Yahoo Tuning Groups Ultimate Backup tuning Re: Occam's Razor or Keenan's Sand Filter?

Hello, there, Dave Keenan, and welcome back to the Tuning List.

Thank you especially for a statement which may highlight a basic
cultural or philosophical difference in perspective between certain
participants on this list, and sometimes manifested in internal
dialogues within individual participants on this list, a difference
maybe more easily negotiated when those of us on both or all sides of
it recognize this divergence.

> Readers are bound to conclude (in the absence of accompanying words
> to the contrary) that the author who uses large prime ratios feels
> that they have some special psychoacoustic significance
> (i.e. "audibly discrete", as you say). So far, I am not aware of any
> evidence of this.

As I will emphasize below, your perspective has a long and honored
history, going back at least to Gioseffo Zarlino, who resisted the use
of quantities, niceties, or visual designs in music which have no
apparent audible consequences. From such a philosophical viewpoint,
one _can_ very reasonably read an implicit claim into the use of large
primes, or other integers, that they are intended to have special
auditory significance.

From my own medievalist perspective, numbers could be used in tunings
or otherwise in the construction of a piece or music without any
actual or implicit claim to "psychoacoustical significance" or an
"audibly discrete" character.

Otherwise I might find myself concluding that Baude Cordier was
claiming special "psychoacoustical significance" for his compositions
notated in the form of a compass or a heart -- two beautiful
compositions, by the way, where the visual design is emblematic of the
text.

Similarly, I might conclude that Josquin was claiming a special
"psychoacoustical significance" when using the solmization syllables
from the name of Duke Hercules of Ferrara, or from the phrase "Vive le
roy!" to write masterful pieces, a Mass and a fanfare.

The use of large prime numbers in tunings seems to me to admit of a
similar kind of technique: picking emblematic or otherwise attractive
numbers, and building a tuning on them. Of course, I would look at the
values in cents to see what the likely effect might be -- with
performance in an actual timbre (another variable) confirming,
denying, or modifying such an estimation.

When someone like Jacky Ligon or Robert Walker finds interesting
mathematical patterns in musically-applied primes, then I would
certainly agree that we can and should ask whether such patterns might
have any special significance, or might be explained by some more
general mathematical principles, as Paul Erlich has done most
skillfully in one very recent instance.

However, the use of a ratio in itself seems to me a matter of art --
be it emblematic, purely arbitrary ("I like the number, or intuit that
somehow it may produce an interesting result"), or done in a carefully
calculated "psychoacoustical" way. A fine point: one can pick numbers
in a "random" kind of way, but then select those that happen to
produce an interesting value in cents for the musical purpose at hand.

While celebrating the emblematic or other artistically arbitrary use
of numbers in music -- as opposed to in an acoustical or other
scientific experiments, where quantities should indeed be explicable
and significant to the design of the experiment -- I would like to
join in due skepticism concerning claims for the "superiority" of
certain tunings, whether defined with familiar mathematical quantities
or otherwise.

For example, while Kornerup's Golden Meantone has the intriguing
property of symmetrical Golden Ratios between various classes of
intervals, and the Noble Fifth tuning curiously has a supraminor sixth
very closely approximating an interval ratio of Phi itself, I would be
cautious about claiming any distinctive acoustical properties for
either tuning not shared by others in the same general regions of the
continuum. Other meantones between 1/4-comma and 2/7-comma, or
neo-Gothic tunings with fifths around 704 cents, may have essentially
similar acoustical qualities (in slightly different shadings, of
course).

Similarly, as myself an enthusiast and promoter of the "e-based
tuning" with a fifth of around 704.61 cents, and a ratio of whole-tone
to diatonic semitone equal to Leonhard Euler's e, I would not
necessarily claim any special psychoacoustical properties for it not
shared by other tunings in the very close vicinity.

More specifically, when it is realized as a 24-note tuning, this
e-based temperament includes a variety of minor seventh (actually a
major sixth plus diesis) almost identical to a pure 7:4 -- most
felicitous, since Euler was an early European champion of this
ratio. However, while the use of e as a generator may be a charming
example of serendipity (or "syncronicity"), we could also produce a
musically similar result simply by calculating a fifth size where a
chain of 15 fifths up generates a pure 7:4 -- ~704.588 cents.

