monzo as vector?

Something has struck me as non-intuitive about how monzo is being used. But
then again I probably missed the moment of creation of the monzo. But in
any case I will expose my contradictory impressions here.

My original impression was that a monzo is a notation for a number. It is
an n-tuple notation for a scalar number in fact, almost always a rational
number. So it is confusing to me that this is called a vector. On the
other hand scalar ratios are ordinarily described as vectors (in a lattice)
around here. But still I was surprised for some reason.

Then it is really confusing to me to talk about monzos vanishing. Because
there was never a monzo in the first place. It is just a notation for a
number. If a monzo can vanish, then it is not just a notation for a number,
but is another kind of object.

There. I think that covers it. Any clarifications?

Thanks,
Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> Something has struck me as non-intuitive about how monzo is being
used. But
> then again I probably missed the moment of creation of the monzo.
But in
> any case I will expose my contradictory impressions here.
>
> My original impression was that a monzo is a notation for a
number. It is
> an n-tuple notation for a scalar number in fact, almost always a
rational
> number. So it is confusing to me that this is called a vector.
On the
> other hand scalar ratios are ordinarily described as vectors (in a
lattice)
> around here. But still I was surprised for some reason.

Yes. I can see how there could be a little cognitive dissonance
there at first. But you've now understood that it is extremely
useful for our purposes, to treat rationals as vectors whose basis
consists of logarithms of the prime numbers. I'm probably not
using "basis" in the correct mathematical sense here, but I think
you know what I mean.

> Then it is really confusing to me to talk about monzos vanishing.
Because
> there was never a monzo in the first place. It is just a notation
for a
> number. If a monzo can vanish, then it is not just a notation for
a number,
> but is another kind of object.

Ah. Now I think you have a very valid point. A monzo is a prime
exponent vector which is a representation of a rational number. It
is not the representation that vanishes but the thing it represents.

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> Something has struck me as non-intuitive about how monzo
> is being used. But then again I probably missed the moment
> of creation of the monzo. But in any case I will expose my
> contradictory impressions here.
>
> My original impression was that a monzo is a notation for
> a number. It is an n-tuple notation for a scalar number

correct.

> in fact, almost always a rational number.

not necessarily correct, especially in my work.

i often use rational exponents the monzos, for example,
when describing fraction-of-a-comma meantones. i realize
that there are mathematical arguments against this
because it is "meaningless", but it does allow me
to draw a meantone as a line on the prime-space lattice.

anyway, Gene invented "monzo" as a term and defined
it as having integer elements, so he says i need to
call these beasties "rational monzos", and i'm OK
with that.

> So it is confusing to me that this is called a vector.
> On the other hand scalar ratios are ordinarily described
> as vectors (in a lattice) around here. But still I was
> surprised for some reason.

the elements in a monzo represent coordinates on a
lattice. those coordinates define a point representing
a pitch, whose distance from the origin (ratio 1/1)
defines the interval notated as that monzo.

the interval itself may be represented by a line-segment
spanning the Euclidian distance between the origin
and the point defined in the monzo, and it's considered
to have a direction emanating outward from 1/1.
isn't that a vector?

as Gene and i have explained, the reason for using
the term "monzo" instead of just calling it a "vector",
is because by definition a monzo specificially:

- expresses a ratio which represents a musical pitch
or interval;

- contains elements which represent the exponents
of the prime-series, in the order of the prime-series,
which derive from the prime-factorization of the
terms in the ratio;

- represent the mapping of those pitches and intervals
onto a tone-space / prime-space lattice.

