Tuning-Math Digests messages 9126 - 9150

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Message: 9126

Date: Tue, 13 Jan 2004 01:14:55

Subject: Re: The Exotemperaments in the dual representation

From: Carl Lumma

Yahoo groups: /tuning_files/files/Erlich/dualxoom.gif *

Beautiful!

-C.


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Message: 9127

Date: Tue, 13 Jan 2004 09:27:40

Subject: Two questions for Gene

From: Paul Erlich

1. When you're done enumerating 12-tone blocks, how about looking at 
5-tone? There are some nice japanese scales that are pelograndpa :)

2. Can you explain, in as non-technical a manner as possible, the 
proof that there are 21.5 :) commas in the 5-limit with epimericity < 
1/2?

.5:)  (1/1) - undefined
1.5:)  2/1 - exo
2.5:)  3/2 - exo
3.5:)  4/3 - exo
4.5:)  5/4 - exo
5.5:)  6/5 - exo
6.5:)  9/8 - exo
7.5:)  10/9 - exo
8.5:)  16/15 - ?
9.5:)  25/24 - ?
10.5:)  27/25 - ?
11.5:)  32/27 - exo
12.5:)  81/80
13.5:)  128/125
14.5:)  135/128 - ?
15.5:)  250/243
16.5:)  256/243
17.5:)  648/625
18.5:)  2048/2025
19.5:)  3125/3072
20.5:)  15625/15552
21.5:)  32768/32805

I think for 5-limit linear we can just cover these in a few pages, 
and separately make brief mention of anything else of "historical 
importance" (531441/524288 and probably that's it?) . . .


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Message: 9128

Date: Tue, 13 Jan 2004 10:34:22

Subject: Question for Dave Keenan

From: Paul Erlich

If a timbre has

2nd partial off by < 10.4 cents
3rd partial off by < 16.5 cents
4th partial off by < 20.8 cents
5th partial off by < 24.1 cents
6th partial off by < 26.9 cents

does it 'hold together' as a single pitch, or does it fall apart into 
multiple pitches?

(I'll try to prepare some examples, playing random scales . . .)

If yes:

If I take any inharmonic timbre with one loud partial and some quiet, 
unimportant ones (very many fall into this category), and use a 
tuning system where

2:1 off by < 10.4 cents
3:1 off by < 16.5 cents
4:1 off by < 20.8 cents
5:1 off by < 24.1 cents
6:1 off by < 26.9 cents

and play a piece with full triadic harmony, doesn't it follow that 
the harmony should 'hold together' the way 5-limit triads should?


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Message: 9130

Date: Tue, 13 Jan 2004 11:05:16

Subject: 7-limit, epi

From: Paul Erlich

I find sixty with n*d<10^18, but maybe there are some more complex 
ones? How  do we know how many?

       16875       16807
        2430        2401
        1728        1715
         360         343
        3125        3087
      703125      702464
        6144        6125
       19683       19600
         256         245
          54          49
         405         392
          50          49
        4000        3969
    78125000    78121827
          36          35
        5120        5103
          15          14
         200         189
          25          21
         225         224
         648         625
         128         125
        2048        2025
          27          25
          16          15
           6           5
          81          80
         256         243
          32          27
           4           3
(          1           1 )
           9           8
          10           9
           5           4
         135         128
       32805       32768
          25          24
         250         243
        3125        3072
       15625       15552
         126         125
           7           5
          21          20
          28          27
           7           6
          35          32
         525         512
         875         864
        4375        4374
       65625       65536
        3136        3125
         392         375
          49          45
          49          48
         245         243
        1029        1000
         686         675
       10976       10935
        1029        1024
        2401        2400


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Message: 9131

Date: Tue, 13 Jan 2004 03:15:26

Subject: Re: summary -- are these right?

From: Carl Lumma

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.

/root/tentop.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


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Message: 9133

Date: Tue, 13 Jan 2004 05:25:30

Subject: Re: summary -- are these right?

From: Carl Lumma

>> 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.
>> 
>> /root/tentop.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


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Message: 9137

Date: Wed, 14 Jan 2004 03:16:55

Subject: Re: summary -- are these right?

From: Carl Lumma

>>> 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


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Message: 9138

Date: Wed, 14 Jan 2004 12:26:35

Subject: Re: summary -- are these right?

From: Carl Lumma

>> Well I don't know what "linear combinations" are
>
>Linear Combination -- from MathWorld *

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

-Carl


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Message: 9139

Date: Wed, 14 Jan 2004 12:55:08

Subject: Re: summary -- are these right?

From: Carl Lumma

>> 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


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Message: 9140

Date: Wed, 14 Jan 2004 16:51:53

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

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

Linear Combination -- from MathWorld *


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Message: 9141

Date: Wed, 14 Jan 2004 16:55:22

Subject: Re: TOP on the web

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@u...> wrote:

> "Tenney complexity"? Is it 
> anything like Graham complexity? 

For a commatic unison vector n/d, the Tenney Complexity is log(n*d). 
I think Tenney used base-2 logs.


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Message: 9143

Date: Wed, 14 Jan 2004 16:59:42

Subject: Re: summary -- are these right?

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx 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.


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Message: 9145

Date: Wed, 14 Jan 2004 13:59:27

Subject: Dave's Way (was Re: Re: summary -- are these right?)

From: Carl Lumma

>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...

A method for optimally distributing any comma *

"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


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Message: 9147

Date: Wed, 14 Jan 2004 19:36:02

Subject: Re: Two questions for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> > 2. Can you explain, in as non-technical a manner as possible, the 
> > proof that there are 21.5 :) commas in the 5-limit with 
epimericity 
> < 
> > 1/2?
> 
> I'm afraid I don't have a proof, and proving it would probably be 
> difficult. The proof is a proof that the number of p-limit commas 
> with epimericity e < 1 is finite. In practice, you get to a point 
> where it seems obvious no futher commas are going to show up; if 
they 
> did, it would be very odd from a number theoretic point of view.

Hmm . . . that severely reduces the appeal of using epimericity to 
define our badness contour. In this case, we should probably use 
something that crosses zero error at some finite complexity.


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Message: 9149

Date: Wed, 14 Jan 2004 19:38:02

Subject: Re: Two questions for Gene

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> The proof is a proof that the number of p-limit commas 
> with epimericity e < 1 is finite.

There are an infinite number of commas with e < 1, but for any 
positive d, there are a finite number of commas with e < 1-d -- 
correct?



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