Combination tone triads, as well as isoharmonic triads, on the triadic HE graph

So looking at the latest HE graph,
/tuning/files/SteveMartin/tnxgc.svg
you will notice that there are a bunch of "valleys" that run through
the chart that correspond randomly to low entropy?

For example, there is a vertical line at l=700, and a vertical line at
l=1200. Pretty obvious why this is, because if one of the dyads in the
chord is 2/1 or 3/2, it's going to be of comparatively low entropy to
the things around it no matter how bad the other dyad is.

There's also a diagonal line going from (0,1200) to (1200,0). This one
is also pretty easy to figure out, because it corresponds to l+u=1200.
If the outer dyad is 2/1, then it's going to be of comparatively low
entropy to the things around it.

But what about the line going from (0,1200) to (1200,702)? That one
follows a line of 200u+83l=240,000. If we assume the line goes through
actually (1200,700), not (1200,702), then a line can be traced through
it that is 12u+5l=14,400. WTF does this mean?

A clue might be given by looking at the points on this line: 1:2:3,
2:3:5, 3:4:7, 4:5:9, 5:6:11, 6:7:13.

So there is a valley of low entropy on the curve for all triads that
follow the form a:b:a+b. Note that a+b is the sum tone for a:b. This
can also be thought of as the lowest note forming a first-order
"difference" tone for the upper two.

There's also an obvious "curve" like structure below this that runs
from 1:1:1 to 4:5:6 to 3:4:5 to 2:3:4 to 1:2:3. This also contains
things like 3:5:7. This curve seems to contain all of the triads that
are "isoharmonic," e.g. for any triad a:b:c, c-b = b-a. So these
triads can be generated by a:a+D:a+2D, where D is the difference.

I'm not sure why this is, but the top a:b:a+b valley (I think actually
a curve) seems to have less entropy, on average, than every other
similar structure on the graph except for:

1) where l=0 or u=0
2) where l=1200
3) where l+u=1200

Can anyone offer any explanation for this?

-Mike

I also notice they're in Paul's original work as well, and on the
triangular axes:

/tuning/files/triad2p40000.jpg

-Mike

On Sat, Dec 4, 2010 at 8:30 PM, Mike Battaglia <battaglia01@...> wrote:
> So looking at the latest HE graph,
> /tuning/files/SteveMartin/tnxgc.svg
> you will notice that there are a bunch of "valleys" that run through
> the chart that correspond randomly to low entropy?
>
> For example, there is a vertical line at l=700, and a vertical line at
> l=1200. Pretty obvious why this is, because if one of the dyads in the
> chord is 2/1 or 3/2, it's going to be of comparatively low entropy to
> the things around it no matter how bad the other dyad is.
>
> There's also a diagonal line going from (0,1200) to (1200,0). This one
> is also pretty easy to figure out, because it corresponds to l+u=1200.
> If the outer dyad is 2/1, then it's going to be of comparatively low
> entropy to the things around it.
>
> But what about the line going from (0,1200) to (1200,702)? That one
> follows a line of 200u+83l=240,000. If we assume the line goes through
> actually (1200,700), not (1200,702), then a line can be traced through
> it that is 12u+5l=14,400. WTF does this mean?
>
> A clue might be given by looking at the points on this line: 1:2:3,
> 2:3:5, 3:4:7, 4:5:9, 5:6:11, 6:7:13.
>
> So there is a valley of low entropy on the curve for all triads that
> follow the form a:b:a+b. Note that a+b is the sum tone for a:b. This
> can also be thought of as the lowest note forming a first-order
> "difference" tone for the upper two.
>
> There's also an obvious "curve" like structure below this that runs
> from 1:1:1 to 4:5:6 to 3:4:5 to 2:3:4 to 1:2:3. This also contains
> things like 3:5:7. This curve seems to contain all of the triads that
> are "isoharmonic," e.g. for any triad a:b:c, c-b = b-a. So these
> triads can be generated by a:a+D:a+2D, where D is the difference.
>
> I'm not sure why this is, but the top a:b:a+b valley (I think actually
> a curve) seems to have less entropy, on average, than every other
> similar structure on the graph except for:
>
> 1) where l=0 or u=0
> 2) where l=1200
> 3) where l+u=1200
>
> Can anyone offer any explanation for this?
>
> -Mike
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So there is a valley of low entropy on the curve for all triads that
follow the form a:b:a+b.
I agree with your analysis (and as you say later, it is a curve not a line).

> There's also ... triads ... a:a+D:a+2D, where D is the difference.
Ditto.

> ... Can anyone offer any explanation for this?
> ... I also notice they're in Paul's original work as well, and on the triangular axes:
/tuning/files/triad2p40000.jpg

I would only point to the fact that in that diagram (and also others like:
/tuning/files/PaulErlich/trimap.jpg)
there is a large space around these ratios, with no rival candidates within the Tenney set. To a certain extent, the Triadic HE graph just turns these diagrams of (discrete) Tenney sets into a continuous graph.

BTW I just noticed this at the harmonic entropy group:
/tuning/files/Erlich/fun.gif