Re: The Golden Mediant: Complex ratios and metastable intervals

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

Dan Stearns wrote [TD810.9]:

>One quick point or question on a first read through... I don't think
>it's always clear that Golden Mediant algorithm and the mediant method
>are essentially asking opposite questions; in other words, I'm seeing
>the Golden Mediant as a wonderful little closed-form shortcut to a
>sort of "eventuality of mediant iterations."

Yes. That's precisely what Pepper's function gives us.

>But isn't the mediant
>method in its usual application generally being asked to give the
>*least* complex (or most stable) point between two others, with only
>successive iterations eventually closing in on maximally complex zone?

Quite right. But we have to be careful here since mathematical complexity
doesn't always correspond with subjective complexity.

>It seemed as though all these points are nicely delineated at one
>point or another but perhaps not consistently clear -- maybe I should
>just read through it a few more times!

No. You're right, it isn't consistently delineated. This is because, once
you get one or two iterations past any ratios that could possibly be simple
enough to be attractors, there really isn't much difference between further
mediants and the golden mediant from a psychoacoustic perspective. However,
this may also represent a compromise of different emphases between authors.
I'm quite happy to ignore 14:11 and 23:18 as irrelevant and go straight to
(5+9phi):(4+7phi), for example.

However it should be noted that nothing special is detected precisely at
the golden mediant (at least _I_ can't detect it). The effect is very broad
and quite unlike the "locking-in" of a precise simple ratio such as 5:4.

>So, what do the significantly larger discrepancies between these two
>mediant methods mean (the 5/4 6/5 example for instance)? That the
>traditional mediant is a non trivial distance from the actual point of
>maximum complexity (i.e., the Golden Mediant)... that would be my
>take.

Yes. That's my take on it too. But in line with what I said above, one
could also claim that 11:9 is a significant attractor.

>Anyway, looking forward to reading it a few more times, and once
>again -- my compliments to the chefs!

Thanks for your kind words Dan.

I must say that working on this paper with Margo, in whatever spare time I
could muster these past two weeks, has been a very enjoyable experience.
She is such a wonderfully "civilised" person. I think I mean that in the
sense that Gandhi took when in answer to the question "What do you think of
Western Civilisation?" he replied "It would be a very good idea".

In fact, you're all wonderful people, and I find this to be a very
civilised list.

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

Oops! Sorry.

An earlier post of mine in this thread, won't be making much sense to
anyone. Silly me. I meant to post it to the thread titled
"Re: Harmonic entropy/complexity: Dave Keenan's Golden Mediant"
as I have now done.

-- Dave Keenan
http://dkeenan.com

David C Keenan wrote,

> Quite right. But we have to be careful here since mathematical
complexity doesn't always correspond with subjective complexity.

Right, though I do see mediants, in their standard
application here, as the most stable point between two others, with
the real problem being at what point is that no longer relevant. But
as you point out, a need to reconcile, or even consistently
differentiate, musical (or subjective) relevance and psychoacoustic
relevance only muddies up the waters... In all likelihood it would
seem that the best we can hope for is ever improved and better
understood/explained rules of thumb.

> I'm quite happy to ignore 14:11 and 23:18 as irrelevant and go
straight to (5+9phi):(4+7phi), for example.

Well oddly enough, I was just in the midst of an off-list "discussion"
of the 14/11... and my gist of my view was that I'd be hard pressed to
call it an "attractant", but I sure couldn't tolerate such a barren
expanse of nothingness between a 5/4 and a 9/7! that's just miles and
miles of real estate to my mind!

> Thanks for your kind words Dan.

Well it's good to see you back Dave, if in fact you are back... you
were always one of my favorite "math guys" at the tuning list, because
you always manage to explain difficult things in a very deliberate and
easy to understand manner.

- dan

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

http://www.egroups.com/message/tuning/12994

> 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?
>
Ummm. Actually, I reread this, and now I'm "getting" the context.
Thanks, David!!!
___________ ____ ___ _
Joseph Pehrson

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

http://www.egroups.com/message/tuning/13002

>
> Well oddly enough, I was just in the midst of an off-list
"discussion" of the 14/11... and my gist of my view was that I'd be
hard pressed to
> call it an "attractant", but I sure couldn't tolerate such a barren
> expanse of nothingness between a 5/4 and a 9/7! that's just miles
andmiles of real estate to my mind!
>

Hi Dan!

Would you mind keeping such tuning discussions "on list..??" You
just make the rest of us curious/yellow.
___________ ____ __ _
Joseph Pehrson

What about the idea of applying The Golden Mediant to temperaments --
specifically MOS generators?

In other words, say we take the 2s5L/5s2L mapping:

1/2 3/5
4/7
5/9 7/12

The Golden Mediant taken (i/j, m/n), or (1/2, 3/5), gives the 2s5L
Golden Mediant generator of ~696.21�, and the following temperament:

0 192 385 504 696 889 1081 1200
0 192 311 504 696 889 1008 1200
0 119 311 504 696 815 1008 1200
0 192 385 577 696 889 1081 1200
0 192 385 504 696 889 1008 1200
0 192 311 504 696 815 1008 1200
0 119 311 504 623 815 1008 1200
0 192 385 504 696 889 1081 1200

The Golden Mediant taken (m/n, i/j), or (3/5, 1/2), gives the 5s2L
Golden Mediant generator of ~672.85�, and the following temperament:

0 146 291 527 673 819 964 1200
0 146 381 527 673 819 1054 1200
0 236 381 527 673 909 1054 1200
0 146 291 437 673 819 964 1200
0 146 291 527 673 819 1054 1200
0 146 381 527 673 909 1054 1200
0 236 381 527 763 909 1054 1200
0 146 291 527 673 819 964 1200

Looks awfully promising to me!