This is a lot of fifths, and your chain-of-fifths paper made the
quite reasonable choice to draw a line at a chain no longer than 14,
although offering tools for those interested in longer chains. If you
had drawn a slightly different line, then your methods would have
identified this region without any need to use e.

http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm
http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

In fact, when I hit on the idea of the e-based tuning, I wasn't sure
of exactly what musical results it might produce; and having explored
them, I have found them in other tunings in the general region also.
In turn, it was through comparing 24-note versions of these tunings
that I came to realize that chain of 13 or 14 fifths were moving
toward 7-based ratios (9:7, 7:6) a bit beyond 46-tET, and then to
discover that 15 e-based fifths yielded a near-pure 7:4.

Having expressed my own viewpoint, I would like fully to acknowledge
yours, which also has a long and honorable tradition, inviting any
correction to my attempted statement of it: "Numbers used in
discussions of tunings should have acoustical significance, and
integers should not be multiplied without purpose."

To quote Zarlino (1558) on notational intricacies appealing to the eye
of the performer rather than the ear of the listener -- and the use of
"audibly not discrete integer ratios" might be placed in much the same
category:

"Why then do some persons work so hard to introduce
so many intricacies into their compositions?....
I know that those who employ these devices, if they
have any sense, will reply that such things have no
value at all, because when their ciphers are reduced
to simple common terms the harmony will be as good
as before. Since they do not contribute to better
harmony and cannot be perceived by the senses, why
burden the singer with such things needlessly?....
My advice is to ignore such ciphers and to
concentrate on those matters that lead to the
production of good sweet harmonies."[1]

Possibly even more Keenanesque is the comment of Zarlino soon
following:

"Thus it may be truthfully said that this method
of composing results only in needlessly multiplying
difficulties without increasing harmoniousness.
Therefore it is useless, for, as the Philosopher
says, it is vain to multiply anything with no
purpose."[2]

Here is a musical application of a rule of parsimony here attributed
to Aristotle (often known simply as "the Philosopher" to late medieval
and Renaissance authors), but which might also be called Occam's Razor
or Keenan's Sand Filter, to invoke a worthy contemporary recycling
technology.

The immediately following sentences also have a Keenanesque ring to
me:

"For music, being a science that deals with sounds and
tones -- particular objects of the sense of hearing --
is concerned only with the sonority (as Ammonius says)
that springs from pitches and tones, and with nothing
else. Therefore it seems to me that all musical
speculations not directed toward this end are vain
and useless."[3]

In responding to your statement, I feel it very important to
acknowledge the tradition from which I perceive you come, not only
understanding it but relishing it, and possibly in the process
clarifying my own sometimes more emblematic leanings.

-----
Notes
-----

1. Gioseffo Zarlino, _The Art of Counterpoint: Part Three
of Le Istitutioni harmoniche, 1558_, trans. Guy A. Marco and
Claude Palisca (W. W. Norton, 1976), ISBN 0-393-00833-9, Chapter 71,
"The Value of These Devices to Good Harmony," at pp. 263-264.

2. Ibid. p. 264.

3. Ibid.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗jpehrson@rcn.com

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

/tuning/topicId_18241.html#18241

> However, the use of a ratio in itself seems to me a matter of art --
> be it emblematic, purely arbitrary ("I like the number, or intuit
that somehow it may produce an interesting result"), or done in a
carefully calculated "psychoacoustical" way. A fine point: one can
pick numbers in a "random" kind of way, but then select those that
happen to produce an interesting value in cents for the musical
purpose at hand.
>

Congrats to Margo Schulter for the VERY interesting commentary which
links some of our recent discussion with similar concerns going back
to antiquity! I always love such material, since those of us living
in the present, as most of us do (that sounds like a "George
Bushism...")tend to be a little on the smug side concerning our
perceptions and accomplishments...

Just as a minor aside, I would also like to mention what is now a
"tradition" although maybe not such a "grand" one of using various
numbers in the compositional process, unlinked to "scientific"
underpinnings. Of course, John Cage comes immediately to mind, with
his cohort Merce Cunningham, but there has been so much of this kind
of work within the last 30 years or so, that it is a minor
"tradition" in a way.

I think Monzo would be the one to comment here, but I believe there
was also rather spurious numerology determining various compositional
factors throughout the Second Viennese school and going back to
Mahler and others... Right, Monz? Perhaps Schumann even used it.
Certainly the works by Bartok and others involving the "golden mean"
really had very little to do with science, although obviously
proportion...

