Octave-reduced TOP error?

Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
interval to really care too much about.

Furthermore, let's say you have a TOP tuning of some temperament such
that 11/1 is 6 cents flat. Let's also say the octaves are 3 cents
sharp. So with that setup, the TOP optimized 11/8 will now be 15 cents
flat.

This seems bad. Is there a way around it?

Sent from my iPhone

>Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
>5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
>interval to really care too much about.

Graham and Igs have both complained about this. I think it's
a legitimate complaint. Of course you can optimize for whatever
list of intervals you like, but in order to get a parsimonious
optimization like TOP they'll need to be the basis of the free
abelian group you want to be parsimonious about...

-Carl

On 4/10/2011 6:29 PM, Carl Lumma wrote:
>> Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
>> 5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
>> interval to really care too much about.
>
> Graham and Igs have both complained about this. I think it's
> a legitimate complaint. Of course you can optimize for whatever
> list of intervals you like, but in order to get a parsimonious
> optimization like TOP they'll need to be the basis of the free
> abelian group you want to be parsimonious about...
>
> -Carl

I've tried variations of TOP that use superparticular intervals (2/1, 3/2, 5/4, 7/6) or ratios of adjacent primes (2/1, 3/2, 5/3, 7/5) (well, 1 isn't a prime, but the others are). The tricky thing is figuring out how to weight them. These days, what I typically use is an unweighted RMS optimization of all the superparticulars up to a certain point (2/1, 3/2, 4/3, 5/4, 6/5, 7/6, 8/7). This tends to bias the results to favor the lower primes without any weighting, and does well with many musically useful intervals (which are either superparticular, or the product of two superparticulars). 7/4 for instance is (3/2) (7/6), but with the usual TOP optimization it's (2/1)^-2 (7/1), and 7/1 is less well approximated to begin with. 9/5 is (3/2) (6/5).

On 11 April 2011 00:27, Mike Battaglia <battaglia01@gmail.com> wrote:
> Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
> 5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
> interval to really care too much about.
>
> Furthermore, let's say you have a TOP tuning of some temperament such
> that 11/1 is 6 cents flat. Let's also say the octaves are 3 cents
> sharp. So with that setup, the TOP optimized 11/8 will now be 15 cents
> flat.

TOP will tend not to optimize like that because the average weighted
deviation ends up small.

> This seems bad. Is there a way around it?

Take the standard deviation instead of the RMS, or equivalent. Don't
optimize the scale stretch. It's all in http://x31eq.com/primerr.pdf

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 11 April 2011 00:27, Mike Battaglia <battaglia01@...> wrote:
> > Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
> > 5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
> > interval to really care too much about.
> >
> > Furthermore, let's say you have a TOP tuning of some temperament such
> > that 11/1 is 6 cents flat. Let's also say the octaves are 3 cents
> > sharp. So with that setup, the TOP optimized 11/8 will now be 15 cents
> > flat.
>
> TOP will tend not to optimize like that because the average weighted
> deviation ends up small.

You mean that the average TOP tuning ends up being underneath that error bound?

> > This seems bad. Is there a way around it?
>
> Take the standard deviation instead of the RMS, or equivalent. Don't
> optimize the scale stretch. It's all in http://x31eq.com/primerr.pdf

Thanks. I guess I'm finally ready to read all of that now.

-Mike

On Sun, Apr 10, 2011 at 6:29 PM, Carl Lumma <carl@lumma.org> wrote:
>
> >Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
> >5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
> >interval to really care too much about.
>
> Graham and Igs have both complained about this. I think it's
> a legitimate complaint. Of course you can optimize for whatever
> list of intervals you like, but in order to get a parsimonious
> optimization like TOP they'll need to be the basis of the free
> abelian group you want to be parsimonious about...

That shouldn't be a problem, because if 2/1 is a basis element of said
group, then you can replace any one of the other basis elements with
its octave-equivalent and still generate the same group, due to
associativity.