- dan

Paul Erlich wrote:
>Fascinating Margo! I assume you are familiar with Pierre's discussions of
>the Stern-Brocot tree, and with its identical twin, Wilson's Scale Tree,
>where Wilson meticulously works out all these "Golden Mediants"?

I was aware of the Stern Brocot tree and Farey series but not a closed-form
expression for the limits. Re Wilson, see below.

>I believe this formula:
>
>> (i + Phi * m)
>> GoldenMediant(i/j, m/n) = -------------
>> (j + Phi * n)
>
>is quite similar to something presented in the appendix to one or both of
>Manfred Schroeder's Springer "pop-math" books,

I'm guessing it's in his 'Number Theory in Science and Communication'. The
whole book sounds great. The review says it's written to be understandable
by people with only (advanced?) high school math (which is of course what
Paul means by "pop-math").

>and Wilson applied it to the
>scale tree, as you'll see at http://www.anaphoria.com/hrgm01.html

Thanks. I hadn't seen this before. And even if I had I probably wouldn't
have deciphered it!

Clearly Wilson predates Schroeder, and who knows where he got it? Some
amncient Arab or Indian probably figured it out first.

Does anyone want to suggest another name for it rather than "golden
mediant"? Would anyone prefer "noble mediant"? Or is there a problem with
overloading the term "mediant" in either of these ways? "Phi-weighted
midpoint" just seems too long to me and fails to make the point that it is
a limit of mediants.

I think it's important to make it very clear that the golden mediant is
being applied to two completely different musical purposes.

1. Melodic or logarithmic (fractions of an octave) (e.g. Wilson, Kornerup,
Lamothe, Pepper, Stearns)

2. Harmonic or linear (interval ratios) (e.g. Schulter, Keenan)

Could someone please explain to me the significance of golden mediants (or
noble numbers) in the first application? At present, it just looks like
numerology to me. Is it somehow related to Rothenberg propriety?

Carl Lumma wrote:
>Paul, I asked you on my first visit what is meant by 'simplicity' when
>we say that the mediant of two ratios is the simplest ratio between them.
>You said perhaps n*d. Does anybody know? I seem to remember reading
>a proof of this property of mediants -- on Cut the Knot? -- link would
>be appreciated.

I think that n*d may be standard, but it seems like just about any function
which is a monotonically increasing function of both n and d would work.
e.g. n+d.

>Anywho, take 4/1 and 1/1. Mediant 5/2 seems more complex than 3/1.
>Ouch!

Mediants (and therefore Golden Mediants) are only defined on (or at least
their interesting properties only occur when they are applied to)
"adjacent" fractions. "Adjacent" here has the technical meaning of
"adjacent in some Farey sequence". A simple way to determine whether two
fractions are adjacent (which doesn't require any mention of Farey
sequences) is to perfoms two "cross-multiplications" as follows.

Two fractions i/j and m/n are _adjacent_ if and only if i*n and j*m differ
by exactly 1.

This is a point I failed to make in Margo's and my paper. Sorry.

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

David C Keenan wrote,

> Could someone please explain to me the significance of golden
mediants (or noble numbers) in the first application? At present, it
just looks like numerology to me. Is it somehow related to Rothenberg
propriety?

Ah, numerology... I haven't heard that in a while -- I even kind of
miss it! The Stern-Brocot tree can easily be seen in terms of
propriety if you substitutes "s" and "L" for "x" and "y" -- see:

<http://206.4.57.253/editorial/knot/SB_tree.html>

2s+L and s+2L being the first mediants to define "proper" scales where
s and L are not equivalent or trivial terms... 3s+2L and 2s+3L being
the first mediants in the "strictly proper" zone; i.e., mediant
iterations falling between 2s+L and s+L, and s+L and s+2L (within a
non-trivial "cap", or condition that distinguishes between
meaningfully strictly proper scales and the two borders)... and 3s+L
and s+3L being the first mediants in the "improper" zone; i.e.,
mediant iterations falling between s and 2s+L, and s+2L and L (again,
within some non-trivial "cap", or condition that distinguishes between
improper scales and their two borders of proper scales, and L=0 and
s=0 "scales").

My gut feeling (numerological intuition...) is that the "golden
mediants" of s, s+L and s+L, L give some "special" proportional
rendering of the basic two stepsize mappings given that they are all
"adjacent" fractions as well. Note that they all cut to the chase, and
shore up interval class distinction as strictly proper, meantonesque
scales.

The results look good... if the logic's more than suspect, well, lay
it on me! These kinds of collisions between linear and logarithmic
designs really turn me on (ahem, so to speak), so it's possible that
I've simply overheated my engines and jumped the gun on all this... I
don't think so though -- let me know.

dan

Are you there Margo? What do you think of changing the name of the function
to "Noble Mediant", since the numbers that result from applying the
function are apparently already called "noble numbers".

Hey, has anyone noticed that Keenan Pepper hasn't been on the list since I
came back. Maybe we really _are_ the same person!

I wrote:
>>Could someone please explain to me the significance of golden mediants (or
>>noble numbers) in the [melodic/logarithmic] application? At present, it
>>just looks like
>>numerology to me. Is it somehow related to Rothenberg propriety?

Paul Erlich replied:
>Yes, it is. More directly, it's related to MOS scales. Look in the archives
>from a few months ago,

I already did that, and failed to understand anything of the following kind:
e.g. "The noble mediant between two fractions of an octave 2^(i/j) and
2^(m/n) gives scales which have special property X(i,j,m,n)." Can't someone
just tell me what X is supposed to be?

>and spend more time staring at Wilson's papers.

I tried that too. It just gives me a headache. How come there's no text to
provide the keys to all these beautiful but obscure diagrams?

