About Noble mediants (was: How to Describe Tempered Intervals in Terms of JI)

Several posts showed much interest lately about the concept of "Noble mediants" published by Dave Keenan and mentionned by Margo Schulter in this passionating article (that covers many other subjects !) :

http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt

The concept of Noble mediant is indeed giving interesting results as a alternative mediant between two intervals in numerous cases.
More generally, it means certain more or less close intervals can be found to belong to precise series, and as such they will point to an infinity of other JI ratios, that will converge toward a "Noble mediant" value.
In a trivial example of 8/5 and 13/8, Dave Keenan's formula :
(8 + 13Phi) / (5 + 8Phi) gives a very precise approximation of Phi itself, (sqrt of 5 +1) / 2.
It indicates that we consider 13/8 to be more advanced in the series and as such has to be given Phi times more credibility, and the same law can be found in any Phi series : that may apply then to lots of intervals, and this formula can be applied indeed to any intervals.
However if we consider other intervals, belonging to other noble numbers series, such as
6/5 and 29/24
It is not certain that Phi will be the most adequate and natural noble mediant, while
12 + 29*2.414 / 5 + 12*2.414 ~ sqrt of 2 + 1 suggests another and more significative noble mediant.
And the following Aksaka converging series :
5 12 29 70 169... where 12/5, 29/12, 35/24, 169/70, etc... converge towards (sqrt 2) + 1,
could seem more pertinent and attractive, when Phi is replaced locally by (sqrt 2) + 1, in Dave's very same formula.
= many noble mediants...
- - - - - - -
Jacques

Igliashon/City of the Asleep wrote :

> Dear Margo,
>
> --- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> > Here's a paper I wrote a few years and revised with some helpful
> > suggestions from David Keenan and Andrew Heathwaite, to whom I
> > express my gratitude while emphasizing that the views expressed
> > are, of course, my responsibility and not theirs:
> >
> > <http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>
>
> Thank you for providing such a superbly-written and most helpful > resource! Your
> writing is clear, concise, and (dare I say) elegant, and this paper > is just what
> I needed.
>
> What fascinates me a great deal is the fact that the "nobly > intoned" mediants
> are essentially local maxima of harmonic entropy. Also, the fact > that there is
> no noble mediant in the major second range is quite telling > of...something. I
> feel as if I'm on the brink of revelation regarding harmony and > musical
> perception, but I can't quite get over the last barrier to > "enlightenment".
> There is something going on with the number phi, both in harmony > and melody. I
> look at Wilson's "scale tree" and notice that something special is > happening
> when, in an MOS scale, L:s = phi...but I can't put my finger on > what it is.
> Some great pattern is staring me in the face, if I could but bring > my focus to
> it. More meditation is in order.
>
> Thank you, though, for giving me another piece of the puzzle!
>
> -Igs

Dear Jacques and Igliashon,

Please forgive me for my delay in addressing this post! I have
been preparing the notes for my first tuning in the Ethno Extras
series, which have proved much longer than I might have guessed.
But your comments on the noble mediant, and some possible
alternatives, are most interesting, and indeed I would not be
surprised were there a variety of possible mathematical models
for finding or approximating these points of "maximum
complexity."

[Jacques]

> The concept of Noble mediant is indeed giving interesting
> results as a alternative mediant between two intervals in
> numerous cases.

> More generally, it means certain more or less close intervals
> can be found to belong to precise series, and as such they
> will point to an infinity of other JI ratios, that will
> converge toward a "Noble mediant" value.

This was indeed the thing that I excitedly realized when Dave
Keenan explained to me his "Noble mediant" concept in September
of 2000. While your following example with Phi itself is indeed a
classic illustration, some of our focus was on the "plateau" or
regoin of complexity between 5/4 and 9/7. I was excited to see
how by taking a series of mediants, or by directly taking a
mediant between these two simpler ratios using different
weightings, I could generate a great variety of just ratios.

> In a trivial example of 8/5 and 13/8, Dave Keenan's formula :
> (8 + 13Phi) / (5 + 8Phi) gives a very precise approximation of
> Phi itself, (sqrt of 5 +1) / 2.

> It indicates that we consider 13/8 to be more advanced in the
> series and as such has to be given Phi times more credibility,
> and the same law can be found in any Phi series : that may
> apply then to lots of intervals, and this formula can be
> applied indeed to any intervals.

When Dave recognized this relationship, prompted in part by
Keenan Pepper's proposal for his regular temperament based on a
logarithmic ratio of Phi between the sizes (e.g. in cents) of the
larger apotome and the smaller diatonic semitone or limma, a kind
of counterpart of Kornerup's famous Golden Meantone, we indeed
fuund that it seemed to fit many situations.

And Igliashon, your remark about Erv Wilson's scales at
logarithmic points of Phi reminds me that Keenan Pepper's
temperament, which was the catalyst to Dave's recognition of the
"Noble mediant," does appear as a Wilson horagram. Also, as if to
connect these two themes, the Wilson/Pepper temperament has an
augmented fifth from eight fifths up (e.g. C-G#) at 832.765
cents, very close to Phi itself at 833.090 cents. It's a happy
coincidence that in this particular "L:s = Phi" scale, Phi is
also closely approximated as an interval ratio: two distinct
things, of course, as Dave would often emphasize.

> However if we consider other intervals, belonging to other
> noble numbers series, such as 6/5 and 29/24
> It is not certain that Phi will be the most adequate and
> natural noble mediant, while
> 12 + 29*2.414 / 5 + 12*2.414 ~ sqrt of 2 + 1 suggests another
> and more significative noble mediant.

While I cannot speak for Dave, it seems intuitively likely to me
that various noble mediants or the like might best apply in
different situations. And I suspect that your experience with
differentially coherent JI might make you especially alert to the
fine points of these different possible models.

> And the following Aksaka converging series :
> 5 12 29 70 169... where 12/5, 29/12, 35/24, 169/70, etc...
> converge towards (sqrt 2) + 1,
> could seem more pertinent and attractive, when Phi is replaced
> locally by (sqrt 2) + 1, in Dave's very same formula. = many
> noble mediants...

One very small point: here, looking at the series, I suspect that
your "35/24" was meant to be "35/29," the octave reduction of
29:70, at 325.562 cents.

Such series might indeed be an opportunity to discover
alternative noble mediants, maybe sooner or later a good topic
for another paper, in which I would warmly encourage you.

Best,

Margo Schulter
mschulter@...