TOP on the web

I've put up a TOP web page. It needs to have, at least, a discussion
of equal and linear temperaments and Tenney complexity and badness
added to it, but it should be valuable as a starter. Here it is:

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

Paul, could you tell me what you want attributed to you?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've put up a TOP web page. It needs to have, at least, a
discussion
> of equal and linear temperaments and Tenney complexity and badness
> added to it, but it should be valuable as a starter. Here it is:
>
> http://66.98.148.43/~xenharmo/top.htm
>
> Paul, could you tell me what you want attributed to you?

My eyes glazed over. I'll have to look at this again some other time.

I came up with the idea of tempering uniformly by length, and then
with the observation that this minimizes maximum weighted error over
all intervals. Strictly codimension one. I also came up with a way to
minimize maximum weighted error over all intervals, for dimension one
(ETs).

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

> I came up with the idea of tempering uniformly by length, and then
> with the observation that this minimizes maximum weighted error
over
> all intervals. Strictly codimension one.

OK.

I also came up with a way to
> minimize maximum weighted error over all intervals, for dimension
one
> (ETs).

I think several people were doing ets; I know I was.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > I came up with the idea of tempering uniformly by length, and
then
> > with the observation that this minimizes maximum weighted error
> over
> > all intervals. Strictly codimension one.
>
> OK.
>
> I also came up with a way to
> > minimize maximum weighted error over all intervals, for dimension
> one
> > (ETs).
>
> I think several people were doing ets; I know I was.

Yes, you and Graham both, but from what I could tell, you were both
using more complicated methods. Though mine was fairly obvious and
I'm sure either of you could have come up with it anyway.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've put up a TOP web page. It needs to have, at least, a
discussion
> of equal and linear temperaments and Tenney complexity and badness
> added to it, but it should be valuable as a starter. Here it is:
>
> http://66.98.148.43/~xenharmo/top.htm

"Because of the transcendence and linear independence of the logs of
odd primes, the coordinates of TOP(S) can never be the same as any of
the coordinates of JIP. TOP(S) therefore retunes every rational
number by some amount, and this includes octaves; hence the
alternative acronym of Tempered Octaves, Please for TOP."

Something must be wrong with either the premise or the inference,
since, for example, Top Beep has pure octaves.

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

> "Because of the transcendence and linear independence of the logs of
> odd primes, the coordinates of TOP(S) can never be the same as any of
> the coordinates of JIP. TOP(S) therefore retunes every rational
> number by some amount, and this includes octaves; hence the
> alternative acronym of Tempered Octaves, Please for TOP."
>
> Something must be wrong with either the premise or the inference,
> since, for example, Top Beep has pure octaves.

I'm getting this for TOP(27/25):

[1214.176 1879.486 2819.230]

What are you getting?

Here are some retunings: 2-->2.016 3/2-->1.469 4/3-->1.373
5/4-->1.253 5/3-->1.721

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

> "Because of the transcendence and linear independence of the logs of
> odd primes, the coordinates of TOP(S) can never be the same as any of
> the coordinates of JIP. TOP(S) therefore retunes every rational
> number by some amount, and this includes octaves; hence the
> alternative acronym of Tempered Octaves, Please for TOP."
>
> Something must be wrong with either the premise or the inference,
> since, for example, Top Beep has pure octaves.

I was implicity assuming all primes were in the facorization of the
comma. This only works for primes which appear in that factorization,
and 2 does not appear in 27/25.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > "Because of the transcendence and linear independence of the logs
of
> > odd primes, the coordinates of TOP(S) can never be the same as
any of
> > the coordinates of JIP. TOP(S) therefore retunes every rational
> > number by some amount, and this includes octaves; hence the
> > alternative acronym of Tempered Octaves, Please for TOP."
> >
> > Something must be wrong with either the premise or the inference,
> > since, for example, Top Beep has pure octaves.
>
> I'm getting this for TOP(27/25):
>
> [1214.176 1879.486 2819.230]
>
> What are you getting?

See /tuning/topicId_50628.html#51193 -- the same 3
and 5, but not the same 2. I guess the tuning of 2 actually has a
range of possibilities while maintaining the same maximum weighted
error for the tuning as a whole.

Hopefully you've seen my formula for TOP tempering of a single
comma . . .

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

> See /tuning/topicId_50628.html#51193 -- the same
3
> and 5, but not the same 2. I guess the tuning of 2 actually has a
> range of possibilities while maintaining the same maximum weighted
> error for the tuning as a whole.

Right, but this is exactly what I wanted to avoid.

> Hopefully you've seen my formula for TOP tempering of a single
> comma . . .

I remember looking at it, and then deciding it would be easier to
start from scratch. I'm glad I did, since formulating things in terms
of normed vector spaces and their duals makes it all crystal clear,
in case you are a mathematician.

Gene,

I've read through this web page, very interesting. Just a few quick
and albeit too-obvious questions: Is TOP(c) the solutions shown on
the right side of the tables? Is this "Tenney complexity"? Is it
anything like Graham complexity?

