(mostly for Carl, Mike B) Update on 3HE

Further to our discussions off-line:
Once again I've been short of time, but now I have an update and I find that my email account seems to be unusable. Carl, Mike, would you pass it on?

First, I note wryly that I seem to have confused Paul - this is some sort of an achievement; perhaps not one to be proud of. I was trying to say that I needed to look at the shape of a "Tenney set" in triad space in order to be certain that I included enough "seed points" in the calculation.

The last file I sent contains good results (I believe) for all triads a:b:c with a*b*c <=256 (free parameter s=1.225%, seed points limit N=1M). I did look at a plot of 3HE vs a*b*c but I couldn't see a significant trend. Perhaps 256 is too small.

Paul then (re-)asked for a*b*c <=10000. This calculation is running now, starting with the triads 1:1:x, but it is taking a loooong time (note that there are 10000 of these alone, and the seed point set contains 20M points even with the "cutoffs" I am applying). I have uploaded a graph of the results *so far*, see:

/tuning/files/SteveMartin/3HE11x.png

Steve.

Hi Steve,

I haven't gotten any e-mails recently. I lost track of
what you and Paul were on about some time ago.

> (free parameter s=1.225%,

Why so large? Or is this 'equivalent' to 1% because
of the distance you've chosen? I don't believe I caught
which distance you did finally choose.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> ... lost track of what you and Paul were on about ...
> > ... s=1.225% ...
> Why so large? ... which distance you did finally choose.

We have still not found a compelling reason to choose one over the others. So, yes, as you guessed, I chose the value of "s" such that whichever distance formula is correct, "s" will still be no smaller than 1%.
(To explain to the group - the 3HE calculation takes place in "triad space" where we know how to calculate distances *relative* to one another but are having trouble assigning *absolute* values in cents to distances (we have three main candidate "multipliers"). This is important because "s" translates to a value in cents. There is no such ambiguity in the 2HE calculation.)

All of my calculations so far have been of triads 0-x-y where 0<=x<=y<=2400cents; this is a fairly small part of triad space. For this purpose, it was OK to restrict the seed points to RI triads that are "near" the area of interest. Paul seemed surprised that I was "cutting off" the seed points in this way - but some of them are literally octaves away from the area of interest, so it seems justified.

(Carl, did you see an email where I made this point? Did you see an email with a file t256b.out attached?)

Paul then asked me to calculate 3HE for all triads a:b:c with a*b*c <= 10K, and I could see the interest in this. However, these points cover a much larger area of triad space (e.g. it includes the triad 1:1:10K which is 13 octaves away from 1:1:1); the seed points also need to cover a correspondingly much larger area. Still, I set up this calculation, which is still running(!). The file that I uploaded (containing partial results) shows some surprises (for me anyway) that need to be explained.

Still, I am happy with the results for 0<=x<=y<=2400cents.

Steve M.

"martinsj013" <martinsj@...> wrote:

> We have still not found a compelling reason to choose one
> over the others.

It seems to me there's a highly compelling reason to
choose one over the others.

> Paul seemed surprised that I was "cutting off" the seed
> points in this way - but some of them are literally octaves
> away from the area of interest, so it seems justified.
> (Carl, did you see an email where I made this point? Did
> you see an email with a file t256b.out attached?)

Yes. I was surprised Paul thought these points would
be significant.

> Paul then asked me to calculate 3HE for all triads a:b:c
> with a*b*c <= 10K, and I could see the interest in this.
> However, these points cover a much larger area of triad
> space (e.g. it includes the triad 1:1:10K which is
> 13 octaves away from 1:1:1); the seed points also need to
> cover a correspondingly much larger area. Still, I set
> up this calculation, which is still running(!). The file
> that I uploaded (containing partial results) shows some
> surprises (for me anyway) that need to be explained.

I'd be interested to hear about these surprises. Are
they coming from the inclusion of more triads within the
region you previously searched, or from the inclusion of
very wide triads you'd previously excluded?

-Carl

On Jul 18, 2011, at 5:45 PM, martinsj013 <martinsj@...> wrote:

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> ... lost track of what you and Paul were on about ...
> > ... s=1.225% ...
> Why so large? ... which distance you did finally choose.

