Tenney harmonic-distance

i've just added a table and lattice-diagram to the bottom
of my "harmonic distance" page:

http://tonalsoft.com/enc/harmonic-distance.htm

i used log(2) here, to conform with Paul's use of
harmonic-distance in his latest paper. notice the
entry for the syntonic-comma about midway down the list:

2,3,5,7,11-monzo ..... ratio ...... ~cents ...... log(2) HD
[ -4 4, -1 0 0 > .... 81 / 80 .... 21.5062896 ... 12.6617781

did i do everything correctly here?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> i've just added a table and lattice-diagram to the bottom
> of my "harmonic distance" page:
>
> http://tonalsoft.com/enc/harmonic-distance.htm
>
>
> i used log(2) here, to conform with Paul's use of
> harmonic-distance in his latest paper. notice the
> entry for the syntonic-comma about midway down the list:
>
> 2,3,5,7,11-monzo ..... ratio ...... ~cents ...... log(2) HD
> [ -4 4, -1 0 0 > .... 81 / 80 .... 21.5062896 ... 12.6617781
>
>
> did i do everything correctly here?

Looks right except the lattice at the end. 2-dimensional 5-limit
lattices assume octave-equivalence, while Tenney harmonic distance
doesn't. Tenney harmonic distance assumes a 3-dimensional 5-limit
lattice. If you insist on doing this anyway, I'd recommend reducing
each pitch to the octave between -600 cents and 600 cents, instead of
the octave between 0 and 1200 cents.

But I think it would be much better if you associated this kind of
lattice (2-dimensional 5-limit), not with Tenney harmonic distance,
but with "expressibility" (Kees van Prooijen's measure):
http://www.kees.cc/tuning/perbl.html

I was positive that
http://tonalsoft.com/enc/ratio-of.htm
had a link to an "expressibility" page, but the reference, link and
the page seem to have vanished, and there's no link to it from any of
the complexity pages either . . .

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

> did i do everything correctly here?

I'd delete everything about Minkowski metrics.

hi Paul,

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > i've just added a table and lattice-diagram to the bottom
> > of my "harmonic distance" page:
> >
> > http://tonalsoft.com/enc/harmonic-distance.htm
> >
> >
> > i used log(2) here, to conform with Paul's use of
> > harmonic-distance in his latest paper. notice the
> > entry for the syntonic-comma about midway down the list:
> >
> > 2,3,5,7,11-monzo ..... ratio ...... ~cents ...... log(2) HD
> > [ -4 4, -1 0 0 > .... 81 / 80 .... 21.5062896 ... 12.6617781
> >
> >
> > did i do everything correctly here?
>
> Looks right except the lattice at the end. 2-dimensional
> 5-limit lattices assume octave-equivalence, while Tenney
> harmonic distance doesn't. Tenney harmonic distance assumes
> a 3-dimensional 5-limit lattice.

right, i knew that ... just didn't have time to make a
nice 3D lattice.

> If you insist on doing this anyway, I'd recommend reducing
> each pitch to the octave between -600 cents and 600 cents,
> instead of the octave between 0 and 1200 cents.

hmm, not a bad idea.

> But I think it would be much better if you associated this
> kind of lattice (2-dimensional 5-limit), not with Tenney
> harmonic distance, but with "expressibility" (Kees
> van Prooijen's measure):
> http://www.kees.cc/tuning/perbl.html
>
> I was positive that
> http://tonalsoft.com/enc/ratio-of.htm
> had a link to an "expressibility" page, but the reference,
> link and the page seem to have vanished, and there's no link
> from any of the complexity pages either . . .

