summary -- are these right?

TM reduction or LLL reduction -> canonical basis

...Which of TM, LLL is preferred these days, and is there
a definition of "basis" somewhere? It's a list of commas,
right?

----

Hermite normal form -> canonical map

...can someone give an algorithm for turning a basis (or
whatever one needs) into a map in Hermite normal form by
hand?

----

Standard val -> canonical val

...the standard val is just the best approximation of each
identity in the ET, right? Are there any other contenders
for canonical val?

----

TOP -> weighted minimax optimum tuning -> canonical temperament

...did Gene or Graham say there's a version of TOP equivalent
to weighted rms? And Paul, have you looked at the non-weighted
Tenney lattice?

----

Thanks,

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> TM reduction or LLL reduction -> canonical basis
>
> ...Which of TM, LLL is preferred these days,

LLL is just use to "set up" for TM, I believe. TM seems like one good
option, but there are probably better or equally good ways to define
things beyond 2 dimensions.

> and is there
> a definition of "basis" somewhere?

You should hang it on your refrigerator. Once you do, you may be able
to understand this: for the kernel of a temperament, it will be a
list of linearly independent commas that don't lead to torsion; for a
temperament, it will be a list of linearly independent intervals that
generate the whole temperament.

> ----

> Standard val -> canonical val
>
> ...the standard val is just the best approximation of each
> identity in the ET, right? Are there any other contenders
> for canonical val?

Yes.

(I'm in a hurry, my apologies)

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> You should hang it on your refrigerator. Once you do, you may be
able
> to understand this: for the kernel of a temperament, it will be a
> list of linearly independent commas that don't lead to torsion; for
a
> temperament, it will be a list of linearly independent intervals
that
> generate the whole temperament.

Do you mean "vals" and not "intervals"?

> > ----
>
> > Standard val -> canonical val
> >
> > ...the standard val is just the best approximation of each
> > identity in the ET, right? Are there any other contenders
> > for canonical val?
>
> Yes.

Now that we've got TOP, we might take the n-division val closest to
JIP, or such that its TOP tuning is closest to the JIP.

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

> Now that we've got TOP, we might take the n-division val closest to
> JIP

By which I meant, such that V/V(2) is closest to the JIP.

> or such that its TOP tuning is closest to the JIP.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > You should hang it on your refrigerator. Once you do, you may be
> able
> > to understand this: for the kernel of a temperament, it will be a
> > list of linearly independent commas that don't lead to torsion;
for
> a
> > temperament, it will be a list of linearly independent intervals
> that
> > generate the whole temperament.
>
> Do you mean "vals" and not "intervals"?

No, I mean intervals -- for example, for meantone temperament, the
list could consist of the meantone fifth and octave.

>> and is there
>> a definition of "basis" somewhere?
>
>You should hang it on your refrigerator. Once you do, you may be able
>to understand this: for the kernel of a temperament, it will be a
>list of linearly independent commas that don't lead to torsion; for a
>temperament, it will be a list of linearly independent intervals that
>generate the whole temperament.

? I can't hang it on my refrigerator if I don't have it!

>> Standard val -> canonical val
>>
>> ...the standard val is just the best approximation of each
>> identity in the ET, right? Are there any other contenders
>> for canonical val?
>
>Yes.
>
>(I'm in a hurry, my apologies)

Paul, please take your time. At your convenience, I'd love to have
your full comments on my message.

And did you see the posts where I compare zeta, gram, and TOP-et
tunings?

Thanks,

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> and is there
> >> a definition of "basis" somewhere?
> >
> >You should hang it on your refrigerator. Once you do, you may be
able
> >to understand this: for the kernel of a temperament, it will be a
> >list of linearly independent commas that don't lead to torsion;
for a
> >temperament, it will be a list of linearly independent intervals
that
> >generate the whole temperament.
>
> ? I can't hang it on my refrigerator if I don't have it!

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

> >> Standard val -> canonical val
> >>
> >> ...the standard val is just the best approximation of each
> >> identity in the ET, right? Are there any other contenders
> >> for canonical val?
> >
> >Yes.
> >
> >(I'm in a hurry, my apologies)
>
> Paul, please take your time. At your convenience, I'd love to have
> your full comments on my message.
>
> And did you see the posts where I compare zeta, gram, and TOP-et
> tunings?

Yup . . .

>> > is there a definition of "basis" somewhere?
//
>http://mathworld.wolfram.com/VectorSpaceBasis.html

Ah, good. That's what I thought.

>> >You should hang it on your refrigerator. Once you do, you may be
>> >able to understand this: for the kernel of a temperament, it will
>> >be a list of linearly independent commas that don't lead to
>> >torsion;

This is the only sense I've ever noticed it used around here, and
it's what I meant by "TM reduction -> canonical basis".

>> >for a temperament, it will be a list of linearly independent
>> >intervals that generate the whole temperament.

Generate the pitches in the temperament. One also needs the map.

>> And did you see the posts where I compare zeta, gram, and TOP-et
>> tunings?
>
>Yup . . .

I've been wondering about working backwards from the technique
to TOP for codimension > 1 temperaments. How would it apply to
a pair of vals? Which commas is it tempering in the single-val
case? etc.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> > is there a definition of "basis" somewhere?
> //
> >http://mathworld.wolfram.com/VectorSpaceBasis.html
>
> Ah, good. That's what I thought.
>
> >> >You should hang it on your refrigerator. Once you do, you may be
> >> >able to understand this: for the kernel of a temperament, it
will
> >> >be a list of linearly independent commas that don't lead to
> >> >torsion;
>
> This is the only sense I've ever noticed it used around here, and
> it's what I meant by "TM reduction -> canonical basis".
>
> >> >for a temperament, it will be a list of linearly independent
> >> >intervals that generate the whole temperament.
>
> Generate the pitches in the temperament. One also needs the map.

If it's a regular tuning, rather than a regular temperament, one
doesn't need a map.

> >> And did you see the posts where I compare zeta, gram, and TOP-et
> >> tunings?
> >
> >Yup . . .
>
> I've been wondering about working backwards from the technique
> to TOP for codimension > 1 temperaments. How would it apply to
> a pair of vals?

A pair of vals -> dimension = 2. How would what apply?

> Which commas is it tempering in the single-val
> case?

Nothing new to TOP here.

> etc.
>
> -Carl

Not sure what these questions mean, but working forwards from my
technique for ETs to dimension>1 seems possible, a linearly-
constrained minimax problem, good for linear programming . . .

>> >> And did you see the posts where I compare zeta, gram, and
>> >> TOP-et tunings?
>> >
>> >Yup . . .
>>
>> I've been wondering about working backwards from the technique
>> to TOP for codimension > 1 temperaments. How would it apply to
>> a pair of vals?
>
>A pair of vals -> dimension = 2. How would what apply?

We're looking for TOP for codimension 2, aren't we?

>> Which commas is it tempering in the single-val case?
>
>Nothing new to TOP here.

TOP is a single-comma technique last I heard. Yet ETs require
more than a single comma in the 5-limit...

Oh, and just in case these got lost...

>...did Gene or Graham say there's a version of TOP equivalent
>to weighted rms? And Paul, have you looked at the non-weighted
>Tenney lattice?

-Carl

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

> I've been wondering about working backwards from the technique
> to TOP for codimension > 1 temperaments. How would it apply to
> a pair of vals? Which commas is it tempering in the single-val
> case? etc.

A pair of vals can be used to define the space for the temperament
with two parameters, which would be a good starting point.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Not sure what these questions mean, but working forwards from my
> technique for ETs to dimension>1 seems possible, a linearly-
> constrained minimax problem, good for linear programming . . .

All this TOP stuff can be set up as a linear programming problem.

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

> TOP is a single-comma technique last I heard.

Where have you been? It applies to any number of commas.

> >...did Gene or Graham say there's a version of TOP equivalent
> >to weighted rms? And Paul, have you looked at the non-weighted
> >Tenney lattice?

I don't recall saying it, but you could do something along those
lines if you wished.

>> TOP is a single-comma technique last I heard.
>
>Where have you been? It applies to any number of commas.

So where are the TOP 7-limit linear temperaments?

>> >...did Gene or Graham say there's a version of TOP equivalent
>> >to weighted rms? And Paul, have you looked at the non-weighted
>> >Tenney lattice?
>
>I don't recall saying it, but you could do something along those
>lines if you wished.

RMS lines, or unweighted lines?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> TOP is a single-comma technique last I heard.
> >
> >Where have you been? It applies to any number of commas.
>
> So where are the TOP 7-limit linear temperaments?

/tuning-math/message/8504

>> >Where have you been? It applies to any number of commas.
>>
>> So where are the TOP 7-limit linear temperaments?
>
>/tuning-math/message/8504

Oh dear. Does anybody besides Gene understand this yet?
Last I heard Paul was trying to use heron's formula to get
around straightness.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >Where have you been? It applies to any number of commas.
> >>
> >> So where are the TOP 7-limit linear temperaments?
> >
> >/tuning-math/message/8504
>
> Oh dear. Does anybody besides Gene understand this yet?
> Last I heard Paul was trying to use heron's formula to get
> around straightness.

Right, but that was pre-TOP. Now I can at least understand how to do
these as a linear programming exercise, if no other way . . .

