Re: Dissonant clusters

Joseph: My recollection is that elsewhere Ivor said that the cluster of
all 19 19-tet notes was the most dissonant one he had tried. He thought
this was important because the triads of 19-tet sounded less dissonant
to him than those of 12.

Ivor had been a 1/4-tone composer until he heard Joel Mandelbaum' tape
from his dissertation.

--John

--- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote:

/tuning/topicId_38179.html#38179

>
> Joseph: My recollection is that elsewhere Ivor said that the
cluster of
> all 19 19-tet notes was the most dissonant one he had tried. He
thought
> this was important because the triads of 19-tet sounded less
dissonant
> to him than those of 12.
>
> Ivor had been a 1/4-tone composer until he heard Joel Mandelbaum'
tape
> from his dissertation.
>
> --John

***Thanks John! Yes, it seems he terms 19-tET "aggressive" but some
on this list apparently disagree...

Joseph

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote [#38179]:
> Here is a comment from Ivor Darreg from his translation of
> Vyshnegradski's _Quartertone Harmony_ in Xenharmonicon #6:
>
> "Investigations conducted in recent years by this translator have
> shown that a cluster of all 24 quartertones is not so dissonant as
> the familiar cluster of all twelve semitones..."
>
> Does this make any sense?? Would there be any acoustical reason
for
> that??
>
> Thanks!
>
> Joseph

--- In tuning@y..., John Chalmers <JHCHALMERS@U...> wrote [#38179]:
>
> Joseph: My recollection is that elsewhere Ivor said that the
cluster of
> all 19 19-tet notes was the most dissonant one he had tried. He
thought
> this was important because the triads of 19-tet sounded less
dissonant
> to him than those of 12.
>
> Ivor had been a 1/4-tone composer until he heard Joel Mandelbaum'
tape
> from his dissertation.
>
> --John

When in doubt, get a second opinion. Since nobody else tried any of
this, I decided to hear it for myself last night on the Scalatron.

I used a simulated-sawtooth waveform with the high end filtered out
and tried clusters of tones in a couple of different octaves, both
above and below middle C. I found a problem in trying to hold down a
whole octave of keys all at once, especially when trying to change
the tuning quickly -- I needed one hand free to change the tuning.
So the best I could do was to spread one hand out over as many keys
as I could, which covered almost an octave.

I tried 12-ET, 17-WT (well-temperament, but close to 17-ET), 19-ET,
22-ET, 24-ET, and 31-ET. Of these, I thought that 12-ET was *least*
dissonant and 17-WT most dissonant (even though it had fewer tones
sounding than most of the rest). The rest came pretty close to 17-WT
in overall dissonance, and I concluded that the difference isn't
significant enough to matter.

I seem to recall Ivor Darreg saying that adding more tones in a
cluster brings you closer to white or pink noise, which would produce
simultaneous beating at many different rates, causing the individual
beats to become obscured in a hodge-podge of noise. The result would
sound less dissonant than overt beating.

On a related note, I once ran an experiment to determine the most
dissonant intervals between the unison and the octave (in a middle
register). I concluded that they were (in order of decreasing
dissonance) around 70 (~24:25), 1145 (~16:31), and 740 cents
(~15:23). The first of these intervals is very close to the single
degree of 17-ET (and the average single degree of 17-WT), which might
explain why 17-tone clusters sound so dissonant. I also thought that
the last of these intervals (very close to a meantone wolf fifth)
used alone would have a more shocking effect in a composition than an
entire cluster of semitones in 12-ET.

