Yahoo Tuning Groups Ultimate Backup tuning-math Response to epimorphicity thing

> 2.4 Epimorphicity / constant structure: A MOS scale is associated with a val v such that every interval subtending n scale steps maps to n under the val (it's epimorphic). Assuming that v can be a factor in the wedge product of the temperament, so that all commas have v(c) = 0, this implies that no interval can appear in more than one class (constant structure).

This is an offshoot to both this and your reply on [tuning].

Whether an interval actually subtends n scale steps is going to be
tuning-dependent. You addressed the case where we're in the 3-limit,
and 2187/2048 is our commatic vector, but we tune the generator to
exactly 700 cents - which means we're left with something that isn't
CS, because we're "accidentally" tempering out the Pythagorean comma,
leaving us with the <12 19| val - and the diatonic scale is neither
convex nor epimorphic in that space, because it'll look like # # ##
# # #, and end up not being surjective. Alright.

However, what if you tune the generator to something irrational that's
between 720 and 750 cents, exclusive? Then the thirds and fourths will
switch places, but you won't get the dimensionality reduction allowing
you to make that argument. You'll still have an epimorphic block, but
your intervals will just end up being reversed, which is to say that
the third scale step will be higher than the fourth scale step. But
the block should still be epimorphic, because should still be
surjective onto that val.

Another way of putting this is that a val like <12 19| doesn't just
have to apply to 12-equal - it can apply to something like
Werckmeister as well. In fact, it can even apply to something that's
so severely warped that the intervals switch places - such as a
12-note "well temperament" generated by a chain of "3/2's" which are
733.9238423 cents wide.

-Mike

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > 2.4 Epimorphicity / constant structure: A MOS scale is associated with a val v such that every interval subtending n scale steps maps to n under the val (it's epimorphic). Assuming that v can be a factor in the wedge product of the temperament, so that all commas have v(c) = 0, this implies that no interval can appear in more than one class (constant structure).
>
> This is an offshoot to both this and your reply on [tuning].
>
> Whether an interval actually subtends n scale steps is going to be
> tuning-dependent. You addressed the case where we're in the 3-limit,
> and 2187/2048 is our commatic vector, but we tune the generator to
> exactly 700 cents - which means we're left with something that isn't
> CS, because we're "accidentally" tempering out the Pythagorean comma,
> leaving us with the <12 19| val - and the diatonic scale is neither
> convex nor epimorphic in that space, because it'll look like # # ##
> # # #, and end up not being surjective. Alright.
>
> However, what if you tune the generator to something irrational that's
> between 720 and 750 cents, exclusive? Then the thirds and fourths will
> switch places, but you won't get the dimensionality reduction allowing
> you to make that argument. You'll still have an epimorphic block, but
> your intervals will just end up being reversed, which is to say that
> the third scale step will be higher than the fourth scale step. But
> the block should still be epimorphic, because should still be
> surjective onto that val.

In the terminology of http://xenharmonic.wikispaces.com/Periodic+scale , the scale would be "weakly epimorphic" but not "epimorphic".

> Another way of putting this is that a val like <12 19| doesn't just
> have to apply to 12-equal - it can apply to something like
> Werckmeister as well. In fact, it can even apply to something that's
> so severely warped that the intervals switch places - such as a
> 12-note "well temperament" generated by a chain of "3/2's" which are
> 733.9238423 cents wide.

I'm not really sure what you mean by that last example, but you definitely have an important point that MOSes are tuning-dependent.

As a more familiar example, consider the MOSes of 7-limit meantone. I think we can all agree that the first MOSes are 5, 7, 12, 19, 31... but what is the next one? Is it 43, or 50? The answer is entirely tuning-dependent.

The number of notes in the 100th MOS in this sequence is more or less meaningless to even talk about, for this reason. So this is a caveat to MOS Property 4.1: MOSes do come in an infinite increasing sequence, but the specific elements of the sequence get more fuzzy and uncertain as you go toward ever greater numbers of notes. For any *exactly* defined generator and period you of course have a well-defined infinite sequence, but in practice nobody cares whether the meantone generator is 696.578 or 696.579 cents.