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> TOP is a single-comma technique last I heard.
> >
> >Where have you been? It applies to any number of commas.
>
> So where are the TOP 7-limit linear temperaments?

They are awaiting my code. I've analyzed ets and codimension one on
my web site, and the next to tackle is linear.

> >> >...did Gene or Graham say there's a version of TOP equivalent
> >> >to weighted rms? And Paul, have you looked at the non-weighted
> >> >Tenney lattice?
> >
> >I don't recall saying it, but you could do something along those
> >lines if you wished.
>
> RMS lines, or unweighted lines?

Both.

Gene,

I don't know how many people are beating down the xenharmony door
with interest, but in any case I can only thank you if you find
the time to humor me, one interested party... Thanks!

>> >> >...did Gene or Graham say there's a version of TOP equivalent
>> >> >to weighted rms? And Paul, have you looked at the non-weighted
>> >> >Tenney lattice?
>> >
>> >I don't recall saying it, but you could do something along those
>> >lines if you wished.
>>
>> RMS lines, or unweighted lines?
>
>Both.

Well if you can do unweighted TOP that gives min. rms over all
intervals, hats off to you sir.

By the way, I'm almost able to construct outlines of the stuff you
write now, which can't be a bad thing.

http://66.98.148.43/~xenharmo/top.htm

Pasting in the mathworld definition of norm is a big help (or did
I read in one of your e-mails...). Anyway, at "3." is "|c|" absolute
value of scalar c?

4. looks like the triangle inequality.

Should the first "these" on the page be changed to "as"?

"A linear functional on a real vector space is a linear mapping from
the space to the real numbers. It is like a val, but its coordinates
can be any real number."

Amazing; I'm still 100% with you!

"JIP = <1 log2(3) ... log2(p)|"

What's the "1" doing in there? Oh, it's log2(2) but since you've
simplified it we have to guess what's going on in this series.
Multiplication?

Let's see, so far we've got a...

() real (un-normed) vector space, call it "JI"
() normed vector space, call it Tenney [which is log-weighted JI]
() linear functional from JI to the reals, JIP [which sounds
like one of Paul's 'stretched' vals]

...and now [drum roll]. . .

"For any finite-dimensional normed vector space ..."

...such as Tenney?...

"... __the space of linear functionals__ is called the dual space.
__It__ ..."

What's "It"? Tenney?

Did you mean "__a__ space of linear functionals is __a__ dual space"?

" ... has a norm induced on it defined by

||f|| = sup |f(u)|/||u||, u not zero"

So now I've lost track of what f is. I'm **guessing** it's a space
defined by a basis given in JIPs, perhaps something that looks like...

< 1194.3343134713434 1910.9349015541493 2786.7800647664676 ]
< 1202.2814046729093 1898.3390600098567 2784.2306213477896 ]

What's "sup"?

"We may change basis in the Tenney space by resizing the elements,
so that the norm is now

|| |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|"

What, so now I'm guessing |v2| is the Euclidean distance along v2?
But with |c| above, I didn't think Euc. distance would be defined
for a scalar. I pray you're not mixing meanings for |x|.

"Each of the basis elements now represents
[big honkin' space]
something of the same size as 2, but that should not worry us."

Uh, ok. . . .

"It is a standard fact that the dual space to L1 is the L infinity
norm, and vice-versa."

Oh, a standard fact. I feel much better.

"We may call the dual space to the Tenney space, with this norm, the
val space. Just as monzos form a lattice in Tenney space, vals form a
lattice in val space."

I'm lost. But wait...

"A regular tuning map T is a linear functional. If c is a comma which
T tempers out, then T(c) = 0. If we have a set of commas C which are
tempered out, then this defines a subspace Null(C) of the val space,
such that for any T in Null(C), T(c)=0 for each comma tempered out."

...I understand this!

"If we have a set of vals V, this defines a subspace Span(V) of the val
space consisting of the linear combinations of the vals in V."

Well I don't know what "linear combinations" are (do you simply mean
pairwise combinations?) but I get the gyst.

"Either way we define this subspace, it corresponds to a regular
temperament."

...Either of two ways, neither of which you've mentioned.

"We may find this minimal distance, and the corresponding point, by
finding the radius where a ball around the JIP first intersects it. In
the val space, the unit ball looks like a measure polytope--which is
to say a rectangle or rectangular solid of whatever the dimension of
the space. It consists of all points v in the val space such that

|| v || <= 1

The corners of this measure polytope, one of which is the JIP, are

<+-1, +-log2(3), ..., +-log2(p)|

If n is the dimension of the Tenney space, and so of the val space,
then there are 2^n such corners."

Wow this is awesome. I understand the glaze.

"The codimension of a regular temperament is the dimension of its
kernel, or the number of linearly independent commas needed to define
it. A temperament of codimension one is defined by a single comma c.
It is a linear temperament in the 5-limit, planar in the 7-limit,
spacial in the 11-limit, and so forth."

Now here's something for monz's encyclopedia. This should have been
written years ago. Thanks, Gene!

.....

It doesn't look like you've gotten around to codimension > 1 TOPs,
unweighted TOPs, or RMS-equivalent-TOPs.

Oh, and try searching your page for "this this".

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> Gene,
>
> I don't know how many people are beating down the xenharmony door
> with interest, but in any case I can only thank you if you find
> the time to humor me, one interested party... Thanks!

I suspect there isn't much interest but I don't track it. Thanks for
the positive comment.

> Well if you can do unweighted TOP that gives min. rms over all
> intervals, hats off to you sir.

Over *all* intervals? I didn't know you were asking for that! First
thing would be to construct your all-intervals norm.

> By the way, I'm almost able to construct outlines of the stuff you
> write now, which can't be a bad thing.
>
> http://66.98.148.43/~xenharmo/top.htm
>
> Pasting in the mathworld definition of norm is a big help (or did
> I read in one of your e-mails...). Anyway, at "3." is "|c|" absolute
> value of scalar c?

It is.

> 4. looks like the triangle inequality.

No normed vector space would be complete without it.

> Should the first "these" on the page be changed to "as"?

If you mean where I defined the Tenney lattice I can see that needs
fixing.

> "A linear functional on a real vector space is a linear mapping from
> the space to the real numbers. It is like a val, but its coordinates
> can be any real number."
>
> Amazing; I'm still 100% with you!
>
> "JIP = <1 log2(3) ... log2(p)|"
>
> What's the "1" doing in there? Oh, it's log2(2) but since you've
> simplified it we have to guess what's going on in this series.
> Multiplication?
>
> Let's see, so far we've got a...
>
> () real (un-normed) vector space, call it "JI"

I didn't introduce that.

> () normed vector space, call it Tenney [which is log-weighted JI]
> () linear functional from JI to the reals, JIP [which sounds
> like one of Paul's 'stretched' vals]
>
> ...and now [drum roll]. . .
>
> "For any finite-dimensional normed vector space ..."
>
> ...such as Tenney?...
>
> "... __the space of linear functionals__ is called the dual space.
> __It__ ..."
>
> What's "It"? Tenney?

No, "it" is the dual space to our original normed vector space.

> Did you mean "__a__ space of linear functionals is __a__ dual space"?

No. There is only one dual to the given space.

> " ... has a norm induced on it defined by
>
> ||f|| = sup |f(u)|/||u||, u not zero"
>
> So now I've lost track of what f is. I'm **guessing** it's a space
> defined by a basis given in JIPs, perhaps something that looks like...

"f" is a linear functional.

> < 1194.3343134713434 1910.9349015541493 2786.7800647664676 ]
> < 1202.2814046729093 1898.3390600098567 2784.2306213477896 ]
>
> What's "sup"?

"Max" will work.

> "We may change basis in the Tenney space by resizing the elements,
> so that the norm is now
>
> || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|"
>
> What, so now I'm guessing |v2| is the Euclidean distance along v2?

I've simply changed the basis so that instead of the second coordinate
being a log2(3), it corresponds to something the same size as 2, but
in the 3 direction; so it;s of size log2(2)=1 now.

> But with |c| above, I didn't think Euc. distance would be defined
> for a scalar. I pray you're not mixing meanings for |x|.

I'm not sure what you mean by Euclidean distance, but I suspect it
isn't what I would mean.

> "Each of the basis elements now represents
> [big honkin' space]
> something of the same size as 2, but that should not worry us."
>
> Uh, ok. . . .
>
> "It is a standard fact that the dual space to L1 is the L infinity
> norm, and vice-versa."
>
> Oh, a standard fact. I feel much better.
>
> "We may call the dual space to the Tenney space, with this norm, the
> val space. Just as monzos form a lattice in Tenney space, vals form a
> lattice in val space."
>
> I'm lost. But wait...

Where did you get lost? We have vals sitting at lattice points, and a
means of measuring distance.

> "A regular tuning map T is a linear functional. If c is a comma which
> T tempers out, then T(c) = 0. If we have a set of commas C which are
> tempered out, then this defines a subspace Null(C) of the val space,
> such that for any T in Null(C), T(c)=0 for each comma tempered out."
>
> ...I understand this!
>
> "If we have a set of vals V, this defines a subspace Span(V) of the val
> space consisting of the linear combinations of the vals in V."
>
> Well I don't know what "linear combinations" are (do you simply mean
> pairwise combinations?) but I get the gyst.