Thanks. I've printed this out and plan to read it until I get it,
hopefully...

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've put up a TOP web page. It needs to have, at least, a
discussion
> of equal and linear temperaments and Tenney complexity and badness
> added to it, but it should be valuable as a starter. Here it is:
>
> http://66.98.148.43/~xenharmo/top.htm
>
> Paul, could you tell me what you want attributed to you?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Gene,
>
> I've read through this web page, very interesting. Just a few quick
> and albeit too-obvious questions: Is TOP(c) the solutions shown on
> the right side of the tables?

Top(c) is the top tuning for the temperament defined by the comma c.

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

"Tenney complexity" would be for example the norm on linear wedgies I
gave, and it is a sort of weighted Graham complexity from a point of
view which gives no special role to octaves.

> Thanks. I've printed this out and plan to read it until I get it,
> hopefully...

I'll be adding to it. Carl pointed out one place where it needed
translation into some known language, for that matter.

--- In tuning-math@yahoogroups.com, "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.

On Mon, 12 Jan 2004 18:13:59 -0000, "Gene Ward Smith" <gwsmith@svpal.org>
wrote:

>I've put up a TOP web page. It needs to have, at least, a discussion
>of equal and linear temperaments and Tenney complexity and badness
>added to it, but it should be valuable as a starter. Here it is:
>
>http://66.98.148.43/~xenharmo/top.htm

I can plug the formulas into a program, and they seem to produce accurate
results, but I have no idea how to generalize them to higher limits. There
seems to be a pattern to them, but it'd take a long time and some guesswork
to figure it out. Could you be more explicit about how this works (in
language a non-mathematician programmer can understand)?

Also, how would this work for more than one comma? I'd like to eventually
be able to figure out things like the TOP for a linear tuning with a
mapping of [(1, 0), (2, -2), (1, 6), (3, -1), (3, 2)] and commas of [-7 3
1], [-4 -1 0 2], and [-5 1 0 0 1] (what I'm calling superpelog), or any
other interesting and unusual things that come up. I'd like to know if the
260.76 cent generator and 1206.55 cent period is still optimal, or whether
the extension to 11 limit changes anything. In fact, I'd like to see the
derivation for this particular example if it isn't too nasty.

(I've noticed there's a new notation for monzos that involves different
symbols like angle brackets and vertical bars, but I'm continuing to use
the old notation with square brackets since I don't know the appropriate
usage of the new notation.)

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:

> I can plug the formulas into a program, and they seem to produce
accurate
> results, but I have no idea how to generalize them to higher
limits. There
> seems to be a pattern to them, but it'd take a long time and some
guesswork
> to figure it out. Could you be more explicit about how this works
(in
> language a non-mathematician programmer can understand)?

I can get formulas like this for linear temperaments directly from
the wedgie, but I found out today that this approach is out of hand
from a practical point of view. It's also out of hand for higher
prime limits anyway. The solution to all of this is to add to the
page a discussion of how to set the thing up as a linear programing
problem, which I plan on doing, including dealing with the special
cases which arise when you are in a subgroup space and dual space,
one which uses only some of the primes in the prime limit.

> (I've noticed there's a new notation for monzos that involves
different
> symbols like angle brackets and vertical bars, but I'm continuing
to use
> the old notation with square brackets since I don't know the
appropriate
> usage of the new notation.)

It distinguishes monzos, |...>, fron vals, <...|. It worked for
physics.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> On Mon, 12 Jan 2004 18:13:59 -0000, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> >I've put up a TOP web page. It needs to have, at least, a
discussion
> >of equal and linear temperaments and Tenney complexity and badness
> >added to it, but it should be valuable as a starter. Here it is:
> >
> >http://66.98.148.43/~xenharmo/top.htm
>
> I can plug the formulas into a program, and they seem to produce
accurate
> results, but I have no idea how to generalize them to higher
limits. There
> seems to be a pattern to them, but it'd take a long time and some
guesswork
> to figure it out. Could you be more explicit about how this works
(in
> language a non-mathematician programmer can understand)?
>
> Also, how would this work for more than one comma?

Oh, so you were talking about one comma above? The formula is very
easy then; my most recent posting of it is here:

/tuning/topicId_51595.html#51762

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've put up a TOP web page. It needs to have, at least, a
discussion
> of equal and linear temperaments and Tenney complexity and badness
> added to it, but it should be valuable as a starter. Here it is:
>
> http://66.98.148.43/~xenharmo/top.htm
>
> Paul, could you tell me what you want attributed to you?

It's the other Paul again:

For a {2,5}-comma, we get |(c2 + c5 p5)/(c2 - c5 p5)|, and for a BP
comma involving only odd numbers, we get |(c3 p3 + c5 p5)/(c3 p3 - c5
p5)|. The corresponding TOP tunings are

What does BP stand for? Thanks

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> What does BP stand for? Thanks

I'm using it to mean music which excludes 2s, but originally it
referred to dividing the 12th into 13 parts.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > What does BP stand for? Thanks
>
> I'm using it to mean music which excludes 2s, but originally it
> referred to dividing the 12th into 13 parts.