-Mike

On Sun, Apr 10, 2011 at 9:01 PM, Herman Miller <hmiller@io.com> wrote:
>
> I've tried variations of TOP that use superparticular intervals (2/1,
> 3/2, 5/4, 7/6) or ratios of adjacent primes (2/1, 3/2, 5/3, 7/5) (well,
> 1 isn't a prime, but the others are). The tricky thing is figuring out
> how to weight them. These days, what I typically use is an unweighted
> RMS optimization of all the superparticulars up to a certain point (2/1,
> 3/2, 4/3, 5/4, 6/5, 7/6, 8/7). This tends to bias the results to favor
> the lower primes without any weighting, and does well with many
> musically useful intervals (which are either superparticular, or the
> product of two superparticulars). 7/4 for instance is (3/2) (7/6), but
> with the usual TOP optimization it's (2/1)^-2 (7/1), and 7/1 is less
> well approximated to begin with. 9/5 is (3/2) (6/5).

This seems interesting... I should code this up.

This also reminds me that it would be nice to host the xenharmonic
wiki on a server that we have control over, so that we can make it
such that for some custom XML tag like <temperament name="Barbados"
limit="2.3.13/10" comma="676/675" />, we can just have it spit out
whatever error metrics we want across the entire wiki by recoding the
display method for it.

-Mike

On 11 April 2011 14:50, battaglia01 <battaglia01@gmail.com> wrote:

>> > Furthermore, let's say you have a TOP tuning of some temperament such
>> > that 11/1 is 6 cents flat. Let's also say the octaves are 3 cents
>> > sharp. So with that setup, the TOP optimized 11/8 will now be 15 cents
>> > flat.
>>
>> TOP will tend not to optimize like that because the average weighted
>> deviation ends up small.
>
> You mean that the average TOP tuning ends up being underneath that error bound?

Is TOP being TOP-max today? That makes it simpler, anyway. It's
optimizing max(E) - min(E) where I is the weighted error. Your
example is 9 cents sharp for 8:1 and 6 cents flat for the 11/1. I
don't know what Tenney weighting says, but it looks unbalanced -- the
8:1 is too sharp. The only reason the TOP optimization would set the
stretch that way is if something has to be even flatter than the 11/1.
So there's an inherent problem that you can't wish away.

Another thing to think about: stretching the scale has more impact on
large intervals than small ones. A one cent octave stretch will be
about half a cent for 11/8 but three cents for 8/1, for example. The
point of the standard deviation error is that you get about the same
error regardless of the stretch ... which may not have been the
question you asked ... I'm not sure, but I'm going home now.

Graham

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
> 5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
> interval to really care too much about.

This is misleading: TOP optimizes *all* intervals in a prime limit, not
just the primes.

Kalle

On Mon, Apr 11, 2011 at 7:03 AM, Graham Breed <gbreed@gmail.com> wrote:
>
> On 11 April 2011 14:50, battaglia01 <battaglia01@gmail.com> wrote:
>
> >> > Furthermore, let's say you have a TOP tuning of some temperament such
> >> > that 11/1 is 6 cents flat. Let's also say the octaves are 3 cents
> >> > sharp. So with that setup, the TOP optimized 11/8 will now be 15 cents
> >> > flat.
> >>
> >> TOP will tend not to optimize like that because the average weighted
> >> deviation ends up small.
> >
> > You mean that the average TOP tuning ends up being underneath that error bound?
>
> Is TOP being TOP-max today? That makes it simpler, anyway. It's
> optimizing max(E) - min(E) where I is the weighted error.

Do you mean where E is the weighted error?

> Your example is 9 cents sharp for 8:1 and 6 cents flat for the 11/1. I
> don't know what Tenney weighting says, but it looks unbalanced -- the
> 8:1 is too sharp. The only reason the TOP optimization would set the
> stretch that way is if something has to be even flatter than the 11/1.
> So there's an inherent problem that you can't wish away.

I just threw out an example to demonstrate the concept that if the
octaves are stretched, and 7/1 and 11/1 are what's optimized, then
error will accumulate when you go down 2-3 octaves to 7/4 and 11/8.
And since I care more about 11/8 and 11/4 than 11/1, I thought there
might be a better way to go about doing it.

