka calls it a "generated tone system." I decided that "generated
system" was silly and wordy and I should just call it a "tone group,"
which is a term Paul used in a conversation on XA once. I was calling
it a "regular tuning system," but I think "tone group" is shorter and
clearer.

For the names of these new things, I'd like to use terminology that's
simple and close to what we have. It seems that in several places on
the wiki, Gene uses the term "abstract" to mean unpitched. Am I
correct that the term is being used consistently in this manner? If
so, then I will call an unpitched tone group an "abstract tone group,"
for instance.

The same question applies to the use of the word "weak" to mean
nonmonotonic, and to the use of the word "regular" to mean generated.

-Mike

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

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

> It's a group that specifically represents musical intervals. For
> instance, the group of vals is not an abstract tone group.

It's a group which represents musical intervals, but you have no idea which ones?

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 5:41 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > It's a group that specifically represents musical intervals. For
> > instance, the group of vals is not an abstract tone group.
>
> It's a group which represents musical intervals, but you have no idea
> which ones?

Right. It's a generic group of musical intervals.

-Mike

From: Carl Lumma (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

>> >I don't mean that naming scheme. I just want to know if every
>> >"abstract regular temperament" is also a "regular temperament."
>>
>> Pardon?
>>
>> Those two terms are used synonymously, by like, everyone.
>
>They're not completely synonymous though. 1/4-comma meantone is a
>regular temperament,

Not according to me!!!!!

>> If you want to talk about scale theory, use "generated tuning"
>> or something like that.
>
>Zabka calls it a "generated tone system."

Also used in the academic music theory circles (Carey, Clampitt etc).
Even "regular tuning" would be good.

>I decided that "generated
>system" was silly and wordy and I should just call it a "tone group,"

Good.

>For the names of these new things, I'd like to use terminology that's
>simple and close to what we have. It seems that in several places on
>the wiki, Gene uses the term "abstract" to mean unpitched. Am I
>correct that the term is being used consistently in this manner? If
>so, then I will call an unpitched tone group an "abstract tone group,"
>for instance.

Gene tacked on "abstract" to appease Paul and George Secor with
their nonsense about historical "temperaments". Nobody actually
writes the "abstract" part in practice.

-Carl

From: Carl Lumma (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

>> It's a group which represents musical intervals, but you have no idea
>> which ones?
>
>Right. It's a generic group of musical intervals.

Another reason it shouldn't have "temperament" in the name. -C.

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
>
> On Thu, Jan 24, 2013 at 5:41 PM, genewardsmith 
> wrote:

> > It's a group which represents musical intervals, but you have no idea
> > which ones?
> 
> Right. It's a generic group of musical intervals.

If it's just a group, it would be far less confusing to call it "a group".

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, Carl Lumma  wrote:

> Gene tacked on "abstract" to appease Paul and George Secor with
> their nonsense about historical "temperaments". Nobody actually
> writes the "abstract" part in practice.

I do when I'm talking about what Mike is calling (M, G), where M is a mapping from a JI group. The idea is a specific tuning for a temperament factors through the abstract temperament. I used to use other terminology for this, but it never caught on, so appeasing Paul, George and Ringo was sort of convenient.

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 5:46 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >They're not completely synonymous though. 1/4-comma meantone is a
> >regular temperament,
>
> Not according to me!!!!!

What do you call 1/4-comma meantone then if it's not a regular temperament??

> Gene tacked on "abstract" to appease Paul and George Secor with
> their nonsense about historical "temperaments". Nobody actually
> writes the "abstract" part in practice.

OK, I just want to know names to give to these types of things:

1) Things like the 5-limit rank-1 temperament eliminating 128/125 and
256/243, with no assigned tuning
2) Things like the 5-limit rank-1 temperament eliminating 128/125 and
256/243, with an 80 cent generator
3) Things like the 5-limit rank-1 temperament eliminating 128/125 and
256/243, tuned to Keenan's marimba well-tuning
4) Things like the unmapped, regularly generated tuning system with
generators {696, 1200}
5) Unmapped regularly generated tuning systems in general, without
regard to specific tunings

Either #1 or #2 should be "regular temperament," and either #4 or #5
should be "tone group." What is a very simple classification system
for all five of these things?

I don't care if we ditch "abstract" and was never that invested in it;
I just want a clear naming scheme for the whole thing.

>>> It's a group which represents musical intervals, but you have no idea
>>> which ones?
>>
>>Right. It's a generic group of musical intervals.
>
> Another reason it shouldn't have "temperament" in the name. -C.

It doesn't; I was referring to an "abstract tone group" there, or an
"unpitched generated tuning system" which hasn't been assigned a
tuning map yet, or whatever we're calling it now.

-Mike

From: Mike Battaglia (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 5:51 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
> >
> > On Thu, Jan 24, 2013 at 5:41 PM, genewardsmith
> > wrote:
>
> > > It's a group which represents musical intervals, but you have no idea
> > > which ones?
> >
> > Right. It's a generic group of musical intervals.
>
> If it's just a group, it would be far less confusing to call it "a group".

The term "tone group" refers to a group of musical tones and not a
group of anything else. The nomenclature applies to the application of
the structure, not an aspect of the structure itself.

-Mike

From: Carl Lumma (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

At 02:57 PM 2013/01/24, you wrote:
>I do when I'm talking about what Mike is calling (M, G), where M is a 
>mapping from a JI group. The idea is a specific tuning for a 
>temperament factors through the abstract temperament.

I know, but I dispute that you really practice this most of
the time. Certainly Wikipedia doesn't

http://en.wikipedia.org/wiki/Regular_temperament

http://en.wikipedia.org/wiki/Miracle_temperament

etc.

And IMO it should be killed completely.

-Carl

From: Carl Lumma (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

>> >They're not completely synonymous though. 1/4-comma meantone is a
>> >regular temperament,
>>
>> Not according to me!!!!!
>
>What do you call 1/4-comma meantone then if it's not a regular temperament??

A tuning of a regular temperament.

>> Gene tacked on "abstract" to appease Paul and George Secor with
>> their nonsense about historical "temperaments". Nobody actually
>> writes the "abstract" part in practice.
>
>OK, I just want to know names to give to these types of things:
>
>1) Things like the 5-limit rank-1 temperament eliminating 128/125 and
>256/243, with no assigned tuning

A temperament.

>2) Things like the 5-limit rank-1 temperament eliminating 128/125 and
>256/243, with an 80 cent generator

A temperament and a tuning.

>3) Things like the 5-limit rank-1 temperament eliminating 128/125 and
>256/243, tuned to Keenan's marimba well-tuning

A temperament and a tuning.

>4) Things like the unmapped, regularly generated tuning system with
>generators {696, 1200}

Don't care. Generated tuning?

>5) Unmapped regularly generated tuning systems in general, without
>regard to specific tunings

Generated tunings?

>Either #1 or #2 should be "regular temperament," and either #4 or #5
>should be "tone group." What is a very simple classification system
>for all five of these things?

I'll accept the modifier "regular" but I'll omit it most of the
time, like most people already do.

-Carl

From: Carl Lumma (2013-01-24)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

>>2) Things like the 5-limit rank-1 temperament eliminating 128/125 and
>>256/243, with an 80 cent generator
>
>A temperament and a tuning.