I wrote
>>[That the Noble Mediant only applies to adjacent fractions] is a point I
>>failed to make in Margo's and my paper. Sorry.

Paul Erlich replied:
>Where is this paper?

http://www.egroups.com/message/tuning/12915
It's the paper that started this thread.

---------------------------------------------------------------------
I think the main message of that paper is that we often mistakenly assume
that what we want in a temperament is simply to minimise the dissonance of
certain intervals by minimising their distance from JI (i.e. simple integer
ratios). But Pythagorean and neoGothic major thirds are not trying to be
4:5, or 7:9 or even 11:14 or 18:23. They want to _maximise_ their
dissonance. They are trying to be (4+7phi):(5+9phi).

Pythagorean and neoGothic minor thirds and major and minor sixths are
trying to be other noble mediants too, but they can't all do it at once.

We now have the _simple_ maths that will let us simultaneously minimise the
dissonance of some intervals while maximising the dissonance of others.

For example, here's a new answer to that hoary old problem of what is the
"true" minor seventh (in ordinary diatonic contexts). Is it 4:7 (969 c)?
5:9 (1018 c)? 9:16 (996 c)?

No. The "true" minor seventh is the limit of that series at
(4+5phi):(7+9phi) or 1002 cents.

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

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> http://www.egroups.com/message/tuning/13178
>
> > [Paul Erlich]
> > and spend more time staring at Wilson's papers.
>
> [Dave]
> I tried that too. It just gives me a headache. How come there's
> no text to provide the keys to all these beautiful but obscure
> diagrams?

Over lunch one day, Erv explained that to me: these papers are
just the notes that accompanied long lectures. So a lot of the
explanation was given orally; the papers are like abstracts.

I too am mystified by a lot of what I see in Wilson's papers,
and thank Paul Erlich and Carl Lumma for explaining a good
deal of his concepts to me when I visited them.

But I do think that if one is able to devote the time to reading
everything that he's published (I certainly haven't yet), the
connections can be made and an glimmer of understanding will
appear.

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

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/13183

> --- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> > http://www.egroups.com/message/tuning/13178
> >
> > > [Paul Erlich]
> > > and spend more time staring at Wilson's papers.
> >
> > [Dave]
> > I tried that too. It just gives me a headache. How come there's
> > no text to provide the keys to all these beautiful but obscure
> > diagrams?
>
>
> Over lunch one day, Erv explained that to me: these papers are
> just the notes that accompanied long lectures. So a lot of the
> explanation was given orally; the papers are like abstracts.
>
> I too am mystified by a lot of what I see in Wilson's papers,
> and thank Paul Erlich and Carl Lumma for explaining a good
> deal of his concepts to me when I visited them.
>
> But I do think that if one is able to devote the time to reading
> everything that he's published (I certainly haven't yet), the
> connections can be made and an glimmer of understanding will
> appear.
>

This situation has "vexed" me a little as well... although the
diagrams become clearer and clearer through list discussion and other
web links.

Still, Kraig should be praised for keeping this entire collection on
the Web. Hear that, Kraig?? Nope. No answer.

BUT, some explanatory links would certainly be appreciated. I also
find that the long Wilson scans of the originals take FOREVER to
materialize, even on fairly fast systems. A presentation of these
concepts more on the lines of "Monzo webpages" would surely
facilitate things.

But, I wouldn't want to criticize the way that Kraig is doing things!
Kraig?? Kraig?? The silence of the Anaphorian ocean...
______________ ___ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
> http://www.egroups.com/message/tuning/13186
>
> BUT, some explanatory links would certainly be appreciated. I also
> find that the long Wilson scans of the originals take FOREVER to
> materialize, even on fairly fast systems. A presentation of these
> concepts more on the lines of "Monzo webpages" would surely
> facilitate things.
>
> But, I wouldn't want to criticize the way that Kraig is doing
> things!

Hmmm... it's very interesting that you should suggest that, Joe.
When Kraig first starting putting the Wilson papers online, I
felt the same way you do, so I actually *did* make "Monzo
webpages" of two of them, for my own benefit. I typed in all
of the text so that it could be formatted as regular HTML, and
put the diagrams right at the point where the text explains
them, for easy reference.

I would love to put them online, so that others could benefit
too, but Kraig has been adamant that Wilson's work stand exactly
as it was originally published. And yes, I do commend Kraig
strongly for his work and devotion in publishing Wilson's work
on the web, and for this attitude as well.

But IMO, as long as the Wilson papers are online at
http://www.anaphoria.com in their original form, then there's no
harm in having 'edited' versions of them somewhere else (like
the Sonic Arts site). But given Kraig's position, I would
never do this without first getting permission from Erv himself,
and since Erv doesn't ordinarily use a computer, I'd probably
have to get him to see what the webpages look like first before
he'd approve them. Hopefully on a visit to Erv in the near
future...

Another bit of synchronicity: just this week I started
'translating' some of Erv's diagrams into Monzo lattices for
inclusion into my book and webpages; so if that actually does
help a little, then 'help is on the way'. (I think the version
of my book that you have has some hand-drawn versions, but I'm
not sure... look in the chapter on '11', right after 'Partch'.)