i never had my own page about "expressibility" ... guess
i should make one. but i've never really understood how
Kee's measure works. care to explain it?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Paul,
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > i've just added a table and lattice-diagram to the bottom
> > > of my "harmonic distance" page:
> > >
> > > http://tonalsoft.com/enc/harmonic-distance.htm
> > >
> > >
> > > i used log(2) here, to conform with Paul's use of
> > > harmonic-distance in his latest paper. notice the
> > > entry for the syntonic-comma about midway down the list:
> > >
> > > 2,3,5,7,11-monzo ..... ratio ...... ~cents ...... log(2) HD
> > > [ -4 4, -1 0 0 > .... 81 / 80 .... 21.5062896 ... 12.6617781
> > >
> > >
> > > did i do everything correctly here?
> >
> > Looks right except the lattice at the end. 2-dimensional
> > 5-limit lattices assume octave-equivalence, while Tenney
> > harmonic distance doesn't. Tenney harmonic distance assumes
> > a 3-dimensional 5-limit lattice.
>
>
> right, i knew that ... just didn't have time to make a
> nice 3D lattice.
>
>
> > If you insist on doing this anyway, I'd recommend reducing
> > each pitch to the octave between -600 cents and 600 cents,
> > instead of the octave between 0 and 1200 cents.
>
>
> hmm, not a bad idea.
>
>
>
> > But I think it would be much better if you associated this
> > kind of lattice (2-dimensional 5-limit), not with Tenney
> > harmonic distance, but with "expressibility" (Kees
> > van Prooijen's measure):
> > http://www.kees.cc/tuning/perbl.html
> >
> > I was positive that
> > http://tonalsoft.com/enc/ratio-of.htm
> > had a link to an "expressibility" page, but the reference,
> > link and the page seem to have vanished, and there's no link
> > from any of the complexity pages either . . .
>
>
>
> i never had my own page about "expressibility" ... guess
> i should make one.

Some stuff that was up there is definitely gone. You lost some
updates somewhere . . .

> but i've never really understood how
> Kee's measure works. care to explain it?

The simplest way to put it is that any ratio of N has an
expressibility of log(N).

http://www.tonalsoft.com/enc/ratio-of.htm

As Kees illustrates on his page, if you have a 5-limit interval
whose "monzo" is [m n> (I mean the 3,5-monzo, of course), then the
expressibility will be equal to

|m|*log(3) + |n|*log(5) when m and n have the same sign and

max( |m|*log(3), |n|*log(5) ) otherwise.

Let me know if this makes sense so far.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> Looks right except the lattice at the end. 2-dimensional 5-limit
> lattices assume octave-equivalence, while Tenney harmonic distance
> doesn't. Tenney harmonic distance assumes a 3-dimensional 5-limit
> lattice.

I would say it *defines* such a lattice, which doesn't exist until
you define a metric. But I just got Mathieu and he thinks twelve
notes of equal temperament are a lattice, so I'm probably doomed.

> But I think it would be much better if you associated this kind of
> lattice (2-dimensional 5-limit), not with Tenney harmonic distance,
> but with "expressibility" (Kees van Prooijen's measure):
> http://www.kees.cc/tuning/perbl.html

Why not ||3^a 5^b|| = sqrt(a^2+ab+b^2)? This is, after all, what
people keep drawing, so they must like something about it.

This reminds me I need to put up something about the Hahn metric on
my web page.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > Looks right except the lattice at the end. 2-dimensional 5-limit
> > lattices assume octave-equivalence, while Tenney harmonic
distance
> > doesn't. Tenney harmonic distance assumes a 3-dimensional 5-limit
> > lattice.
>
> I would say it *defines* such a lattice, which doesn't exist until
> you define a metric. But I just got Mathieu and he thinks twelve
> notes of equal temperament are a lattice, so I'm probably doomed.
>
> > But I think it would be much better if you associated this kind
of
> > lattice (2-dimensional 5-limit), not with Tenney harmonic
distance,
> > but with "expressibility" (Kees van Prooijen's measure):
> > http://www.kees.cc/tuning/perbl.html
>
> Why not ||3^a 5^b|| = sqrt(a^2+ab+b^2)?

As I've argued before in several different ways, and as Partch
implicitly agrees, imposing octave-equivalence gives a concordance
ranking that agrees with "expressibility" much better than with this
measure. When the ratios get too complex, neither measure is any good.

> This is, after all, what
> people keep drawing, so they must like something about it.

I'm dreading addressing this again. My counterargument is
that "people" are seeing the complexity of the relationship between
one note and another as being measured *along* the lines of the
lattice, and not along some as-the-crow-flies paths through the
intervening territory. At least you have no right to simply assume
the latter over the former, unless said "people" say so themselves.