We have still not found a compelling reason to choose one over the others.
So, yes, as you guessed, I chose the value of "s" such that whichever
distance formula is correct, "s" will still be no smaller than 1%.
(To explain to the group - the 3HE calculation takes place in "triad space"
where we know how to calculate distances *relative* to one another but are
having trouble assigning *absolute* values in cents to distances (we have
three main candidate "multipliers"). This is important because "s"
translates to a value in cents. There is no such ambiguity in the 2HE
calculation.)

Can you refresh my memory about what distance metrics we're comparing? Are
we talking about norms?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Can you refresh my memory about what distance metrics we're comparing? Are we talking about norms?

If the two points are given in terms of their lower and upper intervals in cents (i.e. (cents(b/a),cents(c/b)) as (l1,u1) and (l2,u2):

The three alternatives are:
1) Name: "RMS"
Definition: D^2 = (l1-l2)^2 + (u1-u2)^2 + (l1+u1-l2-u2)^2 (call this F^2)
Justification: it *is* simply the RMS of the three intervals found between the notes of the triad, with no multiplying factor
Counter: the distance between two triads "an octave apart" is given as *not* 1200 cents, but 1200*sqrt(2) cents
And it doesn't agree with the coordinate system (see below).

2) Name: "triad space"
Definition: D^2 = 2/3 * F^2
Justification: this definition agrees with the oblique coordinate system in triad space, e..g. any point on the line l=1200 cents is actually 1200 cents from its nearest neighbour on the line l=0 cents
Counter: the distance between two triads "an octave apart" is still not given as 1200 cents, but 1200*sqrt(4/3) cents
And we have introduced a multiplying factor.

3) Name: "subtlety"
Definition: D^2 = 1/2 * F^2
Justification: the distance between two triads "an octave apart" (e.g. any two of 1:1:1, 1:1:2, 1:2:2) is now given as 1200 cents
Counter: it doesn't agree with coordinate system
And we have introduced a multiplying factor.

Paul originally (years ago) derived 2) and shortly afterwards introduced the "subtlety" 3). See:
http://lumma.org/tuning/erlich/2000.10.05.TheGeometryOfTriangularPlots.txt

But in a recent email discussion Paul mentioned 1), as if he was not so sure. FWIW, I started out using 1), then switched to 3), then 2).

Carl seems to have decided which one he favours, but if he has said which it is, I don't remember it!

Steve.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> It seems to me there's a highly compelling reason to
> choose one over the others.

OK, I don't recall you saying which one; and Paul seemed less sure now than when he originally posted on this (see my other post). Don't say it - let's see if Mike B comes up with the same as you.

> [Surprises in the graph of 3HE for triads 1:1:x] ... Are
> they coming from the inclusion of more triads within the
> region you previously searched, or from the inclusion of
> very wide triads you'd previously excluded?

The latter. As expected, the graph starts with its lowest value at 1:1:1 and increases smoothly, with the rate of increase gradually slowing, almost to zero. But then the surprises are:
1) at about 4000 cents it starts to waver, and then starts to look like a sawtooth wave;
2) the values start to increase again, at a rate comparable to that in the vicinity of 1:1:1.

see /tuning/files/SteveMartin/3HE11x.png

The wavering is probably because the set of seed points is limited; I would guess that with a larger product limit for the seed points, the threshold for this effect would move to the right. But I would also expect edge effects to mean that the values start to decrease, not increase. Of course I have not let the calculation run far enough to see if that does eventually happen.

Steve.

> Carl seems to have decided which one he favours, but if he has
> said which it is, I don't remember it!
>
> Steve.

I don't favour one, I favour *deciding* on one:

>> It seems to me there's a highly compelling reason to
>> choose one over the others.

A coin toss will do. -Carl

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> The latter. As expected, the graph starts with its lowest value
> at 1:1:1 and increases smoothly, with the rate of increase
> gradually slowing, almost to zero. But then the surprises are:
> 1) at about 4000 cents it starts to waver, and then starts to
> look like a sawtooth wave;
> 2) the values start to increase again, at a rate comparable to
> that in the vicinity of 1:1:1.
>
> see /tuning/files/SteveMartin
> /3HE11x.png

Weird! -C.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> I don't favour one, I favour *deciding* on one:
> >> It seems to me there's a highly compelling reason to
> >> choose one over the others.