I think I finally am starting to get TOP tuning. One more question:
In the "solution" column, why are certain terms negative? I see that
they correspond with the negative terms in the denominator. Does
it have to do with the "2^n corners" of the "ball" around the JIP?
Do you take absolute values of TOPs when you define the tuning?

Thanx Paul Hj

Oops! These are logarithmic. Negatives come into play. Sorry, once
again, never mind!

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > What does BP stand for? Thanks
> >
> > I'm using it to mean music which excludes 2s, but originally it
> > referred to dividing the 12th into 13 parts.
>
> I think I finally am starting to get TOP tuning. One more question:
> In the "solution" column, why are certain terms negative? I see that
> they correspond with the negative terms in the denominator. Does
> it have to do with the "2^n corners" of the "ball" around the JIP?
> Do you take absolute values of TOPs when you define the tuning?
>
> Thanx Paul Hj

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > What does BP stand for? Thanks
>
> I'm using it to mean music which excludes 2s, but originally it
> referred to dividing the 12th into 13 parts.

Better stick to the original definition, or anything that pertains to
this labyrinthine set of webpages:

http://members.aol.com/bpsite/

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> I can plug the formulas into a program, and they seem to produce
accurate
> results, but I have no idea how to generalize them to higher limits.

I've added something on how to set it up as a linear programming problem.

> Also, how would this work for more than one comma? I'd like to
eventually
> be able to figure out things like the TOP for a linear tuning with a
> mapping of [(1, 0), (2, -2), (1, 6), (3, -1), (3, 2)] and commas of
[-7 3
> 1], [-4 -1 0 2], and [-5 1 0 0 1] (what I'm calling superpelog), or any
> other interesting and unusual things that come up.

Here's some data on superpelog. You can see the top tunings for it
correspond in the 5, 7 and 11 limits, so would you object to simply
calling this "pelog", and saying "11-limit pelog" or "7-limit pelog"
if you want to be more specific? They have the same generators.

TOP 5: [1206.548265, 1891.576247, 2771.109113]
TOP 7: [1206.548264, 1891.576247, 2771.109113, 3358.884653]
TOP 11: [1206.548264, 1891.576247, 2771.109113, 3358.884653, 4141.165078]

Wedgies
pelog7: <<2 -6 1 -14, -4 19||
pelog11: <<2 -6 1 -2 -14 -4 -10 19 16 -9}||

TM comma basis
5 limit: {135/128}
7 limit: {135/128, 49/48}
11 limit: {33/32, 45/44, 49/48, 33/32}

I'd like to know if the
> 260.76 cent generator and 1206.55 cent period is still optimal, or
whether
> the extension to 11 limit changes anything.

Apparently not.

> (I've noticed there's a new notation for monzos that involves different
> symbols like angle brackets and vertical bars, but I'm continuing to use
> the old notation with square brackets since I don't know the appropriate
> usage of the new notation.)

Monzos: |...>
Vals: <...|
Linear temperament wedgies: <<...||
Planar temperament wedgies: <<<...|||

It makes sense to use <...| also for tuning maps, but I didn't above,
as I'm not sure if doing so would sow confusion.

>Monzos: |...>
>Vals: <...|
>Linear temperament wedgies: <<...||
>Planar temperament wedgies: <<<...|||

Thank GOD you wrote this. Where's monz?

>It makes sense to use <...| also for tuning maps, but I didn't above,
>as I'm not sure if doing so would sow confusion.

How could it sow confusion? I didn't know there was a difference
between a map a some number of vals.

-Carl

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

> Here's some data on superpelog. You can see the top tunings for it
> correspond in the 5, 7 and 11 limits, so would you object to simply
> calling this "pelog", and saying "11-limit pelog" or "7-limit pelog"
> if you want to be more specific? They have the same generators.
>
> TOP 5: [1206.548265, 1891.576247, 2771.109113]
> TOP 7: [1206.548264, 1891.576247, 2771.109113, 3358.884653]
> TOP 11: [1206.548264, 1891.576247, 2771.109113, 3358.884653,
4141.165078]
>
> Wedgies
> pelog7: <<2 -6 1 -14, -4 19||
> pelog11: <<2 -6 1 -2 -14 -4 -10 19 16 -9}||
>
> TM comma basis
> 5 limit: {135/128}
> 7 limit: {135/128, 49/48}
> 11 limit: {33/32, 45/44, 49/48, 33/32}

33/32 twice? ;)

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

> > 11 limit: {33/32, 45/44, 49/48, 33/32}
>
> 33/32 twice? ;)

It's a set, Paul. I can list 33/32 as many times as I feel like, and
it doesn't change anything. :)