Or in practice, just a "tuning" or a "system", again, like most
people write in the middle of a paragraph where lengthy, exact
word assemblies are neither needed nor desirable.

-Carl

From: genewardsmith (2013-01-24)
Subject: Re: A useful generalization of epimorphicity

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

> The term "tone group" refers to a group of musical tones and not a
> group of anything else. The nomenclature applies to the application of
> the structure, not an aspect of the structure itself.

If it's neither mapped from Q+, nor mapped to pitches, in what way is it "a group of musical tones"?

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 6:05 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >Either #1 or #2 should be "regular temperament," and either #4 or #5
> >should be "tone group." What is a very simple classification system
> >for all five of these things?
>
> I'll accept the modifier "regular" but I'll omit it most of the
> time, like most people already do.

I don't care if we ditch regular. I think Gene won't like it. If he
doesn't, I'd be happy at this juncture if perhaps he'd tell me what
"regular" actually means, since it also apparently encompasses
circulating temperaments and temperaments with more than one mapping
for 9/1.

-Mike

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 6:37 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > The term "tone group" refers to a group of musical tones and not a
> > group of anything else. The nomenclature applies to the application of
> > the structure, not an aspect of the structure itself.
>
> If it's neither mapped from Q+, nor mapped to pitches, in what way is it
> "a group of musical tones"?

I don't see how the mapping from Q+ is relevant to what we're
discussing, but if you like, you can consider it an equivalence class
of mappings from Q+, such that two mappings are equivalent if an
isomorphism exists between them. Other than that, this isn't really
that abstract. Whenever we talk about scale imprints, such as aabaaab,
we're talking about things which exist in this "group of musical
tones."

-Mike

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> I don't care if we ditch regular. I think Gene won't like it. If he
> doesn't, I'd be happy at this juncture if perhaps he'd tell me what
> "regular" actually means, since it also apparently encompasses
> circulating temperaments and temperaments with more than one mapping
> for 9/1.

Regular means the same interval is the same size everywhere it appears, in other words *not* circulating and not inconsistent.

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> I don't see how the mapping from Q+ is relevant to what we're
> discussing, but if you like, you can consider it an equivalence class
> of mappings from Q+, such that two mappings are equivalent if an
> isomorphism exists between them.

Morphisms between morphisms now?

 Other than that, this isn't really
> that abstract. Whenever we talk about scale imprints, such as aabaaab,
> we're talking about things which exist in this "group of musical
> tones."

WHAT group of musical tones? You've provided no definition for one so far as I can see, which is why I've been complaining for the last few days you are not making sense.

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 6:37 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > The term "tone group" refers to a group of musical tones and not a
> > group of anything else. The nomenclature applies to the application of
> > the structure, not an aspect of the structure itself.
>
> If it's neither mapped from Q+, nor mapped to pitches, in what way is it
> "a group of musical tones"?

I should also add that a periodic scale is nothing more than a
real-valued quasiperiodic function of an integer variable. But, what
you wrote on the wiki is that it's a "musical interval"-valued
quasiperiodic function of an integer variable. But, you complain when
I say that an abstract tone group is a free abelian group whose
elements are "musical intervals." I don't really get what you're
driving at.

-Mike

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 8:07 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > I don't see how the mapping from Q+ is relevant to what we're
> > discussing, but if you like, you can consider it an equivalence class
> > of mappings from Q+, such that two mappings are equivalent if an
> > isomorphism exists between them.
>
> Morphisms between morphisms now?

If an isomorphism exists between their codomains, I mean.

> Other than that, this isn't really
> > that abstract. Whenever we talk about scale imprints, such as aabaaab,
> > we're talking about things which exist in this "group of musical
> > tones."
>
> WHAT group of musical tones? You've provided no definition for one so far
> as I can see, which is why I've been complaining for the last few days you
> are not making sense.

ANY free abelian group whose elements are supposed to represent
"musical intervals" is canned an "abstract tone group," just like ANY
quasi-periodic function of an integer variable mapping to "musical
intervals" is called a "periodic scale." I'm trying my best to
explain, but I don't see what the divide is.

-Mike

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> I should also add that a periodic scale is nothing more than a
> real-valued quasiperiodic function of an integer variable. But, what
> you wrote on the wiki is that it's a "musical interval"-valued
> quasiperiodic function of an integer variable.

Read the definition again. Do you see the word "cents"? I say it's musical interval valued because it is.

 But, you complain when
> I say that an abstract tone group is a free abelian group whose
> elements are "musical intervals." I don't really get what you're
> driving at.

What I'm driving at is that I don't know what the hell you mean. In what sense are the elements musical intervals?

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 8:24 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > But, you complain when
> > I say that an abstract tone group is a free abelian group whose
> > elements are "musical intervals." I don't really get what you're
> > driving at.
>
> What I'm driving at is that I don't know what the hell you mean. In what
> sense are the elements musical intervals?

A scale is a set of musical intervals. All MOS's are scales. 1L1s is
an MOS. Thus, 1L1s is a scale, which means it's a set of musical
intervals. These musical intervals have neither a specified harmonic
mapping, nor a tuning. The group generated by this MOS is an example
of an abstract tone group, and its elements are musical intervals in
exactly the same sense that 1L1s is an MOS, meaning a scale, meaning a
collection of musical intervals.

-Mike

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> A scale is a set of musical intervals. All MOS's are scales. 1L1s is
> an MOS. Thus, 1L1s is a scale, which means it's a set of musical
> intervals. These musical intervals have neither a specified harmonic
> mapping, nor a tuning. The group generated by this MOS is an example
> of an abstract tone group, and its elements are musical intervals in
> exactly the same sense that 1L1s is an MOS, meaning a scale, meaning a
> collection of musical intervals.

1L1s is not what people ordinarily call a scale, but at least it has some connection to music. In what sense does your "musical interval group" have any connection to music?

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 9:18 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > A scale is a set of musical intervals. All MOS's are scales. 1L1s is
> > an MOS. Thus, 1L1s is a scale, which means it's a set of musical
> > intervals. These musical intervals have neither a specified harmonic
> > mapping, nor a tuning. The group generated by this MOS is an example
> > of an abstract tone group, and its elements are musical intervals in
> > exactly the same sense that 1L1s is an MOS, meaning a scale, meaning a
> > collection of musical intervals.
>
> 1L1s is not what people ordinarily call a scale,

Let's be crystal clear here: for any x and y, xLys is an MOS, and all
MOS's are scales. Thus, 5L2s, 2L3s, and 1L1s are all scales.

> but at least it has some connection to music.

Can you please state explicitly what connection to music this has
which satisfies you, so I can just adapt it?

-Mike

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> Can you please state explicitly what connection to music this has
> which satisfies you, so I can just adapt it?

1L1s means that within an octave there is some interval c with 600 < c < 1200 cents.

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 10:37 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > Can you please state explicitly what connection to music this has
> > which satisfies you, so I can just adapt it?
>
> 1L1s means that within an octave there is some interval c with 600 < c <
> 1200 cents.

If you insist on your MOS's having a 1200 cent period, you're gonna
have a bad time. Next you'll insist that the period be some form of
2/1 too.

-Mike

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Thu, Jan 24, 2013 at 10:37 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
>
> > Can you please state explicitly what connection to music this has
> > which satisfies you, so I can just adapt it?
>
> 1L1s means that within an octave there is some interval c with 600 < c <
> 1200 cents.