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

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/13193

> Hmmm... it's very interesting that you should suggest that, Joe.
> When Kraig first starting putting the Wilson papers online, I
> felt the same way you do, so I actually *did* make "Monzo
> webpages" of two of them, for my own benefit. I typed in all
> of the text so that it could be formatted as regular HTML, and
> put the diagrams right at the point where the text explains
> them, for easy reference.
>
> I would love to put them online, so that others could benefit
> too, but Kraig has been adamant that Wilson's work stand exactly
> as it was originally published. And yes, I do commend Kraig
> strongly for his work and devotion in publishing Wilson's work
> on the web, and for this attitude as well.
>

But wait a minute! Don't authors put up pages all the time about
other authors?? As long as there is no plagiarism and the ideas are
correctly cited, one is within the copyright law, correct?? Also
people would be able to learn from this, and there could be clear
links to the original source. I think it would be a little unfair of
Kraig to object to such elaboration... Besides he ain't here no mo...
_____________ _____ __ __
Joseph Pehrson

Joseph Pehrson wrote,

> But wait a minute! Don't authors put up pages all the time about
other authors?? As long as there is no plagiarism and the ideas are
correctly cited, one is within the copyright law, correct?? Also
people would be able to learn from this, and there could be clear
links to the original source. I think it would be a little unfair of
Kraig to object to such elaboration... Besides he ain't here no mo...

I TOTALLY disagree! If you respect somebody you do the right thing by
them. Kraig made his objections to Joe and they seem perfectly
sensible to me. So unless Wilson himself says otherwise I'd abide by
Kraig's wishes (which I would imagine are probably Wilson's anyway).

- dan

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

http://www.egroups.com/message/tuning/13219

> Joseph Pehrson wrote,
>
> > But wait a minute! Don't authors put up pages all the time about
> other authors?? As long as there is no plagiarism and the ideas are
> correctly cited, one is within the copyright law, correct?? Also
> people would be able to learn from this, and there could be clear
> links to the original source. I think it would be a little unfair
of
> Kraig to object to such elaboration... Besides he ain't here no
mo...
>
> I TOTALLY disagree! If you respect somebody you do the right thing
by
> them. Kraig made his objections to Joe and they seem perfectly
> sensible to me. So unless Wilson himself says otherwise I'd abide by
> Kraig's wishes (which I would imagine are probably Wilson's anyway).
>
> - dan

The objections are nonsense. Anything that prohibits someone from
learning something is nonsense...
__________ ____ __ __
Joseph Pehrson

Joseph Pehrson wrote,

> The objections are nonsense. Anything that prohibits someone from
learning something is nonsense...

We'll have to agree to disagree here. The objections are, as I
understand them anyway, that Wilson has his work posted the way he
wants it already... In any event, it's there to learn from as it is
and is being added to on a fairly regular basis... and a forum like
this list is wide open for discussing or going over any aspect of it;
where's the prohibition on learning from it come in? Would I prefer
not to wait nine centuries for eye eroding scans? Sure... but they're
an artistic presentation very particular to the individual who created
them, so if he don't want 'em publicly turned into ASCII facsimiles or
whatnot, well, I understand... and I don't think this is nonsense.

- dan

Joseph Pehrson wrote,

>The objections are nonsense. Anything that prohibits someone from
>learning something is nonsense...

Joe, I can't disagree more. Plato said all virtue is knowledge, and
I agree (though I'll ask what _isn't_ knowledge, from that POV). But
you seem to be implying that all knowledge is virtue, which I think
is nonsense.

-Carl

[Please note that this is being posted after two days offline due to an
interruption of telephone service, and much on-line discussion which I've
just been catching up on now that service has been restored -- hopefully
solving whatever the technical problem was.]

Hello, there, Dan Stearns and everyone, and I'm replying to your
comment in TD 814:

> Well oddly enough, I was just in the midst of an off-list
> "discussion" of the 14/11... and my gist of my view was that I'd be
> hard pressed to call it an "attractant", but I sure couldn't
> tolerate such a barren expanse of nothingness between a 5/4 and a
> 9/7! that's just miles and miles of real estate to my mind!

Dan, as someone who makes my musical home in this charming terrain
much of the time, I must agree with you, and am not sure if anyone
would disagree. At the same time, I would agree with Dave Keenan that
if we are simply trying to estimate the central region of maximum
complexity or ambiguity, then the Golden Mediant might be an
appealingly parsimonious technique.

Before I go on, I would just to like to thank Dave for his most
gracious and insightful comments, and also to note that while my
errant thread heading was indeed posted with the intention of making a
connection with the ongoing "harmonic entropy" discussion, I have duly
returned to the original thread title.

Let us celebrate the contributions of both authors of the "two
Keenans" Golden Mediant function, Keenan Pepper and Dave Keenan, and
also delve into the noble antecedents of this mathematical
application, a most fitting theme which Dave's response has happily
brought into focus.

As to the general plateau region between 5:4 and 9:7, and maybe
specifically the part of the plateau between around Pythagorean and
17-tet, I wonder if Dave and I are addressing two different questions:
locating a cardinal landmark, the rough point of maximum complexity;
and exploring the general area with its various shades of vertical
and cadential color.

One task might be compared to surveying an area and finding its
geographical center; the other to walking about it and getting to know
its many attractions.

For the surveying approach, the Golden Mediant has the advantage of
quick and ready rule of thumb, like finding the area under a curve
simply by computing an antiderivate rather than by going through
numerical integration with all of those rectangles or trapezoids or
whatever.

For _exploring_ the area under that curve, so to speak -- or the area
between Pythagorean and 17-tet or 22-tet or whatever, the Golden
Mediant provides one very useful cardinal point of orientation, along
with distances from clear "valleys" (here 5:4 and 9:7), and some other
landmarks like familiar tunings (e.g. Pythagorean, 29-tet, 17-tet).

The Golden Mediant at around 422.487 cents suggests that the area of
maximum complexity is right around 17-tet, and that Pythagorean is not
quite 15 cents from it on the part of the plateau maybe a bit closer
to 5:4 than to 9:7.

Note that a 23:18 or weighted intermediate at y:x=2 or (5+18):(4+14)
is just a bit larger than a 17-tet major third, while the Pythagorean
81:64 is at y:x=4/9 or (45+36:36+28). Thus if we're interested in the
region between Pythagorean and 17-tet or the 23:18, we can try out any
y:x values between around 4/9 and 2.

