Yahoo Tuning Groups Ultimate Backup metatuning In the bottomless chum bucket of the ATL, a rational discussion

A week or so ago, Carl Lumma deleted one of Gene Ward
Smith's ATL posts (I believe -- can't be sure since it was
deleted and Carl "Master Censor" Lumma only quoted
part of it). Carl Lumma urged everyone "Let's not argue."

This sums up everything that's wrong with the ATL.

The entire problem is that the members of the Alternative Lying List
do NOT argue. Instead, as Johnny Reinhard pointed out in
a recent ATL post, "Flaming is endemic." Plain English translation --
the devotees of numerology who tyrannize the ATL typically
resort to empty name-calling and hysterical lies, rather than
rational arguments.

Carl Lumma's ill-advised injunction "Let's not argue" is pure
poison.

Argumentation and disputation are the very lifeblood of a
democracy. No democracy can function for long without
freewheeling argument. Therefore what the ATL needs is
LOTS of argument -- but _rational_ arguments, based on logic
and facts, using common sense to test claims against reality.
This has seldom been done on the ATL.

Instead, what we typically get on the ATL is the sort of venom
spewed out by Dante Rosati and Graham Breed and Carl Lumma
and Gene Ward Smith...sadistic name-calling wielded like
burning crosses on black peoples' front lawns in a futile
effort to support insupportable superstitions. As we have
seen from the dire fate of the Dante Rosati, this strategy
doesn't work.

Instead of argumentations and disputation, Carl Lumma wants
everyone on the ATL to "not argue." There's only one way
to make absolutely sure that no one argues -- eliminate
open discussion, for open discussion leads inevitably to
argument. But the only way you can be sure of getting rid
of open discussion is by setting up a police state run by
censorship. Otherwise, people will argue in public. There is no
open debate in a dictatorship -- the dictator shouts "Jump!"
and the populace asks "How high?" Anyone who tries to
argue with the dictator gets dragged off to the gulag,
as Carl "Master Censor" has openly threatened to do with me.

By telling us that he wants to get rid of open debate ("Let's
not argue"), Carl Lumma has shown us exactly what he stands
for. That's good. Now that we know Carl Lumma is a mortal
enemy of an open Jeffersonian ATL based on freewheeling
debate, it's much clearer what Carl Lumma is doing and why.
Since I have repeatedly urged open debate and deplored
empty name-calling, it stands to reason that Carl Lumma must
revile me. Did Stalin not revile Osip Mandelstam? People
who loathe an open society and detest open debate are not
likely to react well to those like myself who continually call
for freewheeling rational argumentation and disputation.

Speaking of which--

Jacob Barton makes some excellent points:

Message 7132 of 7143 | Previous | Next
Msg #
From: "piccolosandcheese" <jbarton@r...>
Date: Sun Apr 18, 2004 7:15 am
Subject: Math and music

>Jacob steps into the line of fire.

>For the moment he dodges the flamethrowers and looks solely at the
unanswered
>question, "what is the objective verifiable evidence that math has
any causal
>connection with music?" Actually, the original question was "does
math have a
>connection with music?" This one seems easier.

If you check, Jacob, you'll find that I asked one question as two
sentences (because
the sentence gets too long if I ask 'em as part of one question).

1. What is the objective verifiable evidence that math has any causal
connection
with music?

2. What is the objective verifiable evidence that math has any
explanatory power
for music?

I contend that math has no causal connection with, and no explanatory
power for, music. There is no objective verifiable evidence that it
does. On the contrary -- overwhelming evidence converges from many
different fields to show shows that math has no causal connection
with, and no explanatory power for, music.

The first problem with Jacob's discussion is that he deals with a
subset of my question: does math have any connection with music (at
all)?

Since it's easy enough to dream up a far-fetched connection
between anything and anything else by means of mental gymnastics, the
question "Is X connected to Y?" can always be answered in the
affirmative.

Therefore that question is not meaningful unless it is understood by
a eraonsable person in the context of my repeated queries in the more
elaborate form about a putatitve causal connection..

Let us take an example. I have previously pointed out that a deluded
person could see 3 dogs, note that 3 is the basis of the Pythagorean
tuning, and therefore incorrectly conclude that the 3 dogs are
connected to Pythagorean tuning. The person could go farther and
conclude that the 3 dogs are the basis of Pythagorean tuning, or that
they must howl in Pythagorean major thirds, etc., etc. This is Gene
Ward Smith's methodology, so admirably summed up in the incredible and
unspeakably foolish example of false reification in which Smith crowed
"Sit at the piano and you're confronted with an exponential function."
No, you're confronted with a piano. The exponential function is IN
YOUR HEAD. You can't touch it, you can't taste it, you can't smell it,
you can't see it, and you certainly can't see an expontential
function. You can only hear sounds which might or might not be
representable as exponential functions.

What a classic example of the fallacy of false reification...

The problem here is that by means of sufficient ingenuity absolutely
anything can be connected in some ludicrously far-fetched way to
absolutely anything else. The kidn of garbled reasoning we observe in
Gene "Woolly-Headed Numerology" Smith is endemic on the ATL. Is math
connected to music? Dogs are connected to music. (see above) Hot
fudge sundaes are connected to music because the word "sundae" has 7
letters and the the Western diatonic scale has 7 notes. (This is a
far-fetched connection, but you must admit that it IS a connection.)
Pig bladders are connected to music because if you fill a pig bladder
with air you can make a pitched sound with it, and pitched sounds can
be used to produce music. Cyclotrons are connected to music because
elementary particles travel in circles and experiments have shown that
judgements of octave simlilarity produce a circular judgment of pitch
class in which pitches constantly seem to return to
their starting point in terms of pitch class... And so on.