If v1, v2, ..., vn are n vals, c1 v1 + c2 v2 + ... + cn vn, where the
c's are any real number (they don't need to be integers now) is an
element in the span of {v1, ..., vn}.

> "Either way we define this subspace, it corresponds to a regular
> temperament."
>
> ...Either of two ways, neither of which you've mentioned.

You quoted it--Span(V) or Null(C).

> "We may find this minimal distance, and the corresponding point, by
> finding the radius where a ball around the JIP first intersects it. In
> the val space, the unit ball looks like a measure polytope--which is
> to say a rectangle or rectangular solid of whatever the dimension of
> the space. It consists of all points v in the val space such that
>
> || v || <= 1
>
> The corners of this measure polytope, one of which is the JIP, are
>
> <+-1, +-log2(3), ..., +-log2(p)|
>
> If n is the dimension of the Tenney space, and so of the val space,
> then there are 2^n such corners."
>
> Wow this is awesome. I understand the glaze.
>
> "The codimension of a regular temperament is the dimension of its
> kernel, or the number of linearly independent commas needed to define
> it. A temperament of codimension one is defined by a single comma c.
> It is a linear temperament in the 5-limit, planar in the 7-limit,
> spacial in the 11-limit, and so forth."
>
> Now here's something for monz's encyclopedia. This should have been
> written years ago. Thanks, Gene!

Yer welcome. Your reaction is a distinct improvement on sullen resentment.

>> Well if you can do unweighted TOP that gives min. rms over all
>> intervals, hats off to you sir.
>
>Over *all* intervals? I didn't know you were asking for that!

What did you think I wanted? (Seriously; it might be interesting.)

>First thing would be to construct your all-intervals norm.

hrm...

>> By the way, I'm almost able to construct outlines of the stuff you
>> write now, which can't be a bad thing.
>>
>> http://66.98.148.43/~xenharmo/top.htm
>>
>> Pasting in the mathworld definition of norm is a big help (or did
>> I read in one of your e-mails...). Anyway, at "3." is "|c|" absolute
>> value of scalar c?
>
>It is.
>
>> 4. looks like the triangle inequality.
>
>No normed vector space would be complete without it.

Great, some of this is finally sinking in.

>> Should the first "these" on the page be changed to "as"?
>
>If you mean where I defined the Tenney lattice I can see that needs
>fixing.

Yep. Ctrl+F, by the way, is man's best friend. Don't forget
"this this".

>> Let's see, so far we've got a...
>>
>> () real (un-normed) vector space, call it "JI"
>
>I didn't introduce that.

Hrm, looks like I was wrong then and JIP is a linear functional
from Tenney to the reals (not JI to the reals).

>> "For any finite-dimensional normed vector space ..."
>>
>> ...such as Tenney?...
>>
>> "... __the space of linear functionals__ is called the dual space.
>> __It__ ..."
>>
>> What's "It"? Tenney?
>
>No, "it" is the dual space to our original normed vector space.

What does a basis for this dual space look like?
Does it have the same dimension as Tenney?

>> Did you mean "__a__ space of linear functionals is __a__ dual space"?
>
>No. There is only one dual to the given space.

Only one way to map coordinates in a space to the reals? I find
that counterintuitive but I'll take your word for it.

>> " ... has a norm induced on it defined by
>>
>> ||f|| = sup |f(u)|/||u||, u not zero"
>>
>> So now I've lost track of what f is. I'm **guessing** it's a space
>> defined by a basis given in JIPs, perhaps something that looks like...
>
>"f" is a linear functional.

You're defining a norm on f and I'm unable to imagine at the moment
what a norm on a mapping would be like. A norm on a space seems
much more intuitive.

>> What's "sup"?
>
>"Max" will work.

Check.

>> "We may change basis in the Tenney space by resizing the elements,
>> so that the norm is now
>>
>> || |v2 v3 ... vp> || = |v2| + |v3| + ... + |vp|"
>>
>> What, so now I'm guessing |v2| is the Euclidean distance along v2?
>
>I've simply changed the basis so that instead of the second coordinate
>being a log2(3), it corresponds to something the same size as 2, but
>in the 3 direction; so it;s of size log2(2)=1 now.
>
>> But with |c| above, I didn't think Euc. distance would be defined
>> for a scalar. I pray you're not mixing meanings for |x|.
>
>I'm not sure what you mean by Euclidean distance, but I suspect it
>isn't what I would mean.

If v2 is a vector, what is |v2|? What's the absolute value of a
vector?

>> "Each of the basis elements now represents
>> [big honkin' space]
>> something of the same size as 2, but that should not worry us."
>>
>> Uh, ok. . . .
>>
>> "It is a standard fact that the dual space to L1 is the L infinity
>> norm, and vice-versa."
>>
>> Oh, a standard fact. I feel much better.
>>
>> "We may call the dual space to the Tenney space, with this norm, the
>> val space. Just as monzos form a lattice in Tenney space, vals form a
>> lattice in val space."
>>
>> I'm lost. But wait...
>
>Where did you get lost? We have vals sitting at lattice points, and a
>means of measuring distance.

More is clicking now. You can measure the distance between vals.
Wild.

I think I got lost with "something the same size as 2".

>> "If we have a set of vals V, this defines a subspace Span(V) of the val
>> space consisting of the linear combinations of the vals in V."
>>
>> Well I don't know what "linear combinations" are (do you simply mean
>> pairwise combinations?) but I get the gyst.
>
>If v1, v2, ..., vn are n vals, c1 v1 + c2 v2 + ... + cn vn, where the
>c's are any real number (they don't need to be integers now) is an
>element in the span of {v1, ..., vn}.

Oh, v1 is a val now? At the top of the page v is a vector and this
seems confusing.

>> "Either way we define this subspace, it corresponds to a regular
>> temperament."
>>
>> ...Either of two ways, neither of which you've mentioned.
>
>You quoted it--Span(V) or Null(C).

Aha!

>> "We may find this minimal distance, and the corresponding point, by
>> finding the radius where a ball around the JIP first intersects it. In
>> the val space, the unit ball looks like a measure polytope--which is
>> to say a rectangle or rectangular solid of whatever the dimension of
>> the space. It consists of all points v in the val space such that
>>
>> || v || <= 1
>>
>> The corners of this measure polytope, one of which is the JIP, are
>>
>> <+-1, +-log2(3), ..., +-log2(p)|
>>
>> If n is the dimension of the Tenney space, and so of the val space,
>> then there are 2^n such corners."
>>
>> Wow this is awesome. I understand the glaze.
>>
>> "The codimension of a regular temperament is the dimension of its
>> kernel, or the number of linearly independent commas needed to define
>> it. A temperament of codimension one is defined by a single comma c.
>> It is a linear temperament in the 5-limit, planar in the 7-limit,
>> spacial in the 11-limit, and so forth."
>>
>> Now here's something for monz's encyclopedia. This should have been
>> written years ago. Thanks, Gene!
>
>Yer welcome. Your reaction is a distinct improvement on sullen
>resentment.

Am I prone to sullen resentment? I don't think I've ever resented
anything of yours. I often wish more hand-holdy treatments were
available, and I think the availability of such treatments would
generally help the cause (if there is a cause). But of course the
availability of mathematical treatments is a positive thing.

-Carl

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

> Am I prone to sullen resentment?

No, but I get tired of comlaints that I discuss tuning on the tuning
list and tuning math on the tuning-math list.

>>> Well if you can do unweighted TOP that gives min. rms over all
>>> intervals, hats off to you sir.
>>
>>Over *all* intervals? I didn't know you were asking for that!
>
>What did you think I wanted? (Seriously; it might be interesting.)
>
>>First thing would be to construct your all-intervals norm.
>
>hrm...

By "unweighted" I probably mean a norm without coefficents for
an interval's coordinates. This ruins the correspondence with
taxicab distance on the odd-limit lattice given by Paul's/Tenney's
norm, which Paul thinks has pyschoacoustic import, as on that
lattice intervals do not have unique factorizations and thus a
metric based on unit lengths is likely to fail the triangle
inequality.

In a glossy way I can see how TOP, where intervals relatively
prime to the comma(s) being tempered out unaffected, might
lead to minimax. This suggests that all intervals will be affected
by ROP (RMS-OPtimal). Maybe something akin to drawing a radius
from the origin to the interval in Euclidean space and uniformly
shrinking the sphere so defined, whereas TOP would warp the space
by shrinking it more in some dimensions than others.

Does any of this make any sense?

-Carl

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

> Well I don't know what "linear combinations" are

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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>> Well if you can do unweighted TOP that gives min. rms over all
> >>> intervals, hats off to you sir.
> >>
> >>Over *all* intervals? I didn't know you were asking for that!
> >
> >What did you think I wanted? (Seriously; it might be interesting.)
> >
> >>First thing would be to construct your all-intervals norm.
> >
> >hrm...
>
> By "unweighted" I probably mean a norm without coefficents for
> an interval's coordinates.

?

> This ruins the correspondence with
> taxicab distance on the odd-limit lattice given by Paul's/Tenney's
> norm,

Huh? Which odd-limit lattice and which norm?

