Re: [tuning] Plant cells under a microscope?

Paul,

Just so I know I understand this correctly, I would say:

- 6:7:9 is the largest cell to the "southwest" (along the vien) from the red
cell on the lower right
- 7:9:12 is 3 cells to the north of the green cell on the right
- 9:12:14 is 5 cells to the west of the green cell on the upper left

I would be interested to see the Voronoi plot that represents the 8 by 8
matrix, that is harmonics 8-15 by subharmonics 8 to 15, normalized to an
octave. Is this possible? Although as I look some of the possible triads,
I believe is may be a rather complex task, or one that isn't suited to such
an analysis.

What do you think?

Darren Burgess
SEJIS

> From: "Paul H. Erlich" <PERLICH@ACADIAN-ASSET.COM>
>
> 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.
>

Darren Burgess wrote,

>- 6:7:9 is the largest cell to the "southwest" (along the vien) from the
red
>cell on the lower right

Right.

>- 7:9:12 is 3 cells to the north of the green cell on the right

Right.

>- 9:12:14 is 5 cells to the west of the green cell on the upper left

Right.

>I would be interested to see the Voronoi plot that represents the 8 by 8
>matrix, that is harmonics 8-15 by subharmonics 8 to 15, normalized to an
>octave. Is this possible? Although as I look some of the possible triads,
>I believe is may be a rather complex task, or one that isn't suited to such
>an analysis.

>What do you think?

If I'm understanding you correctly, you'd like to see the triads possible in
the 15-limit tonality diamond, rather than the triads possible in the first
64 harmonics. I think I can do that, although it would require some work,
since I'd have to weed out all the identical triads, which was easy to do
with the frist 64 harmonics using Matlab's gcf (greatest common factor)
function. If I did do it, though, it would no longer have anything to do
with perception or harmonic entropy. For one thing, the graph will come out
symmetrical -- otonal and utonal chords will occupy cells that are exact
mirror images. 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.

Still want me to try it?

Paul,

Thank you for your response.

> If I'm understanding you correctly, you'd like to see the triads possible
in
> the 15-limit tonality diamond, rather than the triads possible in the
first
> 64 harmonics.

Yes

> I think I can do that, although it would require some work

Perhaps I could do the grunt work of assembling data if it can easily be
done by calculation.

> since I'd have to weed out all the identical triads, which was easy to do
> with the frist 64 harmonics using Matlab's gcf (greatest common factor)
> function.

Perhaps it could done by a plot of the triads x:y:z with x, y, and z less
than or equal to W and divisible by any interger between 2 and 15,
inclusive. 'W' is a relatively high figure, perhaps 640. For example the
triad 16/15, 5/4, 10/7 is represented in the harmonic series as 448:525:600,
if my calculations are correct. Limiting W would, I believe, limit the
complexity of the triads. This leaves two questions: Does this limit us to
the triads in the 15 limit tonality diamond? Does it eliminate the
identical triads?

> If I did do it, though, it would no longer have anything to do
> with perception or harmonic entropy. For one thing, the graph will come
out
> symmetrical -- otonal and utonal chords will occupy cells that are exact
> mirror images. 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.

Yes, I figured that my interest had little to do with your original intent.
I found the graph to be fascinating, and wanted to see how the diamond would
look. My primary concern is developing musical resources and compositions
from the 15 limit diamond, and perhaps a Voronoi diagram would provide an
interesting glimpse into the relationships found between the triads of the
diamond. Kind of like a lattice.

Perhaps generating a Voronoi diagram of the diamond would merely generate a
pretty (or not) picture that would be useless for any other purpose. If so,
then lets 86 the idea.

Darren

>Limiting W would, I believe, limit the
>complexity of the triads.

By a certain definition of complexity, yes.

>This leaves two questions: Does this limit us to
>the triads in the 15 limit tonality diamond?

No -- you won't find 1:5:125 in the 15-limit tonality diamond.

>Does it eliminate the
>identical triads?

No, although making that easier was the point of using a harmonic series
representation in the first place, right?

Anyway, I think I can just calculate these things directly from the diamond
and check for repeats, no need for tricks.

>Yes, I figured that my interest had little to do with your original intent.
>I found the graph to be fascinating, and wanted to see how the diamond
would
>look. My primary concern is developing musical resources and compositions
>from the 15 limit diamond, and perhaps a Voronoi diagram would provide an
>interesting glimpse into the relationships found between the triads of the
>diamond.

OK then! I'll e-mail it to you when I'm done.

>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?

You got it!