There are so many different types of TOP optimization that I can't
keep track anymore, so hopefully it'll all make sense when I read the
primerr primer.

> Another thing to think about: stretching the scale has more impact on
> large intervals than small ones. A one cent octave stretch will be
> about half a cent for 11/8 but three cents for 8/1, for example. The
> point of the standard deviation error is that you get about the same
> error regardless of the stretch ... which may not have been the
> question you asked ... I'm not sure, but I'm going home now.

I'm not sure how that works out...

-Mike

On Mon, Apr 11, 2011 at 10:03 AM, Kalle Aho <kalleaho@mappi.helsinki.fi> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
> > 5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
> > interval to really care too much about.
>
> This is misleading: TOP optimizes *all* intervals in a prime limit, not
> just the primes.
>
> Kalle

My understanding of this is - TOP works directly with the primes, and
in so doing optimizes everything, because every ratio is composed of
primes. But there are different ways to optimize all intervals in a
prime limit.

My understanding is that if you're TOPtimizing for 81/80,
geometrically speaking, you trace out the unison vector for 81/80
along the lattice, and then the TOP optimization is that which tempers
the tuning of the axes such that the unison vector is shortened
uniformly along its length. That's the understanding I have of it from
reading a Middle Path. However, if the axes were 2/1, 3/2, and 5/4
instead of 2/1, 3/1 and 5/1, this might lead to a different result. Or
if they were 2/1, 16/15, and 25/24. Or if they were 81/80, 3/2, and
2/1, or if they were 250/243, 10/9, and 2/1, etc.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Apr 11, 2011 at 10:03 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > Is there a variant of TOP that optimizes for intervals like 2/1, 3/2,
> > > 5/4, 7/4, etc instead of 2/1, 3/1, 5/1, 7/1? 7/1 is a pretty wide
> > > interval to really care too much about.
> >
> > This is misleading: TOP optimizes *all* intervals in a prime limit, not
> > just the primes.
> >
> > Kalle
>
> My understanding of this is - TOP works directly with the primes, and
> in so doing optimizes everything, because every ratio is composed of
> primes. But there are different ways to optimize all intervals in a
> prime limit.

No, it doesn't put special interest in primes. TOPtimizing primes is
just an easy way to TOPtimize all the rest. Maybe Paul should explain
this to you, just ask him!

Kalle

On Mon, Apr 11, 2011 at 10:38 AM, Kalle Aho <kalleaho@mappi.helsinki.fi> wrote:
>
> No, it doesn't put special interest in primes. TOPtimizing primes is
> just an easy way to TOPtimize all the rest. Maybe Paul should explain
> this to you, just ask him!

I will ask him again, but I'll also just leave it at this:

1) By convention, if a comma doesn't involve a certain prime, it will
be left unaltered during TOP tempering. Or at least that's what it
seems to say in Middle Path.
2) The "prime numbers" that generate the 5-limit could just as easily
be 81/80, 3/2, and 2/1 as could they be 2/1, 3/1, 5/1. We're talking,
specifically, about the basis elements of a free abelian group, and
both of the sets above function as such elements.
3) If we have the axes mapped to 81/80, 3/2, and 2/1, then 81/80 will
just span one step along the 81/80 axis, and hence if you "uniformly
shorten it," as TOP does, you will only be shortening things along
that one axis.
4) The other two axes, 3/2 and 2/1, aren't involved, and hence by
convention #1 above, you'd leave them alone.