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> On a related note, I once ran an experiment to determine the most
> dissonant intervals between the unison and the octave (in a middle
> register). I concluded that they were (in order of decreasing
> dissonance) around 70 (~24:25), 1145 (~16:31), and 740 cents
> (~15:23).

i looked at this question based on the harmonic entropy formulation
that i use as default nowadays, which uses the tenney seeding and an
s value of 1.2%.

i calculated the discordance for every cents value, and sorted.

the global maximum was at 60 cents, and the next highest values were
all adjacent, in the range 47-90 cents.

then, 1144 cents shows up, followed by more adjacent values in the
range 1109-1157, and at the same time the first range expands to 43-
138 cents.

then, 651 shows up, and while the range 646-654 is established, 139,
140, 1159, and 1108 show up.

not far behind 651, 754 shows up . . .

so it looks like we have fair agreement with your observations.

if you'd like to see a graph, i can make one; it'll look a lot like
the second graph here:

http://www.ixpres.com/interval/dict/harmentr.htm

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > On a related note, I once ran an experiment to determine the most
> > dissonant intervals between the unison and the octave (in a
middle
> > register). I concluded that they were (in order of decreasing
> > dissonance) around 70 (~24:25), 1145 (~16:31), and 740 cents
> > (~15:23).
>
> i looked at this question based on the harmonic entropy formulation
> that i use as default nowadays, which uses the tenney seeding and
an
> s value of 1.2%.
>
> i calculated the discordance for every cents value, and sorted.
>
> the global maximum was at 60 cents, and the next highest values
were
> all adjacent, in the range 47-90 cents.
>
> then, 1144 cents shows up, followed by more adjacent values in the
> range 1109-1157, and at the same time the first range expands to 43-
> 138 cents.
>
> then, 651 shows up, and while the range 646-654 is established,
139,
> 140, 1159, and 1108 show up.
>
> not far behind 651, 754 shows up . . .
>
> so it looks like we have fair agreement with your observations.

In general, yes.

I should have said that the three interval sizes I gave were local
points of least consonance. I made my observations over an entire
octave, and they are summarized in this graph:

/tuning-math/files/secor/consonce.gif

(If viewing this in Microsoft Photo Editor, be sure to set the zoom
control to 100%.)

I calculated numerical values for the peaks of consonance as 1/(m*n)
for each ratio m/n. Where I found specific ratios to conincide with
what I perceived as points of minimum local consonance, I used the
same formula. The rest of the curve is rather subjective, in which I
only intended to give a general impression of what I observed. The y
axis is a logarithmic scale, so as to magnify the differences as
consonance decreases, just to make them easier to see.

I wanted to have something in which the entire curve would be the
result of some sort of calculation, so I am very delighted to see
what you have. I have one question. When I looked here:

> http://www.ixpres.com/interval/td/Erlich/entropy.htm

I saw (about 40% of the way down) some tabulations of local minima
and maxima in which some of the maxima did not quite coincide with
exact ratios and could therefore only be considered approximations of
maximum consonance (or minimum dissonance). So I would surmise that
the peaks would also be approximations, give or take a few cents. Or
is these calculations somewhat different from the graph you
mentioned? (I have not had time to look at all of this in detail.)

> if you'd like to see a graph, i can make one; it'll look a lot like
> the second graph here:
>
> http://www.ixpres.com/interval/dict/harmentr.htm

Don't go to a lot of trouble. What I am primarily interested in is a
method for calculating points of local minimum consonance that fall
between peaks.

I recently did some experiments with chord progressions involving
suspensions in which two tones a small "semitone" apart resolve to a
minor or subminor third. After trying different sizes of semitones
in various keys, I came to the conclusion that 70 cents (~24:25) is
very close (closer than, say, 63 cents, ~27:28) to the most
harmonically dissonant semitone. (This agrees with an earlier
observation I made that an interval of this size is also the most
melodically effective resolution of the leading tone.) This is all
subjective, of course, based on the observations of just one
listener, myself. But my observations were made over a period of
many years, and each new one has confirmed all of the earlier ones.
So take this for what you think it's worth.

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> I should have said that the three interval sizes I gave were local
> points of least consonance. I made my observations over an entire
> octave, and they are summarized in this graph:
>
> /tuning-math/files/secor/consonce.gif

nice, though i'm a bit suspicious about the exact positions of the
local minima . . .

> (If viewing this in Microsoft Photo Editor, be sure to set the zoom
> control to 100%.)
>
> I calculated numerical values for the peaks of consonance as 1/
(m*n)
> for each ratio m/n.

harmonic entropy, for the simplest ratios, turns out to go very much
as tenney harmonic distance = log(n*d), so we have very similar
measures in some respects.

