Tuning and Weaving (was: Set theory)

Paul, I have no problem waiting for further discussion if you're engaged
in other projects that preclude taking this up right now (*believe me*
I've been there!) But I should comment about your hints at where you
might begin a response. I'm afraid this leads to something quite
lengthy...

On 10 Apr 2001 tuning@yahoogroups.com wrote:

Paul wrote:
>
> My problem with [Xenakis' sieve theory] is that these constructions seem
very contrived, not at
> all simple, and about as far from a "derivation" or "explanation" as you can
> get. Much more simple and powerful descriptions of scales, I feel, come from
> considering a few very basic considerations _including_ tuning, and if
> understood correctly, the inherent flexibility of the tuning will come out
> naturally.

In many ways, the "diatonic" is a measure against which all other scales,
historical or synthetic or "natural," must be measured. Why? The easy
reason (and the one most likely to upset a lot of people since it is
highly subjective) is that it is arguably the most successful scale,
having been the basis for a plethora of great (I don't blush at this word)
music. But there is a deeper reason to use the diatonic as a measure.
Many have noted that the diatonic is "over-determined" -- i.e., there are
so many ways of generating it that you start to lose count. One way is by
stacking fifths until you get so close to an octave identity (at 7) that
you can call it quits. Then there's maximal evenness, in whatever form
you wish to express it. Then there are historical derivations based on
writings of earlier theorists (making the assumption that they were
significantly "closer" to the birth of diatony than we are -- not
necessarily correct); tetrachord coupling (building from smaller
constituent molecules); the logical sum of three diatonic triads whose
roots are a fifth apart; etc. Xenakis' sieve theory is only one of the
more recent "derivations."

Now don't flinch at the word "derivation" here. I don't in any way mean
to suggest that the word implies "this, and no other way, is how we got
the diatonic." *A* derivation is just that -- one way of constructing the
object. The amazing thing, and what has to make all scale-builders
jealous (it does me!), is the over-determination: the fact that there is a
multipicity of possible derivations, any one of which gets you to the
same diatonic scale. And this isn't an argument on behalf of the diatonic
scale, since I personally believe that over-use demands that we give it a
good long rest, while still keeping it as a model or measure for other
configurations. Instead of simply throwing out the diatonic, I'd rather
keep trying to surpass it, even if this is a futile effort.

So I'm not defending Xenakis' sieve theory as *the* way any given scale,
historical or otherwise, came into being. But I would defend Xenakis'
fruitful idea against charges that it is inelegant or contrived or not
"simple." You have to understand that to make that charge is to call
Peano's axiomatics of numbers (Xenakis' basis) contrived or too complex to
be useful. While he doesn't state it so baldly, Xenakis' various
formulations can be reduced to this:

v(xm^yn)

where v is logical disjunction ("or"), ^ is conjunction ("and"), and xm
and yn (which may be complemented) are "residual classes" m mod x and n
mod y respectively (e.g., residual class 0 mod 4 is {0,4,8,...}, i.e., the
augmented triad). I'm all for simplicity in its place, which means I
believe only in those simple, elegant formulations which serve as a
*basis* for complexity (which I generally equate with "life"). When you
put the above "normal" formulation into its most general form,
v(xm^yn^...), there is hardly any scale you can't cover. If, on the other
hand, you start throwing things out when they get complex, there are a lot
(most?) scales discussed on this list that you'd have to throw out. I
prefer to keep *all* of them and reserve "argument" for the discussion of
relative merits and demerits for specific compositional applications.
But more importantly, Xenakis' formulation suggests scales you may never
have thought of without that formulation, i.e., it's a creativity trigger.

Are there "more simple and powerful" formulations than Xenakis'? Sure,
DEPENDING on context. But I know of no other that potentially covers so
many (possibly all) scales. And his use of the word "sieve" is
(anthropologically) more suggestive than the generally used term
"background chromatic." Why I say this leads to...

> For example, set theory alone will never be able to explain why 7 notes and
> not some other number (you're not allowed to appeal to a 12-tone master set
> since that was constructed later in history).

As for "why 7?" you might want to look at ethnomusicologist Jay Rahn's
"Coordination of Interval Sizes in Seven-Tone Collections" (JMT 35:1&2) --
there's a bit of set theory in there. But let me take up the issue of
what I am and am not allowed to appeal to.