Also, coming back to this a few hours later, your concept of things
makes no sense to me.

Scales neither require mappings (periodic scale) nor tunings (abstract
Fokker block). Furthermore, scales are sets of musical intervals,
which neither require mappings (600 cents) nor tunings (meantone
fifth).

So you have no problem with the concept of a musical interval being a
thing not requiring a mapping nor a tuning. You also have no problem
with the concept of a set of musical intervals not requiring a mapping
nor a tuning. But, when it comes to a group of musical intervals which
require neither a mapping nor a tuning, you're unable to figure out
what it all means.

In good faith, here's a great paper on the subject, and on how two
MOS's which haven't been assigned tunings nor mappings can be combined
via the "product word" operation to create a Fokker block which has
neither tunings nor mappings:
http://link.springer.com/chapter/10.1007/978-3-642-21590-2_24?no-access=true

This is useful and applies to not only regular temperaments and Fokker
groups, but also to rank-2 scales and rank-3 scales.

Other than that, I'm not sure there's much more I can say on this topic.

-Mike

From: Carl Lumma (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

Mike wrote:
>I don't care if we ditch regular. I think Gene won't like it. If he
>doesn't, I'd be happy at this juncture if perhaps he'd tell me what
>"regular" actually means, since it also apparently encompasses
>circulating temperaments and temperaments with more than one mapping
>for 9/1.

I'm pretty sure it means there has to be a unique mapping from
just intonation.

>Scales neither require mappings (periodic scale) nor tunings (abstract
>Fokker block). Furthermore, scales are sets of musical intervals,
>which neither require mappings (600 cents) nor tunings (meantone
>fifth).

In Gene's definition (which I use), scales do require tunings.
MOSs are scales. A scale is something that can be specified
with a .scl file.

-Carl

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> In good faith, here's a great paper on the subject, and on how two
> MOS's which haven't been assigned tunings nor mappings can be combined
> via the "product word" operation to create a Fokker block which has
> neither tunings nor mappings:
> http://link.springer.com/chapter/10.1007/978-3-642-21590-2_24?no-access=true

It has approximate tunings, just as specifying a MOS in terms of the pattern of large and small steps has approximate tunings. And to prove that you can create Fokker blocks in this way requires more extensive recourse to tuning.

> This is useful and applies to not only regular temperaments and Fokker
> groups, but also to rank-2 scales and rank-3 scales.

Product words are useful, and in fact are connected to defining Fokker blocks in terms of rank two temperaments. That hardly proves taking an abstract group which has no relationship to music of any kind and declaring it to be "musical" in some undefined sense is useful.

Can you try to find one other person (Keenan would be the obvious choice) who understands what you are trying to say?

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

I'm going to come back to the "regular" thing later, as this
conversation is going to get crazy if I start going into that.

On Fri, Jan 25, 2013 at 2:36 AM, Carl Lumma <carl@lumma.org> wrote:
>
> >Scales neither require mappings (periodic scale) nor tunings (abstract
> >Fokker block). Furthermore, scales are sets of musical intervals,
> >which neither require mappings (600 cents) nor tunings (meantone
> >fifth).
>
> In Gene's definition (which I use), scales do require tunings.
> MOSs are scales. A scale is something that can be specified
> with a .scl file.

So if MOS's are scales, then is 5L2s, which has no tuning, not an MOS?
What about the meantone diatonic scale in general, before it's been
assigned a tuning map? If so, what do you call these things?

-Mike

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> So if MOS's are scales, then is 5L2s, which has no tuning, not an MOS?
> What about the meantone diatonic scale in general, before it's been
> assigned a tuning map? If so, what do you call these things?

It's not true that has no tuning; presuming it lies within an octave, the generator range is 4\7 < g < 3\5, and whatever interval of repetition it lies within, that is the range of tunings in terms of fractions of it. I don't think these things have been given a name.

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 5:11 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > So if MOS's are scales, then is 5L2s, which has no tuning, not an MOS?
> > What about the meantone diatonic scale in general, before it's been
> > assigned a tuning map? If so, what do you call these things?
>
> It's not true that has no tuning; presuming it lies within an octave, the
> generator range is 4\7 < g < 3\5, and whatever interval of repetition it
> lies within, that is the range of tunings in terms of fractions of it. I
> don't think these things have been given a name.

The name given to them by everyone I've ever seen refer to them is
"MOS," and insisting otherwise is in the same sort of spirit as Paul
insisting that the space of tuning maps and the space of real vals
must have different "units."

-Mike

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

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

> The name given to them by everyone I've ever seen refer to them is
> "MOS," and insisting otherwise is in the same sort of spirit as Paul
> insisting that the space of tuning maps and the space of real vals
> must have different "units."

Can you point to someone other than you who thinks a 5L2s inside an octave and a 5L2s within 600 cents are the same? I'm sorry, I just don't buy your claim everyone has been using your language.

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, "genewardsmith"  wrote:

> Can you point to someone other than you who thinks a 5L2s inside an octave and a 5L2s within 600 cents are the same? I'm sorry, I just don't buy your claim everyone has been using your language.
>

"An MOS or Moment Of Symmetry is a scale in which every interval except for the period comes in two sizes."
http://xenharmonic.wikispaces.com/MOSScales

"Below are ranges of generators for various L-s patterns of MOS, with the number of steps in the scale from 2 to 22"
http://xenharmonic.wikispaces.com/Generator+ranges+of+MOS

"To summarize, you can design scales by building a chain of one interval (the generator) within a period of another interval -- often, but not always, the octave. When the resulting set of notes has exactly two step sizes, we call the scale a Moment of Symmetry, or MOS, scale."
http://xenharmonic.wikispaces.com/MOS+Cradle

"An MOS specifically consists of:

1. A period "P" (of any size but most commonly the octave or a 1/N fraction of an octave)
2. A generator "g" (of any size, for example 700 cents in 12 equal temperament) which is added repeatedly to make a chain of scale steps, starting from the unison or 0 cents scale step, and then reducing to within the period
3. Exactly two sizes of scale steps (Large and small, often written "L" and "s")
4. Where each number of scale steps, or generic interval, within the scale occurs in no more than two different sizes, and in exactly two if the interval is not a multiple of the period
5. The unison or starting point of the scale is then allowed to be transferred to any scale degree--all the modes of an MOS are legal."
http://xenharmonic.wikispaces.com/Mathematics+of+MOS

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 5:27 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > The name given to them by everyone I've ever seen refer to them is
> > "MOS," and insisting otherwise is in the same sort of spirit as Paul
> > insisting that the space of tuning maps and the space of real vals
> > must have different "units."
>
> Can you point to someone other than you who thinks a 5L2s inside an octave
> and a 5L2s within 600 cents are the same? I'm sorry, I just don't buy your
> claim everyone has been using your language.

What do you mean that they're "the same?" I said that 5L2s refers to
an MOS, even though it's not been assigned a different tuning map. It
hasn't even been assigned a choice of period; it could refer to the
TOP meantone diatonic scale or the TE meantone diatonic scale or the
12-EDO diatonic scale.