> Where I found specific ratios to conincide with
> what I perceived as points of minimum local consonance, I used the
> same formula.

harmonic entropy predicts that the formula breaks down, most likely
already when m*n = 100.

> I wanted to have something in which the entire curve would be the
> result of some sort of calculation, so I am very delighted to see
> what you have.

thanks, and the math is particularly beautiful, i feel. you'll have
to join the harmonic entropy list to find out more :) though i think
gene could contribute a lot if he wanted to.

> I have one question. When I looked here:
>
> > http://www.ixpres.com/interval/td/Erlich/entropy.htm
>
> I saw (about 40% of the way down) some tabulations of local minima
> and maxima in which some of the maxima did not quite coincide with
> exact ratios and could therefore only be considered approximations
of
> maximum consonance (or minimum dissonance).

the local maxima are maxima of dissonance, not consonance.

wait, you must already know that, and be mentally turning the graphs
upside-down.

> So I would surmise that
> the peaks would also be approximations, give or take a few cents.

yes, the harmonic entropy curve, i think, converges to a limit as N
goes to infinity. with N finite, you're only taking the first N terms
in the series, as it were, so there can be some "wiggles" in the
approximation. in the limit, the minima of disssonance should nail
the simple ratios perfectly, though it would be up to someone like
gene to prove this mathematically.

> > if you'd like to see a graph, i can make one; it'll look a lot
like
> > the second graph here:
> >
> > http://www.ixpres.com/interval/dict/harmentr.htm
>
> Don't go to a lot of trouble.

too late -- i already did:

/tuning/files/dyadic/default.gi
f

> What I am primarily interested in is a
> method for calculating points of local minimum consonance that fall
> between peaks.

well, obviously i have such a method!

george, you and i have such a deep convergence of interests and
ideas, it blows me away. we should definitely arrange to meet at some
point, if possible -- or at least collaborate on at least a few
articles. write me off-list if any of this sounds appealing to
you . . .

> I recently did some experiments with chord progressions involving
> suspensions in which two tones a small "semitone" apart resolve to
a
> minor or subminor third. After trying different sizes of semitones
> in various keys, I came to the conclusion that 70 cents (~24:25) is
> very close (closer than, say, 63 cents, ~27:28) to the most
> harmonically dissonant semitone.

isn't this *highly* dependent on register and timbre, though? maybe
on loudness as well?

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_38179.html#38269

> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > On a related note, I once ran an experiment to determine the most
> > dissonant intervals between the unison and the octave (in a
middle
> > register). I concluded that they were (in order of decreasing
> > dissonance) around 70 (~24:25), 1145 (~16:31), and 740 cents
> > (~15:23).
>
> i looked at this question based on the harmonic entropy formulation
> that i use as default nowadays, which uses the tenney seeding and
an
> s value of 1.2%.
>
> i calculated the discordance for every cents value, and sorted.
>
> the global maximum was at 60 cents, and the next highest values
were
> all adjacent, in the range 47-90 cents.
>
> then, 1144 cents shows up, followed by more adjacent values in the
> range 1109-1157, and at the same time the first range expands to 43-
> 138 cents.
>
> then, 651 shows up, and while the range 646-654 is established,
139,
> 140, 1159, and 1108 show up.
>
> not far behind 651, 754 shows up . . .
>
> so it looks like we have fair agreement with your observations.
>
> if you'd like to see a graph, i can make one; it'll look a lot like
> the second graph here:
>
> http://www.ixpres.com/interval/dict/harmentr.htm

***This is really amazing stuff. I'll continue Johnny Reinhards,
*What If...":

Imagine if composers *truly* had versatile control over this entirely
*continuous* range and could really make the most of these
dissonances and consonances easily without having always to be within
a defined step-scale...

JP

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > I should have said that the three interval sizes I gave were
local
> > points of least consonance. I made my observations over an
entire
> > octave, and they are summarized in this graph:
> >
> > /tuning-math/files/secor/consonce.gif
>
> nice, though i'm a bit suspicious about the exact positions of the
> local minima . . .