Here I'll ask something that might seem incredibly naive (relating to my
statement above re the birth of diatony): why do we believe that a
"background chromatic" is a relatively modern invention? Yes, if we look
at this idea in the very limited ways available via surviving texts,
Josquin probably did not think much in terms of a 12-tone master set or
canvass. But does this mean that neither he nor any other composer (or
theorist) did not work with some sort of sieve, whether consciously or
unconsciously? What would happen if we assume that such a sieve was so
deeply imbedded in certain important human activities by the time writing
was invented, that it went either unnoticed or unremarked upon? Assuming
such a (subconscious?) sieve might explain a lot. Now, since this puts us
well outside the realm of conclusive historical evidence, let alone
proofs, it's incumbant on me to make up a plausible story starting from:
Where would a pre-historic chromatic sieve come from? We'll start with
something I assume we're all sitting on right now, something we rarely
give notice to (unless it's missing): a piece of fabric.

Many have assumed that the art of weaving appeared some time in the
neolithic, but archaeologists Olga Soffer and James Adovasio have recently
discovered the remnant of a relatively complex weave that (along with
other artifacts in a Moravian site) dates to 27,000 BC. So we can safely
assume (other examples confirm this as well) that weaving, in its simplest
forms, dates even earlier -- probably much earlier. Adovasio, a textile
specialist (and discoverer of the Meadowcroft site near Pittsburgh that
shattered previous theories dating arrival of homo sapiens in the Western
hemisphere), believes that weaving was one of the most important
"discoveries" of prehistoric peoples, not just for clothing, but for nets
to trap animals as a food source -- a survival strategy every bit as
important as making spearheads. What does this have to do with music?

As an amateur weaver, I can tell you that, while warping a loom or doing
the actual weaving, the weaver MUST keep hundreds if not thousands of
threads straight. To sit at a loom and go "over 2, under 1, over 3, under
2, over ....." countless times with as little error as possible, there are
only two ways I can think of. One is to have a diagram or series of
numbers for the pattern on a piece of paper in front of you and keep
referring to it. A lot of things argue against this approach: you start
to go crosseyed, you lose your place, you get bored (vs the more positive
"meditation" trance), and so on. A much more efficient and natural way to
keep weaving patterns straight is to "sing" or "rhyme" or "dance" the
pattern as you weave. It's much easier to sing "dah da da dahhh dahhh da
dah dah" than to think/count "2-1-1-3-3-1-2-2" -- and the next row has to
go "da da dahhh dahhh da dah dah dah" which is arguably an easier
conceptual switch than "1-1-3-3-1-2-2." But the important thing to take
away is that the "tune" or "scale" sung is the SAME THING as the
arithmetic pattern. When you sing the pattern quickly, your rendition is
probably more rhythmic than melodious, but when you slow it down to weave,
it helps the momory to put the various syllables at different pitch
levels.

But pre-historic weavers didn't have the numbers even available for
expressing any weaving patterns -- so they didn't have a choice. Complex
patterns appeared in weaving well before numbers were invented by kings
and userers. How would you go about teaching a child to weave (passing
the technology along) if neither you nor the child had any concept of
"number" of threads? -- You'd gesture "watch and listen" and then sing out
loud as you weave.

A weaver's "work song" isn't simply a way to make a boring job more
pleasant, it's actually necessary, in some form, to do the job itself for
anything more complex than a simple plain (over-under/mom's potholder)
weave; and we know that these weavers did go beyond plain weave for a
variety of utilitarian, and possibly aesthetic, reasons (and artifacts
show this).

But the main point here is that, IF a connection could somehow be
established between early weaving patterns and musical scale patterns
(including drumming patterns), it would concomitantly suggest that there
is a (subconscious?) background "chromatic" for related melodies,
rhythms, scales, ragas, formulas -- you simply can't weave without the
textile version of such a chromatic canvas, whether you're consciously
aware of it or not. And the repetitive nature of weaving would tend to
culturally instantiate a lot of musical patterns WELL before any Greek
thought of scientifically measuring ratios on a monochord. QED;-)

Now what patterns might show up (no one that I know of has looked yet, but
I'm beginning to ask around) that could make a pre-historic weaving-music
connection? You might want to look for fabric relics that showed a
2212221 pattern -- after all, this shows up in some African rhythmic
patterns as well as our diatonic suggesting some deeply imbedded roots
(musicologists are loath to admit to a mixing of pitch and rhythm patterns
-- but weave patterns jwould get around this objection); or perhaps a
pentatonic 23223 (actually weaving either of these patterns would produce
a kind of twill). But chances are it wouldn't be this easy or direct, and
you'd have to end up guessing from much flimsier evidence. E.g., you
could easily identify shorter weaving patterns such as 21 or 221 and
surmise that repetition of 21 resulted in octatonic patterns and 221 got
variously developed as a basic tetrachord. As flimsy as that might be,
for my money it makes just as much sense as theories that posit ratios or
overtones as a basis which, when you accept them as PART of the whole
story as well (rather than going with either/or arguments), might point to
another over-determination for certain more general patterns.

