Re: TD 861 -- Wilson's footprints on plateau! (for Kraig Graidy)

Hello, Kraig Grady, and everyone.

Kraig, welcome back to the list! Your presence here is a great gift,
and I'd like to thank you so warmly both for making the Wilson
archives available and for calling my attention to one of the most
remarkable and moving diagrams I have seen.

Yes, the page you've made available -- I saved it as sctree19.gif in
MS-DOS, which isn't case sensitive -- shows Erv's footprints prominently
inscribed on the neo-Gothic plateau between 5:4 and 9:7. It also shows the
"Noble Mediant" between these two ratios marked with a large dot or small
circle, and identified as having a ratio of "1.2763932023," which I
quickly replicated with GNU Emacs Calc.

How moving it is to look at that chart and see the familiar ratios of
the region: 14:11, of course, near the top, and many others, some
quickly recognized and others new and enticing. I also see other
"Noble Mediants" between ratios such as 9:7 and 4:3, or 5:4 and
14:11.

It is almost like being in contact with some interstellar
civilization: that chart, at least to me, is a "universal language"
immediately comprehensible without need for linguistic explication.

Kraig, it seems that maybe we have a lot to discuss about larger
integer ratios, on list or off.

Above all, I'd like to confirm that Erv's scale tree indeed maps the
neo-Gothic region, "walks the plateau," and very directly and
beautifully presents the "Noble Mediant" concept.

Thank you for being here, and for this special gift -- and of, course,
my deepest admiration for Erv.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hi all,

Margo Schulter wrote:
>How moving it is to look at that chart and see the familiar ratios of
>the region: 14:11, of course, near the top, and many others, some
>quickly recognized and others new and enticing. I also see other
>"Noble Mediants" between ratios such as 9:7 and 4:3, or 5:4 and
>14:11.
>
>It is almost like being in contact with some interstellar
>civilization: that chart, at least to me, is a "universal language"
>immediately comprehensible without need for linguistic explication.

Margo, I think you are somewhat overstating the case here. An awful lot of
linguistic explication had already gone on before you saw that chart, and
I'm sorry to say you have still misunderstood it.

Note that Wilson calls it a "scale tree", not an "interval tree". The
fractions represent fractions of an octave (logarithmic, melodic), not
frequency ratios (linear, harmonic). Can you confirm this Kraig?

Of course the mathematics of "Noble Mediants" is the same in both cases,
but whether or not any particular noble number has any significance or not,
(and what that significance is), is quite different in the two cases.

In the logarithmic-melodic case (Wilson's case), the noble fractions of an
octave are scale generators. An interval whose frequency ratio is 2^n
(where n is a noble number) is stacked on top of itself (and octave-reduced
as necessary) until it forms a scale which has Myhill's property (that
there are only two interval sizes in any interval class). And so it has
only two sizes of scale step. This is called a "Moment of Symmetry" (MOS)
by Wilson. This property is important for appreciation or recognition of
melody, not harmony.

MOS scales can of course be generated with non-noble fractions of an
octave. A concise statement of the significance of these noble numbers in
the logarithmic-melodic case eluded me (and apparently Paul Erlich) until
Dan Stearns provided it. (I'm assuming it is true. I haven't checked it)

Dan Stearns wrote:
>I'm only considering scales where L/s = phi to be "Golden scales"...
>
>I think the one overriding characteristic of these scales is that they
>put a given Ls index into an arrangement where interval class
>distinction is uniquely "optimized". So if clear, uniquely defined
>interval classes are your thing, these generalized Golden scales would
>seem to be of interest.

One should note that the good melodic properties of noble generators are
very "broadly tuned". i.e. It may be quite acceptable for ratios between
steps of a scale (in cents) L/s may be anywhere from about 1.3 to 2.5
(centered around phi) and still give recognisable scales. The precise best
value for a generator is far more strongly determined by the harmony. i.e.
the desire to approximate simple ratios to obtain consonances. A 1% error
can be intolerable here. As Carl Lumma (approximately) said "Dissonances
are easy to find; consonances you really have to work at."

Margo Schulter wrote:
>Above all, I'd like to confirm that Erv's scale tree indeed maps the
>neo-Gothic region, "walks the plateau," and very directly and
>beautifully presents the "Noble Mediant" concept.

In the linear-harmonic case the significance of _some_ noble numbers, as
direct frequency ratios (n, not 2^n), is that of local maxima of
dissonance. But just as, when looking for local minima of dissonance, whole
number ratios a:b cease to be significant beyond some level of complexity
(a*b), so noble mediants between two ratios cease to give local maxima of
dissonance when any of the ratios go beyond that same level of complexity.