While walking through various integer values for x and y is one way to
explore the plateau and come up with some intriguing intervals with
large integer ratios, I'd emphasize that intervals with irrational
ratios are equal members of this continuum of beauty: for example, the
charming major third of 29-tet.

As to 14:11 or (5+9):(4+7), the classic mediant, I would say that it
is enough that this ratio appeals to the intellect (or at least to
mine) while its sonorous realization is beautiful as one of many fine
shadings of color offered by this general region.

Whether 14:11 is tuneable by ear or recognizable as a "primary color,"
so to speak, is a question on which I am undecided, although Dave's
answer is in the negative based on experiment, and I would be curious
about the experiences of others.

In any event, however, I would say that an interval or tuning need not
be tuneable by ear to be of interest; it may represent a subtle
shading on the continuum with musical appeal, and that is sufficient
reason to tune and enjoy it.

Zarlino's 2/7-comma meantone, for example, is known as a temperament
quite difficult to tune by ear, as Zarlino himself suggested (in 1571,
if I am correct) at least by comparison to 1/4-comma with pure major
thirds. There is evidently no special "locking-in" or other defining
aural event when a major or minor third is precisely 1/7 syntonic
comma from pure, as happens in this tuning.

Yet the tuning is esteemed, and Mark Lindley not too long ago authored
an article espousing it as an ideal temperament for some of the organ
pieces of Andrea Gabrieli, for example, featuring modes with an
emphasis on the arithmetic division of the fifth with minor third
below and major third above. This is getting into the "modern" side of
my repertory, but the point is that the historical meantone continuum
features many temperaments which represent fine gradients or mixtures
rather than the simplest primary colors, so to speak.

Similarly, a neo-Gothic tuning featuring major thirds at or close to
14:11 is at once giving sonorous form to a noble ratio -- and I
emphasize the equal nobility of the 29-tet major third and others with
irrational ratios -- and letting us experience one shade or hue of the
xenharmonic rainbow.

In some ways, maybe, a question of at least equal interest to the
"tuneability by ear" of a ratio such as 14:11 might be the general
discrimination of the ear in distinguishing between these shadings.
For example, can people easily distinguish between 29-tet, and a
regular tuning with pure 14:11's, and 17-tet, etc.? How about shades
of 5-limit meantone, for that matter?

In musical practice, of course, a 14:11 does not occur in isolation:
it has associated with it in a neo-Gothic context, for example, a
certain distance of expansion to a stable fifth (around 285 cents or a
bit more, roughly the size of a 33:28 or 13:11), a diatonic semitone
of around 80 cents, a chromatic semitone or apotome of around 130
cents, etc.

All of these factors give flavor to the musical "vintage" of the
interval, so to speak. Just how fine the aural distinctions are which
can be made by listeners remains an open question.

As someone, maybe Paul Erlich, once commented, intervals with complex
integer ratios such as the typical unstable sonorities of Pythagorean
tuning have a quality akin to that of tempered intervals. Thus from my
point of view, the traditional 81:64 and likewise 19:15, 14:11, 23:18,
as well as the neighboring major thirds of 29-tet and 17-tet, etc.,
are all part of the rainbow.

Most appreciatively,

Margo Schulter
mschulter@value.net

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

http://www.egroups.com/message/tuning/13225

> Joseph Pehrson wrote,
>
> > The objections are nonsense. Anything that prohibits someone from
> learning something is nonsense...
>
> We'll have to agree to disagree here. The objections are, as I
> understand them anyway, that Wilson has his work posted the way he
> wants it already... In any event, it's there to learn from as it is
> and is being added to on a fairly regular basis... and a forum like
> this list is wide open for discussing or going over any aspect of
it;
> where's the prohibition on learning from it come in? Would I prefer
> not to wait nine centuries for eye eroding scans? Sure... but
they're an artistic presentation very particular to the individual
who
created them, so if he don't want 'em publicly turned into ASCII
facsimiles or whatnot, well, I understand... and I don't think this
is
nonsense.
>

Well, I'm still recovering from Dante's "spell," but I think I can
converse reasonably (whew!) about this now. Actually, I thought it
was *ME* having a little trouble with the Anaphoria-Wilson stuff...
but when I found out that DAVID KEENAN couldn't figure it out... and
this guy is a *REAL* mathematician... then I realized something
really was amiss. Yes, and the materials are handwritten, typed and
take tremendously long to load it. NOBODY can vouch that it's optimal
this way...

OK... as an "artistic manifestation"... well MAYBE, but it looks more
like music or number theory to me.

I was just saying that I would hope that Kraig Grady would be open
enought to let people make commentary on these materials, like Monzo
has. It sounds like he is not. I am also not convinced, after his
activities leaving this list in a huff, that he is tolerant to other
people's opinions.... Perhaps he can still convince me otherwise. I
have no reason to assume, thought, that he wants to...

I just think some more "elaborative" links from Monz or somebody...
with the CLEAR understanding that this is NOT the real Wilson
stuff... and then some CLEAR and obvious links to the *real*
materials
would be in order. It would help everybody understand this stuff.

The more we understand it... the more we can PRACTICE IT and the more
we can look at the original artistic materials with "informed" eyes.

Look... there are big books out "explaining" James Joyce's Ulysses.
Should I throw them all in the flamerator just because it's not the
*real* item. No I NEED them. And for Finnegan's Wake, I need a book
that will explain the book that explains the book!!!

