Yahoo Tuning Groups Ultimate Backup harmonic_entropy All along the ridge tops

Now for a longer tour, if you're up for it. This one feels more like an airplane ride -
ascending for a while, cruising around a bit, then descending very gradually for a soft
landing. But it will be easier to describe it as a walk, or a glide, along the ridge tops.
Cutting across the map without regard to ridges, as we did in the first tour, is easy.
Surprisingly, it's a bit scary holding to the ridge tops. If the short tour made you
sea-sick, be sure to take your motion-sickness medicine now.

Looking at the graph "triangle_01," find 3:2:2 around half way up the right side. That's
where we start. We glide right along the top of the major ridge that heads Northwest, all
the way to 3:3:2. Making a hard right, we glide up the major ridge that leads to 4:3:2. Up
until now it's been all uphill - a 3:2 with the other tone ascending all the way from unison
to octave. Then we follow the edge down to 6:4:3. From there we make a soft right and
follow the third major ridge down to it's last big hilltop before the edge. Another hard
right, and we're following one of the minor ridges over to 20:15:12, up to the stack of
fourths near top center (all downhill from here - a continuous 4:3 with other tone descending
from 16:9 to unity), then make our first left hand turn of the day and follow the minor ridge
Southwest, crossing our path at 9:8:6 and continuing on to 4:4:3 on the edge. Now for the
final turn, another hard left, to follow the lowest of the moderately large minor ridges
across to our final destination of 4:3:3.

Along the way, we pause at most humps of any significance, as judged by their height compared
to surrounding heights. Each pause lasts 5 seconds, with 5 seconds of travel time in
between. Actually they're all slightly under 5 seconds. That's because, at Paul's
suggestion, I slightly mistuned each interval to prevent overtone phase lock. These flangey
triads are easier on the ear, but since each 5 second period starts with all partials in
phase and ends in an arbitrary phase relationship, I had to edit out part of a cycle at each
joint to prevent clicks and pops. You'll still hear sudden timbre shifts at many joints;
sometimes they even sound like sudden small pitch shifts but that's an aural illusion. By
the end we lose almost a second in edits. It's still probably better than enduring phase
lock. At least you'll have little trouble discerning the boundary points!

http://www.egroups.com/files/harmonic_entropy/Finnamore/ridgeglide1.mp3

Here's the itinerary. For the sake of brevity, let it be understood that each tetrad holds
for 5 seconds, then is followed by a 5 second glide to the next one.

Time Tetrad
0:00 3: 2: 2
0:10 12: 9: 8
0:20 15:12:10
0:30 6: 5: 4
0:40 9: 8: 6
0:50 3: 3: 2 - hard right
1:00 10: 9: 6
1:10 7: 6: 4
1:20 18:15:10
1:30 15:12: 8
1:40 4: 3: 2 - the top, soft right
1:50 6: 4: 3 - soft right
2:00 15:10:18
2:10 9: 6: 5
2:20 21:14:12
2:30 15:10: 9 - hard right
2:40 20:15:12 - soft right
2:50 16:12: 9 - stack of fourths, first left turn
3:00 5: 4: 3
3:10 9: 8: 6
3:20 4: 4: 3 - hard left
3:30 16:15:12
3:40 20:18:15
3:50 8: 7: 6
4:00 28:24:21
4:10 32:27:24
4:20 20:16:15
4:30 4: 3: 3

I'm not entirely sure I got everything right, especially toward the end. Corrections
welcome. I still have the pieces.

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

🔗Paul H. Erlich <PERLICH@...>

🔗Joseph Pehrson <pehrson@...>

--- In harmonic_entropy@egroups.com, "David J. Finnamore"
<daeron@b...> wrote:

http://www.egroups.com/message/harmonic_entropy/192

> Now for a longer tour, if you're up for it. This one feels more
like an airplane ride - ascending for a while, cruising around a bit,
then descending very gradually for a soft landing. But it will be
easier to describe it as a walk, or a glide, along the ridge tops.