I did this in my pre-Scalatron days, using a retunable electronic
organ and a StroboConn tuner. Using a timbre having both odd and
even harmonics and rich in partials (a combination of diapason and
string stops in unison), I kept one pitch constant (at middle C) and
varied the second pitch from a unison to an octave, measuring the
local maximum and minimum points that I observed with the StroboConn
(with a precision no better than to the nearest cent). While there
was no question about the exact size of the maximum points of
consonance, I was interested in determining which ratios were not
perceived as being more consonant than their mistunings. I was also
interested in the local minimums, for which I could not predict exact
values (more about this below, particularly in regard to the most
dissonant semitone, for which my 70-cent size determination was based
on more than just this one experiment).

It's nice to see that we agree at a local minimum around 1144 cents,
however.

> > (If viewing this in Microsoft Photo Editor, be sure to set the
zoom
> > control to 100%.)
> >
> > I calculated numerical values for the peaks of consonance as 1/
(m*n)
> > for each ratio m/n.
>
> harmonic entropy, for the simplest ratios, turns out to go very
much
> as tenney harmonic distance = log(n*d), so we have very similar
> measures in some respects.
>
> > Where I found specific ratios to conincide with
> > what I perceived as points of minimum local consonance, I used
the
> > same formula.
>
> harmonic entropy predicts that the formula breaks down, most likely
> already when m*n = 100.

I did observe a neutral third (11/9, m*n=99) to be a local maximum,
however. Still, this would indicate that it is not quite as
dissonant as the product of the integers would lead us to believe.

> > I wanted to have something in which the entire curve would be the
> > result of some sort of calculation, so I am very delighted to see
> > what you have.
>
> thanks, and the math is particularly beautiful, i feel. you'll have
> to join the harmonic entropy list to find out more :) though i
think
> gene could contribute a lot if he wanted to.

I'm not as young as you are anymore and have a lot of other
microtonal things that I would like to accomplish that have a higher
priority that this -- particularly in areas that have been largely
neglected and in which I believe that I can make a unique
contribution (e.g., the microtonal accordion design and 19+3
temperament that I just posted). I did the graph for a book I am
writing and would be all too happy to replace it with something less
subjective. So, yes, I would be interested in seeing what has
developed there and in contributing what I can to the dialogue, if
only for a short time.

> > I have one question. When I looked here:
> >
> > > http://www.ixpres.com/interval/td/Erlich/entropy.htm
> >
> > I saw (about 40% of the way down) some tabulations of local
minima
> > and maxima in which some of the maxima did not quite coincide
with
> > exact ratios and could therefore only be considered
approximations
> of
> > maximum consonance (or minimum dissonance).
>
> the local maxima are maxima of dissonance, not consonance.
>
> wait, you must already know that, and be mentally turning the
graphs
> upside-down.

Right. I was also thinking of my graph, which shows consonance. But
it helps to remember which way is up.

> > So I would surmise that
> > the peaks would also be approximations, give or take a few cents.
>
> yes, the harmonic entropy curve, i think, converges to a limit as N
> goes to infinity. with N finite, you're only taking the first N
terms
> in the series, as it were, so there can be some "wiggles" in the
> approximation. in the limit, the minima of disssonance should nail
> the simple ratios perfectly, though it would be up to someone like
> gene to prove this mathematically.
>
> > > if you'd like to see a graph, i can make one; it'll look a lot
like
> > > the second graph here:
> > >
> > > http://www.ixpres.com/interval/dict/harmentr.htm
> >
> > Don't go to a lot of trouble.
>
> too late -- i already did:
>
>
/tuning/files/dyadic/default.gi
f

Something that I am suspicious about in your graph is that it doesn't
shown enough local points of maximum consonance. Between 5:4 and
4:3, for example, I see a local minimum consonance at 448 cents
without any indication that 9:7 is more consonant than a mistuning,
something about which I had no doubt in my observations. (The same
could be said for 10:7.) Now when it comes to an interval such as
11:9 (or even 11:8), the difference between it and a mistuning is
very subtle -- my graph exaggerates the difference (reflecting the
fact that you can hear a lot more under laboratory conditions than
you can in a musical performance) -- and I was completely unable to
hear 16:13 or 17:14 as such -- but rather as points of local minimum
consonance.