It doesn't ruin the original in *ANY* way... especially if provided
by somebody competant like Monzo... and perhaps with the explanatory
materials reviewed by Wilson himself...
___________ ____ __ _
Joseph Pehrson

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

http://www.egroups.com/message/tuning/13235

> Joseph Pehrson wrote,
>
> >The objections are nonsense. Anything that prohibits someone from
> >learning something is nonsense...
>
> Joe, I can't disagree more. Plato said all virtue is knowledge, and
> I agree (though I'll ask what _isn't_ knowledge, from that POV).
But
> you seem to be implying that all knowledge is virtue, which I think
> is nonsense.
>
> -Carl

Well then, Carl... what would you prefer... ignorance and
obstructions
to knowledge??
___________ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
> http://www.egroups.com/message/tuning/13223
>
> --- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
>
> http://www.egroups.com/message/tuning/13219
>
> > Joseph Pehrson wrote,
> >
> > > But wait a minute! Don't authors put up pages all the time
> > > about other authors?? As long as there is no plagiarism and
> > > the ideas are correctly cited, one is within the copyright
> > > law, correct?? Also people would be able to learn from this,
> > > and there could be clear links to the original source. I
> > > think it would be a little unfair of Kraig to object to such
> > > elaboration... Besides he ain't here no mo...
> >
> > [Dan]
> > I TOTALLY disagree! If you respect somebody you do the right
> > thing by them. Kraig made his objections to Joe and they seem
> > perfectly sensible to me. So unless Wilson himself says
> > otherwise I'd abide by Kraig's wishes (which I would imagine
> > are probably Wilson's anyway).
> >
>
> [Joe Pehrson]
> The objections are nonsense. Anything that prohibits someone from
> learning something is nonsense...

Joe, I totally agree with your statement. But at the same time,
as an author with a lot of stuff that I've made available to the
world totally free while I still live in utter poverty, I still
have to agree with Dan too.

Yes, Kraig's position certainly reflects that of Erv himself,
and that's why I respected it. But I also still harbor the
desire to talk to Erv about it and hopefully convince him to
allow me to put up the webpages that are easier to read and
understand.

If he doesn't agree to it, then I'm going to continue doing
exactly what I've been doing all along: study Wilson's work
and write my own explanations of it. Now, NOBODY can prevent
me from doing that!

Then if someone thinks I've misrepresented Erv or they don't
understand what I wrote (or drew, which is the main thing
in this case), they can seek the originals themselves, and
grapple with them the same way I (and everyone else who
understands it) did.

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

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:
> http://www.egroups.com/message/tuning/13225
>
> ... The objections are, as I understand them anyway, that Wilson
> has his work posted the way he wants it already... In any event,
> it's there to learn from as it is and is being added to on a
> fairly regular basis... and a forum like this list is wide open
> for discussing or going over any aspect of it; where's the
> prohibition on learning from it come in? Would I prefer not
> to wait nine centuries for eye eroding scans? Sure... but they're
> an artistic presentation very particular to the individual who
> created them, so if he don't want 'em publicly turned into ASCII
> facsimiles or whatnot, well, I understand... and I don't think
> this is nonsense.

Right you are, Dan.

There's one thing I'm really amazed at: there's absolutely
*nothing* preventing anyone else here from learning Wilson's
work the same way I did, by painstakingly redoing the originals
in a way that's easier for them to understand - not for public
consumption, but just for their own personal benefit. Of course,
it does require... a lot of W O R K !!!!

I guess that's the problem...

... but it hasn't stopped Kraig, and it won't stop *me*...

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

Introductory note:

Anytime you start to criticize criticizing, you're setting
yourself up for trouble (naturally, this note is no exception).
This can even get to the point where the problem is actually
complaining about the problem. I think we saw that today.
What's more damaging to the list: a few angry flames, or 200
messages about what constitutes an appropriate message? My usual
response is silence, but I thought I'd add this intro in case
it could help out in the future: I think the list description
at our egroups page (which was recently quoted on-list) is
all we should need to determine appropriate material for the
list. I would add that anytime you see a message subject in
caps, you should adjust the case if you want to reply. Now,
back to the subject of Wilson's papers...

Joseph Pehrson wrote...

>It doesn't ruin the original in *ANY* way... especially if provided
>by somebody competant like Monzo... and perhaps with the explanatory
>materials reviewed by Wilson himself...

If reviewed by Wilson himself, then fine.

>>>The objections are nonsense. Anything that prohibits someone from
>>>learning something is nonsense...
>>
>>Joe, I can't disagree more. Plato said all virtue is knowledge, and
>>I agree (though I'll ask what _isn't_ knowledge, from that POV).
>>But you seem to be implying that all knowledge is virtue, which I think
>>is nonsense.
>
>Well then, Carl... what would you prefer... ignorance and
>obstructions to knowledge??

Try again. That's denying the antecedent, and attacking the person.

Joseph- as the conceptual movement tried to show, the presentation of
a work has great influence on how the work functions. Wilson's papers
aren't just papers, they're the part of Wilson's work he's decided to
give us, in the way he's decided to give them to us. Monz hit the
nail on the head:

>There's one thing I'm really amazed at: there's absolutely
>*nothing* preventing anyone else here from learning Wilson's
>work the same way I did, by painstakingly redoing the originals
>in a way that's easier for them to understand - not for public
>consumption, but just for their own personal benefit. Of course,
>it does require... a lot of W O R K !!!!
>
>I guess that's the problem...

That's not the problem, that's the point! I'm the first guy to
get up and say I only understand a fraction of Wilson's stuff.
If I work at it, and think I can help explain it to somebody else,
I will. But I _won't_ mark up Wilson's papers and publish the
result without his explicit consent.

-Carl

If Joe Monzo makes it clear on his website that the material
presented is his personal interpretation of Wilson's diagrams, then
everything is OK.

This interpretation may be what Wilson intended; if it is not, than
it is Monzo's creation and it can still be a source of inspiration
for all of us.