> which Paul thinks has pyschoacoustic import,

I think odd-limit has import if a composer wishes to treat all octave-
related interval classes as a single entity.

> as on that
> lattice intervals do not have unique factorizations and thus a
> metric based on unit lengths is likely to fail the triangle
> inequality.

Not following.

>> Well I don't know what "linear combinations" are
>
>http://mathworld.wolfram.com/LinearCombination.html

Thanks! It's sometimes hard to tell when a piece of
language is specialized.

-Carl

>> By "unweighted" I probably mean a norm without coefficents for
>> an interval's coordinates.
>
>?

The norm on Tenney space...

|| |u2 u3 u5 ... up> || = log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|

The 'coefficients on the intervals coordinates' here are log2(2),
log2(3) etc.

>> This ruins the correspondence with
>> taxicab distance on the odd-limit lattice given by Paul's/Tenney's
>> norm,
>
>Huh? Which odd-limit lattice and which norm?

It's the same norm on a triangular lattice with a dimension for each
odd number. The taxicab distance on this lattice is log(odd-limit).
It's also the same distance as on the Tenney lattice, except perhaps
for the action of 2s in the latter (I forget the reasoning there).

>> as on that
>> lattice intervals do not have unique factorizations and thus a
>> metric based on unit lengths is likely to fail the triangle
>> inequality.
>
>Not following.

Hmm, maybe I was wrong. I was thinking stuff like ||9|| = ||3|| = 1
and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems bad
though, since the 3s are pointed in the same direction.

-Carl

>In a glossy way I can see how TOP, where intervals relatively
>prime to the comma(s) being tempered out unaffected, might
>lead to minimax. This suggests that all intervals will be affected
>by ROP (RMS-OPtimal). Maybe something akin to drawing a radius
>from the origin to the interval in Euclidean space and uniformly
>shrinking the sphere so defined, whereas TOP would warp the space
>by shrinking it more in some dimensions than others.
>
>Does any of this make any sense?

Maybe not. It's hard to see how tempering an interval with no
factors in common with any of the commas being tempered out could
help matters.

Let's visit Dave's Method for Optimally Distributing Any Comma...

http://dkeenan.com/Music/DistributingCommas.htm

"The particular kind of optimisation I'm referring to here is the one
where we want to minimise the maximum of the absolute values of the
errors of all the intervals that we care about; assuming they relate
to the comma under consideration"

...this sounds an awful lot like TOP, as I think Graham mentioned.

"The RMS (or sum of squares) error cannot be minimised by this method
but since it is a continuous function its minima may be found by
equating its partial derivatives to zero and solving."

...this sounds exactly like what Paul was telling me to do to get
unweighted RMS the 'old way'.

Here's one thing that puzzles me, though...

"The optimum distribution will occur when the comma is equally
distributed over all those factors in the longest line, with zero
errors for those in the shortest line."

That's not like TOP.

"the real proviso is, it works for ratios where the counts, of
tempered prime factors on each side of the ratio, differ by at most one.

So what do we do to make it work when they differ by more than one? We "weight" the count for the offending prime so they don't differ by more
than 1 for any of the intervals under consideration. In this case, if we
weight the count of 2's by 1/2 then 3:4 = 3:(2 * 2) will have one
tempered prime on top and none on the bottom, 5:8 (= 5:(2 * 2 * 2)) will
have 1.5 tempered primes on the top and 1 on the bottom."

What kind of weighting is this?

-Carl

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

> >> By "unweighted" I probably mean a norm without coefficents for
> >> an interval's coordinates.
> >
> >?
>
> The norm on Tenney space...
>
> || |u2 u3 u5 ... up> || = log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
>
> The 'coefficients on the intervals coordinates' here are log2(2),
> log2(3) etc.

So 'unweighted', 9 has a length of 2 but 11 has a length of 1 . . . :(

> >> This ruins the correspondence with
> >> taxicab distance on the odd-limit lattice given by
Paul's/Tenney's
> >> norm,
> >
> >Huh? Which odd-limit lattice and which norm?
>
> It's the same norm on a triangular lattice with a dimension for each
> odd number.

That's not a desirable norm.

> The taxicab distance on this lattice is log(odd-limit).

No it isn't -- try 9:5 for example.

> It's also the same distance as on the Tenney lattice, except perhaps
> for the action of 2s in the latter (I forget the reasoning there).

Try building up the reasoning from scratch.

> >> as on that
> >> lattice intervals do not have unique factorizations and thus a
> >> metric based on unit lengths is likely to fail the triangle
> >> inequality.
> >
> >Not following.
>
> Hmm, maybe I was wrong. I was thinking stuff like ||9|| = ||3|| = 1
> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems bad
> though, since the 3s are pointed in the same direction.

What lattice/metric was this about?

>> By "unweighted" I probably mean a norm without coefficents for
>> an interval's coordinates.
//
>> The norm on Tenney space...
>>
>> || |u2 u3 u5 ... up> || =
>> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
>>
>> The 'coefficients on the intervals coordinates' here are
>> log2(2), log2(3) etc.
>
> So 'unweighted', 9 has a length of 2 but 11 has a length of
> 1 . . . :(

Unless you use odd-limit.

>> >> This ruins the correspondence with taxicab distance on
>> >> the odd-limit lattice given by Paul's/Tenney's norm,
>> >
>> >Huh? Which odd-limit lattice and which norm?
>>
>> It's the same norm on a triangular lattice with a dimension
>> for each odd number.
>
> That's not a desirable norm.
>
>> The taxicab distance on this lattice is log(odd-limit).
>
> No it isn't -- try 9:5 for example.

This is what you were claiming in 1999.

""
But the basic insight is that a triangular lattice, with
Tenney-like lengths, a city-block metric, and odd axes or
wormholes, agrees with the odd limit perfectly, and so is
the best octave-invariant lattice representation (with
associated metric) for anyone as Partchian as me.
""

>> It's also the same distance as on the Tenney lattice,
>> except perhaps for the action of 2s in the latter (I
>> forget the reasoning there).
>
> Try building up the reasoning from scratch.

Here's what I was trying to remember...

"""
The reason omitting the 2-axis forces one to make the lattice
triangular is that typically many more powers of two will be
needed to bring a product of prime factors into close position
than to bring a ratio of prime factors into close position. So
the latter should be represented by a shorter distance than the
former. Simply ignoring distances along the 2-axis and sticking
with a rectangular (or Monzo) lattice is throwing away
information.
//
... a weight of log(axis) should be applied to all axes, and
if a 2-axis is included, a rectangular lattice is OK. If a
2-axis is not included, a triangular lattice is better.
//
... in an octave-specific rectangular (or parallelogram)
lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In
an octave-specific sense, 7:1 and 5:1 really are simpler than
7:5; the former are more consonant. 7:4 and 5:4 are each three
rungs in the rectangular lattice, but they still come out a little
simpler than 7:5 since the rungs along the 2-axis are so short.
If you can buy that 35:1 is as simple as 7:5, then the octave-
specific lattice really should be rectangular, not triangular.
35:1 is really difficult to compare with 7:5 -- it's much less
rough but also much harder to tune . . .
"""

>> I was thinking stuff like ||9|| = ||3|| = 1
>> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems
>> bad though, since the 3s are pointed in the same direction.
>
> What lattice/metric was this about?

Unweighted odd-limit taxicab. By "3+3" I meant adding two 3
vectors. The equation is 1 < 2. It violates...

""
The city block distance has the very important property that
if an interval arises most simply as the sum of two simpler
intervals, the metric of the first interval is the sum of the
metrics of the other two.
""

...where clearly you were impling log() weighting.

More golden oldies at:
http://lumma.org/tuning/lattice1999.txt

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >> By "unweighted" I probably mean a norm without coefficents for
> >> an interval's coordinates.
> //
> >> The norm on Tenney space...
> >>
> >> || |u2 u3 u5 ... up> || =
> >> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
> >>
> >> The 'coefficients on the intervals coordinates' here are
> >> log2(2), log2(3) etc.
> >
> > So 'unweighted', 9 has a length of 2 but 11 has a length of
> > 1 . . . :(
>
> Unless you use odd-limit.

Please elaborate on how that's 'unweighted' in your view.

> >> >> This ruins the correspondence with taxicab distance on
> >> >> the odd-limit lattice given by Paul's/Tenney's norm,
> >> >
> >> >Huh? Which odd-limit lattice and which norm?
> >>
> >> It's the same norm on a triangular lattice with a dimension
> >> for each odd number.
> >
> > That's not a desirable norm.
> >
> >> The taxicab distance on this lattice is log(odd-limit).
> >
> > No it isn't -- try 9:5 for example.
>
> This is what you were claiming in 1999.
>
> ""
> But the basic insight is that a triangular lattice, with
> Tenney-like lengths, a city-block metric, and odd axes or
> wormholes, agrees with the odd limit perfectly, and so is
> the best octave-invariant lattice representation (with
> associated metric) for anyone as Partchian as me.
> ""

Right -- you need those odd axes, which screws up uniqueness, and
thus most of how we've been approaching temperament. Kees apparently
saw the issue the same way I did, and I'm still puzzling over some
issues with his framework, but no one here seemed able to help (last
year).

