Re: definitions of period, equivalence, etc. (was: Re: graham's line

In-Reply-To: <abul01+9p4m@eGroups.com>
Dave:
> > Note that those with an IoE _can_ be treated mathematically as rank
> > one, provided all arithmetic is modulo the IoE.

Paul:
> i used to think so, but it seems gene was able to convince me
> otherwise. i think that you can't handle torsion properly unless you
> express the unison vectors in IoE-specific, rather than IoE-
> equivalent/IoE-invariant terms.

I've never been convinced of this. Every time it comes up people say they
aren't interested enough to prove it either way. So you should state it
as unproven and not use it to dismiss other ideas.

The biggest outstanding problem is that we don't have an algorithm for
calculating the optimal generator size for a given mapping and target
consonances. But then nobody's seriously looked. There is a
specific problem with contorsion, as different generator sizes give
different, but equally valid results. But hey, we're not counting
contorsion anyway. Adding a range of generator sizes to the definition
solves this one if there's no other way.

There's also the detail that IoE-equivalent algebra gives a
period-equivalent result, but you can get round that if you know the (IoE
equivalent) sizes of the intervals you're approximating. The real problem
here is that the IoE-equivalence doesn't make it any simpler if you won't
accept period-equivalence.

I'm sure the torsion problem will go away when somebody looks at it
properly. It doesn't matter anyway if we're only considering linear
temperaments, because they don't have to be derived from unison vectors.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <abul01+9p4m@e...>
> Dave:
> > > Note that those with an IoE _can_ be treated mathematically as
rank
> > > one, provided all arithmetic is modulo the IoE.
>
> Paul:
> > i used to think so, but it seems gene was able to convince me
> > otherwise. i think that you can't handle torsion properly unless
you
> > express the unison vectors in IoE-specific, rather than IoE-
> > equivalent/IoE-invariant terms.
>
> I've never been convinced of this. Every time it comes up people
say they
> aren't interested enough to prove it either way. So you should
state it
> as unproven and not use it to dismiss other ideas.

i think the burden of proof rests on the other side. we all agree
that torsion is something that needs to be handled separately, and so
far the only method i've seen for detecting torsion uses IoE-specific
representations!

> I'm sure the torsion problem will go away when somebody looks at it
> properly.

again, the burden would seem to "somebody" to show that.
you're "sure", i'm "sure not".

> It doesn't matter anyway if we're only considering linear
> temperaments, because they don't have to be derived from unison
>vectors.

they have to be derivable from unison vectors. the construction
starts with the just lattice and then establishes equivalencies. the
unison vectors simply tell you what the equivalencies were in the
original just lattice.

emotionaljourney22 wrote:

> i think the burden of proof rests on the other side. we all agree
> that torsion is something that needs to be handled separately, and so
> far the only method i've seen for detecting torsion uses IoE-specific
> representations!

Torsion can be detected with IoEE representations. The problem, if there
is one, would be distinguishing torsion that tells you to divide the
octave from torsion that tells you the unison vectors aren't in their
simplest terms.. If you can't detect the difference, it isn't a problem.
If you can detect it, you can deal with it. Why assume otherwise?

> > I'm sure the torsion problem will go away when somebody looks at it
> > properly.
>
> again, the burden would seem to "somebody" to show that.
> you're "sure", i'm "sure not".

Perhaps you could start by showing what the problem is.

> > It doesn't matter anyway if we're only considering linear
> > temperaments, because they don't have to be derived from unison
> >vectors.
>
> they have to be derivable from unison vectors. the construction
> starts with the just lattice and then establishes equivalencies. the
> unison vectors simply tell you what the equivalencies were in the
> original just lattice.

Say what? We've agreed on a definition of linear temperament that says
nothing about unison vectors. That definition is directly applicable to
the octave-equivalent case. How they're derived is secondary.

The unison vectors tell you more than the equivalences. They also tell
you what's considered to be a unison. For example, in twintone (or
whatever it's being called today) 50:49 is a unison vector. In octave
equivalent terms, that is (-2 2). So it follows that half the unison, (-1
1), will also be an equivalence. That's 7:5. As 7:5 isn't half of 50:49,
but much closer to half of an octave, we know the period can't be the
octave. No problem at all. You don't need to consider the
octave-specific case. Give a counter example.