Therefore we must necessarily dismiss Jacob Barton's point that
answering the question "Does math have a connection with music" seems
easier to answer than my full-on question. It seems easier because it
is easier -- in fact anything can be connected in some outlandish way
with music, so that doesn't tell us anything.

When I use words I use them in a common-sense way, not in the
sophistical scheming trick-and-trap casuistry words are used by
Carl Lumma or Gene "Woolly-Headed Numerology" Smith or
Graham Breed. When I say conneciton, I mean the comon sense
meaning of connection -- a meaningful substantive connection,
which is to say, a causal caonnection with epxlanatory power
in the case case of math and music.

Jacob Barton continues:

>Math as I know it is a human creation whose purpose was to make
>some sense of our surrounding world.

We must be careful here. Is mathematics per se actually a pure
confection of the human mind?

This is a very dangerous metaphysical claim. It is extraordinarily
sweeping, and not supported by the available evidence.

This claim really boils down to the assertion that the Pythagorean
theorem, or the Axiom Of Choice, or the Banach-Tarkski Theorem, are
purely artificial confections produced solely by human culture. But
is that true?
Purely artifical and contingent products of human culture, like music,
do not share universals and typically have no basis in objective
reality or science or mathematics. For example, no one has succeeded
in proving that any particular form of poetry, such as Sapphic
hexameters, is based on nature, or can be proven to be "the" basic
form of poetry from mathematical considerations. Indeed, it seems
absurd to attempt to do so.

Likewise, no one has succeeded in proving that any individual style of
painting is based on nature or math. The effort to do so seems so
silly that it would provoke laughter. We could of course imagine
some cockamamey system of thought in which a theorist declares that
all colors are represented by frequencies of light which can be
described as numbers, that the numbers form ratios when different
colors of paints are observed, and so on... But this is clearly
twaddle. For there is no objective verifiable evidence that integer
ratios of wavelengths of light reflected from two different regions of
a painting look any more beautiful, or are any more useful to an
artist, than any other ratios of wavelengths of light reflected from
two different areas of a painting.
Which is to say, that there is no objective verifiable evidence to
show that any given set of colors can be shown by mathematics or
physics to be more useful or more beautiful or more appropriate for
use in a painting than any other given set of colors.

Common sense suggests this, of course. In painting at least.

But when it comes to music, suddenly people starting making
rash claims about integer ratios as pitches. As we have seen,
the overwhelming mass of psychoacoustic evidence does NOT
support these rash claims about small integer ratios as the alleged
"basis" of Western (or any other) music. Nor does the available
psychoacoustic evidence support the claim that integer ratio
pitches are any more musically beautiful or musically useful
or musically appropriate than any other pitches. David Doty has
made the extraordinarily ill-advised claim in a recent 1/1
editorial that small integer ratios are "more beautiful" than
other types of pitches. This was incredibly ill-advised because
it gives us the Holy Grail of mathematical music "theory" (so-
called) -- The Beauty Equation.

The Beauty Equation is my name for an hypothetical
equation into which the measurable parameters of a
musical score are put. After cranking and grinding, the
mathematicians gets a number out of The Beauty
Equation which corresponds to the musical beauty
of the composition.

Now, if you're laughing, you're not alone. The Beauty
Equation is patently absurd. But people like David Doty
and, implicitly, Gene Smith, keep making the mistake
of searching for The Beauty Equation over and over
again. Some even think they've found it, as Doty
apparently did when he penned his editorial.

But if we apply Doty's Beauty Equation, we instantly
discover that the "theory" (so-called) predicts that we
can make Bach more beautiful by tuning all the
pitches to small integer ratios. Of course this introduces
melodic commas and produces a horrible sense of
melodic dissonance. Retuning Bach with 5-limig
diatonic JI produces a truly hideous sound -- it's
melodically dissonant beyond endurance.

But wait -- it gets worse.

We quickly recognize that the smaller the integer,
the greater the beauty accoridng to David Doty --
and so we must get rid of all big-integer intervals in
a Bach composition and replace them with smaller
integer intervlas, like octaves. This should increase
the beauty of Bach's music even more, if we follow
through Doty's thinking. But wait...we can go further.
We can elimiante all 2:1s and replace 'em with 1/1
unisons. Thus, if we strip apart a Bach fugue and
reduce it to a single melodic line of 1/1 unisons
played with many different instruments simultaneously,
this should make Bach's fugure immeasurably
more beautiful, according to David Doty's
"Beauty Equation."

If you've stopped laughing now, you realize
how disastrous it is to attempt to produce
a Beauty Equation for music. But that doesn't
keep people like Gene "Woolly-Headed
Numerology" Smith from trying. Gene's verison
of the Beauty Equation is less simple-minded
than Doty's beauty Equation...but it's just
as silly. Smith's version is called TOPS.
Instead of crudely trying to quantify the
beauty of a musical compositions, Smith's
TOPS numerology tries to quantify some
degree of implicit "goodness" in a musical
tuning. Often this boils down to beats.
But of course the same problem applies.
The amount of beating produced by
intervals in a musical scale is not "good"
or "bad" oustide of a particular musical
context any more than musical compsoitions
are objectively "beautiful" outside of the
cultures which generated 'em. It's much more
complex than that since it all depends on
context, culture, and many other influences.
At bottom TOPS is just as laughable and just
as worthless as Doty's Beauty Equation --
since all Beauty Equations are worthless.
They're a failed and foolish effort to quantify
what cannot be quantified. Beauty (or some
measure of musical usefulness or appropriateness
or goodness of whatever-ness of a msuical
scale) can no more be objectively quantified in
music than it can in art or literature. The arts are
arts for a reason -- because they deal with
that which cannot be meaningfully quantified.
Attempting to apply math to music, either in
the doomed TonalSoft software or TOPs or
in Doty's bizarre Beauty Equation, is an
exercise in futility.

