Rather than screwing around trying to transform taxicab metrics, I

would suggest starting off with a Euclidean metric which works the

way you want it to work. For instance, you can give 3, 5, and 5/3 any

values you like, and this determines the metric.

If we want them all to be the same, as I did for my symmetric metric,

then if q([x, y]) = ax^2 + bxy + cy^2, we may substitute for

q([1,0])=1, q([0,1])=1, and q([-1,1])=1, getting the system of three

equations in three unknowns, {a=1, c=1, a-b+c=1}. Solving this gives

us a=b=c=1. If by some voodoo I determine 3 should have a distance of

1 from the unison, and 5 and 5/3 a distance of sqrt(2), I solve

another set of three equations and get x^2+y^2...and so forth.

In the 7-limit, we can solve six equations for six unknowns, by

specifying distances for 3,5,7,5/3,7/3, and 7/5. Etc.

--- In tuning-math@y..., genewardsmith@j... wrote:

> Rather than screwing around trying to transform taxicab metrics, I

> would suggest starting off with a Euclidean metric which works the

> way you want it to work.

Goodness no! I definitely want a taxicab metric.

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

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

> > Rather than screwing around trying to transform taxicab metrics,

I

> > would suggest starting off with a Euclidean metric which works

the

> > way you want it to work.

> Goodness no! I definitely want a taxicab metric.

Is this so your worms will have holes? I searched on "wormhole" and

came up with nothing, but I wonder what taxicab metrics can do for

you that a well-chosen Euclidean metric could not also do.

--- In tuning-math@y..., genewardsmith@j... wrote:

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

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

>

> > > Rather than screwing around trying to transform taxicab

metrics,

> I

> > > would suggest starting off with a Euclidean metric which works

> the

> > > way you want it to work.

>

> > Goodness no! I definitely want a taxicab metric.

>

> Is this so your worms will have holes? I searched on "wormhole" and

> came up with nothing,

Probably this discussion happened before the list moved off the Mills

server.

> but I wonder what taxicab metrics can do for

> you that a well-chosen Euclidean metric could not also do.

A lot of things. First, consider the Tenney lattice I just described.

Second, see my post on this list where I conjectured that unison

vectors with numbers of size S and difference between numerator and

denominator D imply an amount of tempering proportional to D/(S^2).

This depends on the Kees van Prooijen lattice with a taxicab metric.

Lots of other reasons too, probably best enunciated by Paul Hahn back

in the Mills days.

>>Is this so your worms will have holes? I searched on "wormhole" and

>>came up with nothing,

>

>Probably this discussion happened before the list moved off the

>Mills server.

The discussion you're referring to happened while the list was

hosted on onelist, and those messages are in the Yahoo! archive.

The problem is that Yahoo!'s search only goes back arbitrarily

far... I tried this search three times, and got searches from

message #'s 27xxx-27847, 26xxx-27847, and 22xxx-27847, all

showing no matches.

Unexpectedly, when searching my personal archive to verify the

date of the thread, I did find a Mills post containing the

term "wormholes":

Date: Sat, 30 May 1998 11:43:43 -0400

From: Daniel Wolf <DJWOLF_MATERIAL@compuserve.com>

To: "INTERNET:tuning@eartha.mills.edu" <tuning@eartha.mills.edu>

Subject: TUNING digest 1431: Wilson's nines

Message-ID: <199805301143_MC2-3EA2-A973@compuserve.com>

Very often Erv Wilson will put his nines on a separate axis from

threes. A look at his Xenharmonikon cover art will provide several

examples. (All back issues of XH are available from Frog Peak

Music).

One of the most interesting mappings done by Wilson is his mapping

onto a Penrose tiling, treating it as a two dimensional

representation of a 5 dimensional space. When nines or fifteens

are treated as independent axes from threes and fives,

interesting 'wormholes' in the lattice start to appear, where

alternative representations of the same pitch class occur in

surprisingly different contexts.

Although I have not followed up on this work for some years, I was

getting interesting results in using the Penrose tilings as a

control mechanism over random walks over the lattice.

While treating nine as two steps on the three axis will certainly

be the more efficient means of avoiding redundancies, there can be

musically compelling grounds for preserving such redundancies or

ambiguities.

-Carl

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> >>Is this so your worms will have holes? I searched on "wormhole"

and

> >>came up with nothing,

> >

> >Probably this discussion happened before the list moved off the

> >Mills server.

>

> The discussion you're referring to happened while the list was

> hosted on onelist, and those messages are in the Yahoo! archive.

> The problem is that Yahoo!'s search only goes back arbitrarily

> far... I tried this search three times, and got searches from

> message #'s 27xxx-27847, 26xxx-27847, and 22xxx-27847, all

> showing no matches.

Keep clicking "Next" . . . a whole slew of wormhole messages will be

found. I'm surprised you and Gene both said you couldn't find them!

--- In tuning-math@y..., genewardsmith@j... wrote:

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

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

>

> > > Rather than screwing around trying to transform taxicab

metrics,

> I

> > > would suggest starting off with a Euclidean metric which works

> the

> > > way you want it to work.

>

> > Goodness no! I definitely want a taxicab metric.

>

> Is this so your worms will have holes? I searched on "wormhole" and

> came up with nothing

Gene, you have to keep clicking "Next" in the search dialog. It goes

backwards . . . somewhere around message 4000, they start popping up.

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> One of the most interesting mappings done by Wilson is his mapping

> onto a Penrose tiling, treating it as a two dimensional

> representation of a 5 dimensional space.

Wowsers!

When nines or fifteens

> are treated as independent axes from threes and fives,

> interesting 'wormholes' in the lattice start to appear, where

> alternative representations of the same pitch class occur in

> surprisingly different contexts.

These wormholes will appear no matter what metric you use. They might

be thought of as universal commas--3^2 is "approximated" by 9.

--- In tuning-math@y..., genewardsmith@j... wrote:

> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

>

> > One of the most interesting mappings done by Wilson is his mapping

> > onto a Penrose tiling, treating it as a two dimensional

> > representation of a 5 dimensional space.

Was this posted a while ago? I don't see this post in today's

archives.

>

> Wowsers!

Well, it's a two dimensional _slice_ through a 5-dimensional space,

as you probably know . . .

>

> When nines or fifteens

> > are treated as independent axes from threes and fives,

> > interesting 'wormholes' in the lattice start to appear, where

> > alternative representations of the same pitch class occur in

> > surprisingly different contexts.

>

> These wormholes will appear no matter what metric you use. They

might

> be thought of as universal commas--3^2 is "approximated" by 9.

These are not what I call "wormholes".

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

> Was this posted a while ago? I don't see this post in today's

> archives.

Yahoo waited a few days before letting it appear.

>>One of the most interesting mappings done by Wilson is his mapping

>>onto a Penrose tiling, treating it as a two dimensional

>>representation of a 5 dimensional space.

>

>Wowsers!

Recall that Erv Wilson is the person who gave us MOS and CS... some

examples of his Coxeter-like 2-D projections of tonespace can be

seen at...

http://www.anaphoria.com/dal.PDF

...Unfortunately, they are not very legible in this PDF file.

Hard copies of Wilson's articles can be found in back issues of

Xenharmonikon, available from Frog Peak music.

Lately, Erv has been building 3-D projections of tonespace with

Zometool.

-Carl