For example Noble_Mediant(4:5, 7:9) = (4+7phi):(5+9phi) gives a local
maximum of dissonance but Noble_Mediant(4:5, 11:14) does not. Somewhat
misleadingly, Noble_Mediant(7:9, 11:14) _does_ give a local dissonance
maximum. But that is only because it is exactly the same number as
Noble_Mediant(4:5, 7:9).

Margo, In your excellent articles on Keenan Pepper's neo-Gothic tuning you
make use of noble mediants for both purposes (melodic and harmonic). I
think the reader would benefit from a clear distinction between the two.

Also, part 2 may give the reader the impression that I would consider the
neo-Gothic near 17-tET to be somehow ideal because it maximises the
complexity/dissonance of the pythagorean major third. I think your
excellent spectrum chart could benefit from also showing the points that
correspond to maximally complex minor thirds and major and minor sixths.
This would show that they cannot all be maximised together and would, I
think, suggest a compromise somewhere around 29-tET.

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

David C Keenan wrote,

> Margo, In your excellent articles on Keenan Pepper's neo-Gothic
tuning you make use of noble mediants for both purposes (melodic and
harmonic). I think the reader would benefit from a clear distinction
between the two.

Perhaps it would be best to take Keenan Pepper's original post into
account again, as it was making a distinction between a 3/5, 4/7, ...
Fibonacci expansion, and a 4/7, 3/5, ... Fibonacci expansion; the
first being a "Golden fifth" after Kornerup, and the second being a
"Silver fifth" after Pepper.

After Keenan P's original "The other Nobel Fifth" post (in which he
suggested the term "Silver fifth"), Paul Erlich noted that this was,
"perhaps not a good choice, since the silver numbers are already
defined as the ones with a single integer over and over again in their
continued fraction expansions", and Graham Breed then posted a more
"grammatically" correct "Silver fifth" of [(2-sqrt(2))*1200].

Shortly after this Margo Schulter and Dave Keenan posted their joint
effort "The Golden Mediant: Complex ratios and metastable intervals".
And from Dave K's nice little closed form expression (which he had
extrapolated from Keenan P's original post) and my prior use of Ls
scale indexing from the Stern-Brocot Tree, I in turn put together a
simple "generalized Golden scale algorithm" for scales that have
Myhill's property within a given periodicity. While I was first
posting these ideas of a generalized Kornerup (so to speak), Paul
Erlich and Kraig Grady pointed out that this idea was essentially
synonymous with Erv Wilson's "Golden Horagrams of the Scale Tree".
Having it all framed in my mind somewhat differently I had failed to
see this...

Anyway, from the perspective of Keenan P's original post and all that
followed, maybe it would be nice to have two terms that relate to
mediants (logarithmically or linearly) in a sequential (i.e., low to
high) and reversed (high to low) Fibonacci expansion. The first is
obviously a "Golden" ______ (whatever; generator, mediant, etc.). The
second, if not a "Silver" ______ (whatever; generator, mediant, etc.),
then what?

Despite its friction with the standard definition, I think in the
generalized sense that "Silver" (whatever) works here. Any better
ideas?

In the logarithmic sense, a generalized "Golden generator" would
always give a uniquely optimized strictly proper scale, and a
generalized "Silver generator" would always give a uniquely optimized
improper scale.

In the linear sense, I don't recall Dave or Margo giving any meaning
to the reversed (high to low) Fibonacci mediant expansion, though I
may be mistaken? Personally I'd like to say that one is optimizing
something while the other is optimizing another (similar to the
strictly proper/improper distinction that occurs in the logarithmic
case), but least I hopelessly wander too far off into the dreaded
blear of "numerology" <Hi Dave!>, I'll leave it as it is and see what
the cognoscenti say.

--Dan Stearns

Dear Dan,

There is no need for two functions, such as the sort of "left" and "right"
noble mediants you seem to be wanting. We only need the single noble
mediant function, defined as Margo and I did, as the limit of the series
starting with the two fractions arranged in order of
_increasing_complexity_. Take your 3/5, 4/7 example.

3 / 5
4 / 7
7 /12
11/19
...
(3+4phi)/(5+7phi) = Noble_Mediant(3/5, 4/7) = Noble_Mediant(4/7, 3/5)

No surprises there I hope. Now switch the order.

4 / 7
3 / 5
7 /12
10/17
...
(4+3phi)/(7+5phi) = Noble_Mediant(3/5, 7/12) =/= Noble_Mediant(4/7, 3/5)

We just moved up the series until we found the first pair that _are_ in
order of increasing complexity.

Note also that the first sequence can be run backwards one step (to 1/2)
and still have the first two fractions in order of increasing complexity.
So its limit can also be expressed as Noble_Mediant(1/2, 3/5) =
(1+3phi)/(2+5phi).