This would lead to an absurd scenario where the TOP optimal tuning for
meantone, given "prime" axes of 81/80, 3/2, and 2/1, has a generator
of a just 3/2 and a period of a just 2/1. Making the axes be 2/1, 3/1,
and 5/1, instead, is more sensible, and avoids this situation. Have I
misunderstood any part of this?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Apr 11, 2011 at 10:38 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > No, it doesn't put special interest in primes. TOPtimizing primes is
> > just an easy way to TOPtimize all the rest. Maybe Paul should explain
> > this to you, just ask him!
>
> I will ask him again, but I'll also just leave it at this:
>
> 1) By convention, if a comma doesn't involve a certain prime, it will
> be left unaltered during TOP tempering. Or at least that's what it
> seems to say in Middle Path.
> 2) The "prime numbers" that generate the 5-limit could just as easily
> be 81/80, 3/2, and 2/1 as could they be 2/1, 3/1, 5/1. We're talking,
> specifically, about the basis elements of a free abelian group, and
> both of the sets above function as such elements.
> 3) If we have the axes mapped to 81/80, 3/2, and 2/1, then 81/80 will
> just span one step along the 81/80 axis, and hence if you "uniformly
> shorten it," as TOP does, you will only be shortening things along
> that one axis.
> 4) The other two axes, 3/2 and 2/1, aren't involved, and hence by
> convention #1 above, you'd leave them alone.
>
> This would lead to an absurd scenario where the TOP optimal tuning for
> meantone, given "prime" axes of 81/80, 3/2, and 2/1, has a generator
> of a just 3/2 and a period of a just 2/1. Making the axes be 2/1, 3/1,
> and 5/1, instead, is more sensible, and avoids this situation. Have I
> misunderstood any part of this?

Yes, I think so. That you could choose some other basis doesn't mean
that they are primes or "primes".

BTW, I never thought that the uniform shortening business was
particularly illuminating. Once you have more than one comma it goes
out of the window. Look at the endnote xxvi in Middle Path instead.

Quote from page 14: "The tuning would also remain optimal if we
restricted our attention to intervals no larger than an octave".

Kalle

On Mon, Apr 11, 2011 at 11:48 AM, Kalle Aho <kalleaho@mappi.helsinki.fi> wrote:
>
> > This would lead to an absurd scenario where the TOP optimal tuning for
> > meantone, given "prime" axes of 81/80, 3/2, and 2/1, has a generator
> > of a just 3/2 and a period of a just 2/1. Making the axes be 2/1, 3/1,
> > and 5/1, instead, is more sensible, and avoids this situation. Have I
> > misunderstood any part of this?
>
> Yes, I think so. That you could choose some other basis doesn't mean
> that they are primes or "primes".

What do you mean? If the behavior of the algorithm is as I laid out in
my last method, then all I'm suggesting is we use "primes" like 2/1,
3/2, and 5/4 instead of 2/1, 3/1, and 5/1.

> BTW, I never thought that the uniform shortening business was
> particularly illuminating. Once you have more than one comma it goes
> out of the window. Look at the endnote xxvi in Middle Path instead.
>
> Quote from page 14: "The tuning would also remain optimal if we
> restricted our attention to intervals no larger than an octave".

Hm. I'll ping Paul offlist and report back with what he says. You,
Graham, Carl, and Herman, have all made good points, but it just so
happens that they're opposite points.

I think the source of my confusion is that my proposed improvement is
a different method of weighting the primes, not a different method of
setting up the axes, and I've been lumping the two together. It
doesn't help that there are a million different error metrics that
have been proposed over the years, and my ability to understand linear
algebra enough to compare them wasn't up to par until recently.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Apr 11, 2011 at 11:48 AM, Kalle Aho <kalleaho@...> wrote:
> >
> > > This would lead to an absurd scenario where the TOP optimal tuning for
> > > meantone, given "prime" axes of 81/80, 3/2, and 2/1, has a generator
> > > of a just 3/2 and a period of a just 2/1. Making the axes be 2/1, 3/1,
> > > and 5/1, instead, is more sensible, and avoids this situation. Have I
> > > misunderstood any part of this?
> >
> > Yes, I think so. That you could choose some other basis doesn't mean
> > that they are primes or "primes".
>
> What do you mean? If the behavior of the algorithm is as I laid out in
> my last method, then all I'm suggesting is we use "primes" like 2/1,
> 3/2, and 5/4 instead of 2/1, 3/1, and 5/1.

What I mean is that the convention "if a comma doesn't involve a
certain prime, it will be left unaltered during TOP tempering." holds
for genuine primes but for example 81/80 and 2/1 are not independent:
if you temper 2/1 you'll temper 81/80 too because you have 2 in the
prime factorization of 81/80.