Graham

--- In tuning-math@y..., graham@m... wrote:

> > again, the burden would seem to "somebody" to show that.
> > you're "sure", i'm "sure not".
>
> Perhaps you could start by showing what the problem is.

take the classic example of torsion. the fokker determinant is 24. it
seems like you're getting a system with 24 distinct pitch classes per
octave -- but you aren't. you need the octave-specific
representations to find that true number is 12. right?

> > they have to be derivable from unison vectors. the construction
> > starts with the just lattice and then establishes equivalencies.
the
> > unison vectors simply tell you what the equivalencies were in the
> > original just lattice.
>
> Say what? We've agreed on a definition of linear temperament that
says
> nothing about unison vectors. That definition is directly
applicable to
> the octave-equivalent case. How they're derived is secondary.

i'm referring to the definition of "temperament" that we used to
convince dave keenan that contorsion cases aren't kinds of
temperament. this would seem to be an important part of the
definition.

> The unison vectors tell you more than the equivalences. They also
tell
> you what's considered to be a unison. For example, in twintone (or
> whatever it's being called today) 50:49 is a unison vector. In
octave
> equivalent terms, that is (-2 2). So it follows that half the
unison, (-1
> 1), will also be an equivalence. That's 7:5.

7:5 an equivalence? no way, dude. never, nunca, jamas.

--- In tuning-math@y..., graham@m... wrote:

> As 7:5 isn't half of 50:49,
> but much closer to half of an octave, we know the period can't be
the
> octave. No problem at all. You don't need to consider the
> octave-specific case. Give a counter example.

that wasn't an example of torsion, graham, but anyway . . .

there is a difference in character between an algorithm that has to
make a "much closer to" judgment, and an algorithm like gene's which
gives the answer directly. in the former, you have to stop the
mechanism, fiddle around with stuff, and get your hands dirty. in the
latter, the whole thing is sleek and axiomatic.

In-Reply-To: <ac1g4r+m2vm@eGroups.com>
emotionaljourney22 wrote:

> take the classic example of torsion. the fokker determinant is 24. it
> seems like you're getting a system with 24 distinct pitch classes per
> octave -- but you aren't. you need the octave-specific
> representations to find that true number is 12. right?

That's no problem at all. The temperament mapping comes out correct,
usually as schismic. When you look at the 24 pitch classes you'll see
they aren't distinct.

Me:
> > Say what? We've agreed on a definition of linear temperament that
> says
> > nothing about unison vectors. That definition is directly
> applicable to
> > the octave-equivalent case. How they're derived is secondary.

Paul:
> i'm referring to the definition of "temperament" that we used to
> convince dave keenan that contorsion cases aren't kinds of
> temperament. this would seem to be an important part of the
> definition.

It doesn't seem like that at all to me. You don't need to look at unison
vectors to recognize contorsion. Only take the gcd of the generator
mapping. Unison vectors don't do it anyway -- you have to make a
deliberate attempt to remove torsion. You could equate it with contorsion
instead. If things with contorsion aren't temperaments, it's because they
aren't consonance connected.

> 7:5 an equivalence? no way, dude. never, nunca, jamas.

It's equivalent to the period in twintone. If that isn't an "equivalence"
then unison vectors don't show equivalences, or we have to take torsion at
face value.

Graham

In-Reply-To: <ac1mbm+b027@eGroups.com>
emotionaljourney22 wrote:

> that wasn't an example of torsion, graham, but anyway . . .

Torsion's easy do deal with and divisions of the octave are easy to deal
with. It's only telling which is which that may be a problem. If it is,
it's more with using unison vectors to define temperaments than with
temperaments themselves. I still need an example that causes problems.
Don't you have an algorithm for generating periodicity blocks?

> there is a difference in character between an algorithm that has to
> make a "much closer to" judgment, and an algorithm like gene's which
> gives the answer directly. in the former, you have to stop the
> mechanism, fiddle around with stuff, and get your hands dirty. in the
> latter, the whole thing is sleek and axiomatic.

Oh, it's Gene's algorithm now, is it?

It's part of the definition of a unison vector that it should be close to
a unison. I thought Gene had made a strict definition for that. If we
can't differentiate a unison vector from a period-equivalence, torsion
can't be differentiated for divisions of the octave. So it's GIGO.