But, for the sake of argument and to speed this up, I'll hand it to
you that most people assume implicitly that the period has been
assigned a -mapping-, in this case it's a form of 2/1. Still, if you
always insist that an MOS is a "scale," and a "scale" is a thing which
has specific pitches, then even something like "5L2s with a period
mapped to 2/1" wouldn't be an MOS. More worrying is that the "meantone
diatonic scale" wouldn't be an MOS until it's assigned a tuning map.
Most worrying of all is that if you want the "diatonic scale" isn't
even a scale under your definition!

Surely you can't claim the consensus among everyone is that a "scale"
is a thing which has been assigned a specific tuning.

-Mike

From: Mike Battaglia (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 1:57 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > In good faith, here's a great paper on the subject, and on how two
> > MOS's which haven't been assigned tunings nor mappings can be combined
> > via the "product word" operation to create a Fokker block which has
> > neither tunings nor mappings:
> >
> > http://link.springer.com/chapter/10.1007/978-3-642-21590-2_24?no-access=true
>
> It has approximate tunings, just as specifying a MOS in terms of the
> pattern of large and small steps has approximate tunings. And to prove that
> you can create Fokker blocks in this way requires more extensive recourse to
> tuning.

No, tuning isn't involved at any point. He defines MOS's entirely in
terms of words, which he calls "Rational Mechanical Words," then shows
how you can take the product word of them to derive words which are
basically just the obvious generalization of Fokker blocks to scales,
which he calls "(quasi-)pairwise well-formed scales." I can send you
the paper if you don't have it. Tuning only gets involved when
monotonicity gets brought into it.

A mea culpa from me: I screwed up; the thing he calls an "unpitched
generated tone system" is the name he gives to a the thing that's like
a Fokker block, but which hasn't been assigned pitches or mappings,
not the name he gives to the group the Fokker block generates. It's
actually a bit more general than a Fokker block, but I can't figure
how, since he keeps using the notation Z[X] and R[X] to refer to
something other than the ring of polynomials in an indeterminate.

Since the elements in an "unpitched generated tone system" still have
no pitch nor mapping, they're still things which I guess Gene would
consider "unmusical," though neither Zabka nor I nor Keenan feel the
same way. At any rate, the thing I was calling an "abstract tone
group" is the group generated by the "unpitched generated tone
system," not the unpitched generated tone system itself.

> Product words are useful, and in fact are connected to defining Fokker
> blocks in terms of rank two temperaments. That hardly proves taking an
> abstract group which has no relationship to music of any kind and declaring
> it to be "musical" in some undefined sense is useful.

I'm not sure what sort of thing would serve as "proof" that the
generic concept of a musical interval - not any specific musical
interval with a pitch or mapping, but the concept of a musical
interval in general - has "a relationship to music of any kind."

> Can you try to find one other person (Keenan would be the obvious choice)
> who understands what you are trying to say?

I would be surprised if anyone in the group had serious trouble
"understanding what I'm trying to say." Keenan obviously understands
it, since all of his work on rank-3 scales is in this paradigm. Carl
clearly doesn't care about the name one gives to the concept (he
suggested "generated tunings" above), and though I doubt he cares
about the concept at all, I don't think he "doesn't understand" it.
Keenan doesn't, in general, give a damn what we name anything, and at
this point I've lost interest in the name "abstract tone group"
anyway.

I am, however, really interested in understanding the communication
divide between you and me here, so I can better understand the way
that you look at things. My goal is to generalize "Fokker block" in a
way that applies to both scales and to abstract regular temperaments.
The only way that I can understand how to do so is to construct it
entirely group-theoretically, and say that if we're in any way using
this group-theoretic structure to model the behavior of musical
intervals acting under composition, it's called a Fokker block. If it
makes you happy, I can drop that last bit and just say a Fokker block
is a group-theoretic thing, at which point it would not only apply to
your existing wiki article and to scales, but to things like the group
of vals, the Fokker group for a particular val, etc. But, I'm trying
to understand your perspective. Maybe it's a mathematician thing that
I don't get.

-Mike

From: genewardsmith (2013-01-25)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
More worrying is that the "meantone
> diatonic scale" wouldn't be an MOS until it's assigned a tuning map.

Mike, this is how language works, sometimes even in mathematics.

> Surely you can't claim the consensus among everyone is that a "scale"
> is a thing which has been assigned a specific tuning.

I think that's pretty much the consensus for most uses, but sometimes a range of tunings is possible, as with the diatonic scale aka Meantone[7]. If it's not clear and you need it to be, then clarify.

From: Carl Lumma (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

>> In Gene's definition (which I use), scales do require tunings.
>> MOSs are scales. A scale is something that can be specified
>> with a .scl file.
>
>So if MOS's are scales,

Gah, typo. *Aren't* scales.

>What about the meantone diatonic scale in general, before it's been
>assigned a tuning map? If so, what do you call these things?

It's true that I sometimes refer to "scales" without a tuning.
In that respect, the customary and systematic usage don't match.
Really something like "the diatonic scale" is an equivalence
class of scales. -Carl

From: Carl Lumma (2013-01-25)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

>>So if MOS's are scales,
>
>Gah, typo. *Aren't* scales.

To clarify, the thing in a .scl file can be called a "MOS".
But by MOS I thought you meant things like 5L2s. Those things
can't be in .scl files and aren't "scales" in the rigorous
sense.

-Carl

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> No, tuning isn't involved at any point. He defines MOS's entirely in
> terms of words, which he calls "Rational Mechanical Words," then shows
> how you can take the product word of them to derive words which are
> basically just the obvious generalization of Fokker blocks to scales,
> which he calls "(quasi-)pairwise well-formed scales." I can send you
> the paper if you don't have it. Tuning only gets involved when
> monotonicity gets brought into it.
> 
> A mea culpa from me: I screwed up; the thing he calls an "unpitched
> generated tone system" is the name he gives to a the thing that's like
> a Fokker block, but which hasn't been assigned pitches or mappings,
> not the name he gives to the group the Fokker block generates. It's
> actually a bit more general than a Fokker block, but I can't figure
> how, since he keeps using the notation Z[X] and R[X] to refer to
> something other than the ring of polynomials in an indeterminate.
> 
> Since the elements in an "unpitched generated tone system" still have
> no pitch nor mapping, they're still things which I guess Gene would
> consider "unmusical," though neither Zabka nor I nor Keenan feel the
> same way. At any rate, the thing I was calling an "abstract tone
> group" is the group generated by the "unpitched generated tone
> system," not the unpitched generated tone system itself.

Zabka, for the purposes of his paper, defines a "tone" as an element of of a finite set which maps to a finitely generated abelian group he calls Z[X] (something which usually denotes a ring, but never mind.) This is what you want to call a "tone group", so I see where you are coming from. However, outside of Zabka's paper what he calls "tones" are not called tones, what he calls "commas" are not called commas, and what he calls "intervals" are not called intervals. Trying to force people to use this terminology which was created for a particular purpose and makes no sense outside of that context is a truly bad idea. In fact, I wish Zabka had not chosen the language he did, given your reaction.

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 6:14 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >> In Gene's definition (which I use), scales do require tunings.
> >> MOSs are scales. A scale is something that can be specified
> >> with a .scl file.
> >
> >So if MOS's are scales,
>
> Gah, typo. *Aren't* scales.
//snip
> It's true that I sometimes refer to "scales" without a tuning.
> In that respect, the customary and systematic usage don't match.
> Really something like "the diatonic scale" is an equivalence
> class of scales. -Carl

Actually, according to Gene, they're "abstract" scales. Search for
"abstract" on this page:
http://xenharmonic.wikispaces.com/Fokker+blocks.

