Re: Harmonic entropy/complexity: Dave Keenan's Golden Mediant

Hello, there, and this is a note to celebrate David Keenan's
contribution to the theory of "harmonic entropy/sensory dissonance" or
"complexity," applying a "Golden Mediant" function described here by
another mathematical thinker, Keenan Pepper.

As described in an article which I was privileged to co-author with
Dave Keenan, just posted to the Tuning List this afternoon, the
"Golden Mediant" function permits a quick index of the plateau of
maximal complexity, "harmonic entropy" (Paul Erlich), or "sensory
dissonance" (William Sethares) between two simpler ratios. Note that
the sensory dissonance concept of Sethares is timbre-sensitive, but
that the "Golden Mediant" may serve as a useful approximation over a
wide range of harmonic timbres.

A week ago, after seeing Keenan Pepper's post on "The Other Noble
Fifth" (which deserves much laudatory comment in itself), Dave Keenan
promptly realized that the Phi function there applied to finding a
super-Pythagorean counterpart of Thorwald Kornerup's famous "Golden
Meantone" tuning could also be applied to find regions of maximum
complexity or entropy.

The applications of the "two Keenans" both use Phi, often known as the
Golden Ratio, equal to approximately 1.61803398874989484820459.

As described in our joint article, Dave Keenan's "Golden Mediant"
function to find the region of maximum complexity between the simpler
interval ratios i:j and m:n, where i:j is simpler than m:n, is as
follows:

(i + m * Phi)
GoldenMediant = -------------
(j + n * Phi)

As we found, this Golden Mediant often although not always gives
results very close to the "maximum entropy" regions which Paul Erlich
has calculated using computerized techniques based on a statistical
model.

Sometimes the results may be close to Paul's "limited-weighted
midpoint" function recently posted here, as in the following example
borrowed from Paul's article illustrating both approaches.

Let us suppose we wish to find the point of "maximum entropy" or
"complexity" between a pure 4:3 fourth and a septimal 7:5 diminished
fifth. Using the Golden Mediant, and taking 4:3 as the simpler ratio,
we have:

4 + 7 Phi
--------- = ~560.067 cents
3 + 5 Phi

With Paul's "limit-weighted function," taking the size of the two
simpler ratios in cents (4:3 = ~498.045 cents, 7:5 = ~582.512 cents),
we have:

3 (498.045) + 7 (582.512)
------------------------- = ~557.172 cents
10

Here we multiply the size of each ratio in cents by the interval's
odd-limit (3 for 4:3, 7 for 7:5), take the sum of the two products,
and divide by the sum of the two odd-limits.

As it happens, the results for the 4:3-7:5 pair using the Golden
Mediant and the Erlich "odd-limit weighting" are within 3 cents of
each other.

In contrast, one of Paul's list of "harmonic entropy" maxima and
minima suggests a local maximum near 537 cents -- possibly a different
asnwer to a different question, since this result might be affected
not only by the "mediant" or "midpoint" between 4:3 and 7:5, but also
by the shallower "valley" or weaker "attractant" at 11:8 (~551.318
cents). This 11:8, by the way, is the classic mediant between the two
simpler ratios: (4+7):(3+5).

If we recast the problem to be solved with the Golden Mediant or the
limit-weighted function as finding the point of maximum complexity
between 4:3 and 11:8, then for the Golden Mediant we have:

4 + 11 Phi
----------- = ~541.419 cents
3 + 8 Phi

and for the limit-weighted function we have:

3 (498.045) + 11 (551.318)
-------------------------- = ~539.902 cents
14

These two results are quite close to each other, and either is not far
from Erlich's entropy maximum of 537 cents found with his computerized
statistical model based on probablility distributions.

In celebrating both Dave Keenan's new Phi-based function for maximum
entropy/sensory dissonance/complexity, and Keenan Pepper's catalytic
article on a different application of the same kind of mathematical
procedure, I would to share a humorous incident which may illustrate
the limited scope of my mathematical initiation.

Upon first learning of Dave's new algorithm, I wanted to try it on
some ratios. Choosing the pair 5:4-9:7, I excitedly attempted to
calculate:

5 + 9 * Phi
-----------
4 + 7 * Phi

Upon coming up with the familiar classic mediant at ~417.508 cents, or
14:11 (a favorite neo-Gothic integer ratio), I realized the problem: I
had followed an order of operations from left to right, rather than
recognizing the priority of multiplication over addition. Trying
again, I got the correct answer of ~422.487 cents, virtually identical
to Paul Erlich's entropy maximum at 423 cents.

As one who preferentially programs in the PostScript language and
likewise favors the GNU Emacs Calc program, both of which use Reverse
Polish Notation (RPN), I had not fully internalized the basic
conventions of priority regarding operations, something elementary for
an actual mathematician such as Dave.

More importantly, Dave's creative mathematical acumen and experience
together with his musical sensitivity led to the elegance of the
Golden Mediant of complexity, this concept growing also out of a
treasured contribution to the lore of neo-Gothic tunings by another
mathematical thinker and lover of music, Keenan Pepper.