Finally, transmission of musical patterns between radically different
cultures. If a pre-historic traveler migrated to a distant land for
whatever reason, he (she) might bring his songs with him in his head.
But perhaps someone kills him before he gets to sing them. His killer,
never having seen the kind of garment worn by his victim, takes the fabric
home. He wants to make one like it for himself so he examines the weave.
At his primitive warp-weighted loom, without ever having heard his victim
sing, he begins to weave the stranger's music.

Pleasant dreams!

Steve Soderberg

Stephen wrote,

>In many ways, the "diatonic" is a measure against which all other scales,
>historical or synthetic or "natural," must be measured. Why? The easy
>reason (and the one most likely to upset a lot of people since it is
>highly subjective) is that it is arguably the most successful scale,
>having been the basis for a plethora of great (I don't blush at this word)
>music. But there is a deeper reason to use the diatonic as a measure.
>Many have noted that the diatonic is "over-determined" -- i.e., there are
>so many ways of generating it that you start to lose count.

I agree!

>One way is by
>stacking fifths until you get so close to an octave identity (at 7) that
>you can call it quits.

You get closer at 5 than at 7, but OK, I'll buy that. But where does the
fifth itself come from? Tuning theory or set theory?

>Then there's maximal evenness, in whatever form
>you wish to express it.

That I won't buy -- it fails to explain why 7 out of 12 and not, say, 9 out
of 16. And 7 out of 31 is not maximally even -- oops! (uh-oh, I'm getting
McLarenish today)

>You have to understand that to make that charge is to call
>Peano's axiomatics of numbers (Xenakis' basis) contrived or >too complex to
>be useful.

No, it's not tantamount to that at all. It's useful for proving theorems in
number theory. But music?

>When you
>put the above "normal" formulation into its most general form,
>v(xm^yn^...), there is hardly any scale you can't cover.

Which is exactly my problem with it . . .

>If, on the other
>hand, you start throwing things out when they get complex, >there are a lot
>(most?) scales discussed on this list that you'd have to throw >out.

Which ones wouldn't you throw out?

>But more importantly, Xenakis' formulation suggests scales you may never
>have thought of without that formulation, i.e., it's a creativity trigger.

OK, I applaud all creativity triggers.

>[weaving story deleted]

>As flimsy as that might be,
>for my money it makes just as much sense as theories that >posit ratios or
>overtones as a basis

Well then, you must think positing ratios or overtones as a basis is awfully
flimsy. That serves marvelously to illustrate the point I was making in my
original posts about how the relationship between mathematics and music is
very controversial!

--- In tuning@y..., Stephen Soderberg <SSOD@L...> wrote:

/tuning/topicId_20862.html#20862

Hello Stephen...

Well, you've certainly "woven" an interesting tale with this one! I
would be interested in hearing from Paul Erlich what his specific
objections are to the Xenakis "sieve" theory... besides it's over
generality...if it can be done in layman's terms. Maybe it can't...

_______ ______ _____ ___
Joseph Pehrson

Hi Steve. I very much enjoyed your absorbing post on "tuning
and weaving". Sorry that this response is so belated.

--- In tuning@y..., Stephen Soderberg <SSOD@L...> wrote:

/tuning/topicId_20862.html#20862

> Complex patterns appeared in weaving well before numbers
> were invented by kings and userers. How would you go about
> teaching a child to weave (passing the technology along) if
> neither you nor the child had any concept of "number" of
> threads? -- You'd gesture "watch and listen" and then sing
> out loud as you weave.

It's important to remember that just because there is no *written*
record of something, we should not automatically assume that
it did not exist at some particular moment in history.

The physiological record shows that humans have been able
to speak (and have been fully modern in all other respects)
since about 40,000 BC. The earliest written records date
from no earlier than around 4000 BC.

So there's a gap of about 36,000 years during which people
could speak but could not write. I think it's highly unlikely that
the concept and use of numbers did not develop during that
period.

So there's no need to assume that the only method available
to transmit the knowledge of weaving would be to `gesture "watch
and listen"'. By 27,000 BC (the date you gave for the appearance
of the complicated weaving pattern), humans were fully capable
of transmitting any thoughts they had, by speaking in regular
language.

Note that this is not an argument against your general theory here.
I am in full agreement that musical means would have been the
most likely method for keeping track of the necessary patterns
in weaving.

In fact, I would extend your theory, to say that these patterns
probably manifested themselves in rhyme/rhythm patterns of
poetry along with their use as music, in the passing down of
history during the period of oral tradition.

PS – Thanks a bunch for sending me the Möllendorff book.
I translated the whole thing and featured it in my Microfest
presentation.

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