So "abstract" scales are sets of "unpitched," "mapped" intervals, and
"non-abstract" scales are sets of "pitched," "unmapped" intervals.

The term "scale" is apparently supposed to be something encapsulating
both, as you can see from the fact that "equal temperaments," "MOS's"
and "Fokker blocks" are all "families of scales":
http://xenharmonic.wikispaces.com/Families+of+scales, and that a
"convex scale" is apparently something that exists in the interval
lattice of a regular temperament, tuning be damned:
http://xenharmonic.wikispaces.com/Convex+scale

To be honest, I picked this "abstract tone group" name because I was
trying to keep somewhat in line with the terminology that's on the
wiki, since I saw this term "abstract" being thrown around to mean
"unpitched," as in "abstract scale" and "abstract regular
temperament." But, now I'm starting to see that attempting to fit into
the terminology on the wiki is completely hopeless. These terms aren't
being used in any sort of consistent way.

My goal was to build off the Zabka paper, which is remarkable in how
it unifies scales and regular temperaments. But the only way to fit
into this naming scheme is to add even MORE inconsistent qualifiers,
making it even worse. We'll end up with "abstract Fokker blocks,"
"unmapped Fokker blocks," "weak unmapped Fokker blocks," etc, but one
should always note that there's going to be no "weak abstract Fokker
blocks," and some people aren't quite convinced that "unpitched
unmapped Fokker blocks" have anything to do with music at all, etc.
Now we have seven names for the same thing.

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
>
> On Fri, Jan 25, 2013 at 6:14 PM, Carl Lumma  wrote:
> >
> > >> In Gene's definition (which I use), scales do require tunings.
> > >> MOSs are scales. A scale is something that can be specified
> > >> with a .scl file.
> > >
> > >So if MOS's are scales,
> >
> > Gah, typo. *Aren't* scales.
> //snip
> > It's true that I sometimes refer to "scales" without a tuning.
> > In that respect, the customary and systematic usage don't match.
> > Really something like "the diatonic scale" is an equivalence
> > class of scales. -Carl
> 
> Actually, according to Gene, they're "abstract" scales. Search for
> "abstract" on this page:
> http://xenharmonic.wikispaces.com/Fokker+blocks.

Nope: my abstract scales have specific scale elements, not a range of elements.

From: Carl Lumma (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

>My goal was to build off the Zabka paper, which is remarkable in how
>it unifies scales and regular temperaments.

Indeed. And Keenan came to many of the same conclusions
independently. -Carl

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 7:58 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
> Trying to force people to use this terminology which was
> created for a particular purpose and makes no sense outside of that context
> is a truly bad idea. In fact, I wish Zabka had not chosen the language he
> did, given your reaction.

OK, but not even your objections about naming are consistent. For example:

Let S be a contiguous subset of the integers, T a "tuning" function S
-> R, and M a "mapping" function from S -> any quotient group of Q+*.
Then an "abstract" scale is a tuple (S, M), and a "non-abstract" scale
is a tuple (S, T).

As you can see, the only thing that these two structures have in
common is the set S, which is just a contiguous subset of Z with no
further structure.

If we want to allow the existence of "scales" which are dense, such as
treating the entire set of p-limit rationals as a "scale," then we can
even drop that restriction. Now the only thing that they have in
common is the set S, with no further structure -AT ALL-. Or, as you
apparently like to call it, a "musical interval system":
http://xenharmonic.wikispaces.com/Musical+Interval+Systems

A musical interval system is apparently nothing other than a set are,
in some vague way, intended to represent musical intervals. You have
no problem with this, but heaven forbid this set should happen to be a
free abelian group; then suddenly there's a huge problem calling it an
"abstract tone group." Then it's supposed to just be called a "free
abelian group," or also, apparently, a Musical Interval System.

-Mike

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 8:07 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >My goal was to build off the Zabka paper, which is remarkable in how
> >it unifies scales and regular temperaments.
>
> Indeed. And Keenan came to many of the same conclusions
> independently. -Carl

Yes, he did. He should get credit for that as well. Also, sadly,
Keenan and I had about a year's worth of heavily intense offlist
discussions building on those insights, back when XA chat was in
Facebook, which we were just stupid and lazy and failed to document
anywhere at all (e.g. virtual generators, metamodes, wakalixes as
trihexagonal tilings of the lattice, etc.). I was hoping to get some
of that done this week.

-Mike

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 8:04 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
> >
> > Actually, according to Gene, they're "abstract" scales. Search for
> > "abstract" on this page:
> > http://xenharmonic.wikispaces.com/Fokker+blocks.
>
> Nope: my abstract scales have specific scale elements, not a range of
> elements.

Even this objection isn't consistent, as you previously wrote:

> sometimes a range of tunings is possible, as with the diatonic scale aka Meantone[7].

Your "scale elements" are allowed to have a range of tunings, and also
a range of mappings. So what, pray tell, is a "scale," other than a
well-ordered set, where the well-ordering is no greater than ω, whose
elements are intended to represent musical intervals?

And if you want to allow dense scales, then what is a scale other than
a set whose elements are intended to represent musical intervals? As I
just noted, you had no problem calling that a "musical interval
system." If you have a problem with groups whose elements are intended
to represent musical intervals, why not just eliminate the entirety of
that page and say "A musical interval system is a set."?

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> Let S be a contiguous subset of the integers, T a "tuning" function S
> -> R, and M a "mapping" function from S -> any quotient group of Q+*.
> Then an "abstract" scale is a tuple (S, M), and a "non-abstract" scale
> is a tuple (S, T).

I don't recall making these definitions. Where would that be?

> If we want to allow the existence of "scales" which are dense, such as
> treating the entire set of p-limit rationals as a "scale," then we can
> even drop that restriction. Now the only thing that they have in
> common is the set S, with no further structure -AT ALL-. Or, as you
> apparently like to call it, a "musical interval system":
> http://xenharmonic.wikispaces.com/Musical+Interval+Systems

I call it a musical interval system precisely because I don't want to call such things as 11-limit JI a scale. And since as I defined it, a musical interval system is a set of musical intervals, it's hardly the same as what you are promoting.

> A musical interval system is apparently nothing other than a set are,
> in some vague way, intended to represent musical intervals. 

A set of frequencies in Hz or values given in cents from a reference pitch is "nothing other than a set" related "in some vague way" to music? 

You have
> no problem with this, but heaven forbid this set should happen to be a
> free abelian group; then suddenly there's a huge problem calling it an
> "abstract tone group." Then it's supposed to just be called a "free
> abelian group," or also, apparently, a Musical Interval System.

See above. I talk about pitches, and you don't. And since when is a group which you intend to use, in the end, to refer to pitch classes a "tone group" anyway? And please, do NOT call it a "musical interval system", unless it is.

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> > sometimes a range of tunings is possible, as with the diatonic scale aka Meantone[7].
> 
> Your "scale elements" are allowed to have a range of tunings, and also
> a range of mappings.

My abstract regular temperament elements do not have a range of mappings. The range of tunings is exactly the range, whatever that is, of the temperament.

As I
> just noted, you had no problem calling that a "musical interval
> system." 

As I just noted, you've got it all wrong.