This seems like a digression, but it's not.
For Jacob Barton claims that math is a human
invention. If so, then it belongs to same
class as music and visual art -- but music
and art do not stand outside human culture,
as the incomprehension of countless westerners
faced with countless examples of non-western
music and visual art has proven.

This suggests that Jacob Barton is claiming
that math belongs to the same category --
no eternal verities, just a matter of taste
and acculturation.

Now, we face a serious logical problem at this point in Jacob
Barton's assertion that math is a human invention. Attempts to
declare that mathematics is entirely a contingent confection of
the human mind takes no account of the fact that the Pythagorean
Theorem has been discovered and proven in many different
cultures. This can be called the "all nurture" claim for math.
According to what Jacob Barton has claimed, math is just
something we invent, like jazz or hard rock or African masks
-- so perhaps in one culture pi = 7, while in another culture pi
= 1.2? Since people invent math, why not? Perhaps one culture
uses the relationship A^3 + B^5 = C^71 instead of the Western
version of the Pythagorean Theorem A^2 + B^2 = C^2?

No, that just doesn't happen. Pi is the same in every
modern culture. In ancient cultures the approximations
to pi always fell within the same range. No known ancient
cultures used 16 or 97 or 8.2 as an approximation for pi.
(Although the state legistlature of Wisconsin DID once
pass a law mandating that pi = 4. I have a xerox of that bill
haning on the wall of my study.) The Pythagorean Theorem,
even though it is independently discovered by many
different cultures, is always the same. This suggests that
there is an objective element to math which is outside of
human culture.

On the other hand, the other extreme into which Gene Ward
Smith has fallen as an implied claim, is the "all nature" fallacy.
This is the claim or the implication that mathematics is something
that exists entirely outside culture. According to this
view, math exists as some Platonic "thing" in some mystery realm
oustide our universe. (Exactly how that mystery realm comes
in contact with our universe, like something out of a bad
episode of Star Trek, has never been credibly explained.
Perhaps Klingons...?)

So this simple statement by Jacob Barton is certainly
NOT something "we can all agree on." Mathematicians
hold decidedly different positions on the question of
whether math is actually "a human creation." Moreover,
the controversy continues, and I know of no objectively
verifiable way of deciding it.

Jacob Barton remarks:

>Math as I know it is a human creation whose purpose was to make
>some sense of our surrounding world.

A contrary viewpoint:

"A mathematician, on the other hand, is working with his own
mathematical reality. Of this reality, as I explained in ' 22, I take
a `realistic' and not an `idealistic' view. At any rate (and this was
my main point) this realistic view is much more plausible of
mathematical than of physical reality, because mathematical
objects are so much more what they seem. A chair or a star is not
in the least like what it seems to be ; the more we think of it, the
fuzzier its outlines become in the haze of sensations which surrounds
it ; but `2' or `317' has nothing to do with sensation, and its
properties stand out the more clearly the more closely we scrutinize
it. It may be that modern physics fits best into some framework of
idealistic philosophy---I don't believe it, but there are eminent
physicists who say so. Pure mathematics, on the other hand, seems to
me a rock on which all idealism founders: 317 is a prime, not because
we think so, or because our minds are shaped in one way rather than
another, but *because it is so*, because mathematical reality is built
that way." [Hardy, G. H., "A Mathematician's Apology" Cambridge
University Press, England: 1940]

On the other hand, Hardy's claim can't be taken as
the last word. In fact, the problem with Hardy's claim
that math is completely abstract and useless is that
experience has shown that even the most abstruse
and seemingly arcane crannies of mathematics get
unexpectedly dragged into applied fields like physics.
The recent rise of knot theory in the GUT is one
example, but many other examples abound.

So it seems untenable to make the claim that any
area of math can permenantly inhabit an entirely
abstrct realm totally disconnected from observed
reality.

On the other hand, there now exist so many exotic
subsdisciplines in mathematics with no apparent
connection to observed reality (even today), that
it's hard to imagine how most of 'em would ever
fit into physics or chemistry or molecular biology.
For instance, there are quite a few mathematical
journals today dealing with branches of algebraic
topology or group theory which have only
a few hundred subscribers wordlwide. In some
of these subfields of mathematics, there may
be only a hundred or so people who actually
understand what the others are going -- the
number may he even smaller. Wildly abstract
areas of matheamtics like abstract algebras
that deal with many-dimensional spaces with
50 or 1000 or 10,000 dimensions, do not seem
to have an obvious connection with observed
reality.

Lastly, we may note that the claim that math's
"purpose is to make sense of the surrounding world"
doesn't seem to work at a basic level.

That's actually the definition of physics, not
mathematics.

Mathematics and mathematicians do not, as far
as I know, restrict themselves to making sense
of, or even to depending upon, the surrounding
world. Indeed, making sense of the surrounding
world seems to play esesntially no part in math --
according ot Hardy, math involves making sense
of an interior world of mathematics, with not
relationship to the external world.

A good example here is the Banach-Tarski
theorem.

"One may describe this in the following way:
given any two solid spheres, one the size
of a baseball and the other the size of the
earth, both the ball and the earth can be
divided into a finite number of non-overlapping
little solid pieces, so that each part of one is
congruent to one and only one part of the other.
Alternatively one can state the paradox thus: one
can divide the entire earth up into little pieces
and merely my rearranging them make up a sphere
the size of a ball. A special case of this paradox
discovered in 1914 is that a sphere's surface may
be decomposed into two parts which can be
reassembled to give two complete spherical
surfaces, each of the same radius as the original
sphere. These paradoxes, unlike the ones
encountered in set theory of the early 1900s,
are not contradictions. They are logical
consequences of set theory and the axiom of
choice." [Kline, Morris, "Mathematics:
The Loss Of Certainty, Oxford University Press:
1980, pg. 270]

To see just how incorrect Jacob Barton's
statement is that "math is an attempt to
make sense of the surrounding world," let
us take the Banach-Tarksi theorem a
little further.

