Plant cells under a microscope?

Looks like it, but it's actually a plot of the triads x:y:z with x, y, and z
less than or equal to 64, and the upper and lower intervals each between
250¢ and 550¢. http://www.onelist.com/files/tuning/triads.jpg. The three
inversions of the just major triad (clockwise from lower left, 4:5:6, 3:4:5,
and 5:6:8) are shown in red; the three inversions of the just minor triad
(10:12:15, 15:20:24, 12:15:20) are shown in blue; and the three inversions
of Xavier's minor triad (16:19:24, 12:16:19, 19:24:36) are shown in green.
Not surprisingly, the smaller the numbers in the chord, the larger the
"cell" around it. See if you can identify some of the other large cells.

Notes for those following the harmonic entropy discussion (see
http://www.ixpres.com/interval/td/entropy.htm to catch up):

These are Voronoi cells, each of which encloses all points closer to its
"nucleus" (the enclosed triad) than to any other nucleus on the plot. They
are not ideal -- you can see that surrounding the simplest chords (=largest
cells), and proceeding alongside each of the "veins" connecting them, there
are a bunch of rather large cells with their nuclei off to one side. The
bulk of these cells should really belong to the simplest chords and to the
thin cells comprising the veins, respectively. Also you can see a couple of
(mostly heptagonal) pointy cells at the confluence of lesser veins -- these
shouldn't own their pointy end. A 3-term analogue of a Farey series (whose
elements would be the nuclei), delimited by its mediants (which would
somehow define the cell walls), would capture this better, but I haven't
figured out how to define/calculate such a thing. Matlab has a built-in
Voronoi function, which made plotting this easy.

I took a look at the effect of using Voronoi cells instead of
mediant-to-mediant ranges for calculating harmonic entropy in the
1-dimensional (dyad) case. For N=20, there were some anomalous minima below
20/19 and above 19/10, due to the lack of fractions between those and 1/1
and 2/1, repsectively, leading to exceptionally large Voronoi cells for
these fractions. For N=40, though, the curve was qualitatively the same
shape as the curve using mediants. That makes sense because for large enough
N, each of the anomalously large cells (which contains a ratio of N or N-1
and is next to a very simple ratio) is close enough to its simple neighbor
(considerably less than a standard deviation away) that its largeness merely
contributes to the dip in harmonic entropy around the simple ratio.

Therefore I expect that for large enough N, an entropy calculation using the
Voronoi cells will be able to give a reasonable harmonic entropy surface for
triads. Computing this surface will be a very time-consuming task. One minor
complication is that the axes should really be at 120º angles, not 90º
angles, so as to treat the three intervals in the chords symmetrically as
regards distance on the plot (this is like the plot on the cover of John
Chalmers's _Divisions of the Tetrachord_ (see
http://www.ixpres.com/interval/chalmers/diagrams.htm), but here the
intervals add up to zero instead of 500¢). That would not affect the
relative areas of the cells but would change the height of a symmetrical 2-d
bell curve over most of them, decreasing those along the axis representing
the outer interval and increasing those perpendicular to it, so as to treat
the three intervals equivalently regardless of the order in which they are
taken.

Paul,
I had trouble trying to view it yesterday, but today with Carl's link had no
problem.

It's a beautiful and fascinating figure. (I've always been partial to the
look of Voronoi diagrams.) So, the largish cell near the southwest end of
the main vein would be 5:6:7 ...?

Your limit of 64 seems a reasonable stopping point, but do you really think
all those triads are really distinctly recognizable to listeners, even
well-tranined ones? I would guess not. If you agree, how would you reduce
the number of triads to more distinguisable ones?

> -----Original Message-----
> Message: 3
> Date: Wed, 15 Mar 2000 16:18:10 -0500
> From: "Paul H. Erlich" <PERLICH@ACADIAN-ASSET.COM>
> Subject: RE: Plant cells under a microscope?
>
> One of the points of the graph I created is to show that, as
> regards the central pitch processor (the part of the brain that tries to
> find harmonic series and produces the sensations of virtual pitch at their
> fundamentals), otonal triads are more "special" than utonal ones.
>
This point agrees with my own subjective experience. (And being human, I
tend to favor information that supports my pre-existing biases... ;-)

David Canright http://www.mbay.net/~anne/david/

I must have missed the beginning of this thread, but I'm wondering if it is
a start on the generalization of harmonic entropy to chords. Is that right?

John Link

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Paul Erlich wrote back on 14 March 2000:
> http://www.egroups.com/message/tuning/9241
>
> Subject: Plant cells under a microscope?
>
> Looks like it, but it's actually a plot of the triads
> x:y:z with x, y, and z less than or equal to 64, and the
> upper and lower intervals each between 250¢ and 550¢.
> http://www.egroups.com/files/tuning/triads.jpg

Paul, I've been fascinated with this diagram ever since
I saw it for the first time about 2 weeks ago.