> Then it is really confusing to me to talk about monzos
> vanishing. Because there was never a monzo in the first
> place. It is just a notation for a number. If a monzo
> can vanish, then it is not just a notation for a number,
> but is another kind of object.
>
> There. I think that covers it. Any clarifications?
>
> Thanks,
> Kurt

this is perhaps the main reason why some of us want
to use "promo" to specify projection, and why i also
want "vapro" to specify the vanishing aspect.

if an interval is considered to be a unison-vector,
then, written in ratio form, all of its powers
(both positive and negative) are equivalent to it.

written in monzo form, all of its multiples (both
positive and negative) are equivalent to it.
the primary unison-vector is projected onto a
line which represents the whole infinite set of
equivalents. this concept is what we mean by "promo".

in a temperament, it is not just one monzo which
vanishes, but a whole infinite set of them ...
i.e., a promo. this concept is what i mean by "vapro".

there's a pretty clear explanation of this on
the relevant Encyclopaedia pages. note that when
the multiples in a promo are written out in monzo
form with the coefficient which specifies which
multiple it is, the only thing that differentiates
them is the coefficient. thus, in a sense the
monzo part of the expression only "means" the
primary monzo, and so in a way it really does vanish.

and of course, in a temperament which tempers-out
a certain vapro, it really does physically vanish.

-monz

>My original impression was that a monzo is a notation for a number. It
>is an n-tuple notation for a scalar number in fact, almost always a
>rational number. So it is confusing to me that this is called a vector.
>On the other hand scalar ratios are ordinarily described as vectors (in
>a lattice) around here. But still I was surprised for some reason.

In an odd-limit, there are more monzos than numbers.

>Then it is really confusing to me to talk about monzos vanishing.
>It is just a notation for a number. If a monzo can vanish, then
>it is not just a notation for a number, but is another kind of
>object.

If a monzo vanishes, so does the number (comma).

-Carl

>if an interval is considered to be a unison-vector,
>then, written in ratio form, all of its powers
>(both positive and negative) are equivalent to it.
>
>written in monzo form, all of its multiples (both
>positive and negative) are equivalent to it.
>the primary unison-vector is projected onto a
>line which represents the whole infinite set of
>equivalents. this concept is what we mean by "promo".

Aha! But that's really trivial. I can't imagine
needing a specialized term for it.

-Carl

on 8/14/04 10:36 AM, Carl Lumma <ekin@lumma.org> wrote:

>> My original impression was that a monzo is a notation for a number. It
>> is an n-tuple notation for a scalar number in fact, almost always a
>> rational number. So it is confusing to me that this is called a vector.
>> On the other hand scalar ratios are ordinarily described as vectors (in
>> a lattice) around here. But still I was surprised for some reason.
>
> In an odd-limit, there are more monzos than numbers.
>
>> Then it is really confusing to me to talk about monzos vanishing.
>> It is just a notation for a number. If a monzo can vanish, then
>> it is not just a notation for a number, but is another kind of
>> object.
>
> If a monzo vanishes, so does the number (comma).

One doesn't confuse a series of digits with a number. The digits represent
the number. One says "number" not "string of digits". So the question is
what is the primary referent of "monzo"? It appears it is *not* the number
but implies the full structure (prime space, etc.). Nonetheless it is still
a notation and in fact what it refers to is nothing new at all. And thus I
think it is misplaced to say that a "monzo" vanishes. For example there are
alternate notations besides the prime-space mapping that work equally. The
thing that vanishes does not depend on the choice of primes as the basis.
But the "monzo" clearly does. So the monzo is not the thing that vanishes.

Since this is new terminology, I figure it is worth bringing these things
up.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> My original impression was that a monzo is a notation for a number.

It is.

It is
> an n-tuple notation for a scalar number in fact, almost always a
rational
> number. So it is confusing to me that this is called a vector.

I've been opposing calling it a vector.

> Then it is really confusing to me to talk about monzos vanishing.
Because
> there was never a monzo in the first place. It is just a notation for a
> number. If a monzo can vanish, then it is not just a notation for a
number,
> but is another kind of object.

People are only talking about monzos vanishing in the same context
they are talking about rational numbers such as 81/80 vanishing. It
means the same thing--that |-4 4 -1> vanishes in a temperament means
the temperament maps it to the identity element, which
multiplicitively you would write as 1 but which in terms of period and
generator you would write as [0, 0]. No periods and no generators, or
the number 1, are both ways of saying the unison (in this case, of
meantone.)