Perhaps there is not enough resolution in the vertical direction in
your graph -- 7:6 and 9:5 barely show up as consonances -- and a
higher resolution might make a difference for 8:7 and 9:7. (A bump
in the road isn't going to show up on a topographical map, but I sure
feel it when I drive my car over it.)

> > What I am primarily interested in is a
> > method for calculating points of local minimum consonance that
fall
> > between peaks.
>
> well, obviously i have such a method!
>
> george, you and i have such a deep convergence of interests and
> ideas, it blows me away. we should definitely arrange to meet at
some
> point, if possible -- or at least collaborate on at least a few
> articles. write me off-list if any of this sounds appealing to
> you . . .

So many things, so little time ...

> > I recently did some experiments with chord progressions involving
> > suspensions in which two tones a small "semitone" apart resolve
to a
> > minor or subminor third. After trying different sizes of
semitones
> > in various keys, I came to the conclusion that 70 cents (~24:25)
is
> > very close (closer than, say, 63 cents, ~27:28) to the most
> > harmonically dissonant semitone.
>
> isn't this *highly* dependent on register and timbre, though? maybe
> on loudness as well?

I noted above what sort of timbre I used in my observations with the
electronic organ -- that was of a kind that I thought would be most
valid in a general way. Other timbres (particularly those that lack
even partials or contain inharmonic partials) would give somewhat
different results, but I was not interested in pursuing that
specifically; I was only concerned with generalities.

Now register is another story. Helmholtz's assertion that the
consonance and dissonance of intervals depend on register (e.g.,
major thirds in the bass being muddy, therefore dissonant) is tied to
his observation that maximum dissonance is associated with a beat
rate of about 32 Hz. This means that my electronic organ experiment
might have a different result for the local points of least
consonance if the constant tone were, say, an octave higher. Or it
might not matter much, if you consider the following.

With my most recent experiments involving suspensions (to find the
most dissonant semitone) I varied the timbres and registers somewhat
in reaching my conclusion. I found that there is more to dissonance
that just the beat rate -- there is also a psychological perception
conveyed by a semitone of a given size that conveys an impression of
dissonance that seems to be less influenced by (if not independent
of) the register in which it is heard. In connection with this,
consider our perception in comparing two major thirds, for example,
mistuned by the same number of cents, one wide and the other narrow.
There is a somewhat different psychological perception associated
with each that is a function of size (but independent of beating)
that will influence our perception of dissonance. (Might this be
that an interval's melodic properties also come into play when the
interval is heard harmonically?) I believe that something of this
sort played a part in my conclusion that the 70-cent semitone is the
most dissonant.

These things are rather elusive if your objective is to describe
everything mathematically.

--George

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ***This is really amazing stuff. I'll continue Johnny Reinhards,
> *What If...":
>
> Imagine if composers *truly* had versatile control over this
entirely
> *continuous* range and could really make the most of these
> dissonances and consonances easily without having always to be
within
> a defined step-scale...
>
> JP

with 72-equal, you come pretty darn close, far closer than with 12-
equal anyway . . . that's the idea of the "intervallic continuum" or
maneri's "virtual pitch continuum" . . .

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_38179.html#38300

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ***This is really amazing stuff. I'll continue Johnny Reinhards,
> > *What If...":
> >
> > Imagine if composers *truly* had versatile control over this
> entirely
> > *continuous* range and could really make the most of these
> > dissonances and consonances easily without having always to be
> within
> > a defined step-scale...
> >
> > JP
>
> with 72-equal, you come pretty darn close, far closer than with 12-
> equal anyway . . . that's the idea of the "intervallic continuum"
or
> maneri's "virtual pitch continuum" . . .