According to B-T, an object the size of an
atom could be theoretically disassembled
and put back together into an object
the size of the observed universe. Or,
to go even farther, something much
smaller than the Planck length of roughly
1.6 * 10 exp -35 meters (an object smaller
than which has no objective verifiable
physical meaning in our universe, for below
that distance-scale lies only the quantum
foam in which space and time both
disappear) could be taken apart and
reassembled into an object arbitrarily
larger than our universe with a size
bounded only by infinity above...and,
once again, note well that a physical
object larger than the observable universe
(which has a radius of some 13.4 exp 9
light years according to the latest
data) also has no objective verifiable
meaning in our universe.

Since in our universe the Planck length
on the small end and 13.4 billion light-years
on the large end demarcate the limits
of physically meaningful size, the Banach-
Tarski theorem clearly does not make
any kind of sense in our universe. In fact,
the Banach-Tarski result is so bizarre
that it sounds like something dreamed
up by a person on hard drugs. Alas,
the Banach-Tarski theorem has been
proven true -- irrefutably so, mores
the pity.

So Jacob Barton's basic definition of
mathematics simply doesn't fly. He
has actually defined physics, and
physicists are a shabby shifty
lot who use bad unprovable math
to get things done and proof be
damned. They don't care if the potential
fuction blows up in the center of
waveguide, they just ignore the
fact that the math blows up and
build waveguides. What's more,
the waveguides _work_. Once in
the electronics lab I was building
a circuit in college and a math
major friend of mine came along
and took a look at my resistor
and inductor and capacitor values
and tried to argue with me that
the Barkhausen Criterion meant
the circuit wouldn't oscillate.
I switched it on and showed him.
"Look. It oscillates." The guy had
his head so far up his theory-driven
fundament he wouldn't believe his
eyes. "No! No!" he kept saying,
"The Barkhausen Criterion says
the gain must be EXACTLY 1.0
at a phase shift of 2 pi radians!
And you've got a gain of 1.1! It
can't work! The math SAYS it can't
work!!!"
"Look, goddamit," I told the math
major, "it DOES work. It just has
a lot of second harmonic distortion
in the output. The Barkhausen
Criterion is just an approximation
of reality. The math _isn't_ THE
ACTUAL REALITY ITSELF."
"No, no," the math major kept
muttering. "It C*A*N'T work. It
_shouldn't_ work. And if it DOES
work, it *must* not be allowed to
work!"
The Gene "Woolly-Headed
Numerology" Smith approach. Stick
your head in the sand and recite
the math and then mistake the
math for the reality. And when
the math bites you on the ass
and gives you garbage results,
descend into mindless name-calling
and shriek "baboon" and "imbecile"
and "crackpot" any anyone who dares
compare your garbage math with observed
reality.
Physicists constantly
thumb their noses at rigorous
mathematics, so Jacob Barton's
definition here is not something
mathematicians would seem likely
to accept or agree with.

>At the same time, however, a lot of basic math is just
plain true.

Unfortunately, this is simply not so in the commonly
understood meaning of the word "true." Cohen's 1963 proof
proved the independence of the axiom of choice and
the generalized continuum hypothesis from the other
axioms of set theory. By pressing the eject button on
a seeimngly basic elements of mathematics such as the
axiom of choice, we change the rules of the game
quite drastically. We lose the capacity to prove other
results such as the Loewenheim-Skolem theorem, while
with AC we get some nonintuitive results such as the
existence of non-measurable reals and the aforementioned
Banach-Tarki conjecture. The axiom of choice depends on
extending intuition about finite sets to infinite sets. This
might or might not be reasonable, and yields some results
which radically abuse common sense. The sense in which
a result like the existence of a non-measurable set of reals
or the existence of a non-measurable additive function can be
said to be "true" seems to contradict the commonsense
meaning of the word `true':

"With Cohen's independence proofs, mathematics reached
a plight as distrubing as at the creation of non-
Euclidean geometry. As we know, the fact that the
Euclidean parallel axiom is independent of the other
Euclidean axioms made possible the construction
of several non-Euclidean geometries. Cohen's
results raise the issue: which choice among the
many possible versions of the two axioms should
mathematicians make? Even if one considers
only the set-theoretic approach, the variety of
choices is bewildering." [Kline, Morris, "Mathematics:
The Loss Of Certainty, Oxford University Press:
1980, pg. 269]

NO law of nature requires that mathematicians include the
axiom of choice. We can banish it from our system of
mathematics if we wish. This is a very different situation
from physics, in which laws of nature DO require that
Planck's constant = 6.6 x 10 exp-27 erg-sec and that the
speed of light = 3 x 10 exp 8 m/sec. Aside from the obvious
fact that all measurements everywhere at all times appear
to converge on these values for Plack's constant and the
speed of light, it is a matter of simple physics to prove
that stars and atoms could not exist if Planck's constant
or the speed or light were much different than they are.
If Planck's constant were, say, 6.6 x 10 exp -3 erg-sec,
baseballs would diffract through windows when we
threw them - but if Planck's constant were that large,
there would exist no baseballs and no windows, no
stars and no atoms, no planets and no us.

>This enchanted me as a child. Built on these truths are a lot of
>more human constructions: algebra, calculus, matrices. Such
>things still have truth to them, being built on the basics, and plus,
> the helped us connect the dots some more with physics
>and such things.