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 9:10 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > Let S be a contiguous subset of the integers, T a "tuning" function S
> > -> R, and M a "mapping" function from S -> any quotient group of Q+*.
> > Then an "abstract" scale is a tuple (S, M), and a "non-abstract" scale
> > is a tuple (S, T).
>
> I don't recall making these definitions. Where would that be?

That is completely equivalent to every other definition I've ever seen
given for a scale, and if you don't think so, it would be wise to
specify exactly what a "scale" is and in what way it's different from
what I wrote.

> I call it a musical interval system precisely because I don't want to call
> such things as 11-limit JI a scale. And since as I defined it, a musical
> interval system is a set of musical intervals, it's hardly the same as what
> you are promoting.

In the future, why not just write "you need to say what a musical
interval is and I don't, neener neener neener?"

At any rate, a "musical interval system" which is also a free abelian
group is called an "abstract tone group." One which has been given a
tuning is called a "tone group," and one which has been given a
mapping is called a "

> > A musical interval system is apparently nothing other than a set are,
> > in some vague way, intended to represent musical intervals.
>
> A set of frequencies in Hz or values given in cents from a reference pitch
> is "nothing other than a set" related "in some vague way" to music?

It says that a musical interval system includes "all intervals in a
regular temperament." Is the meantone "abstract regular temperament" a
"regular temperament" or not? If so, then the meantone interval
lattice, which has no specified amount of cents, is a "musical
interval system," despite containing neither a set of frequencies in
Hz nor values given in cents from a reference pitch.

> See above. I talk about pitches, and you don't. And since when is a group
> which you intend to use, in the end, to refer to pitch classes a "tone
> group" anyway?

What do you mean "since when?" That's how I just defined it.

-Mike

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 9:27 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>
> At any rate, a "musical interval system" which is also a free abelian
> group is called an "abstract tone group." One which has been given a
> tuning is called a "tone group," and one which has been given a
> mapping is called a "

damn typos. One which has been given a mapping is called an "abstract
regular temperament."

-Mike

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 9:16 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> My abstract regular temperament elements do not have a range of mappings.

As it stands, you haven't given me any definition for what a scale
actually is, other than a set of completely undefined "musical
intervals" which, I presume, is supposed to be the image of a function
mapping from some contiguous subset of Z.

Carl thinks that a scale is supposed to always have a tuning. If I'm
supposed to divine from your response above that a scale is a set of
elements having either all a specified tuning, OR all a specified
mapping, you should just say that directly. If it makes you happy, we
can say that a "tone group" is a group of musical intervals which have
either a specified tuning or a specified mapping.

Assuming that's what you really mean, I still think that it's a
needlessly restrictive definition, since you apparently have no
problem with the meantone diatonic scale being an scale no matter how
it's tuned, and you have no problem with the 17-EDO diatonic scale
being a scale no matter how it's mapped, but you have a problem with
5L2s with a period of 2/1 being a "scale" for absolutely no reason at
all.

> The range of tunings is exactly the range, whatever that is, of the
> temperament.

Temperaments have no range, as it stands, only an optimal tuning. None
of this is as precise as you seem to be demanding that I be with my
"abstract tone group."

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
>
> On Fri, Jan 25, 2013 at 9:10 PM, genewardsmith 
> wrote:
> >
> > > Let S be a contiguous subset of the integers, T a "tuning" function S
> > > -> R, and M a "mapping" function from S -> any quotient group of Q+*.
> > > Then an "abstract" scale is a tuple (S, M), and a "non-abstract" scale
> > > is a tuple (S, T).
> >
> > I don't recall making these definitions. Where would that be?
> 
> That is completely equivalent to every other definition I've ever seen
> given for a scale, and if you don't think so, it would be wise to
> specify exactly what a "scale" is and in what way it's different from
> what I wrote.

I know you know of the article on periodic scales, which defines things quite differently than the above.

> At any rate, a "musical interval system" which is also a free abelian
> group is called an "abstract tone group." 

By whom?

One which has been given a
> tuning is called a "tone group," and one which has been given a
> mapping is called a "

A musical interval system, by definition, has been given a tuning.

> It says that a musical interval system includes "all intervals in a
> regular temperament." Is the meantone "abstract regular temperament" a
> "regular temperament" or not? 

Not unless you give it a tuning.

 And since when is a group
> > which you intend to use, in the end, to refer to pitch classes a "tone
> > group" anyway?
> 
> What do you mean "since when?" That's how I just defined it.

I was complaining about Zabka, whose commas and tones don't even abstractly correspond to commas and tones, but to pitch classes and octave equivalent commas.

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> As it stands, you haven't given me any definition for what a scale
> actually is, other than a set of completely undefined "musical
> intervals" which, I presume, is supposed to be the image of a function
> mapping from some contiguous subset of Z.

Go to the article on periodic scales, and do something you have not yet done--read it carefully. I say nothing about a contiguous subset of Z, and **I DEFINE INTERVALS**!

Read that again: I define, precisely, the musical intervals.

> Carl thinks that a scale is supposed to always have a tuning. If I'm
> supposed to divine from your response above that a scale is a set of
> elements having either all a specified tuning, OR all a specified
> mapping, you should just say that directly.

Really, Mike, your disinterest in reading what your interlocutors say can be tiresome. My definition more or less amounts to saying a periodic scale is what you find in a Scala scl file.

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 10:30 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia wrote:
> >
> > On Fri, Jan 25, 2013 at 9:10 PM, genewardsmith
> > wrote:
> > >
> > > > Let S be a contiguous subset of the integers, T a "tuning" function
> > > > S
> > > > -> R, and M a "mapping" function from S -> any quotient group of
> > > > Q+*.
> > > > Then an "abstract" scale is a tuple (S, M), and a "non-abstract"
> > > > scale
> > > > is a tuple (S, T).
> > >
> > > I don't recall making these definitions. Where would that be?
> >
> > That is completely equivalent to every other definition I've ever seen
> > given for a scale, and if you don't think so, it would be wise to
> > specify exactly what a "scale" is and in what way it's different from
> > what I wrote.
>
> I know you know of the article on periodic scales, which defines things
> quite differently than the above.

You keep complaining I don't carefully read your stuff, but it works
two ways. Note that I didn't say anything at all about "periodic"
scales above. A (non-abstract) periodic scale is a special case of (S,
T), when S = Z and T is quasiperiodic.

> > At any rate, a "musical interval system" which is also a free abelian
> > group is called an "abstract tone group."
>
> By whom?

I've defined it that way repeatedly throughout the course of this
conversation. However, if you insist that a "musical interval system"
must have assigned pitches, then my definition is wrong, and a musical
interval system which is also a free abelian group is instead the
thing that I called a "tone group."

> > It says that a musical interval system includes "all intervals in a
> > regular temperament." Is the meantone "abstract regular temperament" a
> > "regular temperament" or not?
>
> Not unless you give it a tuning.

I don't know why you didn't answer this question when I posted it
yesterday: http://tech.groups.yahoo.com/group/tuning-math/message/21178

So OK, "abstract regular temperaments" are not "regular temperaments,"
and the two categories are mutually exclusive. Then I take, about, 20%
of what I said back.

-Mike

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Fri, Jan 25, 2013 at 10:39 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
> > It says that a musical interval system includes "all intervals in a
> > regular temperament." Is the meantone "abstract regular temperament" a
> > "regular temperament" or not?
>
> Not unless you give it a tuning.