The algorithm for getting the period part of the mapping is already fairly
dirty. I'm certainly assuming it won't get cleaner by ignoring the
octave, which is why I don't do it that way. For the octave-equivalent
case it'd probably mean trying all consonances and getting them
pitch-ordered, or generating a periodicity block or something like that.
In which case torsion should fall out the same way it did with the
octave-specific algebraic approach.

Or you could always construct the period mapping algebraically, but not
*say* that it's an octave-specific process.

There aren't any "much closer to" judgements anyway. Only higher/lower.
If a unison vector gets bigger when you divide it, you need to divide the
octave as well.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <ac1g4r+m2vm@e...>
> emotionaljourney22 wrote:
>
> > take the classic example of torsion. the fokker determinant is
24. it
> > seems like you're getting a system with 24 distinct pitch classes
per
> > octave -- but you aren't. you need the octave-specific
> > representations to find that true number is 12. right?
>
> That's no problem at all. The temperament mapping comes out
correct,
> usually as schismic. When you look at the 24 pitch classes you'll
see
> they aren't distinct.

messy.

> Me:
> > > Say what? We've agreed on a definition of linear temperament
that
> > says
> > > nothing about unison vectors. That definition is directly
> > applicable to
> > > the octave-equivalent case. How they're derived is secondary.
>
> Paul:
> > i'm referring to the definition of "temperament" that we used to
> > convince dave keenan that contorsion cases aren't kinds of
> > temperament. this would seem to be an important part of the
> > definition.
>
> It doesn't seem like that at all to me. You don't need to look at
unison
> vectors to recognize contorsion.

i didn't say you need to look at the unison vectors -- only that it
needs to be *possible in theory* to do so.

> Only take the gcd of the generator
> mapping. Unison vectors don't do it anyway -- you have to make a
> deliberate attempt to remove torsion.

this sounds neither here nor there.

> You could equate it with contorsion
> instead.

isn't that what i said above?

> If things with contorsion aren't temperaments, it's because they
> aren't consonance connected.

it sounds like we're actually agreeing on this issue, just seeing the
same picture in two different ways.

> > 7:5 an equivalence? no way, dude. never, nunca, jamas.
>
> It's equivalent to the period in twintone. If that isn't
an "equivalence"
> then unison vectors don't show equivalences,

what do you mean? 7:5 is not a unison vector . . .

> or we have to take torsion at
> face value.

??

gene, any reason you're staying out of this?

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <ac1mbm+b027@e...>
> emotionaljourney22 wrote:
>
> > that wasn't an example of torsion, graham, but anyway . . .
>
> Torsion's easy do deal with and divisions of the octave are easy to
deal
> with. It's only telling which is which that may be a problem. If
it is,
> it's more with using unison vectors to define temperaments than
with
> temperaments themselves. I still need an example that causes
problems.
> Don't you have an algorithm for generating periodicity blocks?

sure . . .

> > there is a difference in character between an algorithm that has
to
> > make a "much closer to" judgment, and an algorithm like gene's
which
> > gives the answer directly. in the former, you have to stop the
> > mechanism, fiddle around with stuff, and get your hands dirty. in
the
> > latter, the whole thing is sleek and axiomatic.
>
> Oh, it's Gene's algorithm now, is it?

i thought so, isn't it?

> It's part of the definition of a unison vector that it should be
close to
> a unison. I thought Gene had made a strict definition for that.

gene?
my brain is fried.

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> gene, any reason you're staying out of this?

I'm not sure what the point of it all is. If you leave off octaves and just deal with pitch classes, you need to put the octave information back into the mix in one way or another. Why not just leave it in?

--- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:

> > It's part of the definition of a unison vector that it should be
> close to
> > a unison. I thought Gene had made a strict definition for that.

> gene?
> my brain is fried.

I don't recall trying to introduce something about closeness to unity. The most obvious approach is to say a unison vector is one element of a minimal generating set for the kernel, which has no such requirement.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > gene, any reason you're staying out of this?
>
> I'm not sure what the point of it all is. If you leave off octaves
> and just deal with pitch classes, you need to put the octave
>information back into the mix in one way or another.

this seems to be what graham is questioning.

> Why not just leave it in?

graham, that's your cue.