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> anyway, Gene invented "monzo" as a term and defined
> it as having integer elements, so he says i need to
> call these beasties "rational monzos", and i'm OK
> with that.

I also said the most natural definition for "promo" would be to
include them; each promo would consist of a line through the origin
with rational number coordinates. For instance, the promo for
|-4 4 -1> would be |-4t 4t -t> for every rational value of t. This
defines a projective point in the usual manner. Restricting t to the
integers does the same thing, only not in the way it is ordinarily
managed. It doesn't make a great deal of difference which you choose,
however, as the two definitions are equivalent. Either way, we get a
projective plane, and the promos are not discrete as in a lattice, but
dense.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> In an odd-limit, there are more monzos than numbers.

What in the world do you mean?

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Aha! But that's really trivial. I can't imagine
> needing a specialized term for it.

Projective geometry isn't trivial, and this point of view may end up
being useful. I seem to have given Paul the projective bug, and it
would be nice if he reemerged and explained why he is producing all of
these diagrams over on tuning-math.

>> In an odd-limit, there are more monzos than numbers.
>
>What in the world do you mean?

There is no longer a 1:1 mapping between monzos and
the rational numbers.

-C.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> One doesn't confuse a series of digits with a number. The digits
represent
> the number. One says "number" not "string of digits". So the
question is
> what is the primary referent of "monzo"? It appears it is *not* the
number
> but implies the full structure (prime space, etc.).

Strings of digits, when interpreted as numbers, also carry structure,
since you can add, subtract and multiply them, etc. The monzo carries
the same structure as p-limit rational numbers under multiplication;
ie, an abelian group structure, since they are a way of representing
the p-limit rational numbers.

Nonetheless it is still
> a notation and in fact what it refers to is nothing new at all. And
thus I
> think it is misplaced to say that a "monzo" vanishes.

This is a little like saying it is wrong to call the number "C" in
hexidecimal "twelve", isn't it? The monzo vanishes, the number
vanishes, both mean the same thing. Nothing literally vanishes, and
you could get picky about that also. What is important is to define
your terms clearly enough that loose language can then do no harm.

For example there are
> alternate notations besides the prime-space mapping that work equally.

I would discount the idea that monzos are necessarily talking about
prime-space. First, I'd like to see prime-space clearly defined.

The
> thing that vanishes does not depend on the choice of primes as the
basis.
> But the "monzo" clearly does. So the monzo is not the thing that
vanishes.

If I write "12", interpreting it depends on knowing that it should be
interpreted as a number base ten. That does not make it wrong to say
"12 is an even integer". It is also not wrong to say "the roman
numeral XII is an even integer". It *would* be wrong to think there
are two kinds of numbers--roman and arabic.

> Since this is new terminology, I figure it is worth bringing these
things
> up.

We are getting perilously close to philosophy by this, however. :)

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> In an odd-limit, there are more monzos than numbers.
> >
> >What in the world do you mean?
>
> There is no longer a 1:1 mapping between monzos and
> the rational numbers.

This is false. It is extremely important to realize that there is
always a 1-1 mapping between p-limit monzos and p-limit positive
rational numbers, and in fact that they are the same thing, monzos by
definition carry this mapping and serve as a way of notating p-limit
prime numbers. If you don't understand that, you've failed to
understand the definition of a monzo.

Formally, you could define a monzo as an n-tuple of integers, written
|w1 ... wn>, together with a canonical mapping

monz(|w1 ... wn>) = 2^w1 3^w2 ... p^wn

where p is the nth prime. This is a little like giving a formal
definition of the expression of integers as arabic numerals by
defining a mapping from a string of characters to the integers.
Normally, we don't do this sort of thing, if we keep it up we will end
up trying to write everything in some version of first order
mathematical logic.

on 8/14/04 1:30 PM, Carl Lumma <ekin@lumma.org> wrote:

>>> In an odd-limit, there are more monzos than numbers.
>>
>> What in the world do you mean?
>
> There is no longer a 1:1 mapping between monzos and
> the rational numbers.

