Replies to Dan and David on locally concordant tetrads

Dan Stearns wrote (off-list),

>Hi Paul,

>Hmm, some really interesting surprises! However, I'm still pretty
>unsure exactly what this is actually modeling, or saying... because as
>you said, "the otonal 7-limit tetrad is clearly more concordant than
>this ranking implies, which reflects the fact that this model ignores
>synergies between three or more notes. A true triadic or tetradic
>harmonic entropy model (rather than a sum of dyadic harmonic entropy
>models, which is what this is) would take this into account." So
>exactly what does this sort of a "sum of dyadic harmonic entropy
>models" say/mean?

(1) Since dyadic harmonic entropy looks very much like dyadic roughness for
a certain choice of timbre and register, and since, according to Kameoka and
Kuriyagawa, roughness of chords _is_ a simple sum of the roughnesses of the
dyads, what we have here is a proxy for tetradic roughness.

(2) If you _do_ play the notes of the tetrad only two at a time, and choose
all pairs with equal frequency/probability, you have a good measure of the
expected harmonic entropy of the resulting music. This is the interpretation
that I would lean toward when I find these local minima, as I plan to do
(with different methods), for sets of considerably more than four notes
(i.e., scales rather than chords).

David Finnamore wrote,

>> The next most concordant tetrad was:
>> 0 498 886 1384¢
>> or 9:12:15:20 or 1/1:4/3:5/3:20/9. It is an open-voiced JI minor seventh
>> chord in third inversion, or an open-voiced JI major added sixth chord in
>> second inversion.

>Sound familiar, fellow guitarists? I learned that tetrad at age 7 on the
>ukulele as, "My dog has fleas." But, of course, it's also the
(approximate) >standard tuning of the lowest four strings on the guitar.
How did ye olde >guitar and ukulele players know how to do that without a
supercomputer? Tee >hee.

Unfortunately, if you do tune the lowest (highest in pitch) four strings on
the guitar exactly like that, and tune the other two in successive 4:3
fourths, your lowest string will be a comma off from your highest string.

>Paul, I _love_ the graphic you posted for triads (enttriad.jpg). I just
can't >quit staring at it!

If there's any particular slice through the tetradic solid you'd like me
graph, let me know.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> Dan Stearns wrote (off-list),
>
> >Hi Paul,
>
A true triadic or tetradic
> >harmonic entropy model (rather than a sum of dyadic harmonic
entropy
> >models, which is what this is) would take this into account."

So, would this be possible to do??... or is it too difficult to model
mathematically? And, how different would it be than the
diadically-inspired model that Paul just did??
__________ ____ __ __ __
Joseph Pehrson

[Paul Erlich:]
>>> A true triadic or tetradic harmonic entropy model (rather than a
>>> sum of dyadic harmonic entropy models, which is what this is) would
>>> take this into account."

[Joseph Pehrson:]
>>So, would this be possible to do??... or is it too difficult to model
>>mathematically?

[Paul:]
>A few months ago I posted some ideas and graphs involving Voronoi cells
>(see http://www.egroups.com/files/tuning/triads.jpg) which would be an
>approach to solving this problem. Unfortunately, I can't do the
>calculations at the moment, because, though Matlab can give you the set
>of line segments making up the full spread of Voronoi cells for a set
>of points, it provides no way to then find just the edges of the
>Voronoi cell corresponding to a single, selected point. Perhaps I just
>need the help of a clever programmer on this list . . .

OK, I'll bite, Paul. I'm a programmer, and potentially clever enough,
and I have a native C++ compiler with no arbitrary limitations such as
you seem to have to put up with.

Could you send me some quasi-readable source code for the binary
calculations you do, that I could port over to C++ without having to
study "Farley" thingamagigs in great detail? That'd help. Then, those
"Voronoi cells": got any pseudo-code on that? My fear is that it'd be
a tar-pit of confusion. I don't always delve deeply into these
numerical games on the list, BUT I have to admit I'm intrigued by the
potential results of the task. Surely 4:5:6:7 would float toward the
top compared to the myriad of chords that "beat" it in the binary
summation calculation!

Am I correct in thinking that the more refined method would in effect
calculate difference frequencies and consider them in the more
complete effect?

JdL

Paul H. Erlich wrote:

> Unfortunately, if you do tune the lowest (highest in pitch) four strings on
> the guitar exactly like that, and tune the other two in successive 4:3
> fourths, your lowest string will be a comma off from your highest string.

Of course. That's why I said "approximate." Also, by "lower" I meant the 4 closest to the floor, not the ones lowest in pitch. Always a source of confusion with guitars! So what I meant to point out was that the standard D-G-B-E tuning of the 4 highest-pitched guitar strings approximates the tetrad that was found most consonant by your program. I found that very interesting.

--
David J. Finnamore
Nashville, TN, USA
http://members.xoom.com/dfinn.1
--