David, this is really and totally incredible! I have my 3-D color
map and .mp3 file! I'm still trying to figure out all the details,
but, not matter... this is one of the most interesting things I have
ever done on EITHER tuning list.

HA! The main "Tuning List" posters won't know what they're
missing... (unless, of course, they come over here too)

Joseph

🔗jon wild <wild@...>

Nice one David! Would it be easy to produce an alternative version, where
depending on which direction you're going, different voices "glide",
instead of the bottom one always remaining at a fixed pitch? It seems this
would go along with the symmetry of the Chalmers plot, which doesn't
favour any one of the three intervals.

I can imagine two ways of doing this. First, if you move along the "/"
side (as the triangle appears in
http://www.egroups.com/files/harmonic_entropy/Finnamore/triangle_01dfinn_01.jpg
, i.e.
2:2:1
| \
| \
| \ 2:1:1
| /
| /
| /
1:1:1

) then the interval between lower voices remains the same, so I would hold
them constant and move the top voice. Any motion parallel to this within
the triangle would also move only the top voice. Likewise, motion parallel
to the \ side would move only the middle voice. Both of these already
happen in your mp3 tours. But motion parallel to the | side would move the
bottom voice, making it no more important than the other two, and then
motion along any wavefront would only ever involve one voice moving. The
disadvantage is that if you went around the edge of the triangle
anticlockwise from 1:1:1 back to 1:1:1 you'd end up an octave higher than
you started... Making smaller closed tours within the triangle would
result in smaller absolute shifts, up or down depending on direction. This
could be an interesting effect, but if Paul ever makes his applet (I
thought of something like it too, when I first made a terrain picture, but
I looked into getting Java to order a soundcard driver around, and it
wasn't pretty, at least not to me) then this feature would need a careful
implementation.

The other way I can imagine is to consider that motion *perpendicular* to
one side only affects two voices. Then, only if you moved directly due
'East' on David's version of Paul's triangle (I can't remember if the
original is oriented the same way), the bottom voice would stay constant.
Similarly, motion perpendicular to the / side would hold the top note
still and motion perpendicular to the \ side would hold the middle note
still. For other directions these voice-leading rules would be combined,
and since none of the ridges are perpendicular to any side of the
triangle, ridge-running inside the triangle would result in no voice ever
being held steady.

This was an interesting bit of the longer tour:

David Finnamore wrote:

> 3:50 8: 7: 6
> 4:00 28:24:21 [or 1/6 : 1/7 : 1/8]

Does anyone else hear the utonal version as smoother than the otonal
version, or is this an artifact of the slight detuning David introduced?

cheers --jon

🔗Paul H. Erlich <PERLICH@...>

Jon, I'll stick with my suggestion of keeping the geometric mean of the
three frequencies (arithmetic mean of the cents values) constant.

>> 3:50 8: 7: 6
>> 4:00 28:24:21 [or 1/6 : 1/7 : 1/8]

>Does anyone else hear the utonal version as smoother than the otonal
>version, or is this an artifact of the slight detuning David introduced?

I definitely hear the otonal version as posessing a lot more integrity and
the utonal version as having the vertiginous quality we've been discussing.
But purely in terms of "smoothness", i.e., roughness, they should be equal.

Jon, have you had a go at listening to the tetrads we've been discussing? It
would be interesting to see how you react to them.

🔗Paul H. Erlich <PERLICH@...>

🔗Jon Wild <wild@...>

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

> Jon, have you had a go at listening to the tetrads we've been
> discussing?

Hi Paul - yes, I listened a few weeks ago (at the time, some of what
I was hearing didn't match the labels - I mailed Joseph then and he
told me it was being fixed). I took some notes for myself, but I
can't find them just now... generally I like to listen to chords for
longer, and try them at different pitch levels. I usually use a
really simple Csound set-up for this, but I realise if we want to
compare impressions we should all be listening to exactly the same
soundfiles. When I have time I'll have another listening session with
the mp3s.