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

David C Keenan wrote,

> Dear Dan,

Yes, I do understand this.

thanks,

--Dear Dan

Dear Kraig,

I'll have to take your word for it that Wilson intended those charts to
also be applied linearly, and found that noble mediants of adjacent
dissonance minima give dissonance maxima. But I haven't found anything on
your website that suggests this, apart from (the headache-inducing scan of)
Lorne Temes letter (http://www.anaphoria.com/temes.html) which only points
to the golden ratio itself (a single noble number) as "the" maximum of
dissonnance. Nor can I find anything in the egroups tuning archive that
relates a linear interpretation of noble numbers (or Wilson's diagrams) to
dissonance maxima, until Margo's and my "Golden Mediant" post at
http://www.egroups.com/message/tuning/12915

I must say however, that from where I sit, on the other side of the big
pond, this reminds me of having to take the word of the Apostle Paul as to
the meaning of certain enigmatic things that Jesus (may have) said. :-)

By the way, I had not seen that usage of the term "acoustic" before. I
always thought it just meant "relating to sound".

I certainly can't see a linear application of the scale tree in the first
diagram of http://www.anaphoria.com/tres.html

Dear Margo,

My apologies. You were obviously more perceptive than I.

Dear Paul,

In the "Golden Mediant" thread you wrote, of the logarithmic noble
generator between two adjacent fractions of an octave i/j and m/n :

>It is the generator which will produce MOS scales whenever the number of
>notes belongs to the Fibonacci-like series j, n, j+n, . . . and these MOS
>scales will all be proper (with a possible exception for the first few),
>since the ratio of the small to large steps in these MOS scales always
>approaches Phi.

Sorry Paul, this _was_ concise. I just failed to unpack from it the reason
why it was useful for L/s to be close to Phi, rather than merely proper or
strictly proper. i.e. the fact that this gives maximum distinction between
interval classes.

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

I wrote,

>>Sorry Paul, this _was_ concise. I just failed to unpack from it the reason
>>why it was useful for L/s to be close to Phi, rather than merely proper or
>>strictly proper. i.e. the fact that this gives maximum distinction between
>>interval classes.

Paul Erlich replied:

>Well, if you're going to create an infinite number of intervals, then in a
>sense this is true. But for a single MOS scale along the path, is there any
>way you can define "distinction between interval classes" such that L/s =
>Phi implies "maximum distinction between interval classes"?

A very good question. I should have written "the claim that" rather than
"the fact that". I believe it was Dan's claim. Do you have such a
definition Dan? (Anyone remember the Monty Python sketch about Oscar Wilde.
"It was one of Wilde's") :-)

At first glance I thought it was possible, but now it seems to me that the
greatest distinction between interval classes occurs when all the scale
steps are the same size. So what is it we are really trying to optimise here?

Kraig, what does Wilson claim is special about logarithmic noble
generators? What is the musical significance of the ratio of large to small
step sizes (in cents) being phi rather than say 1.5 or 1.7?

What does Kornerup claim for his golden meantone?

Regards,

-- Dave Keenan
http://dkeenan.com

David C Keenan wrote,

> A very good question. I should have written "the claim that" rather
than "the fact that". I believe it was Dan's claim. Do you have such a
definition Dan?

No, not really (unless a hunch qualifies as a definition). I see it as
a unique and fascinating way to articulate any scale that has Myhill's
property within a given periodicity.

But it would seem to me that one must ask the same types of questions
of the linear use of Golden or Nobel mediants: is ~833� really any
better point of complexity between 1:1 and 1:2 (etc., etc.) than any
other; what's the "proof"?

--Dan Stearns

Kraig wrote:

>I think that Erv's experience is that these are not in fact maximally
dissonant when tuned up.
>This being only the point of departure that lead to his investigation. I
cannot articulate much more w/o speculating as to what he thinks. too
dangerous and not on solid ground. He might in fact have trouble with m
first statement.
>

All,

In our paper, Margo and I invited others to tune them up and listen for
themselves. We don't claim that _all_ noble numbers give dissonance maxima,
nor are _all_ dissonance maxima near the noble mediant of the dissonance
minima on either side. Notable exceptions appear near the unison, octave
and fifth. Because the maxima are so broad you will probably need to find
two places either side where you notice that the dissonance has dropped
slightly, and interpolate.

Dan,

Maybe the above is the answer to your challenge too. Good on you Dan. I was
rather disappointed that no-one questioned anything when we first posted it.

Or maybe you are asking for the purely mathematical proof that the noble
mediant is the most complex "ratio" between two simple ones? That looks
very tricky, involving infinities as it does. How could one irrational be
said to be more complex than another? Good question. I might simply end up
defining complexity for irrationals in such a way that the statement comes
out true.