This is just not correct. At one time mathematicians thought
that mathematical proofs represented some form of objective
truth, which is to say, a result which does not depend in
any way on human choice or human belief. But ever since
Goedel's 2nd incompleteness theorem and Cohen's 1963
independence proofs, the commonsense notion that there
exist mathematical truths independent of human belief or
choice has simply been shattered:

"The decision as to which of the many choices [of axioms
to include as part of the substructure of mathematics]
cannot be made lightly because there are both positive
and negative consequences in each case. (..) There are
then many mathematics. There are numerous directions
in which set theory (apart from other foundations of
mathematics) can go. moreover, one can use the axiom
of choice for only a finite number of sets, or a
denumerable number of sets, or of course for any number
of sets. Mathematicians have taken each of these
positions on the axiom." [Kline, Morris, "Mathematics:
The Loss Of Certainty, Oxford University Press:
1980, pg. 269]

The commonsense meaning of the term "true" in which
Jacob Barton (and most of the people on the ATL, who
seem to have no more mathematical education than
a sand dab despite their endless oracular proncouncements
about math in music) simply does not gibe with
a situation in which "whatever you prefer is what
you choose, and it's a matter of individual taste."
Truth in the common parlance means objective truth --
certainly in the sense Jacob Barton is using it,
and in the sense the innumerates who infest the
ATL use it. But objective truth becomes very hard
to reconcile with the situation today in mathematics.
Nowadays, you can order your mathematics a la carte
since Cohen -- with Banach-Tarski or without, without
non-measurable reals or without, with a discontinuous
additive functions or without.

Moreover, the naive Platonist position (which is what
Jacob Barton and people like Joe Monzo espouse) has
always been only one viewpoint among mathematicians.
Some mathematicians hold to constructivism, which says
that the only acceptable proofs are those that can
be constructed by the human mind, and the axiom of choice
gives no method for constructing a choice set, so constructivists
reject the AC. Some mathematicians believe in Formalism,
though this has been altered (some would say destroyed) by
Goedel's incompleteness theorems. However there's argument
about what David Hilbert meant when he talked about a
"finitary" consistency proof, so even that question isn't
settled.
Of one thing we can be certain:
The claim that mathematics is built on eternal "truths" outside
of the axiomatic system can't be built on attempts to reduce math to
logic. Those efforts failed, so it must be accepted that mathematics
is a process of manipulating symbols. Whether those symbols actually
stand for anything objectively real seems debatable. Certainly
mathematicians debate this issue...and people working in the
foundations of mathematics have recent found other apparent
inconsistencies in set theory. What this will mean for the future
the term "mathematical truth" is anyone's guess. Formalism and
logicism have not been strengthened by Goedel's theorem, and
Brouwer's intuitionism now looks like the Last Man Standing.

Note well, Jacob Barton, that intuitionism is a mathematical
belief system wholly at odds with what you are saying above.
The intuitionist position is closer to a touchy-feelie
fuzzy-wuzzy New Age quasi-religion than anything else --
propositions are known to be true, Brouwer claimed, because
we intuit that they are true. Exactly how we gain mathmeatical
knowledge by intution has not been explained. ESP? Aliens
sending us messages from outside our universe? Dogs
yodelling at ultrarsonic frequencies?

Bizarrely, Brouwer's intuitionist posiition seems to be the
one that has held up best at the 20th century crashed to
a close in mathematics.

>Specific to tuning, we have several ways of describing intervals.
> The frequency ratio 6/5 makes as much sense as the prime
>vector [1 1 -1] and the logarithmic 315.6 cents; they are all just
>attempts to make sense of a particular sensation in our
>ear/brain.

Unfortunately that statement is not correct. In fact, your
statement is provably false.

Permit me to demonstrate:

Integer ratios as representations of audible musical
intervals have basic problems.

One of the most basic problems is that there exist an infinite
quantity of positive prime integers A,B,C where
1.0 < B/C, A/(2^N) < 2.0 such that:

ABS[A/(2^N) - B/C] <= epsilon, where epsilon is
an arbitrarily small positive quantity close to, but
greater than, 0.

As a practical matter, this means that for any given
small prime integer ratio such as 3/2 there exist infinitely
many prime integers A such that A/(2^N) approaches
3/2 arbitrarily closely.

Let us take 10 particularly obvious examples to
see how self-evidently true this is:

The integer 196613 is not small by comparison with the
integer 3. Like 3, the integer 196613 is prime.

In cents, 196613/(2^17) = 701.999 cents.

For comparison, the just 3/2 = 701.955 cents.

Can you verifiably hear a difference of 0.044 cents in
real microtonal music in the real world?

But wait -- it gets even worse.

There exists not just one prime number audibly and for
all practical purposes absolutely musically identical
to 3...there exist an INFINITE NUMBER OF THEM.

Here are 8 more prime integers musically absolutely
identical to 3 in any music anyone is likely to hear
or play in the real world:

786433 is also prime. In cents, 786433/2^19 = 701.992 cents.

1572871 is prime. In cents 1572871/2^20 = 701.963 cents.

3145739 is prime. In cents 3145939/2^21 =
701.96105464311335 cents.

6291457 is prime. In cents 6291457/2^22 =
701.95857810082955 cents.

12582917 is prime. In cents 12582917/2^23 =

701.9551384515026408104501 This one is one
of my particular favorites because it's anomalously
close to the just 3/2 even though as a prime it's
extremely small. In terms of beats, I calculate
that at A 110 Hz, the prime ratio above A 110
gives by 12582917/2^23 =
165.0000131130218505859375

If you played this E above A 110 Hz together with
the E a just 3/2 above A 110 Hz, you would hear
0.000013 beats per second. Since there are 3600
seconds per hour, this works out to roughly one
beat every 3 hours.

Do you honestly believe, Jacob Barton and Joe
Monzo and Gene "Woolly-Headed Numerology" Smith
and every other worshipper of small integers
in music, that you can truly actually H*E*A*R
1 beat every 3 hours...?

Please.

Even the benighted members of the ATL can't be
_that_ deluded.