> >> It's also the same distance as on the Tenney lattice,
> >> except perhaps for the action of 2s in the latter (I
> >> forget the reasoning there).
> >
> > Try building up the reasoning from scratch.
>
> Here's what I was trying to remember...

citation?

> """
> The reason omitting the 2-axis forces one to make the lattice
> triangular is that typically many more powers of two will be
> needed to bring a product of prime factors into close position
> than to bring a ratio of prime factors into close position. So
> the latter should be represented by a shorter distance than the
> former. Simply ignoring distances along the 2-axis and sticking
> with a rectangular (or Monzo) lattice is throwing away
> information.
> //
> ... a weight of log(axis) should be applied to all axes, and
> if a 2-axis is included, a rectangular lattice is OK. If a
> 2-axis is not included, a triangular lattice is better.
> //
> ... in an octave-specific rectangular (or parallelogram)
> lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In
> an octave-specific sense, 7:1 and 5:1 really are simpler than
> 7:5; the former are more consonant. 7:4 and 5:4 are each three
> rungs in the rectangular lattice, but they still come out a little
> simpler than 7:5 since the rungs along the 2-axis are so short.
> If you can buy that 35:1 is as simple as 7:5, then the octave-
> specific lattice really should be rectangular, not triangular.
> 35:1 is really difficult to compare with 7:5 -- it's much less
> rough but also much harder to tune . . .
> """
>
> >> I was thinking stuff like ||9|| = ||3|| = 1
> >> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems
> >> bad though, since the 3s are pointed in the same direction.
> >
> > What lattice/metric was this about?
>
> Unweighted odd-limit taxicab.

In which 9 has its own axis . . . so the following:

> By "3+3" I meant adding two 3
> vectors. The equation is 1 < 2.

does not apply.

>> >> By "unweighted" I probably mean a norm without coefficents for
>> >> an interval's coordinates.
>> //
>> >> The norm on Tenney space...
>> >>
>> >> || |u2 u3 u5 ... up> || =
>> >> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
>> >>
>> >> The 'coefficients on the intervals coordinates' here are
>> >> log2(2), log2(3) etc.
>> >
>> > So 'unweighted', 9 has a length of 2 but 11 has a length of
>> > 1 . . . :(
>>
>> Unless you use odd-limit.
>
>Please elaborate on how that's 'unweighted' in your view.

On a unit-length odd-limit lattice both 9 and 11 have length 1.
I'm not claiming anything necessarily good about this, note.
I am asking for comments about it however.

>> >> >> This ruins the correspondence with taxicab distance on
>> >> >> the odd-limit lattice given by Paul's/Tenney's norm,
//
>> >> It's the same norm on a triangular lattice with a dimension
>> >> for each odd number.
>> >
>> > That's not a desirable norm.

Why not?

>> >> The taxicab distance on this lattice is log(odd-limit).
>> >
>> > No it isn't -- try 9:5 for example.
>>
>> This is what you were claiming in 1999.
>>
>> ""
>> But the basic insight is that a triangular lattice, with
>> Tenney-like lengths, a city-block metric, and odd axes or
>> wormholes, agrees with the odd limit perfectly, and so is
>> the best octave-invariant lattice representation (with
>> associated metric) for anyone as Partchian as me.
>> ""
>
>Right -- you need those odd axes, which screws up uniqueness,
>and thus most of how we've been approaching temperament.

But does the metric agree with log(odd-limit) or not?
For 9:5, log(oddlimit) is log(9). If you run it through
the "norm" you get... 2log(3) + log(5). Not the same,
it seems. However if you followed the
lumma.org/stuff/latice1999.txt link, apparently Paul Hahn
did present a metric that agrees with log(odd-limit).

>> >> It's also the same distance as on the Tenney lattice,
>> >> except perhaps for the action of 2s in the latter (I
>> >> forget the reasoning there).
>> >
>> > Try building up the reasoning from scratch.
>>
>> Here's what I was trying to remember...
>
>citation?

You wrote it in 1999. I'm afraid I can't tell you anything
more specific than that.

>> """
>> The reason omitting the 2-axis forces one to make the lattice
>> triangular is that typically many more powers of two will be
>> needed to bring a product of prime factors into close position
>> than to bring a ratio of prime factors into close position. So
>> the latter should be represented by a shorter distance than the
>> former. Simply ignoring distances along the 2-axis and sticking
>> with a rectangular (or Monzo) lattice is throwing away
>> information.
>> //
>> ... a weight of log(axis) should be applied to all axes, and
>> if a 2-axis is included, a rectangular lattice is OK. If a
>> 2-axis is not included, a triangular lattice is better.
>> //
>> ... in an octave-specific rectangular (or parallelogram)
>> lattice, 7:1 and 5:1 are each one rung and 7:5 is two rungs. In
>> an octave-specific sense, 7:1 and 5:1 really are simpler than
>> 7:5; the former are more consonant. 7:4 and 5:4 are each three
>> rungs in the rectangular lattice, but they still come out a little
>> simpler than 7:5 since the rungs along the 2-axis are so short.
>> If you can buy that 35:1 is as simple as 7:5, then the octave-
>> specific lattice really should be rectangular, not triangular.
>> 35:1 is really difficult to compare with 7:5 -- it's much less
>> rough but also much harder to tune . . .
>> """

To my mind the good thing about the Tenney/rectangular approach
is that it gives log(n*d).

>> >> I was thinking stuff like ||9|| = ||3|| = 1
>> >> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems
>> >> bad though, since the 3s are pointed in the same direction.
>> >
>> > What lattice/metric was this about?
>>
>> Unweighted odd-limit taxicab.
>
>In which 9 has its own axis . . . so the following:
>
>> By "3+3" I meant adding two 3
>> vectors. The equation is 1 < 2.
>
>does not apply.

Sure it does. As you say, the 9 appears in two places. If
the metric comes out the same either way, I don't see how this
fact would "screws up uniqueness, and thus most of how we've
been approaching temperament."

When I said "ok" above, I meant it does not violate the
triangle inequality. But it does 'screw up uniqueness'.

-Carl

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

> On a unit-length odd-limit lattice both 9 and 11 have length 1.

Doesn't 9 have length 2?

>> On a unit-length odd-limit lattice both 9 and 11 have length 1.
>
>Doesn't 9 have length 2?

9 occurs in two places on an odd-limit lattice of odd-limit >= 9.
With unit lengths, it has length 1 on the 9 axis or length 2 on
the 3 axis. You can allow this, or impose log weighting which
makes them the same length on a rectangular lattice.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> On a unit-length odd-limit lattice both 9 and 11 have length 1.
> >
> >Doesn't 9 have length 2?
>
> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.

Is this "lattice" one of those goofy things people insist on calling
lattices even though they are not? What in the world do you mean?

9 is always 3^2, so it necessarily is twice as far from the origin as
3, and in the same direction. If 9 and 11 are the same length, 3 is
half that length.

>> >> On a unit-length odd-limit lattice both 9 and 11 have length 1.
>> >
>> >Doesn't 9 have length 2?
>>
>> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.
>
>Is this "lattice" one of those goofy things people insist on calling
>lattices even though they are not?

I don't know.

>What in the world do you mean?

It's a rectangular Thing with an axis for each odd number.

>9 is always 3^2, so it necessarily is twice as far from the origin as
>3, and in the same direction.

Right.

>If 9 and 11 are the same length, 3 is half that length.

There are two different routes to 9 in this Thing, with two
different lengths. Maybe you can tell me what nice properties
such a setup violates.

Only if you impose log-weighting is the above true.

-Carl

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

> There are two different routes to 9 in this Thing, with two
> different lengths. Maybe you can tell me what nice properties
> such a setup violates.

Uniqueness and linearity.

>> There are two different routes to 9 in this Thing, with two
>> different lengths. Maybe you can tell me what nice properties
>> such a setup violates.
>
>Uniqueness and linearity.

And does this interfere with having...

() a lattice
() a metric
() a norm

?

With log weighting we restore linearity, I'm guessing. Then
do we have...

() a lattice
() a metric
() a norm

?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> By "unweighted" I probably mean a norm without coefficents for
> >> >> an interval's coordinates.
> >> //
> >> >> The norm on Tenney space...
> >> >>
> >> >> || |u2 u3 u5 ... up> || =
> >> >> log2(2)|u2|+log2(3)|u3|+ ... + log2(p)|up|
> >> >>
> >> >> The 'coefficients on the intervals coordinates' here are
> >> >> log2(2), log2(3) etc.
> >> >
> >> > So 'unweighted', 9 has a length of 2 but 11 has a length of
> >> > 1 . . . :(
> >>
> >> Unless you use odd-limit.
> >
> >Please elaborate on how that's 'unweighted' in your view.
>
> On a unit-length odd-limit lattice both 9 and 11 have length 1.
> I'm not claiming anything necessarily good about this, note.
> I am asking for comments about it however.

OK . . . this, I suppose, is Paul Hahn's lattice or something?