In-Reply-To: <acjitt+tn6m@eGroups.com>
Gene:
> > I'm not sure what the point of it all is. If you leave off octaves
> > and just deal with pitch classes, you need to put the octave
> >information back into the mix in one way or another.

Paul:
> this seems to be what graham is questioning.

You need to put the octaves in eventually, before you hear actual music.
So you can make a credible case for what Gene said there making sense.
Existing mathematical theory, however, all seems to be within an
octave-equivalent system. We don't have to change that, although it does
make some things clearer to put the octaves in. Which would be why Gene
and me both independently did that.

> > Why not just leave it in?
>
> graham, that's your cue.

What, the bit about nobody saying they care, but the subject keeping on
coming up anyway? As this thread was dormant for so long, I thought
nobody cared after all.

Graham

--- In tuning-math@y..., graham@m... wrote:

> You need to put the octaves in eventually, before you hear actual music.
> So you can make a credible case for what Gene said there making sense.

By "putting the octaves back in" I meant you must do this to define the temperament. If you define a temperament by the octave-equivalent mapping [0,1,4], how do you know it isn't the 160/81 temperament, and not the 81/80? You need explicitly or implicitly to do a further calculation, using real arithmetic and not just algebra, to make this system workable. It seems obviously better to me to simply define the temperament unambiguously in the first place.

> Existing mathematical theory, however, all seems to be within an
> octave-equivalent system.

This makes no sense at all to me.

genewardsmith wrote:

> By "putting the octaves back in" I meant you must do this to define the
> temperament. If you define a temperament by the octave-equivalent
> mapping [0,1,4], how do you know it isn't the 160/81 temperament, and
> not the 81/80? You need explicitly or implicitly to do a further
> calculation, using real arithmetic and not just algebra, to make this
> system workable. It seems obviously better to me to simply define the
> temperament unambiguously in the first place.

The octave equivalent mapping is [1, 4] not [0, 1, 4]. Both 160/81 and
81/80 are tempered out. What difference does it make? How do you know if
it's 81/80 or 80/81 tempered out in an octave-specific meantone?

> > Existing mathematical theory, however, all seems to be within an
> > octave-equivalent system.
>
> This makes no sense at all to me.

I forgot that Karp left the octaves in. But Fokker and Rothenberg are all
octave-equivalent.

Graham

--- In tuning-math@y..., graham@m... wrote:
> genewardsmith wrote:
> The octave equivalent mapping is [1, 4] not [0, 1, 4].

It's [0, 1, 4] after you put the octave info in.

Both 160/81 and
> 81/80 are tempered out. What difference does it make? How do you know if
> it's 81/80 or 80/81 tempered out in an octave-specific meantone?

You don't--they are the same. However, 160/81 defines a completely different, and very bad, temperament; you propose to ignore it, which certainly makes sense, but you still need octave information even to see it is 81/80, and not 160/81, we are looking at. You can do some of the math on pitch classes, but then you can't tell the difference between a small interval and a large one, which is important for commas, obviously.

> > > Existing mathematical theory, however, all seems to be within an
> > > octave-equivalent system.
> >
> > This makes no sense at all to me.
>
> I forgot that Karp left the octaves in. But Fokker and Rothenberg are all
> octave-equivalent.

Does it matter?

Me:
> > The octave equivalent mapping is [1, 4] not [0, 1, 4].

Gene:
> It's [0, 1, 4] after you put the octave info in.

In which case it isn't octave equivalent, is it?

Me:
> Both 160/81 and
> > 81/80 are tempered out. What difference does it make? How do you
> > know if it's 81/80 or 80/81 tempered out in an octave-specific
> > meantone?

Gene:
> You don't--they are the same. However, 160/81 defines a completely
> different, and very bad, temperament; you propose to ignore it, which
> certainly makes sense, but you still need octave information even to
> see it is 81/80, and not 160/81, we are looking at. You can do some of
> the math on pitch classes, but then you can't tell the difference
> between a small interval and a large one, which is important for
> commas, obviously.

[1, 4] uniquely defines an octave-equivalent temperament. Give an example
of this "completely different" temperament you say exists. You can tell
the difference between a large and small interval if you take all
intervals modulo the octave, as Dave suggested at the start of this
thread.

> Does it matter?