For those who were curious about your challenge to figure
out the proportions of the triads for some the larger cells,
but who never bothered to figure them out, here are a
few of the less obvious ones, beyond those uncovered by
Darren Burgess in http://www.egroups.com/message/tuning/9243
and Dave Canright in http://www.egroups.com/message/tuning/9269:

Triads in this group are all rectangular cells on the 'main vein':

7:9:11 the largest cell midway between 4:5:6 and 3:4:5

11:13:15 the largest cell at the southwestern edge of
the 'main vein', below the '300' y-axis label
(a little more than half of it is off the graph)

9:11:13 the largest cell midway between 5:6:7 and 4:5:6

10:13:16 the largest cell midway between 7:9:11 and 3:4:5

8:11:14 the largest cell off the top of the graph on
the 'main vein'

This one is on the 'secondary vein':

9:11:15 11 cells northeast of 5:6:8

The rest of the ones I found were all intersections of
two of the smaller veins:

8:10:13 12 cells northwest of 5:6:8 and
10 cells southwest of 7:9:12

12:14:17 10 cells southeast of 5:6:7 and
10 cells southwest of 10:12:15

11:13:16 4 cells northeast of 12:14:17
6 cells southwest of 10:12:15

10:13:15 6 cells southwest of 9:12:14
the center of the cell is just off the graph
a little above the '450' label on the y-axis

8:11:13 off the top of the graph directly above 12:16:19

9:11:14 6 cells northeast of 10:12:15
6 cells southwest of 8:10:13
10 cells north of 6:7:9

9:12:16 7 cells directly north of 7:9:12

8:11:15 off the top of the graph in the far northeast corner

6:8:11 off the right side of the graph
7 cells east of 9:12:16
8 cells southeast of 8:11:15

Isn't the size of any given cell simply an artifact of
how many times the triad's proportions are repeated within
your given limit?

For example, 3:4:5, which has the largest cell on the plot,
also has the smallest sum-total 'integer complexity' (= 12),
and therefore is also represented within your 64-integer-limit
by the greatest number of redundant identical proportions:
6:8:10, 9:12:15, 12:16:20, 15:20:25, 18:24:30, 21:28:35, 24:32:40,
27:36:45, 30:40:50, 33:44:55, and 36:48:60. The other triads
have less redundancy, and correspondingly smaller cells.

Can you explain how this diagram 'works'? Your color graph
of total diadic harmonic entropy (damn... what's the URL?)
clearly dispalys a symmetrical structure, which gives equal
weight to both otonal and utonal perception, and which most
of us agree is not truly representative of the ear/brain system;
but the Voronoi diagram doesn't. It does indeed seem to portray
the 'specialness' of otonal perception. How?

I did notice some interesting properties that may explain
what's happening:

All the triads on the 'main vein' always have the same
integer-space between the terms in the proportion.
For example, 3:4:5, 4:5:6, 5:6:7 all have a difference of 1
between the terms; 7:9:11, 11:13:15, and 9:11:13 all have
a difference of 2, and 10:13:16 and 8:11:14 have a difference
of 3.

The triads along the other veins do not have this property,
but the larger of them do exhibit a *similar* (but not
identical) sort of regularity.

So, the triads along the 'secondary vein' (along a southwest
to northeast axis, in the bottom right part of the diagram)
always have an integer-space in the upper interval that is
double that in the lower interval. For example, 5:6:8 and
6:7:9 both have a difference of 1 in the lower and 2 in the
upper; 9:11:15 has a difference of 2 in the lower and 4 in
the upper.

I think I'm on to something here...

(unless, of course, my observation about the possibility
of all this simply being artifacts of the numbers and limits
is true...)

I agree with you that this approach seems to be the best way
to model 'tonalness'. Let's work on it some more!!

You also wrote:
> http://www.egroups.com/message/tuning/9279
>
> Subject: more Voronoi plots

that are supposed to be at Carl Lumma's website. But I can't
access them, and would really like to see them. Can you upload
them to your eGroups files folder?

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

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

> Isn't the size of any given cell simply an artifact of
> how many times the triad's proportions are repeated within
> your given limit?
>
> For example, 3:4:5, which has the largest cell on the plot,
> also has the smallest sum-total 'integer complexity' (= 12),
> and therefore is also represented within your 64-integer-limit
> by the greatest number of redundant identical proportions:
> 6:8:10, 9:12:15, 12:16:20, 15:20:25, 18:24:30, 21:28:35, 24:32:40,
> 27:36:45, 30:40:50, 33:44:55, and 36:48:60. The other triads
> have less redundancy, and correspondingly smaller cells.

You've observed some interesting and relevant facts, but I don't
think you've really
'explained' why 3:4:5 is largest -- in the diadic case, Joseph was
righfully surprised to
learn that the simplest ratios occupy the largest "slices" of the
real number line, and this
is analogous.
>
> Can you explain how this diagram 'works'? Your color graph
> of total diadic harmonic entropy (damn... what's the URL?)
> clearly dispalys a symmetrical structure, which gives equal
> weight to both otonal and utonal perception, and which most
> of us agree is not truly representative of the ear/brain system;
> but the Voronoi diagram doesn't. It does indeed seem to portray
> the 'specialness' of otonal perception. How?

By comparing chords in their full otonal representations.

> All the triads on the 'main vein' always have the same
> integer-space between the terms in the proportion.
> For example, 3:4:5, 4:5:6, 5:6:7 all have a difference of 1
> between the terms; 7:9:11, 11:13:15, and 9:11:13 all have
> a difference of 2, and 10:13:16 and 8:11:14 have a difference
> of 3.

That's just a by-product of how different interval-sizes are arranged
(in order, of course)
along the axes.

I'll e-mail you the other Voronoi diagrams privately -- they were
intended for Daniel
Wolf and Darren Burgess and don't really bear on harmonic entropy.