****I'm totally happy with it as a scale *and* a notation! I'm so
glad I was "enlightened!"

JP

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> > > I calculated numerical values for the peaks of consonance as 1/
> (m*n)
> > > for each ratio m/n.
> >
> > harmonic entropy, for the simplest ratios, turns out to go very
> much
> > as tenney harmonic distance = log(n*d), so we have very similar
> > measures in some respects.
> >
> > > Where I found specific ratios to conincide with
> > > what I perceived as points of minimum local consonance, I used
> the
> > > same formula.
> >
> > harmonic entropy predicts that the formula breaks down, most
likely
> > already when m*n = 100.
>
> I did observe a neutral third (11/9, m*n=99) to be a local maximum,
> however.

do you mean a local maximum of consonance?

i can tune 17:13 by ear, but i don't think that makes it consonant,
or even necessarily locally consonant. i do have an alternate set of
curves (equally simple to compute) that show many, if not all,
rational numbers at a local minimum of dissonance, for example:

/tuning/files/dyadic/margo2.gif

> Still, this would indicate that it is not quite as
> dissonant as the product of the integers would lead us to believe.

exactly. back to my original, default, "smooth" harmonic entropy
curve formulation, these graphs point that fact out in vivid detail:

/tuning/files/perlich/stearns.jpg
/tuning/files/perlich/stearns2.jpg
/tuning/files/perlich/stearns4.jpg

/tuning/files/dyadic/default.gi
f

> Something that I am suspicious about in your graph is that it
doesn't
> shown enough local points of maximum consonance. Between 5:4 and
> 4:3, for example, I see a local minimum consonance at 448 cents
> without any indication that 9:7 is more consonant than a mistuning,
> something about which I had no doubt in my observations. (The same
> could be said for 10:7.) Now when it comes to an interval such as
> 11:9 (or even 11:8), the difference between it and a mistuning is
> very subtle -- my graph exaggerates the difference (reflecting the
> fact that you can hear a lot more under laboratory conditions than
> you can in a musical performance) -- and I was completely unable to
> hear 16:13 or 17:14 as such -- but rather as points of local
minimum
> consonance.

> Perhaps there is not enough resolution in the vertical direction in
> your graph -- 7:6 and 9:5 barely show up as consonances -- and a
> higher resolution might make a difference for 8:7 and 9:7. (A bump
> in the road isn't going to show up on a topographical map, but I
sure
> feel it when I drive my car over it.)

it sounds like you'd probably prefer the "pointy" formulation as in
margo2.gif above -- let me know. there's also the s (resolution)
parameter which can always be tweaked to give more or less importance
to more complex ratios (the better your hearing resolution, the more
easily you can identify the complex ratios "as such", because the
complex ratios are in more "crowded" areas amongst all the ratios).

> > > I recently did some experiments with chord progressions
involving
> > > suspensions in which two tones a small "semitone" apart resolve
> to a
> > > minor or subminor third. After trying different sizes of
> semitones
> > > in various keys, I came to the conclusion that 70 cents
(~24:25)
> is
> > > very close (closer than, say, 63 cents, ~27:28) to the most
> > > harmonically dissonant semitone.
> >
> > isn't this *highly* dependent on register and timbre, though?
maybe
> > on loudness as well?

> With my most recent experiments involving suspensions (to find the
> most dissonant semitone) I varied the timbres and registers
somewhat
> in reaching my conclusion. I found that there is more to
dissonance
> that just the beat rate -- there is also a psychological perception
> conveyed by a semitone of a given size that conveys an impression
of
> dissonance that seems to be less influenced by (if not independent
> of) the register in which it is heard. In connection with this,
> consider our perception in comparing two major thirds, for example,
> mistuned by the same number of cents, one wide and the other
narrow.
> There is a somewhat different psychological perception associated
> with each that is a function of size (but independent of beating)
> that will influence our perception of dissonance. (Might this be
> that an interval's melodic properties also come into play when the
> interval is heard harmonically?) I believe that something of this
> sort played a part in my conclusion that the 70-cent semitone is
the
> most dissonant.

well, can you give a sense of *how much*, if at all, this conclusion
is dependent on timbre and register?

Paul,

Sorry, but I missed your message #38302 (27 Jun 02) until this
morning when I saw it in the digest. In it you referred (among other
things) to this curve:

/tuning/files/dyadic/margo2.gif

Let's continue this discussion on the harmonic entropy list, where I
will reply.

--George