I think the best way to address this issue in the MOS generalization thread is to add property 1.0 Monotonicity, which simply says the notes of the scale are numbered in ascending order. That way, if you were looking at some weakly epimorphic lattice structure, e.g. a Fokker block, for some range of tunings it would be monotonic and therefore (strictly) epimorphic, but for other tunings it would be non-monotonic and fail property 1.0.

Keenan

🔗Keenan Pepper <keenanpepper@gmail.com>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Keenan Pepper <keenanpepper@gmail.com>

🔗Mike Battaglia <battaglia01@gmail.com>

On Sun, Sep 25, 2011 at 4:13 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> In the terminology of http://xenharmonic.wikispaces.com/Periodic+scale , the scale would be "weakly epimorphic" but not "epimorphic".

Weakly epimorphic with respect to what? The only reason it would ever
become "weakly epimorphic" is if the ratio of the generator to the
octave exceeds or falls short of some bound that ensures monotonicity,
but if we're working in an octave-equivalent space, the octave might
as well not exist. If it doesn't exist, then our scale is the same as
a segment of a chain of generators arranged in generator-ascending
order. This will form a cyclic group, and the homomorphism mapping
from Z^n onto this group should be epimorphic no matter what the lone
generator is tuned to with respect to the nonexistent octave.

Put another way, and this is a restatement of my example that you
"didn't get," we don't even need to be dealing with higher-dimensional
Fokker blocks for this to be an issue - you can apply a single rank-1
val to the notes of a circulating temperament whose pitches are simply
considered to not be in monotonic ascending order. Or a strict equal
temperament whose generator is considered to be negative. If "weakly
epimorphic" is defined as surjective and nonmonotonic, and
monotonicity has to do with tuning, how can you define "weakly
epimorphic" in an entirely abstract sense?

> As a more familiar example, consider the MOSes of 7-limit meantone. I think we can all agree that the first MOSes are 5, 7, 12, 19, 31... but what is the next one? Is it 43, or 50? The answer is entirely tuning-dependent.

Sure, although my interpretation is that even 5, 7, 12, 19, and 31 are
tuning-dependent as well. The 17c val implies meantone, and I happen
to like it a lot. And 26-equal's error in the 5-limit isn't too much
worse than 12.

The question of what MOS's that a temperament supports is equivalent
to the question of how much error a temperament can handle before it's
too far gone, or to what range of tunings the generators can take
that's considered "valid." One way to do it is to bound the generator
sizes by the region over which the order of the n-limit tonality
diamond is preserved. So for meantone, that gives us 4\7 as a lower
bound (flatter than that and the thing we're calling "6/5" becomes
larger than the thing we're calling "5/4"), and 3\5 as an upper bound
(sharper than that and "4/3" becomes flatter than "5/4"). So this is
equivalent to stating that meantone must produce MOS's of 5, 7, and
12, or alternatively, that it has to produce 5L2s scales, or that it
has to produce 7a5b scales, without stating the relative size of a and
b.

I don't really prefer to think of it this way, however, because I
prefer to think of mappings and tunings as two completely separate
things, with the mapping being a harmonic schema that we're applying
to a "melodic" structure. But whatever floats your boat.

> I think the best way to address this issue in the MOS generalization thread is to add property 1.0 Monotonicity, which simply says the notes of the scale are numbered in ascending order. That way, if you were looking at some weakly epimorphic lattice structure, e.g. a Fokker block, for some range of tunings it would be monotonic and therefore (strictly) epimorphic, but for other tunings it would be non-monotonic and fail property 1.0.

Well, whatever properties we decide that higher-dimensional MOS should
have, any Fokker block of whatever MOS-producing shape we come up with
can be made to have them by simply picking the proper tuning, although
it might be of obscenely high error.