> > BTW, I never thought that the uniform shortening business was
> > particularly illuminating. Once you have more than one comma it goes
> > out of the window. Look at the endnote xxvi in Middle Path instead.
> >
> > Quote from page 14: "The tuning would also remain optimal if we
> > restricted our attention to intervals no larger than an octave".
>
> Hm. I'll ping Paul offlist and report back with what he says. You,
> Graham, Carl, and Herman, have all made good points, but it just so
> happens that they're opposite points.
>
> I think the source of my confusion is that my proposed improvement is
> a different method of weighting the primes, not a different method of
> setting up the axes, and I've been lumping the two together. It
> doesn't help that there are a million different error metrics that
> have been proposed over the years, and my ability to understand linear
> algebra enough to compare them wasn't up to par until recently.

All I'm saying that if you optimize (minimax) the entire n-prime
limit lattice and not just some subset of it and weight errors by
Tenney Harmonic Distance you must get the same result no matter what
the axes are.

Kalle

On Mon, Apr 11, 2011 at 12:30 PM, Kalle Aho <kalleaho@mappi.helsinki.fi> wrote:
>
> > What do you mean? If the behavior of the algorithm is as I laid out in
> > my last method, then all I'm suggesting is we use "primes" like 2/1,
> > 3/2, and 5/4 instead of 2/1, 3/1, and 5/1.
>
> What I mean is that the convention "if a comma doesn't involve a
> certain prime, it will be left unaltered during TOP tempering." holds
> for genuine primes but for example 81/80 and 2/1 are not independent:
> if you temper 2/1 you'll temper 81/80 too because you have 2 in the
> prime factorization of 81/80.

Yes, but the whole concept of "prime factorization" is what is being
generalized to the abelian group/basis vector structure. So instead of
factorizing ratios into constituent primes, you "factor" them into
combinations of basis vectors. Or if that's not how it currently
works, I assume you could make it work that way, which may be what I'm
suggesting. I really don't have time to read the two papers right now,
but you keep responding... :)

> > I think the source of my confusion is that my proposed improvement is
> > a different method of weighting the primes, not a different method of
> > setting up the axes, and I've been lumping the two together. It
> > doesn't help that there are a million different error metrics that
> > have been proposed over the years, and my ability to understand linear
> > algebra enough to compare them wasn't up to par until recently.
>
> All I'm saying that if you optimize (minimax) the entire n-prime
> limit lattice and not just some subset of it and weight errors by
> Tenney Harmonic Distance you must get the same result no matter what
> the axes are.

Is this still TOP error that we're talking about? I thought that
minimax error was something else.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Apr 11, 2011 at 12:30 PM, Kalle Aho <kalleaho@...> wrote:
> >
> > > What do you mean? If the behavior of the algorithm is as I laid out in
> > > my last method, then all I'm suggesting is we use "primes" like 2/1,
> > > 3/2, and 5/4 instead of 2/1, 3/1, and 5/1.
> >
> > What I mean is that the convention "if a comma doesn't involve a
> > certain prime, it will be left unaltered during TOP tempering." holds
> > for genuine primes but for example 81/80 and 2/1 are not independent:
> > if you temper 2/1 you'll temper 81/80 too because you have 2 in the
> > prime factorization of 81/80.
>
> Yes, but the whole concept of "prime factorization" is what is being
> generalized to the abelian group/basis vector structure. So instead of
> factorizing ratios into constituent primes, you "factor" them into
> combinations of basis vectors. Or if that's not how it currently
> works, I assume you could make it work that way, which may be what I'm
> suggesting. I really don't have time to read the two papers right now,
> but you keep responding... :)

Yes, you can generate the lattice with different bases but you still
can't change the temperings of things like 81/80 and 2/1
independently.

> > > I think the source of my confusion is that my proposed improvement is
> > > a different method of weighting the primes, not a different method of
> > > setting up the axes, and I've been lumping the two together. It
> > > doesn't help that there are a million different error metrics that
> > > have been proposed over the years, and my ability to understand linear
> > > algebra enough to compare them wasn't up to par until recently.
> >
> > All I'm saying that if you optimize (minimax) the entire n-prime
> > limit lattice and not just some subset of it and weight errors by
> > Tenney Harmonic Distance you must get the same result no matter what
> > the axes are.
>
> Is this still TOP error that we're talking about? I thought that
> minimax error was something else.