> This reminds me I need to put up something about the Hahn metric on
> my web page.

Cool.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> I'm dreading addressing this again. My counterargument is
> that "people" are seeing the complexity of the relationship between
> one note and another as being measured *along* the lines of the
> lattice, and not along some as-the-crow-flies paths through the
> intervening territory. At least you have no right to simply assume
> the latter over the former, unless said "people" say so themselves.

Eh? I have no right to assume people, often with no mathematical
sophistication, mean what they draw when they draw a picture unless
they say otherwise? I don't buy it, and I think telling someone
they "really mean" something they can't draw when they draw a picture
is elitist nonsense.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > I'm dreading addressing this again. My counterargument is
> > that "people" are seeing the complexity of the relationship
between
> > one note and another as being measured *along* the lines of the
> > lattice, and not along some as-the-crow-flies paths through the
> > intervening territory. At least you have no right to simply
assume
> > the latter over the former, unless said "people" say so
themselves.
>
> Eh? I have no right to assume people, often with no mathematical
> sophistication, mean what they draw when they draw a picture unless
> they say otherwise?

You're doing more than that. You're assuming that these people intend
to portray complexity as as-the-crow-flies distance in the lattice.
That's a huge assumption, which for some reason you're extremely
cavalier about making.

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

> I don't buy it, and I think telling someone
> they "really mean" something they can't draw

You're pulling this out of nowhere. If they've drawn a lattice with
connectors, then the distance along those connectors is something
they've already drawn, not something they can't draw!

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > Eh? I have no right to assume people, often with no mathematical
> > sophistication, mean what they draw when they draw a picture
unless
> > they say otherwise?
>
> You're doing more than that. You're assuming that these people
intend
> to portray complexity as as-the-crow-flies distance in the lattice.

I'm not assuming they intend to portray complexity at all. That is
another case of you telling people what they should be thinking,
because it is what you think when you draw a picture.

I'm assuming that unless people tell me otherwise, what they draw
depicts what they think. You are assuming that unless Paul Erlich
tells them what they really think, they don't know.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > I don't buy it, and I think telling someone
> > they "really mean" something they can't draw
>
> You're pulling this out of nowhere. If they've drawn a lattice with
> connectors, then the distance along those connectors is something
> they've already drawn, not something they can't draw!

Right, and assuming they are what they look like--equilateral
triangles--leads to the conclusion that they are in a Euclidean
space. If you want to count distance along the streets, you can do
that, or if you want to go as the crow flies, you can do that too. To
assume the triangles live inside a space with a metric few people
have heard of, which they would find confusing if it were explained,
and which I doubt you yourself could clearly explain, is assuming way
too much. How many musicians who draw a picture are thinking about
metric spaces?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > > Eh? I have no right to assume people, often with no
mathematical
> > > sophistication, mean what they draw when they draw a picture
> unless
> > > they say otherwise?
> >
> > You're doing more than that. You're assuming that these people
> intend
> > to portray complexity as as-the-crow-flies distance in the
lattice.
>
> I'm not assuming they intend to portray complexity at all.

But complexity is all that was being discussed here -- that's the
context in which you inserted "everyone means sqrt . . ." comment.

> That is
> another case of you telling people what they should be thinking,
> because it is what you think when you draw a picture.

No, you were the one who came in assuming some unavoidable aspect of
people's drawings were meant to portray complexity, not me.

> I'm assuming that unless people tell me otherwise, what they draw
> depicts what they think.

So do I. The only difference is that when you do it, you're imposing
a very individual, personal interpretation on unavoidable aspects of
such drawings, and far-fetchedly imposing your idea of what they mean
onto those who drew them.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > > I don't buy it, and I think telling someone
> > > they "really mean" something they can't draw
> >
> > You're pulling this out of nowhere. If they've drawn a lattice
with
> > connectors, then the distance along those connectors is something
> > they've already drawn, not something they can't draw!
>
> Right, and assuming they are what they look like--equilateral
> triangles--leads to the conclusion that they are in a Euclidean
> space. If you want to count distance along the streets, you can do
> that, or if you want to go as the crow flies, you can do that too.

That was my point. Unlike you, I didn't arbitrarily attach meaning to
aspects of a drawing which the drawer has no control over.