On the other hand, Dave Keenan's clearheaded logic makes one want to
run out right away and take some kind of science class...

A very interesting discussion!

________ _____ ____ __
Joseph Pehrson

🔗MONZ@JUNO.COM

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_18241.html#18247

> I think Monzo would be the one to comment here, but I believe
> there was also rather spurious numerology determining various
> compositional factors throughout the Second Viennese school and
> going back to Mahler and others... Right, Monz? Perhaps Schumann
> even used it.
>
> Certainly the works by Bartok and others involving the "golden
> mean" really had very little to do with science, although
> obviously proportion...

Yes, Joe, apparently there was lots of numerology and Caballa
(originating in the theosophic interpretation of Hebrew Scripture)
going on among the work of Schoenberg and Berg, and J. S. Bach.

No evidence for Mahler's use of any of this... and given my
knowledge of Mahler, I'd say with confidence that he never did
any of this.

Schoenberg seems to have been very intrigued by this stuff.
There's even a book called _Schoenberg: the Composer as
Numerologist_, in which many Schoenberg compositions are
subjected to numerological analysis. I leave it up to the
reader to decide how convincing the author's argument is.

But there is no doubt in my mind that both Schoenberg and
Berg used Cabbala in their pieces. Many motives and tone-rows
use pitches whose sequential letter-names spell out (at least
partially) the names of the composers or other people closely
associated with them. Berg wrote his famous _Lyric Suite_ as
a secret love-gift to his secret lover, and it's filled with
this kind of thing. There have been quite a few articles
published about this aspect of their work.

The practice in this case apparently stems from Schoenberg,
and he apparently knew (or possibly even realized thru his
own study) that Bach did it. Schoenberg also had a deep and
complex involvement with his original religion of Judaism, so
it's not at all surprising to me that he would know about and
make use of Cabbala.

Some of Webern's pieces are based on a four-note motive of the
notes Bb-A-C-B, which is B-A-C-H in German musical notation,
which is the same thing Bach did in his _Art of Fugue_. Other
than that, I don't know of Webern using Cabbala.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗jpehrson@rcn.com

🔗ligonj@northstate.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:
>
> From my own medievalist perspective, numbers could be used in
tunings
> or otherwise in the construction of a piece or music without any
> actual or implicit claim to "psychoacoustical significance" or an
> "audibly discrete" character.

Margo,

I want to personally thank you for this amazing post!

Perhaps we never truly escape the considerations of "psychoacoustical
significance", because what it takes to compose music that will
endure repeated listening, comes together in the synthesis of our
knowledge of the mathematical; the audible; and the musical
properties at work. When I'm writing a melody played on a choice
selection of intervals (no matter what their method of derivation), a
part of my musical judgment comes into play, which is not always
thinking about the numbers, yet this act of creation brings the most
personal and audible "psychoacoustical significance" to my ears in
the "tangible" form of *music*. The creative process of making all
this become manifest in the reality of music, is something which
ultimately can defy mathematical analysis (especially the process of
what it takes to compose compelling melodies), because
personal "choice", and the sum of who you are, comes into play. This
diversity conveniently provided by good old Mother Nature, in this
regard, gives us an infinity of wonderful and valid possibilities,
which are invariably tempered by our own musical tastes.

> The use of large prime numbers in tunings seems to me to admit of a
> similar kind of technique: picking emblematic or otherwise
attractive
> numbers, and building a tuning on them. Of course, I would look at
the
> values in cents to see what the likely effect might be -- with
> performance in an actual timbre (another variable) confirming,
> denying, or modifying such an estimation.

This I agree with, and would add that to chose large prime ratios is
yet but another valid method of tuning/pitch measurement. And yes,
the resulting cents values - along with their unique audible
properties must be the goal (as in my MOS scales built from prime
ratios, where "finding" a selection of choice thirds was the primary
goal.).

> When someone like Jacky Ligon or Robert Walker finds interesting
> mathematical patterns in musically-applied primes, then I would
> certainly agree that we can and should ask whether such patterns
might
> have any special significance, or might be explained by some more
> general mathematical principles, as Paul Erlich has done most
> skillfully in one very recent instance.