I wrote:

>>I certainly can't see a linear application of the scale tree in the first
>>diagram of http://www.anaphoria.com/tres.html

Kraig replied:

>It seems clear that in this case he is focusing on using the tree to
generate epimoric ratios. Logarithmic formulas have no special use of
epimores. It is in relation to the tetrachord. I find it hard to read as
logrithmic. which is possible but not fruitful!
>

Kraig,

Thank you for your kind words (in a subsequent post). And thank you for
hosting Wilson's stuff on your website. Any scan is better than no scan.

What does "epimoric" mean?

I should have said that I could not find a logarithmic interpretation of it
either. I simply cannot figure out what it is trying to show. What is the
significance of not reducing the 4/4 and so ending up with a tree that does
not seem to be a section of the Stern-Brocot tree as it claims, since
fractions connected by lines are not adjacent. I have to admit it seems
more likely that the fractions relate to frequency ratios rather than
fractions of an octave, but it does not make any use of noble numbers. So
let's drop it.

My question is not so much _whether_ Wilson has applied the Stern-Brocot
tree to both fractions of an octave and frequency ratios, but what musical
or psycho-acoustic significance he gives to the noble numbers in each case.

Carl,

My question is not "What is special about MOS?" but "What is special about
MOS with noble generators?".

Paul has answered it fairly well I think. But only for scales with an
infinite number (or very large number) of notes, which doesn't seem all
that relevant.

I suggest that there's nothing special about noble generators for scales
with a reasonable number of notes. It is far more important to choose a
generator that maximises consonance or minimises beats. You can always find
a log noble generator that closely approximates whatever you like. The one
in meantone just happens to be a fairly low-order one.

All,

Thanks to Erv and Kraig, Kornerup on golden meantone is at
http://www.anaphoria.com/korn.html

I had to have several goes at it, with rests in between to pluck up enough
psycho-visual fortitude for those scans. :-) It doesn't seem to establish
anything beyond what Paul has said, but seems to be claiming more.

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

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> http://www.egroups.com/message/tuning/14128
>
> I wrote:
>
> >> I certainly can't see a linear application of the scale tree
> >> in the first diagram of http://www.anaphoria.com/tres.html
>
> Kraig replied:
>
> > It seems clear that in this case he is focusing on using the
> > tree to generate epimoric ratios. Logarithmic formulas have no
> > special use of epimores. It is in relation to the tetrachord.
> > I find it hard to read as logrithmic. which is possible but not
> > fruitful!
> >
>
>
> What does "epimoric" mean?

Hmmm... I thought I had covered this and all the related
terms in my Dictionary a few months ago, but on investigating
now, I see that there was a dead link that broke the whole
hypertext chain. <groan again> It's fixed now:

http://www.ixpres.com/interval/dict/epimorios.htm

It directs the reader to the original Greek term 'epimorios':

> Term for ratios of the form (n+1)/n.
>
> [from John Chalmers, _Divisions of the Tetrachord_]

In a nutshell, it's the Greek version of 'superparticular'.

Along the way, I spruced up Carl Lumma's definition of
'efficiency' and added that too:

http://www.ixpres.com/interval/dict/efficiency.htm

As always, feedback is appreciated, especially detailed
excursions to which I can link from the Dictionary.

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

David C Keenan wrote,

> How could one irrational be said to be more complex than another?
Good question.

See the wonderful, action packed "The Family Of Metallic Means" link
that Paul gave yesterday (section 3 for a sort of "Golden Mean is the
most irrational of all irrational numbers" proof, and section 8 for
what I think is a nice/fair aesthetic summary):

<http://members.tripod.com/vismath1/spinadel/>

> I suggest that there's nothing special about noble generators for
scales with a reasonable number of notes. It is far more important to
choose a generator that maximises consonance or minimises beats.

In a way, I both agree and disagree with that last sentence. Right
from the start, I just intuitively felt that L/s = Phi scales might be
a good candidate for globally striking a similar balance. (Similar in
that a hoped for, generalized "compromise" that leans towards a
balance of consonance and proportion would result.) Naive? Sure, I
would imagine so. But from where I stand that's OK... and often times
intuitive "naivete" is the best way to go about certain things (in my
opinion).

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

http://www.egroups.com/message/tuning/14146

> See the wonderful, action packed "The Family Of Metallic Means" link
> that Paul gave yesterday (section 3 for a sort of "Golden Mean is
the
> most irrational of all irrational numbers" proof, and section 8 for
> what I think is a nice/fair aesthetic summary):
>
> <http://members.tripod.com/vismath1/spinadel/>
>

This is a very intriguing page... however, I believe my math ability
is not entirely up to the task! :(

I'll keep trying...

I did have a question, though concerning the diagram. The "golden
mean" is linked to pentagonal symmetry in a diagram which contains,
within it, a hexany figure...

So, could somebody explain to me, in layman's terms, whether the
hexany CPS process is linked to the golden mean?? It rather looks
like it might be...
________ ____ __ __ _
Joseph Pehrson