> To
> assume the triangles live inside a space with a metric few people
> have heard of, which they would find confusing if it were
explained,
> and which I doubt you yourself could clearly explain, is assuming
way
> too much. How many musicians who draw a picture are thinking about
> metric spaces?

You've made some kind of leap here. You were the one assuming that
certain aspects of a drawing were meant to reflect perfectly the
drawer's conception of the complexity of all ratios, when the drawing
could (and often does) look the way it does for other reasons, and
fail to represent the drawer's conceptions of complexity of all
ratios. In fact, the drawer may have no such conceptions at all. I'd
rather stop ask the person doing the drawing to determine precisely
what they think, rather than drawing presumptive conclusions and
proclaiming to know what they meant.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
> >
> > > > Eh? I have no right to assume people, often with no
> mathematical
> > > > sophistication, mean what they draw when they draw a picture
> > unless
> > > > they say otherwise?
> > >
> > > You're doing more than that. You're assuming that these people
> > intend
> > > to portray complexity as as-the-crow-flies distance in the
> lattice.
> >
> > I'm not assuming they intend to portray complexity at all.
>
> But complexity is all that was being discussed here -- that's the
> context in which you inserted "everyone means sqrt . . ." comment.

No, that was *not* the context. The context was lattices. That
lattices can be in a real vector space using a norm other than the
Euclidean norm is a very sophisticated notion, and assuming people
really mean to say that, and really are thinking it, when they don't
even have a clue what it means and can't visualize it strikes me as
absurd and elitist. You are proposing to tell everyone who posts
lattice diagrams around here, or someone like Mathieu when he draws a
picture on a page, that they are talking about lattices in metric
spaces with unfamiliar norms. I don't think so.

> > That is
> > another case of you telling people what they should be thinking,
> > because it is what you think when you draw a picture.
>
> No, you were the one who came in assuming some unavoidable aspect of
> people's drawings were meant to portray complexity, not me.

I was and am talking about lattices in connection with the diagrams
people draw. Where, pray tell, do I discuss complexity? You don't
*need* to introduce complexity measures via a normed vector space,
leading to a lattice.

> > I'm assuming that unless people tell me otherwise, what they draw
> > depicts what they think.
>
> So do I. The only difference is that when you do it, you're imposing
> a very individual, personal interpretation on unavoidable aspects of
> such drawings, and far-fetchedly imposing your idea of what they mean
> onto those who drew them.

Please! Your un-Euclidean normed vector spaces are what everyone is
really thinking, while my high school geometry is far-fetched and
personal. Get real.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> In fact, the drawer may have no such conceptions at all. I'd
> rather stop ask the person doing the drawing to determine precisely
> what they think, rather than drawing presumptive conclusions and
> proclaiming to know what they meant.

This is my point. Your assumption that they mean distance along
streets, measured I presume using the picture in question, has now
been rejected by you as well as me. All I was and am saying is that if
someone draws a picture, the prima facie assumption should be that it
depicts what it appears to depict unless we know it really doesn't.
You can talk about
un-Euclidean lattices all you like but can hardly force the assumption
on people that this must be what they really meant to draw.

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

> That
> lattices can be in a real vector space using a norm other than the
> Euclidean norm is a very sophisticated notion, and assuming people
> really mean to say that, and really are thinking it, when they don't
> even have a clue what it means and can't visualize it strikes me as
> absurd and elitist.

People know that to get from 1st street and 1st avenue to 6th street
and 6th avenue requires a distance of 10 blocks, not 7.07. Is this
elitist?

But that's beside the point -- you were the one shoving unintended
implications into people's mouths, not me.

Plus, you're the one making it complicated/"elitist" (I wouldn't use
that word, actually) with talks of norms and such.

> You are proposing to tell everyone who posts
> lattice diagrams around here, or someone like Mathieu when he draws
a
> picture on a page,

Ha! That's a funny example. You obviously need to read more of his
book, to see what norm he *actually* intends! It's where he talks
about the lattice looking unequally-spaced from any particular spot
within it, and shows the more and more distant connections getting
further and further apart from one another. You might be surprised --
it sure ain't Euclidean, or even Euclidean-embeddable!