Actually, this is completely inconsistent with the Abstract Regular
Temperament page, which starts with "an abstract regular temperament
is a regular temperament considered apart from any consideration of
tuning". If it's not supposed to be inconsistent, it's completely
unclear how that can be reconciled with that the meantone "abstract
regular temperament" is not a "regular temperament."


> Go to the article on periodic scales, and do something you have not yet
> done--read it carefully. I say nothing about a contiguous subset of Z, and
> **I DEFINE INTERVALS**!
>
> Read that again: I define, precisely, the musical intervals.

I don't know how many times I have to say this: YOU DO NOT USE THE
TERM "INTERVAL" CONSISTENTLY ON THE WIKI.

You define it that way in the "periodic scale" article, and then in
the Wiki article on Abstract Regular Temperament, you say the
following things:

- "The intervals of the abstract temperament may be defined via
multiplication by the projection map, leading to fractional monzos
which are actually the tunings of these intervals in Frobenius
tuning."

- "The intervals of the abstract temperament may be found in the same
way, by applying the mappings (which are fractional vals) to monzos"

- "The intervals of the [abstract] temperament, as an abstract group,
may be defined by the interior product of a wedgie for a p-limit
temperament with the p-limit monzos."

> > Carl thinks that a scale is supposed to always have a tuning. If I'm
> > supposed to divine from your response above that a scale is a set of
> > elements having either all a specified tuning, OR all a specified
> > mapping, you should just say that directly.
>
> Really, Mike, your disinterest in reading what your interlocutors say can
> be tiresome. My definition more or less amounts to saying a periodic scale
> is what you find in a Scala scl file.

I didn't ask you what your definition of a "periodic scale" is. I
asked you what your definition of a **SCALE** is, which is consistent
with all of the ways which you've used it on the wiki.

For example, you talk about "abstract periodic scales" on the Fokker
block page, and write "It follows that the abstract periodic scale
Wk∨S represents a MOS of the temperament defined by Wk." The phrase
"periodic scale" is a link to the periodic scale page, where it's
clear that your abstract periodic scale is NOT A TYPE OF PERIODIC
SCALE. You then go onto say that Wk∨S is a MOS, despite that it is NOT
a MOS, since you've defined MOS specifically to be a type of "periodic
scale." Finally, you say that it's a MOS of the temperament defined by
Wk, but as you just wrote above, Wk doesn't define a "temperament"
until it's been given a tuning.

NOTHING ABOUT THE WAY YOU USE THESE TERMS IS CONSISTENT. Please either
fix it so that it is, or if you don't mind the terms "scale" and
"interval" being more generic and referring to both, stop complaining
about groups of "intervals".

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> Actually, this is completely inconsistent with the Abstract Regular
> Temperament page, which starts with "an abstract regular temperament
> is a regular temperament considered apart from any consideration of
> tuning". 

No, that's just your lack of language skills again, to be blunt about it. "Considered apart from" means you take that part out of the definition. 

apart from
1 : not including (something) : with the exception of (something) 

> I don't know how many times I have to say this: YOU DO NOT USE THE
> TERM "INTERVAL" CONSISTENTLY ON THE WIKI.

Oh, for God's sake! I was talking about one article, not the whole damned wiki.

> You define it that way in the "periodic scale" article, and then in
> the Wiki article on Abstract Regular Temperament, you say the
> following things:

Gosh, I probably define apples and oranges differently also.

> NOTHING ABOUT THE WAY YOU USE THESE TERMS IS CONSISTENT. 

Mike, sorry, I am not going to "fix" what isn't broken.

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Sat, Jan 26, 2013 at 12:03 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> apart from
> 1 : not including (something) : with the exception of (something)

Thanks. I knew what "apart" meant, but I get confused about "from" sometimes.

> > You define it that way in the "periodic scale" article, and then in
> > the Wiki article on Abstract Regular Temperament, you say the
> > following things:
>
> Gosh, I probably define apples and oranges differently also.

You do, but you seem awfully resistant to defining "fruit."

> > NOTHING ABOUT THE WAY YOU USE THESE TERMS IS CONSISTENT.
>
> Mike, sorry, I am not going to "fix" what isn't broken.

Great! I'm going to resume writing about groups of "intervals" then.
Please note that the term interval means the same thing it means all
over the entire wiki.

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> Great! I'm going to resume writing about groups of "intervals" then.
> Please note that the term interval means the same thing it means all
> over the entire wiki.

I'll hold you to that if it ends up not making sense.

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, "genewardsmith"  wrote:
>
> 
> 
> --- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
> 
> > Great! I'm going to resume writing about groups of "intervals" then.
> > Please note that the term interval means the same thing it means all
> > over the entire wiki.
> 
> I'll hold you to that if it ends up not making sense.

If you want to say, rather than that the elements of the group "are" in some magical sense musical intervals, that they represent unspecified or not yet specified intervals, why not say so?

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, "genewardsmith"  wrote:
>
> 
> 
> --- In tuning-math@yahoogroups.com, "genewardsmith"  wrote:
> >
> > 
> > 
> > --- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
> > 
> > > Great! I'm going to resume writing about groups of "intervals" then.
> > > Please note that the term interval means the same thing it means all
> > > over the entire wiki.
> > 
> > I'll hold you to that if it ends up not making sense.
> 
> If you want to say, rather than that the elements of the group "are" in some magical sense musical intervals, that they represent unspecified or not yet specified intervals, why not say so?

You could call it a surrogate group or something of that sort.

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Sat, Jan 26, 2013 at 10:17 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> If you want to say, rather than that the elements of the group "are" in
> some magical sense musical intervals, that they represent unspecified or not
> yet specified intervals, why not say so?

From the fifth post in the thread:
http://tech.groups.yahoo.com/group/tuning-math/message/21153

"It's just a roundabout way of saying to look at Z^n, and treat it as a
lattice of intervals that is regularly generated, but for which we
haven't specified a mapping nor a tuning."

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

--- In tuning-math@yahoogroups.com, Mike Battaglia  wrote:
>
> On Sat, Jan 26, 2013 at 10:17 AM, genewardsmith
>  wrote:
> >
> > If you want to say, rather than that the elements of the group "are" in
> > some magical sense musical intervals, that they represent unspecified or not
> > yet specified intervals, why not say so?
> 
> From the fifth post in the thread:
> http://tech.groups.yahoo.com/group/tuning-math/message/21153
> 
> "It's just a roundabout way of saying to look at Z^n, and treat it as a
> lattice of intervals that is regularly generated, but for which we
> haven't specified a mapping nor a tuning."

Yes, but you are still saying it's musical intervals here somehow, not surrogates.

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Sat, Jan 26, 2013 at 3:47 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> Yes, but you are still saying it's musical intervals here somehow, not
> surrogates.

The only reason I was going with this "abstract tone group" name was
that I was trying to fit into this "abstract" thing. I was going to go
back to calling it a "tone group" or a "group of intervals" or
something like that. I can call it a "surrogate interval group" or
something if you want.