Oh, I see. I meant that as yet we have recognized no compelling argument that has led us to choose one over the others. (I suspect you realized that, and were having a dig at my imprecise wording, but maybe not? :-)

Steve.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> > I don't favour one, I favour *deciding* on one:
> >
> > >> It seems to me there's a highly compelling reason to
> > >> choose one over the others.
>
> Oh, I see. I meant that as yet we have recognized no compelling
> argument that has led us to choose one over the others. (I suspect
> you realized that, and were having a dig at my imprecise wording,
> but maybe not? :-)

Your wording was fine. I made mine deliberately provocative.

Seriously, what are you waiting for? If some better system
comes along, you can change over. But that seems unlikely.
We know all three distances have drawbacks. Flip a coin and
pick one so people can know what the heck you're talking
about! The value of a standard greatly exceeds the differences
between the choices.

It's like in 2001, when at the movies I confided to my friend
Stephen that I was worried about a difficult choice I had to
make. He replied this meant it was in fact an easy choice.
And I was enlightened. Then we watched the movie (O Brother,
Where Art Thou?) and I was even more enlightened.

-Carl

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
> > [Surprises in the graph of 3HE for triads 1:1:x] ...
> ... wavering is probably because the set of seed points is limited; I would guess that with a larger product limit for the seed points, the threshold for this effect would move to the right. But I would also expect edge effects to mean that the values start to decrease, not increase. ...

I am looking into this further. One thing that I have realized is that as x increases, the points 1:1:x get closer together, so that we are seeing more and more of the entire curve (as opposed to just joining together the "most rational" points on it). And as we know, the curve is not smooth! So perhaps the wavering is not so weird.

However, we also know that, at least near 1:1:1, it doesn't have an upward trend (Paul chose to use "Tenney" sets, not Farey sets, for this reason I think). So the upward trend above 4000 cents is still a surprise. I'll report back soon.

Steve.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> Seriously, what are you waiting for? If some better system
> comes along, you can change over. But that seems unlikely.
> We know all three distances have drawbacks. Flip a coin and
> pick one so people can know what the heck you're talking
> about! The value of a standard greatly exceeds the differences
> between the choices.

OK, point taken. But I won't flip a coin, I'll choose the one that matches the coordinate system, for three main reasons:

1) it, ahem, matches the coordinate system; seems to me therefore it is best placed to generalize for use in 4HE :o)

2) I have re-read Paul's "subtlety" text and don't quite agree with it; I think that moving one note in a triad by x cents can be said to move the triad by more than x cents.

3) my code already uses it, and my recent results are correctly labelled as such (but on the down side, they use too large an "s")

Steve.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
> ... as x increases, the points 1:1:x get closer together ... So perhaps the wavering is not so weird ... the upward trend above 4000 cents is still a surprise ...

1) I have uploaded a new graph, see:
/tuning/Files/SteveMartin/3HEsel.png

I eliminated the wavering by only including points like 1:1:2^n in one data series, and in separate series, 1:1:3*2^n, etc. These rational triads are expected to have smaller 3HE values, so I also included separate series for triads shifted by 30 cents in triad space, to illustrate higher 3HE values (the shorthand 1:1+e:2^n-e is intended to indicate this).

In summary, up to about 7000cents, we see the behaviour we've seen before - a steady range of values depending on how "just" the triad is. But the new graph confirms the result of my previous one - that above 7000cents the values start to increase with the size of the outer interval (and there seems no longer to be a clear difference between more and less "just" triads).

Note that these results are only for the upper interval growing large, with the lower remaining small. I will do a set with the lower growing large, and the upper remaining small next.

I'm out of time to write more now - I think the results are explained by the shape of the "Tenney" set in triad space - comparatively "nice" near its centre (the unison point), but not so nice at its edges.

Steve.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
> I have uploaded a new graph ...
Sorry, typo in the URL, correction:
/tuning/files/SteveMartin/3HEsel.png

--- "martinsj013" <martinsj@...> wrote:

> OK, point taken. But I won't flip a coin, I'll choose the one
> that matches the coordinate system, for three main reasons:
> 1) it, ahem, matches the coordinate system; seems to me therefore
> it is best placed to generalize for use in 4HE :o)
> 2) I have re-read Paul's "subtlety" text and don't quite agree
> with it; I think that moving one note in a triad by x cents can
> be said to move the triad by more than x cents.
> 3) my code already uses it, and my recent results are correctly
> labelled as such (but on the down side, they use too large an "s")

Great choice! Can you provide a brief write-up on the order
of Paul's subtly text? If so, I'll add it to the page.
You could start with what you posted the other day. Also it'd
be good to get s scaled to match (at least going forward...
maybe Paul will even re-do the existing graphics).