-Mike

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Sep 25, 2011 at 4:13 PM, Keenan Pepper <keenanpepper@...> wrote:
> >
> > In the terminology of http://xenharmonic.wikispaces.com/Periodic+scale , the scale would be "weakly epimorphic" but not "epimorphic".
>
> Weakly epimorphic with respect to what? The only reason it would ever
> become "weakly epimorphic" is if the ratio of the generator to the
> octave exceeds or falls short of some bound that ensures monotonicity,
> but if we're working in an octave-equivalent space, the octave might
> as well not exist. If it doesn't exist, then our scale is the same as
> a segment of a chain of generators arranged in generator-ascending
> order. This will form a cyclic group, and the homomorphism mapping
> from Z^n onto this group should be epimorphic no matter what the lone
> generator is tuned to with respect to the nonexistent octave.

I don't understand what point you're trying to make here.

I'm not interested in making music with a chain of 3/2s with no pitches shifted by 2/1s, and I don't know of anyone else who is.

> Put another way, and this is a restatement of my example that you
> "didn't get," we don't even need to be dealing with higher-dimensional
> Fokker blocks for this to be an issue - you can apply a single rank-1
> val to the notes of a circulating temperament whose pitches are simply
> considered to not be in monotonic ascending order. Or a strict equal
> temperament whose generator is considered to be negative. If "weakly
> epimorphic" is defined as surjective and nonmonotonic, and
> monotonicity has to do with tuning, how can you define "weakly
> epimorphic" in an entirely abstract sense?

No, you've misunderstood. "Weakly epimorphic" has nothing to do with monotonicity at all.

A scale is strongly epimorphic if and only if it is both weakly epimorphic and monotonic. Every strongly epimorphic scale is also weakly epimorphic.

> Sure, although my interpretation is that even 5, 7, 12, 19, and 31 are
> tuning-dependent as well. The 17c val implies meantone, and I happen
> to like it a lot. And 26-equal's error in the 5-limit isn't too much
> worse than 12.

Of course, you're absolutely correct, but if you use 17c as an example people can respond with something like "Oh, that's just a pathological case some tuning-math people made up. Everybody knows 17edo isn't *really* meantone."

So although you're absolutely correct that a "meantone" temperament might not have a 5-note MOS, because technically the 3/2 generator could be tuned wider than 800 cents, that's only going to confuse most people.

"Technically correct --- the best kind of correct."
-some Futurama character

> Well, whatever properties we decide that higher-dimensional MOS should
> have, any Fokker block of whatever MOS-producing shape we come up with
> can be made to have them by simply picking the proper tuning, although
> it might be of obscenely high error.

True.

Keenan

🔗Mike Battaglia <battaglia01@gmail.com>

On Mon, Sep 26, 2011 at 3:49 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Weakly epimorphic with respect to what? The only reason it would ever
> > become "weakly epimorphic" is if the ratio of the generator to the
> > octave exceeds or falls short of some bound that ensures monotonicity,
> > but if we're working in an octave-equivalent space, the octave might
> > as well not exist. If it doesn't exist, then our scale is the same as
> > a segment of a chain of generators arranged in generator-ascending
> > order. This will form a cyclic group, and the homomorphism mapping
> > from Z^n onto this group should be epimorphic no matter what the lone
> > generator is tuned to with respect to the nonexistent octave.
>
> I don't understand what point you're trying to make here.
>
> I'm not interested in making music with a chain of 3/2s with no pitches shifted by 2/1s, and I don't know of anyone else who is.

What does that have to do with the mathematical definition of
"epimorphic?" You started talking about scales themselves being
epimorphic. When you say talk about the epimorphicity of a scale, I
assume that this is because you're viewing the scale as a homomorphism
from some group to some other group. But I'm not seeing how you're
defining the homomorphism in such a way that a scale can be "weakly
epimorphic" vs "strongly epimorphic" in a way that is
tuning-independent.