> that they are talking about lattices in metric
> spaces with unfamiliar norms.

No, that's only your very convoluted intepretation that leads to that
conclusion. I think complexity is a question each lattice-drawer may
answer, or choose to answer, for themselves, and not have it dictated
by an aspect of their drawing which they can have no control over.

> > > That is
> > > another case of you telling people what they should be
thinking,
> > > because it is what you think when you draw a picture.
> >
> > No, you were the one who came in assuming some unavoidable aspect
of
> > people's drawings were meant to portray complexity, not me.
>
> I was and am talking about lattices in connection with the diagrams
> people draw. Where, pray tell, do I discuss complexity?

I was discussing a complexity measure, and you jumped in with a
different one, saying it's what people "mean" when they draw
triangular lattices. I think that's stuffing your own conceptions
into other people's mouths.

> You don't
> *need* to introduce complexity measures via a normed vector space,
> leading to a lattice.

The lattice can arise from a simple desire to portray symmetrical
configurations built up with the most concordant intervals as your
building blocks (portrayed as "sticks" or "connectors"). Due to the
reality of Euclidean geometry, such a drawing will end up with
certain distance relations whether the drawer likes them or not.
Therefore, it's going too far to imply that these distance relations
portray some essential component of the drawer's model of intervallic
complexity (they might not have any such model to begin with).

> > > I'm assuming that unless people tell me otherwise, what they
draw
> > > depicts what they think.
> >
> > So do I. The only difference is that when you do it, you're
imposing
> > a very individual, personal interpretation on unavoidable aspects
of
> > such drawings, and far-fetchedly imposing your idea of what they
mean
> > onto those who drew them.
>
> Please! Your un-Euclidean normed vector spaces are what everyone is
> really thinking, while my high school geometry is far-fetched and
> personal. Get real.

Now you're just putting words in my mouth -- I've explained myself
enough times, and this will be one of my last attempts.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > In fact, the drawer may have no such conceptions at all. I'd
> > rather stop ask the person doing the drawing to determine
precisely
> > what they think, rather than drawing presumptive conclusions and
> > proclaiming to know what they meant.
>
> This is my point.

Glad we're agreed!

> Your assumption that they mean distance along
> streets, measured I presume using the picture in question, has now
> been rejected by you as well as me.

If it suits you to believe I've changed my mind, go right ahead -- as
long as we reach some agreement, that's progress, and is more
important than rehashing the past.

> All I was and am saying is that if
> someone draws a picture, the prima facie assumption should be that
it
> depicts what it appears to depict unless we know it really doesn't.

I was trying to point out that you were bringing a whole lot of
assumptions into interpreting "what it appears to depict" and that
this hypothetical "someone" deserves more respect than that --
especially when they have absolutely no alternative, due to the
structure of 2D space, but to "appear to depict" a particular formula
under this interpretation of yours.

> You can talk about
> un-Euclidean lattices all you like but can hardly force the
assumption
> on people that this must be what they really meant to draw.

See above -- if you must believe that this is what I did here, fine,
as long as we're getting past it. Communication need to improve,
though. Otherwise, I don't plan to remain on your cast of fictional
characters much longer.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> People know that to get from 1st street and 1st avenue to 6th street
> and 6th avenue requires a distance of 10 blocks, not 7.07. Is this
> elitist?

It becomes so when you start saying the streets aren't in a normal
Euclidean world, and attacking people who assume otherwise.

> But that's beside the point -- you were the one shoving unintended
> implications into people's mouths, not me.

No, all I said really was that there must be something about the
choice of equlateral triangles that people liked; the symmetry, I
presume, some of the time at least is likely to be what they have in
mind. You jumped in with a lot of your own obsessions, put words in my
mouth, dumped on me, and in general acted in a way you were going to
tell me, over on metatuning, how not to act.

> Plus, you're the one making it complicated/"elitist" (I wouldn't use
> that word, actually) with talks of norms and such.

You are the one who went bonkers when I used Euclidean distance, so I
presume you had a reason. Do you even know what in hell you want to
chew me out over?