I think you may be missing that a monzo is by definition involved in a prime
mapping, not an "odd mapping".

-Kurt

on 8/14/04 4:56 AM, monz <monz@tonalsoft.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> Something has struck me as non-intuitive about how monzo
>> is being used. But then again I probably missed the moment
>> of creation of the monzo. But in any case I will expose my
>> contradictory impressions here.
>>
>> My original impression was that a monzo is a notation for
>> a number. It is an n-tuple notation for a scalar number
>
> correct.
>
>> in fact, almost always a rational number.
>
> not necessarily correct, especially in my work.

That's why I said "almost always". ;)

> i often use rational exponents the monzos, for example,
> when describing fraction-of-a-comma meantones. i realize
> that there are mathematical arguments against this
> because it is "meaningless", but it does allow me
> to draw a meantone as a line on the prime-space lattice.
>
> anyway, Gene invented "monzo" as a term and defined
> it as having integer elements, so he says i need to
> call these beasties "rational monzos", and i'm OK
> with that.

Note that it is the scalar value which I was labelling "rational". So the
term "rational monzos" sounds pretty confusing because it is not clear
whether the scalar number being referred to is rational or whether the
scalar elements of the monzo are rational.

If the scalar elements are integral, which I take to be the ordinary case,
then the scalar number the monzo represents is rational.

If the scalar elements are rational, that seems like a strange restriction
and you might as well allow the scalar elements to be simply real.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> If the scalar elements are rational, that seems like a strange
restriction
> and you might as well allow the scalar elements to be simply real.

These are quite different in some respects; with rational elements you
define a unique real number.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >if an interval is considered to be a unison-vector,
> >then, written in ratio form, all of its powers
> >(both positive and negative) are equivalent to it.
> >
> >written in monzo form, all of its multiples (both
> >positive and negative) are equivalent to it.
> >the primary unison-vector is projected onto a
> >line which represents the whole infinite set of
> >equivalents. this concept is what we mean by "promo".
>
> Aha! But that's really trivial. I can't imagine
> needing a specialized term for it.
>
> -Carl

I totally agree.

on 8/14/04 1:43 PM, Gene Ward Smith <gwsmith@svpal.org> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>> One doesn't confuse a series of digits with a number. The digits
> represent
>> the number. One says "number" not "string of digits". So the
> question is
>> what is the primary referent of "monzo"? It appears it is *not* the
> number
>> but implies the full structure (prime space, etc.).
>
> Strings of digits, when interpreted as numbers, also carry structure,
> since you can add, subtract and multiply them, etc. The monzo carries
> the same structure as p-limit rational numbers under multiplication;
> ie, an abelian group structure, since they are a way of representing
> the p-limit rational numbers.
>
> Nonetheless it is still
>> a notation and in fact what it refers to is nothing new at all. And
> thus I
>> think it is misplaced to say that a "monzo" vanishes.
>
> This is a little like saying it is wrong to call the number "C" in
> hexidecimal "twelve", isn't it? The monzo vanishes, the number
> vanishes, both mean the same thing. Nothing literally vanishes, and
> you could get picky about that also. What is important is to define
> your terms clearly enough that loose language can then do no harm.

I'll respond to that below...

> For example there are
>> alternate notations besides the prime-space mapping that work equally.
>
> I would discount the idea that monzos are necessarily talking about
> prime-space. First, I'd like to see prime-space clearly defined.

Right, prime space as used in the context of a monzo is simply rational
space. In some contexts prime space is purely multiplicative and so is
identical with the set of positive integers. But yes a definition and
consistent use would be good.

> The
>> thing that vanishes does not depend on the choice of primes as the
> basis.
>> But the "monzo" clearly does. So the monzo is not the thing that
> vanishes.
>
> If I write "12", interpreting it depends on knowing that it should be
> interpreted as a number base ten. That does not make it wrong to say
> "12 is an even integer". It is also not wrong to say "the roman
> numeral XII is an even integer".