> >> >> The taxicab distance on this lattice is log(odd-limit).
> >> >
> >> > No it isn't -- try 9:5 for example.
> >>
> >> This is what you were claiming in 1999.
> >>
> >> ""
> >> But the basic insight is that a triangular lattice, with
> >> Tenney-like lengths, a city-block metric, and odd axes or
> >> wormholes, agrees with the odd limit perfectly, and so is
> >> the best octave-invariant lattice representation (with
> >> associated metric) for anyone as Partchian as me.
> >> ""
> >
> >Right -- you need those odd axes, which screws up uniqueness,
> >and thus most of how we've been approaching temperament.
>
> But does the metric agree with log(odd-limit) or not?
> For 9:5, log(oddlimit) is log(9). If you run it through
> the "norm" you get... 2log(3) + log(5).

No, because 9 has its own axis.

> Not the same,
> it seems. However if you followed the
> lumma.org/stuff/latice1999.txt link,

The page cannot be found.

> apparently Paul Hahn
> did present a metric that agrees with log(odd-limit).

Can you re-present it?

>
> >> >> I was thinking stuff like ||9|| = ||3|| = 1
> >> >> and thus ||3+3|| < ||3|| + ||3|| but that's ok. It seems
> >> >> bad though, since the 3s are pointed in the same direction.
> >> >
> >> > What lattice/metric was this about?
> >>
> >> Unweighted odd-limit taxicab.
> >
> >In which 9 has its own axis . . . so the following:
> >
> >> By "3+3" I meant adding two 3
> >> vectors. The equation is 1 < 2.
> >
> >does not apply.
>
> Sure it does. As you say, the 9 appears in two places. If
> the metric comes out the same either way,

It doesn't -- see above.

> I don't see how this
> fact would "screws up uniqueness, and thus most of how we've
> been approaching temperament."

Even if the 'metric came out the same either way', every note would
appear in an infinite number of places in the lattice, since for
every integer n the ratio p/q can also be expressed as
p/q*3^(-2n)*9^n
.

>> On a unit-length odd-limit lattice both 9 and 11 have length 1.
>> I'm not claiming anything necessarily good about this, note.
>> I am asking for comments about it however.
>
>OK . . . this, I suppose, is Paul Hahn's lattice or something?

For his diameter measure I think it is.

>> >> >> The taxicab distance on this lattice is log(odd-limit).
>> >> >
>> >> > No it isn't -- try 9:5 for example.
>> >>
>> >> This is what you were claiming in 1999.
>> >>
>> >> ""
>> >> But the basic insight is that a triangular lattice, with
>> >> Tenney-like lengths, a city-block metric, and odd axes or
>> >> wormholes, agrees with the odd limit perfectly, and so is
>> >> the best octave-invariant lattice representation (with
>> >> associated metric) for anyone as Partchian as me.
>> >> ""
>> >
>> >Right -- you need those odd axes, which screws up uniqueness,
>> >and thus most of how we've been approaching temperament.
>>
>> But does the metric agree with log(odd-limit) or not?
>> For 9:5, log(oddlimit) is log(9). If you run it through
>> the "norm" you get... 2log(3) + log(5).
>
>No, because 9 has its own axis.

It's still different than log(odd-limit), and in fact
log(5) + log(9) = 2log(3) + log(5).

>> Not the same,
>> it seems. However if you followed the
>> lumma.org/stuff/latice1999.txt link,
>
>The page cannot be found.

Typo here; "lattice".

>> apparently Paul Hahn
>> did present a metric that agrees with log(odd-limit).
>
>Can you re-present it?

See the above url.

>> I don't see how this
>> fact would "screws up uniqueness, and thus most of how we've
>> been approaching temperament."
>
>Even if the 'metric came out the same either way', every note would
>appear in an infinite number of places in the lattice, since for
>every integer n the ratio p/q can also be expressed as
>p/q*3^(-2n)*9^n
>.

Hmm.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> On a unit-length odd-limit lattice both 9 and 11 have length 1.
> > >
> > >Doesn't 9 have length 2?
> >
> > 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.
>
> Is this "lattice" one of those goofy things people insist on calling
> lattices even though they are not? What in the world do you mean?
>
> 9 is always 3^2, so it necessarily is twice as far from the origin
as
> 3, and in the same direction. If 9 and 11 are the same length, 3 is
> half that length.

What if you're dealing with a world where all sine wave components
are constrained to be in 768-equal? And '9' is not the same as '3^2'?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> On a unit-length odd-limit lattice both 9 and 11 have length
1.
> >> >
> >> >Doesn't 9 have length 2?
> >>
> >> 9 occurs in two places on an odd-limit lattice of odd-limit >= 9.
> >
> >Is this "lattice" one of those goofy things people insist on
calling
> >lattices even though they are not?
>
> I don't know.
>
> >What in the world do you mean?
>
> It's a rectangular Thing with an axis for each odd number.

It's actually triangular, and is what Erv Wilson uses to map out his
CPSs.

>> It's a rectangular Thing with an axis for each odd number.
>
>It's actually triangular, and is what Erv Wilson uses to map out his
>CPSs.

The Thing I was referring to here was most certainly rectangular.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> On a unit-length odd-limit lattice both 9 and 11 have length 1.
> >> I'm not claiming anything necessarily good about this, note.
> >> I am asking for comments about it however.
> >
> >OK . . . this, I suppose, is Paul Hahn's lattice or something?
>
> For his diameter measure I think it is.
>
> >> >> >> The taxicab distance on this lattice is log(odd-limit).
> >> >> >
> >> >> > No it isn't -- try 9:5 for example.
> >> >>
> >> >> This is what you were claiming in 1999.
> >> >>
> >> >> ""
> >> >> But the basic insight is that a triangular lattice, with
> >> >> Tenney-like lengths, a city-block metric, and odd axes or
> >> >> wormholes, agrees with the odd limit perfectly, and so is
> >> >> the best octave-invariant lattice representation (with
> >> >> associated metric) for anyone as Partchian as me.
> >> >> ""
> >> >
> >> >Right -- you need those odd axes, which screws up uniqueness,
> >> >and thus most of how we've been approaching temperament.
> >>
> >> But does the metric agree with log(odd-limit) or not?
> >> For 9:5, log(oddlimit) is log(9). If you run it through
> >> the "norm" you get... 2log(3) + log(5).
> >
> >No, because 9 has its own axis.
>
> It's still different than log(odd-limit), and in fact
> log(5) + log(9) = 2log(3) + log(5).

You're forgetting that 5:3 has its own rung in this lattice, with
length log(5), since the 'odd-limit' of 5:3 is 5 (more correctly, 5:3
is a ratio of 5).

>
> >> Not the same,
> >> it seems. However if you followed the
> >> lumma.org/stuff/latice1999.txt link,
> >
> >The page cannot be found.
>
> Typo here; "lattice".

The page cannot be found.

> >> apparently Paul Hahn
> >> did present a metric that agrees with log(odd-limit).
> >
> >Can you re-present it?
>
> See the above url.

The page still cannot be found.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> It's a rectangular Thing with an axis for each odd number.
> >
> >It's actually triangular, and is what Erv Wilson uses to map out
his
> >CPSs.
>
> The Thing I was referring to here was most certainly rectangular.
>
> -Carl

Well then it's no Thing that I've ever thought about or talked about
or heard of before!

>> The Thing I was referring to here was most certainly rectangular.
>>
>> -Carl
>
>Well then it's no Thing that I've ever thought about or talked about
>or heard of before!

This was a different thing from our thread.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> The Thing I was referring to here was most certainly rectangular.
> >>
> >> -Carl
> >
> >Well then it's no Thing that I've ever thought about or talked
about
> >or heard of before!
>
> This was a different thing from our thread.

You were talking about odd-limit thing:

/tuning-math/message/8662

When and where did you switch to a rectangular thing?

>> >> >> But the basic insight is that a triangular lattice, with
>> >> >> Tenney-like lengths, a city-block metric, and odd axes or
>> >> >> wormholes, agrees with the odd limit perfectly, and so is
>> >> >> the best octave-invariant lattice representation (with
>> >> >> associated metric) for anyone as Partchian as me.
>> >> >
>> >> >Right -- you need those odd axes, which screws up uniqueness,
>> >> >and thus most of how we've been approaching temperament.
>> >>
>> >> But does the metric agree with log(odd-limit) or not?
>> >> For 9:5, log(oddlimit) is log(9). If you run it through
>> >> the "norm" you get... 2log(3) + log(5).
>> >
>> >No, because 9 has its own axis.
>>
>> It's still different than log(odd-limit), and in fact
>> log(5) + log(9) = 2log(3) + log(5).
>
>You're forgetting that 5:3 has its own rung in this lattice, with
>length log(5), since the 'odd-limit' of 5:3 is 5 (more correctly, 5:3
>is a ratio of 5).

I guess so. Can you demonstrate how to get length log(9) out
of 9/5?

>> >> Not the same,
>> >> it seems. However if you followed the
>> >> lumma.org/stuff/latice1999.txt link,
>> >
>> >The page cannot be found.
>>
>> Typo here; "lattice".
>
>The page cannot be found.

Geez, I'm so sorry, it's

http://lumma.org/tuning/lattice1999.txt

-Carl

>> This was a different thing from our thread.
>
>You were talking about odd-limit thing:
>
>/tuning-math/message/8662
>
>When and where did you switch to a rectangular thing?

Let's start over.