It may be worth noting, that the method of scale construction shown
in my initial article;

/tuning/topicId_17399.html#17399

and the use of prime series ratios displayed in the "2001: A MOS
Odyssey" posts;

/tuning/topicId_17744.html#17744

/tuning/topicId_18025.html#18025

as well as the speculative post about mediants;

/tuning/topicId_18060.html#18060

all stand as three mutually exclusive, and perhaps disconnected
investigations into the use of prime series ratios.

> However, the use of a ratio in itself seems to me a matter of art --
> be it emblematic, purely arbitrary ("I like the number, or intuit
that
> somehow it may produce an interesting result"), or done in a
carefully
> calculated "psychoacoustical" way. A fine point: one can pick
numbers
> in a "random" kind of way, but then select those that happen to
> produce an interesting value in cents for the musical purpose at
hand.

The use of the term "psychoacoustical" in these discussions (as
above), carries for me an equivalent meaning to "musical". The true
psychoacoustical value (an validation) of any tuning proposition, is
manifest in the final process of music making. Before this, the
better part of theory is speculative, cold and unproven - as well as
being absolutely "inaudible".

> While celebrating the emblematic or other artistically arbitrary use
> of numbers in music -- as opposed to in an acoustical or other
> scientific experiments, where quantities should indeed be explicable
> and significant to the design of the experiment -- I would like to
> join in due skepticism concerning claims for the "superiority" of
> certain tunings, whether defined with familiar mathematical
quantities
> or otherwise.
>
> However, while the use of e as a generator may be a charming
> example of serendipity (or "syncronicity"), we could also produce a
> musically similar result simply by calculating a fifth size where a
> chain of 15 fifths up generates a pure 7:4 -- ~704.588 cents.

Precisely the kinds of "synchronicity" I sought with the use of the
prime generators.

> In fact, when I hit on the idea of the e-based tuning, I wasn't sure
> of exactly what musical results it might produce; and having
explored
> them, I have found them in other tunings in the general region also.
> In turn, it was through comparing 24-note versions of these tunings
> that I came to realize that chain of 13 or 14 fifths were moving
> toward 7-based ratios (9:7, 7:6) a bit beyond 46-tET, and then to
> discover that 15 e-based fifths yielded a near-pure 7:4.

I began to explore the prime series ratios in a framework without
preconceptions about what I might have *wanted* to find, and only
sought to discover what portion of it all would reveal things of
musical value.

> In responding to your statement, I feel it very important to
> acknowledge the tradition from which I perceive you come, not only
> understanding it but relishing it, and possibly in the process
> clarifying my own sometimes more emblematic leanings.

I will join you in this, by saying that I also revere
Dave's "tradition", as well as other "purely mathematical" approaches
to tuning theory, but reserve the right to strike out on my own into
other explorative possibilities as well. It is a great honor to
participate in the multifaceted and simultaneous investigations
underway here, and it is my hope that we maintain an environment in
which many more will come forward to share the treasures of their
particular research. I find value in all of it.

Thanks,

Jacky Ligon

P.S. Dave Keenan, please direct me to some of your compositions so
that I may appreciate the tangible reality of your
musical/mathematical research. Do you have any MP3s of your
compositions or CDs that I may purchase, so that I may hear what
you've done? I'll be very eager to hear this!

🔗jpehrson@rcn.com

🔗ligonj@northstate.net

--- In tuning@y..., jpehrson@r... wrote:

> Jacky, Dave Keenan is not a composer, nor has he ever claimed to be
> one.

Joseph,

I was completely unaware of this. But I'm sure there must be a number
of fine composers who have used his findings in compositions. Perhaps
these will become revealed. I would like to hear this too! This would
be a great "collaborative" effort.

> Different people on this list all have different approaches and
> accomplishments regarding music. You and I happen to think that
> COMPOSITION is the most important one, but "it ain't necessarily
> so...!"

Although I would never seek to diminish any of the achievements of
anyone's purely theoretical work, by which have been personally
enriched (by many things of this nature), I must apologize in advance
for finding myself in perpetual disagreement with this. Tuning theory
without music, is like trying to walk after having your spinal column
removed. When the theorist's work is taken and given life in the form
of actual music, it then obtains is final validation. This is all a
part of what must include a creative cycle which terminates with an
audible product.

*Music* is, and must forever be the goal. I hope we never lose sight
of this.