But wait. It keeps getting better. Because each
new prime I throw at you roughly improves the
approxiation to the 3/2 by a binary order of
magnitude. That means that every 10 such
primes we find improve the approximation by
roughly a factor of 1000. And we are already
down to 1 beat every 3 hours different from
the just 3/2!

Let's continue enumerating primes audibly identical
to 3:

25165843 is prime. In cents 25165843/2^24 =
701.956307933040562232157346279

50331653 is prime. In cents 50331653/2^25 =
701.955172848029738083009

100663297 is prime. In cents, 100663319/2^26 =
701.95539642543921254 In rough back-of-the
envelope calculations I guesstimate that if
played against a E a just 3/2 above A 110 Hz,
this pitch would produce about 1 beat every
2 months.

But we have still scarcely begun.

Mathematica makes it simple and easy to march
upward, finding prime after prime, and getting
closer and closer to the just 3/2. There is,
in point of fact, absolutely no limit to how
close we can get to the just 3/2. We need only
program Mathematica with the following entirely
trivial algorithm:

Test PrimeQ[3/2*(Y + 2^X)] where Y = 1...N, N
some small number not usually much above
50 or 100, and X = 1,2,3..M For values of
M above 16 the approximation to the just
3/2 drops considerably below 0.01 cent,
and for even trivially small values of
X = 20...100, the approximation to the
just 3/2 becomes so ridiculously close
that it becomes completely musically
meaningless to speak of any difference.

Okay, Jacob Barton.

So where are your small integers now?

You said that small integers like 3
are "musically and mathematically special."
You have now learnt that exactly the
opposite is the case. On the contrary --
there exist an infinite quantity of
primes musically identical and
mathematically within a hair's breadth
of the 3/2. And when I say "a hair's
breadth," we're talking difference
around 1 part per million or less.

Naturally you will *never* learn this stuff
hanging around the ATL, Jacob Barton.
All the liars and character assassins
on the ATL can do is chant "small
integers, small integers" -- because
they're so arrogant and so ignorant
so appallingly incompetent when it
comes to microtonal that the people on
the ATL have no idea that there actually
exist an infinite number of primes
musically identical to 3.

Notice some things here.

First, you can drop any of these large
primes right into any JI ratio in
place of 3.

Here we see a drastic example of the
radical difference twist math and
music. Mathematically 3 is not even
remotely equal to 100663319. They're
both primes, but mathematically not
even Gene "Woolly-Headed Numerology"
Smith could possibly write down the
equation 3 = 100663319 with a straight
face. Mathematically that equation is
absolutely and unequivocally false.

But musically, 3 = 100663319. And
it gets worse. Musically 3 = 50331649,
which means that musically 50331649 =
100663319. And on and on. We have
entered a hall of mirrors, in which
in musical terms an _infinite number_
of prime integers are musically
absolutely audibly identical (unless
you genuinely believe that you can
hear a difference of one beat every
2 months).

And the same is true for every other
small integer. Musically 5 = an
infinite quantity of prime integers,
7 = an infinite quantity of prime
integers, 11 = an infinite quantity
of prime integers.

Mathematically this would make no
sense at all -- but musically, it
is absolutely and provably true,
and we can quickly and easily prove
with with simple listening tests.
Unless one of the liars and character
assassins on this forum wants to try
to claim that he can reliably and
repeatably and objectively verifiably
distinguish an interval which beats
at a rate of 1 beat every 2 months
with the 3/2 from the 3/2 itself,
there is no escaping the brutal
conclusion -- small integers are
nothing special, in musical terms.

In fact, for every small prime
like 3 or 5 or 7 or 11 there exist
an infinite boundless wealth of
primes marching upwards, all
approximating with ever greater
accuracy 3 and 5 and 7 and 11.
And the accuracy of the approximation
becomes so great above 30 digits or
so (which is still extremely small
as far as primes go) that it
would become wholly impossible to
construct any imaginable measuring
equipment capable of measuring
such beats, since the beat period
(beats per X, where X = months)
increases along with the prime
by roughly a factor of 2 with
each new prime. Thus, 10 primes
up we get 1 beat every 2000 months,
10 more primes up we get 1 beat
every 2*10 exp 6 months, 10 more
primes up we get 1 beat every 2* 10exp9
months...well, you get the idea.
Pretty soon we reach at point at
which the estimated lifetime of
the univese is not sufficient to
complete one beat with the just
3/2.

So we see from the most elementary considerations of
prime number theory that in fact the assertion is
false that "small integers are musically...special" --
as is Joe Monzo's claim that "each prime integer has
a different sound," and as are the impliicit assertions
A. D. Fokker made when the made the ill-advised
decision to start using ratio-space lattices.

Ratio space lattices are musically meaningless for
the same reason. There exist an infinite number
of points in ratio space audibly identical to
the pitch between 1/1 and 2/1 represented by
any given point. Distances in ratio space are
musically meaningless for the same reason.

As we study ratio space lattices, we soon
discover that there is no audible property of
a musical tone which corresponds in any
meaningful way to points or distances in
ratio space.

Let us put this more clearly:

Ratio space diagrams are musically
meaningless because for any given
audible musical pitch there is no set of
coordinates in ratio space which can be
objectively verified to correspond with
that audible musical pitch.

Ratio space diagrams are musically
meaningless because for any given audible
musical interval there is no operation
which can be performed on the individual
coordinates in ratios space of the two
pitches which make up that musical
itnerval which can be objectively verified
to correspond with that audible musical
interval.

To put it even more bluntly, it is simple
and easy to prove that ratio space lattices
_appear_ to contain musical information,
but actually do not. No one has ever
succeeded in objectively providing
verifiable proof that any aspect of
music corresponds to either a
coordinate or a distance in ratio
space, and there is vast amount of
evidence against it.