I'm fishing for something we can use to weed down the
number of "lattices" we're interested in. Am I correct
that you think log(odd-limit) is the best octave-equivalent
concordance heuristic, and that it constitutes a norm
on the triangular odd-limit lattice with log weighting?
Am I correct that you believe log(n*d) is the best
octave-specific concordance heuristic and that it
constitutes a norm on the Tenney lattice?

The two obvious variations are rectangular odd-limit
and triangular octave-specific. What say you about those?

Finally, for each of these four lattice types, we can
inquire about what happens when we use no weighting,
("unit lengths").

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> >> But the basic insight is that a triangular lattice, with
> >> >> >> Tenney-like lengths, a city-block metric, and odd axes or
> >> >> >> wormholes, agrees with the odd limit perfectly, and so is
> >> >> >> the best octave-invariant lattice representation (with
> >> >> >> associated metric) for anyone as Partchian as me.
> >> >> >
> >> >> >Right -- you need those odd axes, which screws up uniqueness,
> >> >> >and thus most of how we've been approaching temperament.
> >> >>
> >> >> But does the metric agree with log(odd-limit) or not?
> >> >> For 9:5, log(oddlimit) is log(9). If you run it through
> >> >> the "norm" you get... 2log(3) + log(5).
> >> >
> >> >No, because 9 has its own axis.
> >>
> >> It's still different than log(odd-limit), and in fact
> >> log(5) + log(9) = 2log(3) + log(5).
> >
> >You're forgetting that 5:3 has its own rung in this lattice, with
> >length log(5), since the 'odd-limit' of 5:3 is 5 (more correctly,
5:3
> >is a ratio of 5).
>
> I guess so. Can you demonstrate how to get length log(9) out
> of 9/5?

9/5 is a ratio of 9.

> >> >> Not the same,
> >> >> it seems. However if you followed the
> >> >> lumma.org/stuff/latice1999.txt link,
> >> >
> >> >The page cannot be found.
> >>
> >> Typo here; "lattice".
> >
> >The page cannot be found.
>
> Geez, I'm so sorry, it's
>
> http://lumma.org/tuning/lattice1999.txt

OK, which part were we talking about?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> This was a different thing from our thread.
> >
> >You were talking about odd-limit thing:
> >
> >/tuning-math/message/8662
> >
> >When and where did you switch to a rectangular thing?
>
> Let's start over.
>
> I'm fishing for something we can use to weed down the
> number of "lattices" we're interested in. Am I correct
> that you think log(odd-limit) is the best octave-equivalent
> concordance heuristic,

That or something very similar to it, like perhaps
log(2*odd-limit - 1)
or
log(2*odd-limit + 1)
etc.

> and that it constitutes a norm
> on the triangular odd-limit lattice with log weighting?

Technically, it can't, because you don't have uniqueness, etc.

> Am I correct that you believe log(n*d) is the best
> octave-specific concordance heuristic and that it
> constitutes a norm on the Tenney lattice?

Yes.

> The two obvious variations are rectangular odd-limit

How can odd-limit be rectangular? Makes no sense to me.

> and triangular octave-specific.

Then the metric is not log(n*d) anymore.

>> Can you demonstrate how to get length log(9) out
>> of 9/5?
>
>9/5 is a ratio of 9.

I meant on the lattice.

>> http://lumma.org/tuning/lattice1999.txt
>
>OK, which part were we talking about?

You were looking for Paul Hahn's algorithm, which is
like the 2nd or 3rd message in there. It isn't that
long in any case.

-Carl

>> The two obvious variations are rectangular odd-limit
>
>How can odd-limit be rectangular? Makes no sense to me.

One can certainly have a rectangular lattice with a 9-axis.

>> and triangular octave-specific.
>
>Then the metric is not log(n*d) anymore.

We actually haven't specified how to find the lengths of
rungs like 9:5...

-Carl

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

> >> Can you demonstrate how to get length log(9) out
> >> of 9/5?
> >
> >9/5 is a ratio of 9.
>
> I meant on the lattice.

Yes, that's how this 'lattice' is defined, isn't it?

> >> http://lumma.org/tuning/lattice1999.txt
> >
> >OK, which part were we talking about?
>
> You were looking for Paul Hahn's algorithm, which is
> like the 2nd or 3rd message in there. It isn't that
> long in any case.

OK -- that's the algorithm when each consonance in a given odd-limit
is given a rung of length 1. So going back to the above, if the given
odd-limit is less than 9, 9/5 will have to be constructed out of 3
and 3/5, thus has a length of 2, per Paul Hahn's lattice. No logs get
involved there.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> The two obvious variations are rectangular odd-limit
> >
> >How can odd-limit be rectangular? Makes no sense to me.
>
> One can certainly have a rectangular lattice with a 9-axis.

A 'lattice'-like thing, yes. But then it has nothing to do with odd-
limit. And is there a 2-axis too?

> >> and triangular octave-specific.
> >
> >Then the metric is not log(n*d) anymore.
>
> We actually haven't specified how to find the lengths of
> rungs like 9:5...

True, but if you use something different from what Tenney gives,
you'll be hard pressed to get all the consonant intervals within a
given range (say, 260-500 cents) in the correct order of consonance.

>> >> Can you demonstrate how to get length log(9) out
>> >> of 9/5?
>> >
>> >9/5 is a ratio of 9.
>>
>> I meant on the lattice.
>
>Yes, that's how this 'lattice' is defined, isn't it?

I was asking for any way it could be defined to make it
equal odd-limit, but this seems like cheating because
you require odd-limit infinity, and thus you're never
taking any multi-stop routes.

>> >> http://lumma.org/tuning/lattice1999.txt
>> >
>> >OK, which part were we talking about?
>>
>> You were looking for Paul Hahn's algorithm, which is
>> like the 2nd or 3rd message in there. It isn't that
>> long in any case.
>
>OK -- that's the algorithm when each consonance in a given
>odd-limit is given a rung of length 1.

Right.

>So going back to the above, if the given
>odd-limit is less than 9, 9/5 will have to be constructed out of 3
>and 3/5, thus has a length of 2, per Paul Hahn's lattice. No logs
>get involved there.

Right. It's easy. But it doesn't correspond to the "ratio-of"
the ratio. My point, if any, is that I think this will be impossible
with odd-limit < inf. on a triangular lattice.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> Can you demonstrate how to get length log(9) out
> >> >> of 9/5?
> >> >
> >> >9/5 is a ratio of 9.
> >>
> >> I meant on the lattice.
> >
> >Yes, that's how this 'lattice' is defined, isn't it?
>
> I was asking for any way it could be defined to make it
> equal odd-limit, but this seems like cheating because
> you require odd-limit infinity, and thus you're never
> taking any multi-stop routes.

OK -- but without 'cheating', how can one do in the octave-equivalent
case what Tenney does in the octave-specific case?

> >> >> http://lumma.org/tuning/lattice1999.txt
> >> >
> >> >OK, which part were we talking about?
> >>
> >> You were looking for Paul Hahn's algorithm, which is
> >> like the 2nd or 3rd message in there. It isn't that
> >> long in any case.
> >
> >OK -- that's the algorithm when each consonance in a given
> >odd-limit is given a rung of length 1.
>
> Right.
>
> >So going back to the above, if the given
> >odd-limit is less than 9, 9/5 will have to be constructed out of 3
> >and 3/5, thus has a length of 2, per Paul Hahn's lattice. No logs
> >get involved there.
>
> Right. It's easy. But it doesn't correspond to the "ratio-of"
> the ratio.

Right -- no logs, so no log(odd-limit) or log("ratio-of").

> My point, if any, is that I think this will be impossible
> with odd-limit < inf. on a triangular lattice.

Well, that's exactly what this:

http://www.kees.cc/tuning/lat_perbl.html

was attempting to address, at least for a prime limit of 5.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> > I'm fishing for something we can use to weed down the
> > number of "lattices" we're interested in. Am I correct
> > that you think log(odd-limit) is the best octave-equivalent
> > concordance heuristic,
>
> That or something very similar to it, like perhaps
> log(2*odd-limit - 1)
> or
> log(2*odd-limit + 1)
> etc.
>
> > and that it constitutes a norm
> > on the triangular odd-limit lattice with log weighting?
>
> Technically, it can't, because you don't have uniqueness, etc.

Here's something you might try. Take every consonance in a given odd
limit, expressed as a monzo. Multiply this by the log of the odd limit
of that consonance. In this way, get a collection of points, and take
the convex hull. This gives a convex solid containing the origin, and
therefore defines a metric in the usual way, where the usual way is to
call this the ball of radius one, and then find the norm of a point by
scaling the ball so that the point is on the boundry; the scale factor
is the norm.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > > I'm fishing for something we can use to weed down the
> > > number of "lattices" we're interested in. Am I correct
> > > that you think log(odd-limit) is the best octave-equivalent
> > > concordance heuristic,
> >
> > That or something very similar to it, like perhaps
> > log(2*odd-limit - 1)
> > or
> > log(2*odd-limit + 1)
> > etc.
> >
> > > and that it constitutes a norm
> > > on the triangular odd-limit lattice with log weighting?
> >
> > Technically, it can't, because you don't have uniqueness, etc.
>
> Here's something you might try. Take every consonance in a given odd
> limit, expressed as a monzo. Multiply this by the log of the odd
limit
> of that consonance. In this way, get a collection of points, and
take
> the convex hull. This gives a convex solid containing the origin,
and
> therefore defines a metric in the usual way, where the usual way is
to
> call this the ball of radius one, and then find the norm of a point
by
> scaling the ball so that the point is on the boundry; the scale
factor
> is the norm.