-Kami

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/13259

>
>
> There's one thing I'm really amazed at: there's absolutely
> *nothing* preventing anyone else here from learning Wilson's
> work the same way I did, by painstakingly redoing the originals
> in a way that's easier for them to understand - not for public
> consumption, but just for their own personal benefit. Of course,
> it does require... a lot of W O R K !!!!
>
> I guess that's the problem...
>
> ... but it hasn't stopped Kraig, and it won't stop *me*...
>
>
So basically, Joe, everybody has to do the work *over and over*
again, each person for himself... Isn't that wonderful. If there is
something that might keep somebody from composing, this is the kind
of thing.

Hopefully, you will be able to display your *own*
clarifications/elaborations publicly as you indicated in your former
post...
______________ ___ __ _ _ _
Joseph Pehrson

M. Schulter wrote,

> One task might be compared to surveying an area and finding its
geographical center; the other to walking about it and getting to know
its many attractions.

Yes, that seems like an apt analogy to me as well... and as you've
said before, this really is sort of the heartland of your interests!

Another "hiking" option in this Pythagorean region might be the
logarithmic Phi trail...

The 3/5 trail -- as it's commonly referred to in the great complex of
improper Phi trails -- is a relatively short one, as it quickly leads
one to the edge of a vast expanse of 5-tET wilderness, but it is an
interesting one all the same.

The 3/5 trail begins at the beautiful 4/7 Golden Meantone center where
L/s = phi:

0 119 192 311 385 504 577 623 696 815 889 1008 1081 1200

After an invigorating trek through many breathtaking vistas and the
initial unfolding of the great Pythagorean complex we approach the
first resting stop, the lovely 7/12 stop where L/s = phi+1:

0 80 208 288 416 496 625 575 704 784 912 992 1120 1200

Our next rest area is the rugged yet beguiling 10/17 stop where L/s =
phi+2:

0 60 216 276 432 492 648 552 708 768 924 984 1140 1200

Our now ever-increasingly disorienting journey next rests at the
stunning 5-tET overlook, the 13/22 stop where L/s = phi+3:

0 48 221 269 442 490 663 537 710 758 931 979 1152 1200

The last stop for all but the most intrepid is the 16/27 5-tET
junction, where L/s = phi+4:

0 40 224 264 448 488 672 528 712 752 936 976 1160 1200

At the wilds of the 5-tET junction you broach a roughhewn portal of
sorts, a place at the very outer edges of the logarithmic Pythagorean
complex where a vast expanse of dense 5-tET wilderness rests.

on a clear day you can see for miles,

- dan

[David Keenan wrote in Tuning 823:2 on 21 September 2000:]

> Are you there Margo? What do you think of changing the name of the
> function to "Noble Mediant", since the numbers that result from
> applying the function are apparently already called "noble numbers".

Hello, there, Dave and everyone, and it's a pleasure to be back online
after a two-day interruption of telephone service, or at least of
outgoing calls.

Please let begin by saying, Dave, especially in response to your most
gracious words in this forum, that your skills as a mathematician and
educator, as well as your most moving patience and understanding as a
friend and colleague, are truly wonderful.

"Noble Mediant" sounds fine to me, and I would abide by your
mathematical judgment and sense of precedent here.

> Hey, has anyone noticed that Keenan Pepper hasn't been on the list
> since I came back. Maybe we really _are_ the same person!

Please let me add to your remarks in a previous article that I
sometimes refer to the Noble Mediant as the "two Keenans" function, or
maybe even the "Keenan-squared" function, to suggest the power of
sharing and cooperation of the kind which can happen in a worldwide
community such as this one.

Also, thank you very much for some comments which may serve to
illustrate some nuances of emphasis regarding the Noble Mediant
and plateau regions of complexity generally. As you remarked in an
earlier post (TD 813:25), there can be "different emphases" in how one
approaches such regions, coupled with a hearty consensus that
complexity can be a "beauty mark" rather than a "defect" (as you
observed in another article).

Please let me begin by affirming our very strong area of agreement:

> I think the main message of that paper is that we often mistakenly
> assume that what we want in a temperament is simply to minimise the
> dissonance of certain intervals by minimising their distance from JI
> (i.e. simple integer ratios).

Indeed this is what I see as the main message of our paper also: a
Pythagorean third in medieval European polyphony is a beautiful and
complex interval, not an "out-of-tune" version of a 4:5 or some other
simpler ratio.

Complexity does not have to be an "error," it can be precisely what is
aesthetically ideal.

> But Pythagorean and neoGothic major thirds are not trying to be 4:5,
> or 7:9 or even 11:14 or 18:23. They want to _maximise_ their
> dissonance. They are trying to be (4+7phi):(5+9phi).

Here we come to the matter of our somewhat "different emphases,"
different variations on our common theme that "complexity is
beautiful" in the right musical context. It might be curious to
consider how much this difference is one of theoretical inclinations
or actual musical perception.

As I see it, the Noble Mediant is indeed a cardinal landmark in
exploring a plateau such as that between 4:5 and 7:9, and the distance
of an interval from that "geodesic center" of the plateau becomes a
very helpful reference in navigating the area.

Thus having identified the Noble Mediant for 4:5 and 7:9 as around
422.487 cents, we can place the Pythagorean 81:64 as a bit less than
15 cents on the slope of the plateau a bit closer to 4:5, and the
major third of 17-tet (~423.53 cents) or the slightly larger 23:18
at very close to the central Noble Mediant.

A 14:11 at ~417.51 cents, whether or not as the Classic Mediant
(5+9):(4+7) it might represent any kind of "microdell" in the terrain,
is situated about 5 cents from the center of the plateau, and thus
might have a slightly different "flavor" of complexity.[1]

As an advocate of equal opportunity for intervals with irrational
ratios also, I would add that the 29-tET major third at ~413.79 cents,
for example, is a tad less than 9 cents from the Noble Mediant,
suggesting yet another subtly different shade of color or flavor.