When I perform these ratio space listening tests
on people and they find themselves forced
to give up each claim, the facts become clear.

Listeners cannot tell at better than chance
rates which of two radically different ratio
space coordinate-sets correspond to a
given pitch, which leads us to the inescapable
conclusion James Randi arrived at when
testing astrological charts. Randi tested
believers in astrology by asking them
which of two astrology predictions corresponded
to their astrological sign. They were unable
to answer at better than chance levels.

Randi concluded that astrological charts
have no connection with reality, and certainly
no connection with individual people of a
given astrological sign. (Of course there is
always some far-fetched connection, if we
wishto abuse and misuse the common
meaning of the word "connection" as liars
like Graham Breed love to do in futile
effort to escape from their own lies) but I
am using common sense here in a shocking
break from ATL tradition. And reasonable
folks know that the term "connection"
here indicates a material meaningful
connection, or if you wish, a causal
connection with explanatory power.)

Likewise, listeners cannot tell at better
than coin-flip levels which of two radically
different ratio space distances correspond to
a given musical interval. This leads us tot
the inescapable conclsion James Randi arrived
at with astrology charts. Once again, there
is no material meaningful connection between
the width of a musical inteval and a distance
in ratio space. In common parlance, there
is no connection between the two.

Lastly, listeners cannot tell at better than
coin-flip rates which of two radically different
ratio space distances corresopnds to
a given acoustic roughness between two
musical notes with harmonic series timbres.
This leads us once again to the analog
with astrology. Just as astrology charts
ahve no meaningful connection with the
real world, ratio space distances have
no connection with the acoustic roughness
which occurs between two musical notes
which use harmonic series timbres.

To see why, let us consider another
elementary fact of basic prime number
theory.

For every prime pair A,B there exist
infinitely many prime pairs C,D
such that C/B approximates A/B
arbitrarily closely.

Let us consider some simple and obvious
examples to see how startlingly self-evident
this is:

7 is approximately equal to 11^6/(3^2*5*19*37)

In cents, 11^6/(3^2*5*19*37) = 968.826883705
Compare with 7/4. You will find a difference
of roughly 9/10,000 of a cent. Can you
hear a difference of 9/10,000 of a cent?

11 is musically equal to 19/(13*17)

In cents, 19/(13*17) = 552.029994486
while the just 11/8 = 551.3179423, a difference
of about 0.7 cents.

13 is musically equal to 17^3/(2^5*3^3*7)

In cents, 17^3/(2^5*3^3*7) = 840.175319435
while the just 13/8 = 840.5276617693.

19 is musical equal to 3^9/(7*37)

In cents, 3^9/(7*37) = 297.425062562219
while the just 19/16 = 297.513016132

What of smaller primes?

5 is musically equal to 11^6/(3^2*7*19*37)

In cents, 11^6/(3^2*7*19*37) = 386.314691101
whereas the just 5/4 = 386.31371386483

3 is musically equal to 7^4/5^2

In cents, 7^4/5^2 = 702.67617814683
Compare with the just 3/2 at 701.955000865
for a difference of about 0.7 cents.

I could continue typing out an endless
quantity of these musical equivalencies,
but my fingers get tired, and the arrogant
incompetent ignorami who tyrannize the
ATL would only respond by screaming more
hysterical insults ("imbecile," "lunatic")
and indulging in my infantile name-calling
("crackpot," "baboon," ad nauseum) rather
than discussing these issues rationally
by using facts and logic together with
common sense. And common sense, as any
reasonable person instantly realizes,
must certainly inform us that a musical
note which beats with another musical note
only once every 8 months is in musical
terms the same musical note.

Since an infinite number of musically
nearby points in ratio space audibly
counterfeit every other coordinate,
this means that in ratio space
[1,0,0,0] musically = [0,-2,4,0]

Moreover, in any one of the coordinates
in ratio space we can drop in the
other coordinates as a substitute --
thus, for 13 written at
[0,0,0,0,0,1,0] we can substitute
[-3,0,-1,0,0,3].

This makes points in ratio space
musically meaninglesss since it
is obviously impossible for any
listener to tell which coordinates
in ratio space correspond to a
given just interval. The
coordinates for the 13/8 correspond
to both [0,0,0,0,0,1,0] and
[-3,0,-1,0,0,3], and unless the
listener can reliably and verifiably
hear a difference of 0.4 cents
in real microtonal music in the
real world (which no one can)
there is absolutely no way for
any human being to tell the
difference. Moreover, there exists
not just one duplicate of the 13/8
in ratio space, there are infinitely
many, most of them quite close to
the origin -- which is to say,
whithin no more than 20 axis
clicks out.

(Later we shall see the simple and
obvious reason why ratio space is
so utterly musically meaningless.
As always, the liars and character
assassins on the ATL cannot tell us,
since they spend their lives telling
lies and screaming insults precisely
because they don't know any of the
most basic facts about microtonality.)

Accordingly, it becomes reasonable
to ask the question -- What musical
information does ratio space convey?

Ratio sxpace does not convey infomration
about audible pitch height.

Ratio space does not convey information
about audible interval width.

Ratio space does not convey information
about audible roughness twixt two sustained
picthes which both use harmonic series timbres.
(For conveience let's agree that we're talking
about equiamplitude steady-state harmonics
1 through 6. But it could be any buzzy harmonic
series tone made up non-gapped low order
overtones.)

And the proof is clear. If ratio space conveyed
information about any of these aspects of
music, listeners would be able to quickly and
easily be able to distinguish twixt two radically
different ratio-space lattices by ear. They cannot.

So if listeners cannot audibly distinguish twixt
two raidcally sets of points and distances in
ratio space by ear, then what musical information
does ratio space convey?