There is much more to say, but I would like to express my deep
indebtedness to Dave Keenan and Keenan Pepper, "the two Keenans," for
having an opportunity to participate in these developments.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@egroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

http://www.egroups.com/message/tuning/12918

>
> The applications of the "two Keenans" both use Phi, often known as
the Golden Ratio, equal to approximately 1.61803398874989484820459.
>
> As described in our joint article, Dave Keenan's "Golden Mediant"
> function to find the region of maximum complexity between the
simpler interval ratios i:j and m:n, where i:j is simpler than m:n,
is as follows:
>
> (i + m * Phi)
> GoldenMediant = -------------
> (j + n * Phi)
>
> As we found, this Golden Mediant often although not always gives
> results very close to the "maximum entropy" regions which Paul
Erlich has calculated using computerized techniques based on a
statistical model.
>

Regarding the Margo Schulter post, which propitiously ties in with
Paul Erlich's harmonic entropy studies...

Isn't the application of this constant a little "odd," or, rather, I
should probably say "peculiar...??" How is it that the universe is
behaving in such tidy ways... a nice constant like that "doing the
trick."

There *IS* a Phi in the sky. I would enjoy a little "layman's"
cornerup on why this is working so well....??
__________ ____ __ __ _
Joseph Pehrson

Joseph Pehrson wrote:

> Isn't the application of this constant a little "odd," or, rather, I
> should probably say "peculiar...??" How is it that the universe is
> behaving in such tidy ways... a nice constant like that "doing the
> trick."
>
> There *IS* a Phi in the sky. I would enjoy a little "layman's"
> cornerup on why this is working so well....??

It relates to this bit from Margo and David's original post:

>If the sides of the ratios are considered separately, each may be seen to
>be a series of integers where every number after the second is the sum of
>the preceding two numbers. We say they are Fibonacci-like.

The ratio of adjacent terms in the Fibonacci series tends towards the
golden ratio. So it's not such a surprise that ratios of ratios of
Fibonacci-like series should tend towards something that also involves the
golden ratio.

Greetings all,

Long time no post.

I must, with respect, object to the title of this thread on two grounds:

---------
1. It isn't _my_ Golden Mediant. The purely mathematical result is Keenan
Pepper's. Despite his claim (in private email) not to have seen the general
result implied by his post "The Other Noble Fifth",
http://www.egroups.com/message/tuning/12592
I think it is clear that he would have seen it eventually, whereas I would
not in a million years have come up with a single one of his specific results.

It seems likely to me that this formula for the limit of these iterated
mediants (these other "noble numbers") has been published elsewhere. Has
anyone seen it?

I'd also like to give credit to Paul Erlich for supplying a piece of the
puzzle when he wrote, on 16 Mar 1999, in TD 105.10, regarding 13:8:

>It is very close to the Golden Ratio, and the golden ratio has
>fewer coinciding partials than any other ratio (for any given tolerance
>for what "coinciding" means, you have to go higher up in the partials of
>the Golden Ratio to find that degree of coincidence than you do for any
>other ratio).

2. My other objection to the title of this thread is that it is different
from the title of the original post to which it relates, which is
http://www.egroups.com/message/tuning/12915
and appeared in TD 810.3.

Joseph, I wonder if you missed the explanation of "why Phi" because of this?

Margo, I assume you used the present title in the hope of connecting with
those people following the very interesting harmonic entropy discussion.
This is a laudable aim which I think has now been achieved.
----------

So may I respectfully suggest that any future posts on this topic be posted
under the heading
"Re: The Golden Mediant: Complex ratios and metastable intervals".

Future readers who come into this thread, will at least find this message
directing them back to the original.

Regards,
-- Dave Keenan
http://dkeenan.com

--- In tuning@egroups.com, graham@m... wrote:

http://www.egroups.com/message/tuning/12960

> Joseph Pehrson wrote:
>
> > Isn't the application of this constant a little "odd," or,
rather, I should probably say "peculiar...??" How is it that the
universe is behaving in such tidy ways... a nice constant like that
"doing the trick."
> >
> > There *IS* a Phi in the sky. I would enjoy a little "layman's"
> > cornerup on why this is working so well....??
>
> It relates to this bit from Margo and David's original post:
> If the sides of the ratios are considered separately, each may be
seen to be a series of integers where every number after the second
is the sum of the preceding two numbers. We say they are
Fibonacci-like.
>
> The ratio of adjacent terms in the Fibonacci series tends towards
the golden ratio. So it's not such a surprise that ratios of ratios
of Fibonacci-like series should tend towards something that also
involves the golden ratio.

Thank you, Graham, for your commentary. I guess that would make
logical sense. My question, thought, I think has as much to do with
why this has any application to Paul Erlich's entropy curves... those
"curvey" things... but maybe I will learn more of this in subsequent
reading....

Thanks!!!
________ ____ __ _ _
Joseph Pehrson

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/12998

> Greetings all,
>
> Long time no post.
>

> 2. My other objection to the title of this thread is that it is
different
> from the title of the original post to which it relates, which is
> http://www.egroups.com/message/tuning/12915
> and appeared in TD 810.3.
>
> Joseph, I wonder if you missed the explanation of "why Phi" because
of this?
>

Whoops! This is funny... because I did the same thing, Dave, and
posted to the "wrong" subject line just like *you* did! I realized
it seconds after I hit the "send" button, and that you had requested
a change in the subject line!

All I have to say, which isn't much, is that I get the context now,
after re-reading (and Bookmarking) your (and Ms. Schulter's)
article...

Thanks!
___________ ___ __ __ _
Joseph Pehrson