In TOP (Paul's TOP, not Graham's TOP-RMS stuff) you minimize the
maximum damage over all intervals in a prime limit. The damage of an
interval is error(n/d)/log2(n*d).

Kalle

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> This also reminds me that it would be nice to host the xenharmonic
> wiki on a server that we have control over, so that we can make it
> such that for some custom XML tag like <temperament name="Barbados"
> limit="2.3.13/10" comma="676/675" />, we can just have it spit out
> whatever error metrics we want across the entire wiki by recoding the
> display method for it.

I could send you a copy of the xenwiki if you wanted to experiment.

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> In TOP (Paul's TOP, not Graham's TOP-RMS stuff) you minimize the
> maximum damage over all intervals in a prime limit. The damage of an
> interval is error(n/d)/log2(n*d).

Mike's point is that you can change what you mean by n and d. You have 5/1, numerator 5, denominator 1. Change your basis, so that 5 is written
(3/2)^4 * (81/80)^(-1), and the "numerator" is 81/16 and the "denominator" 81/80. Now you are measuring damage differently.

Hi Kalle,

>Yes, you can generate the lattice with different bases but you still
>can't change the temperings of things like 81/80 and 2/1
>independently.

Sure you can, with the appropriate basis. We're talking about
changing the entire number system, not just the vector space
sitting on top of it. -Carl

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > In TOP (Paul's TOP, not Graham's TOP-RMS stuff) you minimize the
> > maximum damage over all intervals in a prime limit. The damage of an
> > interval is error(n/d)/log2(n*d).
>
> Mike's point is that you can change what you mean by n and d. You
> have 5/1, numerator 5, denominator 1. Change your basis, so that 5
> is written (3/2)^4 * (81/80)^(-1), and the "numerator" is 81/16 and
> the "denominator" 81/80. Now you are measuring damage differently.

That is Mike's point, really?

So, let's say we used octave-reduced primes as the basis. Wouldn't
5/4 and (gasp!) 11/8 have smaller harmonic distance than 3/2? Hardly
useful.

Kalle

Kalle wrote:

>That is Mike's point, really?
>So, let's say we used octave-reduced primes as the basis. Wouldn't
>5/4 and (gasp!) 11/8 have smaller harmonic distance than 3/2? Hardly
>useful.

Firstly, it was my point not Mike's. Secondly, yes, it is a challenge
to come up with a basis and harmonic distance that make sense.
As I said, I don't believe anyone's achieved it.

-Carl

On Mon, Apr 11, 2011 at 6:19 PM, Kalle Aho <kalleaho@mappi.helsinki.fi> wrote:
>
> That is Mike's point, really?
>
> So, let's say we used octave-reduced primes as the basis. Wouldn't
> 5/4 and (gasp!) 11/8 have smaller harmonic distance than 3/2? Hardly
> useful.
>
> Kalle

I guess I just want 5/4 to have a smaller badness rating than 5/1.
That's all. Really wide intervals are sometimes perceived as being two
disparate notes, not one fused note. Thus they are less concordant
than a sound that is perceived as just one note. At least that's how I
see it.

-Mike

On Mon, Apr 11, 2011 at 4:25 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > This also reminds me that it would be nice to host the xenharmonic
> > wiki on a server that we have control over, so that we can make it
> > such that for some custom XML tag like <temperament name="Barbados"
> > limit="2.3.13/10" comma="676/675" />, we can just have it spit out
> > whatever error metrics we want across the entire wiki by recoding the
> > display method for it.
>
> I could send you a copy of the xenwiki if you wanted to experiment.

What? How?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > I could send you a copy of the xenwiki if you wanted to experiment.
>
> What? How?

Either I upload it to a file-sharing location which can deal with 200 MB files, or you ask Barton to make you an "Organizer".