Triadic topography

In response to Jon Wild having found my old plots of total dyadic
discordance of triads, I thought I'd make an updated plot that uses the
Chalmers-style triangular coordinates and the Tenney- rather than
Farey-based dyadic discordance model (in order to eliminate the bias toward
larger intervals). The result is here:

http://www.egroups.com/files/tuning/perlich/triangle_01.jpg

I labeled all the otonal triads with geometric mean < 7. These do not
correspond to all the deepest local minima in this plot -- the utonal triads
and mirror-symmetrical triads are missed, for example. However, when I make
the same kind of plot using a true triadic harmonic entropy model, I expect
that the otonal triads below a certain geometric mean _will_ correspond to
the deepest local minima.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14282

That's a great graphic, Paul. Now if we could only combine that with
the kind of 3-dimensional stuff that Jon Wild has been doing, we'd
have a great masterwork!
_________ ___ __ __
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson"
<josephpehrson@c...> wrote:
> --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/14282
>
> That's a great graphic, Paul. Now if we could only combine that
with
> the kind of 3-dimensional stuff that Jon Wild has been doing, we'd
> have a great masterwork!
> _________ ___ __ __
> Joseph Pehrson

I'll try to produce a 3-d rendering tomorrow
. . . but the colors and contours should
already give you a good sense of the "lay of
the land" on this planet where otonal and
utonal are equal.

Joseph Pehrson wrote,

>http://www.egroups.com/message/tuning/14282
>That's a great graphic, Paul. Now if we could only combine that with
>the kind of 3-dimensional stuff that Jon Wild has been doing, we'd
>have a great masterwork!

Done: http://www.egroups.com/files/tuning/terrain/waves.jpg. I decided that
for this dyadic model, much more detail would be visible if consonance
corresponded to mountains and dissonance corresponded to depressions. Since
we are just summing the dyadic concordance values, there is a noticeable
ridge corresponding to each consonant interval, and peaks on each ridge
where the other two intervals in the triad are consonant. Also, because we
are summing, mathematically the surface is equivalent to what would happen
if we had three linear wave patterns, corresponding to each of the dyadic
concordance functions, with wave fronts parallel to the corresponding side
of triangle. So I chose a blue-green colormap to make the picture more
suggestive of waves.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

> Done: http://www.egroups.com/files/tuning/terrain/waves.jpg.

That is so cool... and it's all shiny! Did Matlab produce this?

>> Done: http://www.egroups.com/files/tuning/terrain/waves.jpg.

>That is so cool... and it's all shiny! Did Matlab produce this?

Yes sir -- though you can't see it, the surface is composed of 360,000 tiny
triangles (which look equilateral when viewed from above). In Matlab you can
specify the "material" of the surface, which affects the
ambient/diffuse/specular strength, specular exponent and specular color
reflectance of the objects, and though I used the default here, Matlab also
provides "shiny", "dull" (good for earth), and "metal" materials, or you can
plug your own numbers in for all these parameters.

Unfortunately (or perhaps fortunately), Matlab does not seem to take into
account the fact that in the real world, parts of the surface would cast
shadows on other parts . . .

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14318

> Done: http://www.egroups.com/files/tuning/terrain/waves.jpg.

Wow, Paul, this is a terrific pizza! But what did you put on it??

Just joking, it's a great graphic, the best I've seen so far!!

___________ ____ __ _ _
Joseph Pehrson

>> Done: http://www.egroups.com/files/tuning/terrain/waves.jpg.

>Wow, Paul, this is a terrific pizza! But what did you put on it??

Psilocybian mushrooms, of course!

I would have made the whole pie (six slices in a hexagon, like my
map-of-Paris graphics) but my machine couldn't handle that at this level of
resolution (2 cents).

Paul,

> Psilocybian mushrooms, of course!

In a delicate white wine sauce?

> I would have made the whole pie (six slices in a hexagon, like my
> map-of-Paris graphics) but my machine couldn't handle that at this
level of
> resolution (2 cents).

But seriously folks (Paul): what kind of monitor and resolution are
you running? Even at 1024x768, which is still larger than the
majority of people surfing, the entire image doesn't fit in the
window. I scaled the image in half and squeezed the file a bit; it's
now up there as waves_small.jpg.