The point I was "trying to make" was that this entire concept of
monotonicity only emerges when we throw octaves back in at the last
minute, having neglected them consistently until this point. So I
don't see how you're officially defining the domain and image of the
homomorphism between higher-limit octave-equivalent space and Z^2 in
order to state whether or not some MOS has a homomorphism that is
surjective. What I suggested was to ignore octaves consistently, even
at the end, which means that you end up you end up with a chain of
generators, and that your scale is an epimorphism from Z to Z/nZ
(chain of generators). Then, when you want to use octaves, you just
use them, and nothing changes in the underlying homomorphism. In
short, I've been treating it like it's an adaptive temperament in
which 2/1 vanishes, you can say. How would you define it differently?

> > Put another way, and this is a restatement of my example that you
> > "didn't get," we don't even need to be dealing with higher-dimensional
> > Fokker blocks for this to be an issue - you can apply a single rank-1
> > val to the notes of a circulating temperament whose pitches are simply
> > considered to not be in monotonic ascending order. Or a strict equal
> > temperament whose generator is considered to be negative. If "weakly
> > epimorphic" is defined as surjective and nonmonotonic, and
> > monotonicity has to do with tuning, how can you define "weakly
> > epimorphic" in an entirely abstract sense?
>
> No, you've misunderstood. "Weakly epimorphic" has nothing to do with monotonicity at all.
>
> A scale is strongly epimorphic if and only if it is both weakly epimorphic and monotonic. Every strongly epimorphic scale is also weakly epimorphic.

OK, I guess we can define it that way. So but then it's "strongly
epimorphic" that can't be defined on a purely abstract level, because
monotonicity itself can't be defined on a purely abstract level. So I
don't understand where the word "epimorphic" is coming from. Perhaps
there's some kind of assumed group homomorphism here that I'm not
seeing between the "abstract temperament" level and the "specifically
tuned temperament" level.

> > Sure, although my interpretation is that even 5, 7, 12, 19, and 31 are
> > tuning-dependent as well. The 17c val implies meantone, and I happen
> > to like it a lot. And 26-equal's error in the 5-limit isn't too much
> > worse than 12.
>
> Of course, you're absolutely correct, but if you use 17c as an example people can respond with something like "Oh, that's just a pathological case some tuning-math people made up. Everybody knows 17edo isn't *really* meantone."
>
> So although you're absolutely correct that a "meantone" temperament might not have a 5-note MOS, because technically the 3/2 generator could be tuned wider than 800 cents, that's only going to confuse most people.

Whenever I play in 17-equal, I always intuitively assume the 17c val.
And, as far as I'm concerned, when I do, and when I play the diatonic
scale, I'm playing in meantone, and I've never been hesitant to refer
to "meantone" or "dominant" in 17-equal. The fact that people seem to
have this notion that 700 cents is the cutoff between meantone and
superpyth, or that 400 cents is the cutoff between 5/4 and 9/7, is I
think pretty arbitrary. And there's no consensus on it, because it
once again boils down to your personal voodoo beliefs about ratio
tolerance.

I certainly think it's possible to perceive 424 cents "as 5/4," and I
also think it's possible to perceive it "as 9/7," and also "as 14/11,"
but only the people who are firmly entrenched in the latter camp will
take any objection to labeling 17-equal or 17c a meantone temperament.
Whatever you feel, there's obviously a certain sense in which the
meantone mapping becomes "pathologic" when applied to certain tunings,
like tunings for which the generator is 15023948 cents, but 17-equal
isn't quite as absurd as that.

If you did want to generalize your intuitive objection to labeling 17c
as a "valid" meantone temperament, it should be noted that it's a
meantone temperament that happens to have a positive tuning with
respect to the 5L2s MOS, rather than a negative one. (I'm using
positive in the sense that generalizes Bosanquet's usage of the term,
it just means that L/s > 2 for some MOS). So an algorithm could be

1) pick an albitonic scale for the temperament (assuming you find some
canonical way to do this)
2) find out whether or not, given the POTE tuning for that
temperament, the albitonic scale is positive or negative
3) The valid range is the one that preserves the positivity or
negativity of this MOS

-Mike

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> What does that have to do with the mathematical definition of
> "epimorphic?" You started talking about scales themselves being
> epimorphic. When you say talk about the epimorphicity of a scale, I
> assume that this is because you're viewing the scale as a homomorphism
> from some group to some other group. But I'm not seeing how you're
> defining the homomorphism in such a way that a scale can be "weakly
> epimorphic" vs "strongly epimorphic" in a way that is
> tuning-independent.