Whatever it's called, I think your idea of defining an entirely
group-theoretic version of "Fokker block" might be a good idea. This
enables us to talk about things like Fokker blocks in the group of
vals, but that might not be so bad. We might consider a 5-limit Fokker
block which is epimorphic under the monzo 81/80, which if <1 0 0| is
your equivalence val, will be a set of vals which map 81/80 to ...,
-1, 0, 1, ... . The Fokker group of a monzo is a collection of 5-limit
subgroups which have 81/80 in them. What use this is, I don't know,
but it's there.

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> The only reason I was going with this "abstract tone group" name was
> that I was trying to fit into this "abstract" thing. I was going to go
> back to calling it a "tone group" or a "group of intervals" or
> something like that. I can call it a "surrogate interval group" or
> something if you want.

I would much prefer "surrogate intervals" to intervals. The intervals of an abstract regular temperament are not surrogates, they are musical notation in a sense, things like G' or Ab. If you have a piece from the meantone era, say by Couperin or Palestrina, you can of course write it in terms of notes such as Ab. You can equally well write it in terms of other ways of specifying the notes of an abstract temperament, such as vals obtained by taking interior products with <<1 4 4||. But everyone agrees, I hope, that Ab refers to music, even if not in a precisely quantified sense in terms of cents.

> Whatever it's called, I think your idea of defining an entirely
> group-theoretic version of "Fokker block" might be a good idea. This
> enables us to talk about things like Fokker blocks in the group of
> vals, but that might not be so bad.

Remember, the vals in question all temper out the commas of the temperament, they are not just any old vals.

From: Mike Battaglia (2013-01-26)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Sat, Jan 26, 2013 at 4:21 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> I would much prefer "surrogate intervals" to intervals. The intervals of
> an abstract regular temperament are not surrogates, they are musical
> notation in a sense, things like G' or Ab. If you have a piece from the
> meantone era, say by Couperin or Palestrina, you can of course write it in
> terms of notes such as Ab. You can equally well write it in terms of other
> ways of specifying the notes of an abstract temperament, such as vals
> obtained by taking interior products with <<1 4 4||. But everyone agrees, I
> hope, that Ab refers to music, even if not in a precisely quantified sense
> in terms of cents.

Part of what I was trying to get at is that, conceptually, the same
"interval" might correspond to more than one actual interval, and also
more than one ratio, and even also to more than one equivalence class
of ratios. The way in which people cognitively group collections of
these into a single mental construct is non-trivial, and I dunno if
it's fully understood.

But, however it works, the defining characteristic of one of these
sorts of conceptual intervals is that the "same" interval can appear
in multiple places, between different pairs of notes, and furthermore
that intervals can combine under composition. And, for all of the
tuning systems that I really care about, the way that they behave
under composition is exactly the same that a free abelian group
behaves. So I wanted to call this a "regularly generated interval
system" or "abstract tone group" or whatever, and prove theorems about
the behavior of the sorts of interval systems that behave exactly like
a free abelian group under composition. But if you think it's best to
just recognize this as just another naturally-occurring example of a
group, and just prove theorems about groups directly, I don't
necessarily object.

> > Whatever it's called, I think your idea of defining an entirely
> > group-theoretic version of "Fokker block" might be a good idea. This
> > enables us to talk about things like Fokker blocks in the group of
> > vals, but that might not be so bad.
>
> Remember, the vals in question all temper out the commas of the
> temperament, they are not just any old vals.

When you say the "vals in question" here, do you mean the vals in the
Fokker block? I don't think this is correct, but rather that they'll
all be epimorphic under some monzo.

-Mike

From: genewardsmith (2013-01-26)
Subject: Re: A useful generalization of epimorphicity

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

> Part of what I was trying to get at is that, conceptually, the same
> "interval" might correspond to more than one actual interval, and also
> more than one ratio, and even also to more than one equivalence class
> of ratios. The way in which people cognitively group collections of
> these into a single mental construct is non-trivial, and I dunno if
> it's fully understood.
> 
> But, however it works, the defining characteristic of one of these
> sorts of conceptual intervals is that the "same" interval can appear
> in multiple places, between different pairs of notes, and furthermore
> that intervals can combine under composition. And, for all of the
> tuning systems that I really care about, the way that they behave
> under composition is exactly the same that a free abelian group
> behaves. So I wanted to call this a "regularly generated interval
> system" or "abstract tone group" or whatever, and prove theorems about
> the behavior of the sorts of interval systems that behave exactly like
> a free abelian group under composition. But if you think it's best to
> just recognize this as just another naturally-occurring example of a
> group, and just prove theorems about groups directly, I don't
> necessarily object.

You said before that your group had no other structure than its group structure, but the above says you have some not-yet-identified connection to music in mind. I think you need to identify it and explain it or none of this conversation is going to make sense.

> When you say the "vals in question" here, do you mean the vals in the
> Fokker block? I don't think this is correct, but rather that they'll
> all be epimorphic under some monzo.

I meant the vals which were notes in an abstract periodic Fokker block scale.

From: Mike Battaglia (2013-01-27)
Subject: Re: [tuning-math] Re: A useful generalization of epimorphicity

On Sat, Jan 26, 2013 at 6:57 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> You said before that your group had no other structure than its group
> structure, but the above says you have some not-yet-identified connection to
> music in mind. I think you need to identify it and explain it or none of
> this conversation is going to make sense.

Say that a "real interval" is the quotient of two actual pitches,
specified in Hz, or the difference if the pitches are being
represented logarithmically. For instance, "702 cents" is a real
interval, as is 3/2. Also, say that a "real note" is an actual pitch.

In practice, numerous "real intervals" can be subsumed into the
sensation of a single perceived interval. For instance, if a string
quartet is playing a rendition of some piece of music, they might
flatten or sharpen the major third in various contexts. Sometimes they
might make it closer to 5/4, or sometimes they might make it closer to
9/7, to deliberately make it sit better in a short. Sometimes they'll
deliberately sharpen it to make it resolve better when it's supposed
to play the role of the "leading tone." And these are the obvious
things that we all know about; there are probably dozens of
unconscious, or perhaps unintentional, things that a string quartet
does to adaptively reintone the notes that they're playing. And then
there's vibrato too.

Nonetheless, there's something about the melodic, and to some extent
harmonic content of what's being played if you listen to a string
quartet vs a piano rendition of the same tune. And by "something," I
mean that it's damn obvious to everyone who's not tone-deaf that
they're playing the same composition. The string quartet might lend
itself to a bit more expressive of a rendition than the piano, but
it's not like they're suddenly playing totally different pieces of
music. The same even applies to the same string quartet playing the
same piece of music in separate performances, where they might change
the intonation differently at different times.

I'm not going to continue for now, because although it's completely
obvious to me that the piano and orchestral versions of Gershwin's
Rhapsody in Blue are the same piece of music, and that some abstract
level of musical information is shared between the two, I'm fairly
convinced that you're going to have some objection, or some random
comment that this only applies if you have AP, or something along
those lines which generally will make no sense to me, so we might as
well get it overwith before I proceed any further.

> > When you say the "vals in question" here, do you mean the vals in the
> > Fokker block? I don't think this is correct, but rather that they'll
> > all be epimorphic under some monzo.
>
> I meant the vals which were notes in an abstract periodic Fokker block
> scale.

You're referring to a "Fokker block" whose elements are vals, and
which forms an abstract "periodic scale" of vals, correct? This is the
sort of thing which would, by definition, be epimorphic under some
element in the dual Z-module, and which would hence be a monzo?

-Mike