Please note that in this forum, one will likely hear mathematics
invoked along side of music more than about any place on earth (it is
quite our necessary purpose), but that this isolated reality exists
here, says little about the fact the most of the world never thinks
about all of this when they experience the joy of music making. What
we put under the microscope of tuning analysis, comes naturally to
most of the world. Perhaps we gain some refined and intimate
understanding of all this with our study, but these things are
internalized in the musical and cultural hearts of the musical
practitioners.

Music is the roots, and the tree that was there in the beginning -
the primal thing that all discussion on this list draws its very life
blood from. This fact, I will never let be diminished in my mind. The
ultimate final *proofs*, IMHO and experience, are indeed the ones
which are manifestedly made audible in the reality of MUSIC.

Thanks,

Jacky Ligon

🔗jpehrson@rcn.com

🔗ligonj@northstate.net

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., ligonj@n... wrote:
>
> /tuning/topicId_18241.html#18274
>
> Thank you, Jacky, for your interesting response.
>
> However, I still think you are seeing it a bit through your "music
> colored" glasses, which I happen also to be wearing!
> Still, there are some people who find the beauty of mathematics,
for
> instance, to be superior to music! Maybe they're tone deaf! :)
> There are lots of different people in the world.

I would humbly disagree with your assumption here Joseph. When I look
at the collaboration between Kraig Grady and Erv Wilson (not a
composer), I derive a wellspring of inspiration from the beautiful
union between the worlds of mathematics and music. To hear Erv
Wilson's theory used in this wonderful music, completes the circle of
creation and collaboration that I revere so much and have been
repeatedly referring to here. We all would be able to appreciate many
more treasures of tuning bliss, if more of this kind of
theorist/composer collaboration was taking place on a grander scale.

> Look at Marcel Duchamp. Finally he decided that chess was a finer
> pursuit than art! Maybe he liked the competitive or "definite"
> nature
> of it...
> Anyway, as you know, he quit doing art and devoted the rest of his
> live to chess.

I love this game just about as much as music too. I will alternately
give it priority in my life from time to time. And I totally
understand the pull toward it - been playing since age 7.

> There are lots of different people in the world, and many different
> kinds of beauty!!!!

It is amazing to me how frequently I find myself in complete
resonance with you Joseph! My mind is open and respectful to all,
and only hope that the door swings both ways for all parties involved.

Jacky Ligon

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., ligonj@n... wrote:
>
> Although I would never seek to diminish any of the achievements of
> anyone's purely theoretical work, by which have been personally
> enriched (by many things of this nature), I must apologize in advance
> for finding myself in perpetual disagreement with this. Tuning theory
> without music, is like trying to walk after having your spinal column
> removed. When the theorist's work is taken and given life in the form
> of actual music, it then obtains is final validation. This is all a
> part of what must include a creative cycle which terminates with an
> audible product.
>
> *Music* is, and must forever be the goal. I hope we never lose sight
> of this.

Jacky,

My interest in tuning mathematics springs 99% from my desire to be a musician who
understands music better. But that does not mean that others can't contribute to my
understanding by working on the math. Much of Dave Keenan's work (including the
single-chain-of-fifth article, in which he anticipated many of your MOS odyssey results) was
partially inspired by my asking him whether there could be any generalized-diatonic scales that I
missed in my paper. His work makes me much more confident as to what the possibilities are,
and what near-misses might be interesting to explore.

In other words, it's "precompositional" work -- and if you compose at the keyboard, the
precompositional decision of how to tune your keyboard can be a difficult mathematical
problem, for which I am grateful there are mathematicians willing to address. Indeed, virtually all
musicians should be grateful to the mathematicians who were involved in figuring out 12-tone
equal temperament and how to implement it on their instruments, though this is usually taken for
granted.

Further, Jacky, you earlier statement to the effect that psychoacousical _is_ the musical, plus
your statement above that music is and must forever be the goal, leads me to wish to repeat all
the reactions I (and Dave) had to your MOS odyssey post. In my opinion, those reactions were
completely in line with the philosophy you seem to be expressing, though I'd be happy to
drop the issue if you wish.