> > You are proposing to tell everyone who posts
> > lattice diagrams around here, or someone like Mathieu when he draws
> a
> > picture on a page,
>
> Ha! That's a funny example. You obviously need to read more of his
> book, to see what norm he *actually* intends! It's where he talks
> about the lattice looking unequally-spaced from any particular spot
> within it, and shows the more and more distant connections getting
> further and further apart from one another. You might be surprised --
> it sure ain't Euclidean, or even Euclidean-embeddable!

Mathieu starts out by saying a lattice is composed out of two sets of
parallel lines, then mutates this to it consisting of 12 notes, and
then wants to space things unevenly. Assuming he means something
precise and mathematical is a lot more than I would want to assume.

> > that they are talking about lattices in metric
> > spaces with unfamiliar norms.
>
> No, that's only your very convoluted intepretation that leads to that
> conclusion.

Yet *you* were the one who hit the roof when I used a Euclidean
metric. You can't very well have it both ways--either a Euclidean
metric is a thought crime, and we should be using some other metric,
or it is OK, and you should climb the hell off my back. What's your
point, and why do you think you need to jump on me when I use what
amounts to high school geometry?

> Now you're just putting words in my mouth -- I've explained myself
> enough times, and this will be one of my last attempts.

Next time, try politeness and respect.

Gene and Paul,

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

> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > Now you're just putting words in my mouth -- I've
> > explained myself enough times, and this will be one
> > of my last attempts.
>
> Next time, try politeness and respect.

can you guys *please* refrain from posting any more of
this kind of thing to the list, and instead work out
your personal differences in private email?

try to be friends again ... but leave the rest of
us out of it until you are. thanks.

-monz

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> I was trying to point out that you were bringing a whole lot of
> assumptions into interpreting "what it appears to depict" and that
> this hypothetical "someone" deserves more respect than that --
> especially when they have absolutely no alternative, due to the
> structure of 2D space, but to "appear to depict" a particular formula
> under this interpretation of yours.

Obviously any claim that there is no alternative to drawing 3/2, 5/4,
and 6/5 so that they make an equilateral triangle is simply false.
There *are* alternative kinds of geometries in which they also make an
equilateral triangle, and in which you can define a lattice; this
requires an abelian group structure and hence can only be something
like a normed vector space. We could be talking, therefore, about an
L1 style norm over a hyperreal field, but somehow I doubt we want to
be. If we make life simple for ourselves, it is a real vector space;
but if you reject that as still too complicated you are running out of
reasons to tell me what an idiot I am for using a Eucidean lattice.
You've left yourself with no alternative.

My formula is simply Eucliean distance, in other words, under the
assmption that the equalateral triangles someone appears to be drawing
are in fact equilateral; reject it, and you are in one of those other
kinds of space you now want to tell me are too complicated!

> See above -- if you must believe that this is what I did here, fine,
> as long as we're getting past it. Communication need to improve,
> though. Otherwise, I don't plan to remain on your cast of fictional
> characters much longer.

You put all kinds of words in my mouth, so I'd think you would have a
little more sympathy for my objections. I don't like being told I said
something I did not say, and then being dumped on for saying it.

>Obviously any claim that there is no alternative to drawing 3/2, 5/4,
>and 6/5 so that they make an equilateral triangle is simply false.
>There *are* alternative kinds of geometries in which they also make an
>equilateral triangle, and in which you can define a lattice; this
>requires an abelian group structure and hence can only be something
>like a normed vector space. We could be talking, therefore, about an
>L1 style norm over a hyperreal field, but somehow I doubt we want to
>be. If we make life simple for ourselves, it is a real vector space;
>but if you reject that as still too complicated you are running out of
>reasons to tell me what an idiot I am for using a Eucidean lattice.
>You've left yourself with no alternative.

Hi Gene,

Sometimes it's not what you talk about, but how you talk about it.
The Erlich-Tenney lattice can be easily explained to people without
using the word "norm" because they have the everyday experience of
driving in a city, and wanting to find the shortest routes.

You can say, the city streets will be consonances, and any interval
in JI will be a route through the city. Further, we want the total
distance traveled on that route to represent the consonance of that
interval.