In other words, I might suggest that we may be dealing with shades
analogous to those of temperament in the traditional sense, each shade
having its own place of beauty and honor on the continuum.

Please note a distinction I have made in writing some of the above
ratios: 4:5 and 7:9, for example, but 81:64 and 14:11. Writing an
interval as a frequency ratio with the terms in ascending order, the
convention you eloquently advocate, may intimate an aurally
recognizable "locking-in" of the partials, etc.

Writing a ratio such as 81:64 or 14:11 in the traditional medieval
fashion as a string ratio, in contrast, may avoid such implications,
while identifying a ratio which an "attractant" to the intellect and a
pleasure to the ear, whether tuned by measuring the proportions of
strings or by defining a synthesizer table.

While certainly accepting the Noble Mediant as a cardinal reference
point, I would say that a 14:11 wants to be a 14:11, a 23:18 a 23:18,
and likewise a major third of 10/29 octave wants to be 10/29 octave.
They may all have a certain family resemblance, and very fine
distinctions along the plateau may be recognized more by the intellect
than by the senses; I celebrate the richness of the continuum.

Curiously, after hearing a neo-Gothic improvisation in more or less
14th-century style using a tuning with pure 14:11 thirds (a poetic figure
of speech? -- and please note that I wrote "14:11" rather than "11:14"), a
friend compared this kind of tuning to my penchant for geometric or
typographical designs featuring complexly nuanced shades of gray. Choosing
a 14:11 seems a bit like choosing 14% gray for a background or for one
side of a polyhedron -- not necessarily a primary color, but a pleasant
number which may have a beautiful visual effect.

Please let me also duly note that when encountering a new large
integer ratio such as 33:26, my natural next step is to determine its
size in cents (here ~412.75 cents, or about a cent smaller than in
29-tET), and to place it on the continuum in reference to familiar
landmarks such as other intervals or tunings -- and also, now, the
Noble Mediant. The expression "33:26" serves as an auspicious _name_
for this location at once unique and closely akin to its immediate
surroundings.

Maybe there is a certain "local patriotism" involved here, as well as
a neo-medieval passion for the lore of large integer ratios: each
millimeter of the plateau, so to speak, has its own place.

Your comments raise a wider philosophical question, maybe applying a
bit more dramatically to certain meantone tunings near attractants
such as 4:5 and 5:6 than to fine neo-Gothic distinctions in the
gently sloping highlands between Pythagorean and 17-tET.

Might we say that all meantone major and minor thirds, for example,
want to be 4:5 or 5:6; or might the thirds of Zarlino's 2/7-comma
temperament be exactly the right size in the right musical context?
Mark Lindley, for example, has suggested that the contrast between the
pure 4:5 and the somewhat "beatful" 5:6 in 1/4-comma meantone may give
a very pleasing variety to certain passages in parallel thirds.

Having offered these odd remarks, I would like to express appreciation
above all both for our common affirmation of the musical value of
complex intervals, and for this precious opportunity to exchange
viewpoints with such an exponent of mathematics, musical sensitivity,
and respect and empathy treasured beyond words.

-----
Notes
-----

1. If one applies the definitional test discussed in recent posts by
Paul Erlich and others, then 14:11 or (5+9):(4+7) meets what I might
call the cross-product test that (9*4)-(7*5) = 1. However, I would
regard the question of "aural recognizability" as one to be decided by
experiment, and Dave Keenan at least has tried this experiment and
reached a negative conclusion under usual timbral conditions.

Most appreciatively,

Margo Schulter
mschulter@value.net

In Tuning Digest 827, Dan Stearns wrote:

> Another "hiking" option in this Pythagorean region might be the
> logarithmic Phi trail...

Hello, there, Dan, and congratulations on a really breathtaking post,
at once a "magical mystery tour" for the musical imagination and a
tour de force.

You really communicate the spirit of "walking the plateau," as I call
it in a post on Keenan Pepper's tuning; really a beautiful exposition,
maybe a bit like those books which travel through the universe in
powers of ten (for macroscopic or microscopic size, temperament,
etc.).

From the viewpoint of my coming article, your post is especially apt, for
the "lovely 7/12 stop where L/s = phi+1," is also Keenan Pepper's "noble
tuning."

One curious observation from the point of view of world musics in
general. While I might consider the "diatonic borderland" where
regular major thirds are expanding past 450 cents, and diatonic
semitones are contracting to smaller and smaller than 50 cents, to be
"rugged," the 5-tet landscape itself suggests to me the opulent
richness of gamelan.

This is not to say that slendro is identical to 5-tet -- it varies
with each gamelan -- but only to say that at this point we have, in
place of a "5T/2S" or "5L/2s" diatonic structure more and more
"stretched," a symmetrical kind of pentatonic scale which, with the
right timbres, can have very pleasant and concordant fifths and
fourths.

Back in the late 1960's and early 1970's, my reaction to gamelan was:
"Wow, here's another beautiful music based on fifths and fourths," and
I wouldn't have guessed that these intervals were actually rather far
from 3:2 or 4:3. To my ears, they were just fifths and fourths, my
special favorites in anything from music of the Dan of Africa to the
polyphonic songs of Georgia in the U.S.S.R. to the shape-note music of
Georgia in the U.S.A.

Anyway, bravo on a great musical tour, the kind of article which I
hope will inspire more such excursions.

Most appreciatively,

Margo Schulter
mschulter@value.net

M. Schulter wrote,

> From the viewpoint of my coming article, your post is especially
apt, for the "lovely 7/12 stop where L/s = phi+1," is also Keenan
Pepper's "noble tuning."

Yes -- "Pepper's Point" on the trail guide!

thanks Margo,

- dan