The roughness I was hearing in the 6:7:8 on David's tour was from the
virtual fundamental messing with these crappy computer speakers, I
believe now.

Paul, I can see qualitatively what restricting or loosening the value
of s does to entropy. But can you explain exactly how it figures into
the entropy calculations, in the pairwise case?

🔗Jon Wild <wild@...>

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
> Jon Wild wrote,
>
> >Then, only if you moved directly due
> >'East' on David's version of Paul's triangle (I can't remember if
> >the original is oriented the same way), the bottom voice would
> > stay constant. Similarly, motion perpendicular to the / side
> > would hold the top note still and motion perpendicular to the \
> > side would hold the middle note still.
>
> Aha -- I think that is identical to my proposal!

I think so too - I somehow missed a bunch of messages and just saw
your proposal. Egroups usually sends me a digest, but they were
having some problems yesterday, I believe.

🔗Paul H. Erlich <PERLICH@...>

🔗Jon Wild <wild@...>

--- In harmonic_entropy@egroups.com, "Paul H. Erlich" <PERLICH@A...>
wrote:
> Jon wrote,
>
> >Paul, I can see qualitatively what restricting or loosening the
> >value of s does to entropy. But can you explain exactly how it
> >figures into the entropy calculations, in the pairwise case?
>
> s is the standard deviation of the normal error ("bell") curve that
> represents the probablity distribution of the heard interval along
> the (logarithmic) interval axis. The actual heard interval is at
> the center and peak of this distribution.
>
> Is that exact enough?

Almost - I knew it was the standard deviation of your bell curve, but
I'm curious to see how you go about an actual numerical example,
starting with a series of ratios up to a certain product limit, and a
value of s, and deriving an entropy value for one point. Would you
mind?

🔗Paul H. Erlich <PERLICH@...>

Jon wrote,

>I'm curious to see how you go about an actual numerical example,
>starting with a series of ratios up to a certain product limit, and a
>value of s, and deriving an entropy value for one point. Would you
>mind?

Not at all. OK, to keep this manageable, let's use a product limit of 10,
and let's use 500 cents as our point, and let's use 10% as our s.

The ratios are

num den
0 1
1 10
1 9
1 8
1 7
1 6
1 5
1 4
1 3
2 5
1 2
2 3
1 1
3 2
2 1
5 2
3 1
4 1
5 1
6 1
7 1
8 1
9 1
10 1
1 0

The cumulative probabilities (the area of the bell curve from minus infinity
up to each of the mediants) are

0
0
0
0
0
0
0
0
0
0
0
6.6613e-016
4.0845e-009
0.4955
0.9868
1
1
1
1
1
1
1
1
1
1

The probabilities are the differences of successive entries above

0
0
0
0
0
0
0
0
0
0
6.6613e-016 -- 1/2
4.0845e-009 -- 2/3
0.4955 -- 1/1
0.4913 -- 3/2
0.013205 -- 2/1
1.1692e-008 -- 5/2
2.2553e-012 -- 3/1
0
0
0
0
0
0
0

The entropy is -sum(p*log(p)), so it's

2.3278e-014
+ 7.8896e-008
+ 0.34793
+ 0.34917
+ 0.05714
+ 2.1355e-007
+ 6.0482e-011
-------------
= 0.75424

How's that?

All along the ridge tops

🔗David J. Finnamore <daeron@...>

🔗Paul H. Erlich <PERLICH@...>

🔗Joseph Pehrson <pehrson@...>

🔗jon wild <wild@...>

🔗Paul H. Erlich <PERLICH@...>

🔗Paul H. Erlich <PERLICH@...>

🔗Jon Wild <wild@...>

🔗Jon Wild <wild@...>

🔗Paul H. Erlich <PERLICH@...>

🔗Jon Wild <wild@...>

🔗Paul H. Erlich <PERLICH@...>

🔗Jon Wild <wild@...>