I'm interested in this approach.

Also (NB Carl), these alternatives to log(odd-limit) don't work:

> > log(2*odd-limit - 1)
> > or
> > log(2*odd-limit + 1)

since the length for 1-limit ratios must be log(1)=0 on the lattice.
But I still wonder whether anything else might make sense here.

>> >> >> Can you demonstrate how to get length log(9) out
>> >> >> of 9/5?
>> >> >
>> >> >9/5 is a ratio of 9.
>> >>
>> >> I meant on the lattice.
>> >
>> >Yes, that's how this 'lattice' is defined, isn't it?
>>
>> I was asking for any way it could be defined to make it
>> equal odd-limit, but this seems like cheating because
>> you require odd-limit infinity, and thus you're never
>> taking any multi-stop routes.
>
>OK -- but without 'cheating', how can one do in the octave-equivalent
>case what Tenney does in the octave-specific case?

My question exactly.

>> My point, if any, is that I think this will be impossible
>> with odd-limit < inf. on a triangular lattice.
>
>Well, that's exactly what this:
>
>http://www.kees.cc/tuning/lat_perbl.html
>
>was attempting to address, at least for a prime limit of 5.

Hmm...

-Carl

>> >> The two obvious variations are rectangular odd-limit
>> >
>> >How can odd-limit be rectangular? Makes no sense to me.
>>
>> One can certainly have a rectangular lattice with a 9-axis.
>
>A 'lattice'-like thing, yes.

If we're going to be going over to the mathematical definition
of lattice, we should come up with a term that means "anything
with rungs".

>But then it has nothing to do with odd-limit. And is there a
>2-axis too?

What would happen either way?

>> >> and triangular octave-specific.
>> >
>> >Then the metric is not log(n*d) anymore.
>>
>> We actually haven't specified how to find the lengths of
>> rungs like 9:5...
>
>True, but if you use something different from what Tenney gives,
>you'll be hard pressed to get all the consonant intervals within a
>given range (say, 260-500 cents) in the correct order of consonance.

So summing up, can we say that we're happy with our
octave-specific concordance heuristic and associated
lattice/metric, and that we have an octave-equivalent
concordance heuristic but *no* associated lattice/metric?

On the other hand, given Gene's recent post, "we" might not
include him...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> The two obvious variations are rectangular odd-limit
> >> >
> >> >How can odd-limit be rectangular? Makes no sense to me.
> >>
> >> One can certainly have a rectangular lattice with a 9-axis.
> >
> >A 'lattice'-like thing, yes.
>
> If we're going to be going over to the mathematical definition
> of lattice, we should come up with a term that means "anything
> with rungs".

A graph (as in graph-theory) but with lengths for each rung?

> >But then it has nothing to do with odd-limit. And is there a
> >2-axis too?
>
> What would happen either way?

If there is a 2-axis, a 9-axis in rectangular lattice seems
superfluous (it doesn't change anything in terms of the taxicab
distances you get, but adds an infinite number of copies of each
pitch), unless you have a reason for treating '9' as different
from '3*3' (and therefore '9/3' different than '3'), etc., such as a
constraint to 768-equal partials.

If there is no 2-axis, you get bad consonance evaluation, for the
usual reasons.

> So summing up, can we say that we're happy with our
> octave-specific concordance heuristic and associated
> lattice/metric, and that we have an octave-equivalent
> concordance heuristic but *no* associated lattice/metric?

I'd prefer not to say 'concordance heuristic', but yes.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> I'm interested in this approach.

I just found out Maple's convex hull finder only works in two
dimensions, which seems terribly lame. Maybe Mathematica or Matlab can
do better?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > I'm interested in this approach.
>
> I just found out Maple's convex hull finder only works in two
> dimensions, which seems terribly lame. Maybe Mathematica or Matlab
can
> do better?

Yes:

CONVHULLN N-D Convex hull.
K = CONVHULLN(X) returns the indices K of the points in X that
comprise the facets of the convex hull of X. X is an m-by-n array
representing m points in n-D space. If the convex hull has p
facets
then K is p-by-n.

[K,V] = CONVHULLN(X) also returns the volume of the convex hull
in V.

CONVHULLN is based on Qhull.

See also CONVHULL, QHULL, DELAUNAYN, VORONOIN, TSEARCHN, DSEARCHN.

QHULL Copyright information for Qhull.

%%%% Qhull Copyright information copied from: %%%%%%%%%%%%%%%%%%
%%%% http://www.geom.umn.edu/software/download/COPYING.html %%%%

Copyright (c) 1993

The National Science and Technology Research Center for
Computation and Visualization of Geometric Structures
(The Geometry Center)
University of Minnesota
400 Lind Hall
207 Church St. SE
Minneapolis, MN 55454 USA

email: software@geom.umn.edu

The software distributed here is copyrighted as noted above. It
is free
software and may be obtained via anonymous ftp from
ftp.geom.umn.edu. It
may be freely copied, modified, and redistributed under the
following
conditions:

1. All copyright notices must remain intact in all files.

2. A copy of this file (COPYING) must be distributed along with
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that you redistribute; this includes copies that you have
modified, or
copies of programs or other software products that include this
software.

3. If you modify this software, you must include a notice giving
the name
of the person performing the modification, the date of
modification,
and the reason for such modification.

4. When distributing modified versions of this software, or other
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products that include this software, you must provide notice
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>> If we're going to be going over to the mathematical definition
>> of lattice, we should come up with a term that means "anything
>> with rungs".
>
>A graph (as in graph-theory) but with lengths for each rung?

That could be a "directed graph" I think. But all the flavors
of graph I'm aware of lack orientation, fixed dimensionality,
and so forth. Maybe "space" would work here?

>> >But then it has nothing to do with odd-limit. And is there a
>> >2-axis too?
>>
>> What would happen either way?
>
>If there is a 2-axis, a 9-axis in rectangular lattice seems
>superfluous (it doesn't change anything in terms of the taxicab
>distances you get, but adds an infinite number of copies of each
>pitch), unless you have a reason for treating '9' as different
>from '3*3' (and therefore '9/3' different than '3'), etc., such
>as a constraint to 768-equal partials.
>
>If there is no 2-axis, you get bad consonance evaluation, for the
>usual reasons.

So rectangular must have 2, and triangular probably shouldn't...

>> So summing up, can we say that we're happy with our
>> octave-specific concordance heuristic and associated
>> lattice/metric, and that we have an octave-equivalent
>> concordance heuristic but *no* associated lattice/metric?
>
>I'd prefer not to say 'concordance heuristic', but yes.

What would you say?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> If we're going to be going over to the mathematical definition
> >> of lattice, we should come up with a term that means "anything
> >> with rungs".
> >
> >A graph (as in graph-theory) but with lengths for each rung?
>
> That could be a "directed graph" I think.

Directed means each rung has a specific beginning point and ending
point.

> But all the flavors
> of graph I'm aware of lack orientation, fixed dimensionality,
> and so forth. Maybe "space" would work here?

A space has an infinite number of points between any two points.

> >> So summing up, can we say that we're happy with our
> >> octave-specific concordance heuristic and associated
> >> lattice/metric, and that we have an octave-equivalent
> >> concordance heuristic but *no* associated lattice/metric?
> >
> >I'd prefer not to say 'concordance heuristic', but yes.
>
> What would you say?

concordance function?

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> If we're going to be going over to the mathematical definition
> >> of lattice, we should come up with a term that means "anything
> >> with rungs".
> >
> >A graph (as in graph-theory) but with lengths for each rung?
>
> That could be a "directed graph" I think.

A directed graph can have both ways or one way streets between the
nodes. A multigraph allows for multiplicity in the connection, which
is a little like a shorter length. Nodes connected by lines at various
distances sounds most like a polytope.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > >> If we're going to be going over to the mathematical definition
> > >> of lattice, we should come up with a term that means "anything
> > >> with rungs".
> > >
> > >A graph (as in graph-theory) but with lengths for each rung?
> >
> > That could be a "directed graph" I think.
>
> A directed graph can have both ways or one way streets between the
> nodes. A multigraph allows for multiplicity in the connection, which
> is a little like a shorter length. Nodes connected by lines at
various
> distances sounds most like a polytope.

?

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

>> >A graph (as in graph-theory) but with lengths for each rung?
>>
>> That could be a "directed graph" I think.
>
>Directed means each rung has a specific beginning point and ending
>point.

Whoops, got my mix crossed up. I meant "network".

>> But all the flavors
>> of graph I'm aware of lack orientation, fixed dimensionality,
>> and so forth. Maybe "space" would work here?
>
>A space has an infinite number of points between any two points.

Crap.

>> >> So summing up, can we say that we're happy with our
>> >> octave-specific concordance heuristic and associated
>> >> lattice/metric, and that we have an octave-equivalent
>> >> concordance heuristic but *no* associated lattice/metric?
>> >
>> >I'd prefer not to say 'concordance heuristic', but yes.
>>
>> What would you say?
>
>concordance function?

Ok.

-Carl