Looks fun, though.

Cheers,
Jon

--- In tuning@egroups.com, "Jonathan Szanto" <JSZANTO@A...> wrote:

http://www.egroups.com/message/tuning/14329

I scaled the image in half and squeezed the file a bit; it's
> now up there as waves_small.jpg.
>
> Looks fun, though.
>
> Cheers,
> Jon

Thanks, Jon, for scaling Paul's "pizza slice." It's just perfect now
for the "overview..." Now, I have to understand exactly what I'm
looking at...
_____________ _____ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
>
> Thanks, Jon, for scaling Paul's "pizza slice." It's just perfect
now
> for the "overview..." Now, I have to understand exactly what I'm
> looking at...

Did you look at it next to the 2-d color triangle that I produced?
All
the bands of color are exactly the same as the 3-d one -- it's like a
topographic map -- and the main otonal peaks are labeled.

The three sides of the pizza are where two of the notes of the triad
are the same. The near corner is 1:1:1, and the other two are 1:1:2
and 1:2:2. The big ridge in front of you is where the outer interval
is a fifth -- its four big peaks are 6:8:9, 10:12:15, 4:5:6, and
8:9:12. Keep exploring!

Joseph wrote,

>Thanks, Jon, for scaling Paul's "pizza slice." It's just perfect now
>for the "overview..." Now, I have to understand exactly what I'm
>looking at...

As Jon mentioned in his description for this file on the website, the 2-d
version of the pizza is at
http://www.egroups.com/files/tuning/perlich/triangle_01.jpg. Before I
endeavor to label the 3-d pizza, see if this 2-d map of it can't give you
the clues (well, it labels some triads outright, but if you see the patterns
you can easily label many more) you need to understand it.

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14358

>
> Did you look at it next to the 2-d color triangle that I produced?
> All the bands of color are exactly the same as the 3-d one -- it's
like a topographic map -- and the main otonal peaks are labeled.
>

I'm sorry, Paul... I'm having trouble with this. So, all the points
are otonal numbers, but could you please illustrate to me how a
"ridge" is constructed of the numbers on either side. Sorry to be so
dense about it!

I wrote,

>> Did you look at it next to the 2-d color triangle that I produced?
>> All the bands of color are exactly the same as the 3-d one -- it's
>>like a topographic map -- and the main otonal peaks are labeled.

>I'm sorry, Paul... I'm having trouble with this. So, all the points
>are otonal numbers, but could you please illustrate to me how a
>"ridge" is constructed of the numbers on either side. Sorry to be so
>dense about it!

Each ridge is a given consonant interval at a given position (lower, upper,
outer) position in the chord. That is, each ridge is composed of the set of
chords with a given consonant interval at a given position (lower, upper,
outer) position in the chord. I labeled some otonal members, but this
process is the "sum of dyadic discordances", as you can read in the title
above the 2-d one -- so it treats otonal and utonal as equals.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14431

>
> Each ridge is a given consonant interval at a given position
(lower, upper,outer) position in the chord. That is, each ridge is
composed of the set of chords with a given consonant interval at a
given position (lower, upper, outer) position in the chord. I labeled
some otonal members, but this process is the "sum of dyadic
discordances", as you can read in the title above the 2-d one -- so
it
treats otonal and utonal as equals.

This graphic is pretty incredible, particularly the colors... but I'm
not "getting" the numbers, yet. For example, one ridge has 3:2:2 on
the bottom and and 6:4:3 on the top. How are these numbers related,
and how can I find what intervals this is??

I must be missing something pretty obvious...

Thanks!

Hi Joseph,

>This graphic is pretty incredible, particularly the colors... but I'm
>not "getting" the numbers, yet. For example, one ridge has 3:2:2 on
>the bottom and and 6:4:3 on the top. How are these numbers related,
>and how can I find what intervals this is??

Joseph,

3:2 and 6:4 are the same interval. Hence this must be the rigde where the
lower interval of each chord is 3:2. Right?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/14465

>
> 3:2 and 6:4 are the same interval. Hence this must be the rigde
where the
> lower interval of each chord is 3:2. Right?

Oh... gotcha!