-Carl

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> I eliminated the wavering by only including points like 1:1:2^n
> in one data series, and in separate series, 1:1:3*2^n, etc.
> These rational triads are expected to have smaller 3HE values,
> so I also included separate series for triads shifted by
> 30 cents in triad space, to illustrate higher 3HE values (the
> shorthand 1:1+e:2^n-e is intended to indicate this).
>
> In summary, up to about 7000cents, we see the behaviour we've
> seen before - a steady range of values depending on how "just"
> the triad is. But the new graph confirms the result of my
> previous one - that above 7000cents the values start to
> increase with the size of the outer interval (and there seems
> no longer to be a clear difference between more and less
> "just" triads).

These "more just" traids are only more just in the prime-limit
sense... one of the coords is still a big odd number, right?

> Note that these results are only for the upper interval growing
> large, with the lower remaining small. I will do a set with
> the lower growing large, and the upper remaining small next.
> I'm out of time to write more now - I think the results are
> explained by the shape of the "Tenney" set in triad space -
> comparatively "nice" near its centre (the unison point), but
> not so nice at its edges.

I think this makes sense. 6 octaves is really wide. If
you imagine taking these from a harmonic series, it isn't
a particularly obvious thing to do.

-Carl

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> ... results ... for the upper interval growing large, with the lower remaining small ... /tuning/files/SteveMartin/3HEsel.png

I've now done the results with the lower interval growing large, and the upper remaining small:
/tuning/files/SteveMartin/3HEsel2.png

Notice that the 3HE value drops to zero after 12000 cents - because after that there are no more "seed points" in this direction (1:1K:1K is the last one because we have a product limit of 1M). In the first graph, this happens after 24000 cents (1:1:1M is the last one). These features are artificial and will change as the product limit changes. On the other hand, the point where the values start to increase may or may not be artificial - I'll need to explore this.

Two more graphs, what do you make of these?
3HE values plotted against a*b*c in cents (instead of c/a in cents):
/tuning/files/SteveMartin/3HEsel3.png

Tenney set in triad space (axes at 90deg - should be 60deg):
/tuning/files/SteveMartin/3HE10Kts.png

Steve.

--- "martinsj013" <martinsj@...> wrote:

> Two more graphs, what do you make of these?
> 3HE values plotted against a*b*c in cents (instead of c/a in cents):
> /tuning/files/SteveMartin
> /3HEsel3.png
>
> Tenney set in triad space (axes at 90deg - should be 60deg):
> /tuning/files/SteveMartin
> /3HE10Kts.png

Nothing's jumping out at me... -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
> > Two more graphs, what do you make of these?
> Nothing's jumping out at me... -Carl

In that case (no disrespect!) I'll get back onto email and send them to Paul :-) I have refined those graphs by doing "cross-sections" of:

(l,u) in cents:
(a) (0,x) for x=0 to 12000 step 25
(b) (30,x) for x=0 to 12000 step 25
(c) (x,0) for x=0 to 12000 step 25
(d) (x,30) for x=0 to 12000 step 25

but can't upload them now - will do.

Steve M.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> (a) (0,x) for x=0 to 12000 step 25
> (b) (30,x) for x=0 to 12000 step 25
/tuning/files/SteveMartin/g11x.png

> (c) (x,0) for x=0 to 12000 step 25
> (d) (x,30) for x=0 to 12000 step 25
/tuning/files/SteveMartin/g1xx.png

shape of Tenney set in triad space (oblique coordinates):
/tuning/files/SteveMartin/tenney.png

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> > (a) (0,x) for x=0 to 12000 step 25
> > (b) (30,x) for x=0 to 12000 step 25
> /tuning/files/SteveMartin/g11x.png
>
> > (c) (x,0) for x=0 to 12000 step 25
> > (d) (x,30) for x=0 to 12000 step 25
> /tuning/files/SteveMartin/g1xx.png

I'm not so sure about the fluctuations on the right,
but I'm very comfortable with the greater divergence
on the left when the lower interval is 30 cents.

-Carl