David Finnamore's tour guide

The following is David Finnamore's guide to the 2-D "green pizza" of
Harmonic Entropy. It goes along with the graphic:

triangle_01dfinn_01.jpg

and the accompanying sound file:

ridgeglide1.mp3

Both these files reside in the directory:

http://www.egroups.com/files/harmonic_entropy/Finnamore/

This is truly a GREAT multimedia harmonic entropy experience!

HOWEVER, "Density 21.5" (me) found the following explanation David
wrote up to be quite useful:

1) You're probably used to looking at harmonic
lattices, and may be trying to interpret this one
similarly, as I did at first. But it actually has a
completely different basis. For one thing, a triad on
a harmonic lattice forms a triangle between three
points; on this triangular discordance map,
ironically, each triad is located at a single point,
_not_ between 3 points. Each unique point on the map
(there is a theoretically infinite number) is a unique
triad whose outer interval is no more than 2:1, and
all possible such triads are represented on the map.

2) We're all used to seeing triads listed from the
bottom up. For some reason I have not been able to
divine, Paul chose to list them top down on this map.
So at a glance, 6:5:4 looks like the minor 5-limit
triad but it's really the major 5-limit one. It's the
one that is normally displayed 4:5:6 on the Tuning
List. It would be less confusing for me if we did it
that way. I guess I should ask Paul about it. He may
have a good reason.

3) The map can be split into otonal and utonal halves
by drawing a line from the 1:1:1 to the center of the
side that opposes the 1:1:1 corner, half-way between
4:3:2 and 6:4:3. Rotating the map so that 1:1:1 faces
straight South, the left side is otonal and the right
side is its utonal mirror image. Once Paul is able to
map actual triadic harmonic entropy, the utonal side
will have lower hill tops than the otonal side.
Tenney discordance ranks them equally.

4) Everything is relative, which is why I was able to
keep the bottom note the same for convenience (a
convenience which was displeasing to Paul and Jon!
8-). Here's what I mean by that. Let's call the 3
numbers C:B:A (since they're listed "backwards" to us
and I'm hoping Paul is willing to switch in the
future). Using the beginning of the ridgetop tour as
an example, when we move from 3:2:2 to 12:9:8 (which
is labeled on my version of the map), the outer tones,
C and A, hold their pitches steady in a 3:2
relationship and only the middle one, B, moves. B
starts in unison with A (2:2) and slides upward as we
move Northwest along the ridge. You might think that
the next chord should be some form of 3:B:2. So
what's up with the 12 and 8? Well, actually 12:8 is
the same thing as 3:2, right? Look across that whole
line from 3:2:2 to 3:3:2 - notice that in all six
chords labeled on my version, C:A is reducible to
3:2. In fact, no matter where you stop on that line,
even where C:B and B:A are irrational, C:A maintains a
3:2 relationship, even though it may be spelled 6:4,
9:6, or 15:10; it could be spelled 8991:5994 - it's
the same interval. In this example we stop where B
has climbed upward by 204 cents, from 2:2 (=1:1) to
9:8 with respect to C. Obviously, if C:B originally
had a 3:2 relationship, and B climbed by 204 cents, it
now has a 4:3 relationship. So at this point on the
map, C:A = 3:2, B:A = 9:8, and C:B = 4:3. The
smallest whole number relationship that represents all
three of those intervals at once is 12:9:8.

Now notice that on all lines parallel to the 3:2:2 -
3:3:2 line, C:A remains constant. 4:3:3 to 4:4:3 - so
all points on that line represent triads in which C:A
= 4:3 and B is somewhere in between. In fact, all
possible such triads lie on that line. 5:3:3 to 5:5:3
- all points on this line have a 5:3 relationship
between C and A, and B lies somewhere in between.
Along the top line, C and A remain an octave apart and
B slides from unison with A up to unison with C
traveling from the East corner to the Northwest
corner.

If you cross the map in any other direction than
parallel to those lines, C:A will change relationship
as you move. Look at the Northeast running line from
4:4:3 to 6:4:3 - what's the same there? Right, and
notice that in all triads on that line, B:A has a 4:3
ratio. Moving along that line or any line parallel to
it, the C is the only tone in motion with respect to
the other two.

Similarly, lines running straight North-South
represent triads in which C:B remains constant and
only the A moves with respect to the other two. Since
it's all relative, you can also keep A constant and
move C and B together, keeping their relationship to
each other constant but changing with respect to A.
The latter is what happens in my "tours" to date.

Moving in any direction other than parallel to the
sides, all three tones will change with respect to
each other. That happens only in the "Short Tour."

Hope this helps.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html

--

>3) The map can be split into otonal and utonal halves
>by drawing a line from the 1:1:1 to the center of the
>side that opposes the 1:1:1 corner, half-way between
>4:3:2 and 6:4:3. Rotating the map so that 1:1:1 faces
>straight South, the left side is otonal and the right
>side is its utonal mirror image. Once Paul is able to
>map actual triadic harmonic entropy, the utonal side
>will have lower hill tops than the otonal side.

This is not correct. There are some otonal chords on both sides, and some
utonal chords on both sides. But it is correct that any chord on one side is
the otonal/utonal twin of its mirror image on the other side.

>Tenney discordance ranks them equally.

Uh . . . dyadic discordance ranks them equally.

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

http://www.egroups.com/message/harmonic_entropy/244

David...

Could you please get me a "definitive" rewrite of your explanation,
after Paul makes the corrections?? I'm getting confused...

Joseph

Joseph wrote,

>after Paul makes the corrections??

I think I already did, right?

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
> This is not correct. There are some otonal chords on both sides,
and some
> utonal chords on both sides

Thank you.

> >Tenney discordance ranks them equally.
>
> Uh . . . dyadic discordance ranks them equally.

I must confess, I don't yet understand the mathematical methods used
to measure dis-/con- cordance and generate the graphs. Thus my
confusion about the terminology. Since the equation on the graph
includes the term "Tenney," I jumped to the conclusion that the
graph represented concordance by his method of measure. It seems
like the term "dyadic" would be as opposed to "triadic," and "Tenney"
would be an alternative method to some other mathemetician's. Is it
possible to use Tenney's method to measure either dyadic or triadic
concordance? Or is there no such thing as "Tenney concordance"?
Thanks for keeping it straight.

David

Doh! It says, "Sum of dyadic discordances" right on the top! Sorry.

Hi David.

"Tenney" refers to the fact that I seeded the dyadic harmonic entropy
calculation with all ratios below a certain Tenney complexity. As opposed to
the Farey or other series I've tried, this choice produces a harmonic
entropy curve with no overall trend as one moves from smaller to larger
intervals.

I'm not sure if Tenney does or does not have a formula for triadic
complexity. But I thought it prudent to emphasize that the otonal/utonal
symmetry comes from the fact that we simply added the dyadic discordances,
and has nothing to do with whether or not we used Tenney.

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" wrote:
> http://www.egroups.com/message/harmonic_entropy/252
>
> ... I thought it prudent to emphasize that the otonal/utonal
> symmetry comes from the fact that we simply added the dyadic
> discordances, and has nothing to do with whether or not we used
> Tenney.

I have a question about *why* summed dyadic discordance gives
a symmetrical layout. Is it because the numbers give all the
complementary pairs of intervals within the 'octave'... or what
exactly is it that's causing the symmetry?

-monz
http://www.ixpres.com/interval/monzo/homepage.html

>I have a question about *why* summed dyadic discordance gives
>a symmetrical layout.

It's because any chord and its mirror inverse have the same three dyads!