It's not. It is tuning-dependent, because the monotonicity part is tuning-dependent.

I'm using the definitions of "weakly epimorphic" and "epimorphic" given at http://xenharmonic.wikispaces.com/Periodic+scale , which does not mention homomorphisms.

> The point I was "trying to make" was that this entire concept of
> monotonicity only emerges when we throw octaves back in at the last
> minute, having neglected them consistently until this point. So I
> don't see how you're officially defining the domain and image of the
> homomorphism between higher-limit octave-equivalent space and Z^2 in
> order to state whether or not some MOS has a homomorphism that is
> surjective. What I suggested was to ignore octaves consistently, even
> at the end, which means that you end up you end up with a chain of
> generators, and that your scale is an epimorphism from Z to Z/nZ
> (chain of generators). Then, when you want to use octaves, you just
> use them, and nothing changes in the underlying homomorphism. In
> short, I've been treating it like it's an adaptive temperament in
> which 2/1 vanishes, you can say. How would you define it differently?

It has nothing to do with "throwing octaves back in at the last minute". Everything also makes sense if you keep octaves in the whole time.

I have two generators, one that represents 2/1 and one that represents 3/2. I define a periodic scale with the construction given at http://xenharmonic.wikispaces.com/Fokker+blocks with the chroma 2187/2048.

If I tune 2/1 as 1200 cents and 3/2 as 700 cents I get [...,-300,-100,0,200,400,600,700,900,1100,1200,1400,...], which is weakly epimorphic and monotonic, so it is "epimorphic".

If I tune 2/1 as 1200 cents and 3/2 as 730 cents I get
[...,-210,50,0,260,520,780,730,990,1250,1200,1460,...], which is weakly epimorphic, but not monotonic, so it is not "epimorphic".

> > No, you've misunderstood. "Weakly epimorphic" has nothing to do with monotonicity at all.
> >
> > A scale is strongly epimorphic if and only if it is both weakly epimorphic and monotonic. Every strongly epimorphic scale is also weakly epimorphic.
>
> OK, I guess we can define it that way. So but then it's "strongly
> epimorphic" that can't be defined on a purely abstract level, because
> monotonicity itself can't be defined on a purely abstract level. So I
> don't understand where the word "epimorphic" is coming from. Perhaps
> there's some kind of assumed group homomorphism here that I'm not
> seeing between the "abstract temperament" level and the "specifically
> tuned temperament" level.

To be clear, note that I'm not the one who defined "weakly epimorphic" and "epimorphic" this way. I'm not sure who did, and I'm just using their definitions.

These definitions indeed mean that "weakly epimorphic" can be defined on a purely abstract leve, but "epimorphic" cannot.

Keenan

🔗Mike Battaglia <battaglia01@gmail.com>

On Mon, Sep 26, 2011 at 5:29 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> It's not. It is tuning-dependent, because the monotonicity part is tuning-dependent.

I don't disagree with anything you wrote in this reply, because I
didn't realize that you were viewing it as though things were weakly
epimorphic by default and just "epimorphic" only if monotonic. I
thought you were saying things were epimorphic by default, and weakly
epimorphic only if non-monotonic. So I wasn't sure how you were going
to define monotonicity in a completely non-abstract way.

> I'm using the definitions of "weakly epimorphic" and "epimorphic" given at http://xenharmonic.wikispaces.com/Periodic+scale , which does not mention homomorphisms.

If the definition of "epimorphic" as we're using it doesn't have to do
with homomorphisms, then that means that it also doesn't have to do
with epimorphisms. I don't really care what we do call it, although I
admit I find it confusing to call something "epimorphic" that doesn't
involve surjective homomorphisms. I guess I'll leave it as an open
question to Gene to clarify.

-Mike