In the 5-limit, for example, the Avenues could be ratios of 5 and
the Streets ratios of 3. Because most people (or Partch, or whatever)
feel that consonance decreases as harmonic limit goes up, the
"cross-town" blocks (on the Avenues) should therefore be longer than
the blocks along the Streets.

But what should their lengths be? A convenience answer suggested
by famous music theorist and composer James Tenney is that they
should be scale with the log of their largest odd factor.

Then you draw a picture, and mention that this can be extended to
higher limits, and you're done. If you want, at the end you can
even tell people what this sort of thing is called in math lingo.

-Carl

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

> Sometimes it's not what you talk about, but how you talk about it.
> The Erlich-Tenney lattice can be easily explained to people without
> using the word "norm" because they have the everyday experience of
> driving in a city, and wanting to find the shortest routes.

They have this experience driving along steets in (what appears to be)
a Eucildean space, and certainly nothing like Tenney space. If you
simply want to talk about distance along streets, you never need to
introduce a new kind of metric.

> In the 5-limit, for example, the Avenues could be ratios of 5 and
> the Streets ratios of 3. Because most people (or Partch, or whatever)
> feel that consonance decreases as harmonic limit goes up, the
> "cross-town" blocks (on the Avenues) should therefore be longer than
> the blocks along the Streets.

So far Euclidean geometry is working fine.

> But what should their lengths be? A convenience answer suggested
> by famous music theorist and composer James Tenney is that they
> should be scale with the log of their largest odd factor.

So you have streets of length log(3), others of length log(5), and
tunnels down the manholes of length log(2). To get to 3/2, you have to
go down a steet of length log(3), and then down a tunnel of depth
log(2), so you go a total distance of log(6). Still Euclidean!

> Then you draw a picture, and mention that this can be extended to
> higher limits, and you're done.

Are you done by assuming the space is Euclidean, assuming it is Tenney
space, or assuming it doesn't matter? You don't yet know how far apart
things are, unless they are between intersections of the
roads/tunnels. Technically, you don't have a lattice, but a network:

http://mathworld.wolfram.com/Network.html

If you want, at the end you can
> even tell people what this sort of thing is called in math lingo.

So far as I can tell, in math lingo it could just be a Euclidean
lattice, raising once again the question of why Paul is or was up in
arms over my use of same.

>> Then you draw a picture, and mention that this can be extended to
>> higher limits, and you're done.
>
>Are you done by assuming the space is Euclidean, assuming it is
>Tenney space, or assuming it doesn't matter? You don't yet know
>how far apart things are,

I thought I'd implied that you must travel along the streets, not
over the buildings or through the woods. I also gave the scaling
for the street lengths. This should be enough for average folks
to know what's going on -- they probably won't want to calculate
anything anyway, but now you can discuss results and they'll have
an idea what you're talking about.

>Technically, you don't have a lattice, but a network:
>
>http://mathworld.wolfram.com/Network.html

Ok, whatever, it's fine to point this out but not important at
this level of explanation.

>> If you want, at the end you can
>> even tell people what this sort of thing is called in math lingo.
>
>So far as I can tell, in math lingo it could just be a Euclidean
>lattice, raising once again the question of why Paul is or was up in
>arms over my use of same.

I thought the formula you gave was not taxicab. Can you please
confirm this? Thanks.

-Carl

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

> I thought the formula you gave was not taxicab. Can you please
> confirm this? Thanks.

The formula I gave was as-the-crow-flies, but if you fly like a crow
down one street and then turn and fly down another, that's the
distance you wanted.

>> I thought the formula you gave was not taxicab. Can you please
>> confirm this? Thanks.
>
>The formula I gave was as-the-crow-flies, but if you fly like a
>crow down one street and then turn and fly down another, that's
>the distance you wanted.

Yes, I understand that you're claiming we're still in Euclidean
space, but again, average person doesn't care. Average person
maybe doesn't even known Euclidean from non-Euclidean. Myself
included.*

-Carl

* I know stuff about, what is it, Euclid's 5th axiom
about parallel lines that you can violate with either positive
or negative curvature to get Riemannian or hyperbolic geometry
instead. I know this subsequently effects the angles of a
triangle. But I've never worked with any non-Euclidean
geometry and I'm certainly not qualified to identify it in
obscure situations like JIPs and soforth.