No. Nobody cares. But we keep arguing about it anyway.

Graham

--- In tuning-math@y..., graham@m... wrote:

> [1, 4] uniquely defines an octave-equivalent temperament.

What's your definition of an octave-equivalent temperament?

Give an example
> of this "completely different" temperament you say exists.

You simply stick 160/81 into the usual machine and turn the crank:

Mapping

[ 1 0]
[ 0 1]
[-5 4]

rms = 231 g = 2.94 badness = 5897 generator = 2219 cents

You can tell
> the difference between a large and small interval if you take all
> intervals modulo the octave, as Dave suggested at the start of this
> thread.

That fails to distinguish 160/81 (large) from 80/81 (small.) You need a standard reduction; but then you are back to my point--you are using real arithmetic to get back the information you first left out, hardly a sensible proceedure.

genewardsmith wrote:

> What's your definition of an octave-equivalent temperament?

A temperament where notes separated by an octave are considered
equivalent.

> You simply stick 160/81 into the usual machine and turn the crank:
>
> Mapping
>
> [ 1 0]
> [ 0 1]
> [-5 4]
>
> rms = 231 g = 2.94 badness = 5897 generator = 2219 cents

But there you're giving an octave-specific definition of a temperament you
say won't work in octave-equivalent space. Obviously an octave equivalent
system can't do octave-specific things, but it works fine on its own
terms.

> You can tell
> > the difference between a large and small interval if you take all
> > intervals modulo the octave, as Dave suggested at the start of this
> > thread.
>
> That fails to distinguish 160/81 (large) from 80/81 (small.) You need a
> standard reduction; but then you are back to my point--you are using
> real arithmetic to get back the information you first left out, hardly
> a sensible proceedure.

Yes, it fails to distinguish them. That's because they're the same. We
told it we wanted octave equivalence, so we can hardly be surprised if
that's what we get. So what's the problem?

Graham

--- In tuning-math@y..., graham@m... wrote:
> genewardsmith wrote:

> > What's your definition of an octave-equivalent temperament?

> A temperament where notes separated by an octave are considered
> equivalent.

That seems to be saying the tone group is a circle group, R/1200R if we use cents. This means that the group is not ordered and its image under the log map does not embed into a field, both of which don't help you. On the plus side, it is a topological group with an invariant metric, which gives us a notion of closeness.

I'd say from a mathematician's point of view, having made things harder in this way, we would want a payoff of some kind.

> But there you're giving an octave-specific definition of a temperament you
> say won't work in octave-equivalent space.

I give octave-specific definitions of everything; you are the one saying it might be better not to.

Obviously an octave equivalent
> system can't do octave-specific things, but it works fine on its own
> terms.

It works when it works? It seems to me it works because you can lift it to octave-specific in cases of practical interest. Why bother to do the heavy lifting? What's the payoff?

genewardsmith wrote:

> That seems to be saying the tone group is a circle group, R/1200R if we
> use cents. This means that the group is not ordered and its image under
> the log map does not embed into a field, both of which don't help you.
> On the plus side, it is a topological group with an invariant metric,
> which gives us a notion of closeness.

Um, right.

> I'd say from a mathematician's point of view, having made things harder
> in this way, we would want a payoff of some kind.

It doesn't matter if there's a payoff or not, so long as we can do it.

> > But there you're giving an octave-specific definition of a
> > temperament you say won't work in octave-equivalent space.
>
> I give octave-specific definitions of everything; you are the one
> saying it might be better not to.

You're being totally obtuse here. I don't even know how to answer that.
I asked for an example that wouldn't work in octave-equivalent terms, and
you give me an octave-specific one. How much less relevant could that be?

> Obviously an octave equivalent
> > system can't do octave-specific things, but it works fine on its own
> > terms.
>
> It works when it works? It seems to me it works because you can lift it
> to octave-specific in cases of practical interest. Why bother to do the
> heavy lifting? What's the payoff?

How many times to I have to repeat that neither I nor anybody else who has
expressed an opinion cares about this? It doesn't matter. Nobody thinks
it would be an improvement. You can talk in octave-equivalent terms and
add the octaves to do the calculations. All I'm saying is that the octave
equivalent algebra+whatever works fine in it's own terms and you do *not*
have to add in the octaves.

Graham