It could be misleading though, if the term "roman numeral" were not
introduced merely to disambiguate "XII", and that's my point. The evenness
has nothing to do with it being roman, which relates to your point...

> It *would* be wrong to think there
> are two kinds of numbers--roman and arabic.

Since in our culture "12" needs little disambiguation, I think it would be
misleading to say "the arabic number 12 is an even integer". The statements
made about monzo's sound like they are describing attributes of the monzo (a
new notation). Maybe it is the anti-"promo" in me (referring to Dave's
joke) that doesn't want a whole branch of tuning math associated
unnecessarily with a particular notation. I think it *is* misleading and it
makes people forget that the attributes being ascribed to things are
completely independent of the monzo notational construct. Rather it sounds
like the whole branch of knowledge belongs to the monzo construct. If
indeed you have a kind of number whose only notation is the monzo then it is
more appropriate to equate them, but that is not the case.

>> Since this is new terminology, I figure it is worth bringing these
> things
>> up.
>
> We are getting perilously close to philosophy by this, however. :)

Ah, having fun yet?

-Kurt

>> >> In an odd-limit, there are more monzos than numbers.
>> >
>> >What in the world do you mean?
>>
>> There is no longer a 1:1 mapping between monzos and
>> the rational numbers.
>
>This is false. It is extremely important to realize that there is
>always a 1-1 mapping between p-limit monzos and p-limit positive
>rational numbers, and in fact that they are the same thing, monzos by
>definition carry this mapping and serve as a way of notating p-limit
>prime numbers. If you don't understand that, you've failed to
>understand the definition of a monzo.

I said odd-limit. Are you blind? You don't define monzos for
odd-limit? Fine, I'll call them shoewax.

-Carl

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 8/14/04 1:43 PM, Gene Ward Smith <gwsmith@s...> wrote:

> > I would discount the idea that monzos are necessarily talking about
> > prime-space. First, I'd like to see prime-space clearly defined.
>
> Right, prime space as used in the context of a monzo is simply rational
> space.

That doesn't help any since I don't know what rational space is
either. I'm perfectly willing to define p-limit rational space as the
real topolgical vector space of dimension pi(p) in which the p-limit
rational numbers reside as a topological lattice, but are we there
yet? Does anyone want to do this?

Rather it sounds
> like the whole branch of knowledge belongs to the monzo construct.

I often talked about wedgies and vals, and it was only until the |>
notation for monzos and <| for vals was introduced that understanding
seemed to blossom. In effect, for some people, it looks as if the
knowledges does belong to the monzo construct--or notation, which
would be a better word. That's how people got it.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I said odd-limit. Are you blind?

A 9-limit monzo is the same as a 7-limit monzo. If you don't know
that, it is not a reason to wax Roth.

You don't define monzos for
> odd-limit? Fine, I'll call them shoewax.

Call them anything you like, but first define them and explain why
they are significant.

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> Ah. Now I think you have a very valid point. A monzo is a prime
> exponent vector which is a representation of a rational number. It
> is not the representation that vanishes but the thing it
represents.

Actually, there's another level of representation in there, the
rational number represents something physical that we can hear or
measure. It's really the physical thing (the comma) that is
distrubuted and is said to "vanish", irrespective of whether we
represent it by a rational number or a monzo or nothing at all.

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
> <snip>
>
> > For example there are alternate notations besides
> > the prime-space mapping that work equally.
>
> I would discount the idea that monzos are necessarily
> talking about prime-space.

me too.

> First, I'd like to see prime-space clearly defined.

Gene, i would welcome your help with that ... perhaps
best done over on tuning-math.