🔗ligonj@northstate.net

--- In tuning@y..., PERLICH@A... wrote:
> In other words, it's "precompositional" work -- and if you compose
at the keyboard, the
> precompositional decision of how to tune your keyboard can be a
difficult mathematical
> problem, for which I am grateful there are mathematicians willing
to address. Indeed, virtually all
> musicians should be grateful to the mathematicians who were
involved in figuring out 12-tone
> equal temperament and how to implement it on their instruments,
though this is usually taken for
> granted.

Apparently you have not read or internalized (or perhaps do not care
to) all I've had to say about the above.

> Further, Jacky, you earlier statement to the effect that
psychoacousical _is_ the musical, plus
> your statement above that music is and must forever be the goal,
leads me to wish to repeat all
> the reactions I (and Dave) had to your MOS odyssey post. In my
opinion, those reactions were
> completely in line with the philosophy you seem to be expressing,
though I'd be happy to
> drop the issue if you wish.

If it pleases anyone to further attribute things to me for which I
have never intended, I honor their right to do so. But since it has
proven futile over the course of many days to make *any* of
the "alleged" stuff I supposedly intended stick, then I feel we are
achieving nothing of positive worth to further continue along such a
fruitless path. All I can add, is that each individual must do what
is right for their own music - and I feel that it should be obvious
there are more than just a couple of styles represented here in this
forum. What may be appropriate for one individual's style and tastes,
may seem alien or exotic to another. These differences, I am full
well able to respect and I hope that all will evolve in a direction
which is all inclusive, for the betterment of all that will come.
This means a personal effort though. I have not well grasped the
point of all that has came from the intolerant and inferential
treatment of my posts of late, and really it is meaningless to me why
it would have to persist, because I just don't have time for anything
this draining and negative in nature. Really what will it achieve? I
have seen nothing in it all of worth. I say lets move on and
celebrate what all bring to this forum in a positive light, and
banish the negative impulses to the remote past. We all benefit from
an environment which is open to myriad of different creative
approaches. Any other reminds me of our Senator Helms in NC, who
didn't like certain kinds of art, and played his cards to damage the
NEA. This jack-boot approach really threatens all that is about
positive growth in the arts.

So for whatever levels of imagined discomfort and nuisance factor my
participation on the list of late has caused, I publicly apologize to
all those injured and annoyed parties, and request we seek more
sublime levels of *positivity intoned* musical interaction and
brother/sisterhood.

Jacky Ligon

🔗MONZ@JUNO.COM

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_18241.html#18259

> Personally, I would be interested in hearing more about how
> Berg used numerology...
>
> If this is too OT, please send it privately... although since
> the use of numbers in art came up on this list and is certainly
> related to tuning... I think it is an ON-T....

Hi Joe,

I don't think this is OT at all. If Schoenberg really was using
numerology in his "free atonality" pieces as that book I mentioned
strongly argues, then that practice had a definite influence on
Schoenberg's later formulation of the 12-tone serial method. The
evolution of this method shows a clear association in Schoenberg's
mind between pitches and the 12-integer sequence.

The point I'm emphasizing (which makes this on-topic) is the
corellation of numbers - and all that they signify, to both
mathematicians and numerologists - with discrete equal divisions
of the pitch-continuum.

As I will be outlining in my upcoming presentation, Schoenberg
and his students did explore microtones a bit, and ultimately
rejected using them. I have a hunch that numerology was at
least partly responsible for Schoenberg's decision, because
12 is a powerful number in both numerology and mathmatics,
since it is so readily divisible (into 2, 3, 4, and 6).

(Despite my own fascination with numbers, I really don't know
much about numerology. Those who do are encouraged to jump in
here.)

I'm sorry I don't actually have the numerology/Caballa info
regarding Berg at hand. IIRC, the articles appeared in British
music journals (_Musical Quarterly_ and/or _Music and Letters_).

-monz
http://www.monz.org
"All roads lead to n^0"

Re: Occam's Razor or Keenan's Sand Filter?

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗jpehrson@rcn.com

🔗MONZ@JUNO.COM

🔗jpehrson@rcn.com

🔗ligonj@northstate.net

🔗jpehrson@rcn.com

🔗ligonj@northstate.net

🔗jpehrson@rcn.com

🔗ligonj@northstate.net

🔗PERLICH@ACADIAN-ASSET.COM

🔗ligonj@northstate.net

🔗MONZ@JUNO.COM

🔗Daniel Wolf <djwolf1@matavnet.hu>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗jpehrson@rcn.com