Chris (my partner at Tonalsoft) and i have already begun
writing up a whole set of "space" definitions, but we
put it on the back-burner for right now. maybe you
can help us clarify things.

i think we conceive of prime-space as a part of
vector-space ... but the only ways i currently have
of dealing with >3-dimensional space are my imagination,
and the algorithms we've built into Musica. i don't
know a lot of the mathematics that deal with this stuff,
so i have a hard time with it until i grok certain
concepts and operations.

-monz

hi Gene and Kurt,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>
> > If the scalar elements are rational, that seems like
> > a strange restriction and you might as well allow the
> > scalar elements to be simply real.
>
> These are quite different in some respects; with rational
> elements you define a unique real number.

in fact, i've also used other numbers besides rationals
as elements in my monzos.

a great example is "golden meantone", where _phi_ is
a part of every monzo:

http://tonalsoft.com/enc/index2.htm?golden.htm

another is LucyTuning, where _pi_ is in every monzo:

http://tonalsoft.com/enc/index2.htm?lucy.htm

again, using primes as the bases, allowing numbers
like this as monzo elements allows me to plot these
types of tunings as straight lines on a "flat"
lattice.

by "flat" i mean that the lattice has not been warped
by employing a vapro in the geometry. if the lattice
is warped, the straight lines for all the examples
i've tried have turned into spirals.

-monz

hi Dave and Carl,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > > if an interval is considered to be a unison-vector,
> > > then, written in ratio form, all of its powers
> > > (both positive and negative) are equivalent to it.
> > >
> > > written in monzo form, all of its multiples (both
> > > positive and negative) are equivalent to it.
> > > the primary unison-vector is projected onto a
> > > line which represents the whole infinite set of
> > > equivalents. this concept is what we mean by "promo".
> >
> > Aha! But that's really trivial. I can't imagine
> > needing a specialized term for it.
> >
> > -Carl
>
> I totally agree.

it's OK ... no-one and nothing is forcing either of
you to use "promo". but *i* will be ...

my intuition, which could be very wrong, is telling
me that promos will be useful in dealing with
higher-dimensional constructions.

-monz

>> I said odd-limit. Are you blind?
>
>A 9-limit monzo is the same as a 7-limit monzo. If you don't know
>that, it is not a reason to wax Roth.

Gene, I'm really sorry for this unprovoked outburst. I'm
going down the tubes. :(

Don't know the wax Roth reference, though.

>> You don't define monzos for
>> odd-limit? Fine, I'll call them shoewax.
>
>Call them anything you like, but first define them and explain why
>they are significant.

I think we've discussed odd-limit n-tuples before, which have
a 1:1 mapping with paths through the 9-limit network. This is
what I meant, with "odd-limit" intended to modify "monzo". You
just read it at face value, and I went crazy. Sorry about
that.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Gene, I'm really sorry for this unprovoked outburst. I'm
> going down the tubes. :(

I hate those tubes.

> Don't know the wax Roth reference, though.

http://www.filmsite.org/hors2.html

I wrote...

>I think we've discussed odd-limit n-tuples before, which have
>a 1:1 mapping with paths through the 9-limit network.

Dang, this isn't even true, since for example 5/4->3/2 is a
different path than 3/2->5/4. But I should hang it up for
tonight...

zzzz. -C.

I wrote...

>I wrote...
>
>>I think we've discussed odd-limit n-tuples before, which have
>>a 1:1 mapping with paths through the 9-limit network.
>
>Dang, this isn't even true, since for example 5/4->3/2 is a
>different path than 3/2->5/4. But I should hang it up for
>tonight...
>
>zzzz. -C.

Maybe this fixes things:

The o-limit Hahn-style n-tuples have a 1:1 mapping with all
pairs of points in the o-limit lattice.

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> The o-limit Hahn-style n-tuples have a 1:1 mapping with all
> pairs of points in the o-limit lattice.

Not hardly. What are these pairs of points?

>> The o-limit Hahn-style n-tuples have a 1:1 mapping with all
>> pairs of points in the o-limit lattice.
>
>Not hardly. What are these pairs of points?

Reply at:

/tuning-math/message/11337

-Carl