As with astrology, the answer is: no information.
Just as astrological chart _seem_ to convey
information but actually don't, points and distances
in ratio space (which is to say, Fokker periodicity
blocks, Monzo's TonalsSoft software, Erv Wilson's
ratio-space diagrams and all the other pseudoscience
involved with drawing diagrams of and making
calculations in ratio space) _seem_ to convey
information about music...but actually do _not_
convey any musically meaningful information at all.

Ratio space diagrams and calculations involving
ratio space diagrams are as musically meaningless
as the numbers on the back of a soda bottle.

This simple fact alone destroys well over 70% of the
numerology bandied about on the ATL.

Jacob Barton continues:

>If we zoom in on people in the Western world who call themselves
>composers, we find that in order to make some sense of the infinite
world of
>sound that composers could create, composers limit their parameters.
They
>impose mathematical systems - rhythmic grids, pitch grids, loudness
grids,
>overall architectures. The construct an environment in which they can
be creative.

This is the fallacy of hasty generalization. Some Western composers
do.
Others do not. To generalize from one instance or even some in which
Westerns composers choose grids does not necessarily provide us with
a valid picture of Western music as it exists today.
Example: Bebe and Cece Barron constructed electronic circuits with
vacuum tubes which created feedback loops and consequently complex
behavior. The tubes would usually burn out. The Barrons recorded
the electronic signals produced by these self-destructing vacuum
tube feedback circuits and used the tapes to create electronic music.
Their music, including the score for Forbidden Planet, is some of the
best ever created.

The Barrons did not "limit their parameters" nor did they "impose
mathematical systems -- rhythmic grids, pitch girds, loudness
grids, overall architectures." Their electronic music remains some
of the best I've heard.

Moreover, even among those composers who do choose to quantize
pitch, there is typically a good deal of unquantized stuff as well.
For
isntance, composers often include glissandi. Some composers have
produced pieces in which all performers produce continuous glissandi
and no note is ever fixed. As for rhythmic grids, it's quite common
in certain styles of Western music to abandon notating time entirely.
Morton Feldman's later music gave up the notation of any kind of
rhythmic grid and Penderecki wrote a quartet in which all 4 string
instruments play the melody lines freely, without bar lines or meter
and without consideration for what the other instruments are
playing at any given moment. Virtually all electronic musique
concrete, and much computer music, abandons the idea of a
rhythmic grid altogether.
Perhaps composers in the 18th century thought of the musical
universe in clockwork terms, but in the 20th (let alone the
21st!) century that is a musical prejudice considered
superfluous by many contemporary composers.

Loudness grids were never used in any musically meaningful
way, to my knowledge. Babbitt attempted to quantify and
serialize loudness and his effort failed spectacularly. There is
no musically meaningful grid of loudness implied by the
Western system of ppppp to fffff, so that's simply a non-starter.
These are vague indications at best. As Leonard Meyer remarks:

"Innate cognitive constraints do not, however, segment other
parameters of sound into discrete, proportional relationships.
For instnace, there is no relationship in the realm of syntactics
that corresponds to, say, a minor third or dotted rhythm. And
the same is true of tempo, sonority, and timbre. Dynamics
may beconme louder or softer, tempo may be faster or slowed,
sonoroties thinner or thicker, and so on. But they cannot be
segmented into perceptually discrete relationships. Because they
are experienced and ocnceptualized in terms of amount, rather
than in terms of kinds of classlike remationships (such as
`major third' or `antecedent-consequent'), I have called these
parameters `statistical..'" [Meyer, Leonard B., "A Universe Of
Universals," The Journal of Musicology, Winter 1998, Vol.
XVI, No. 1, pg. 9]

Pitch, by contrast with loudness, is mapped logarithmically and
tonotopically to a semicircular arc of the auditory cortex. (When
Gabor denies this documented fact, I shall provided references
raising serious questions about his credibility.)

The concept that grids are necessary or even necessarily
useful for music is legacy of the 18th century. AS Meyer
points out, statistical parameters serve equally well as
means of organization. Amounts, rather than kinds, havebeen
used in painting as well as in music and literature in many
different ways. Grids are simply not necessary.

Jacob Barton continues:

>What, then, of the connection? There is no clear objective
generalization
>explaining how we LISTEN to music

In actual fact, at the extremes, psychoacoustics comes into play.
No listener hears very loud sustained beats within 1/4 of the critical
bandwidth as "calming" and "restful." Measurements of galvanic
skin reponse, blood pressure, heart rate and evoked potentials
in EEGs all converge on the conclusion that regions of the brain
associated with negative emotions are activated by extremely
loud prologned beats within 1/4 of the critical bandwidth.

Results like these cited in a recent conference on the biological
bases of musical response, indicate that some extremely
elementary listener responses are not entirely learnt.

On the other hand, the extraordinarily wide range of musical
scales and musical responses and musical styles worldwide
seems to prove that outside of the very most extreme outer
limits of musical simtuli, learning takes precedence over
the biological structure of the human auditory system in
determining a listener's response to music.

So we have a situation much more complex than the either
"nature" (Rameau, Zarlino, Doty, Rosati) advocates or
"nurture" (the Parisian Kook, the Darmstadt kook, the
Princeton kook, the coin-flipping kook, the Viennese
kook) have claimed.

> but typically you have the intuitive-leap reactions [to music]
>(emotional, sensory response and all of that) and the intellectual
>reactions (conscious-thought-driven).

EEG studies have shown that this is by and large not correct.
Non-musicians typically use primarily the right brain hemisphere
when listening to music, while only musicians use both brain
hemispheres. The right brain hemisphere is primarly responsible
for our emotional responses, and thus the experimental evidence
suggets that non-musician listeners primarily have emotional
reactions to music. This is confirmed by studies in which listeners
were asked why they listened to music. An overwhelming majority
of litseners reported "For the emotion." No listeners responded
"For the mathematics."

These references are contained in Eric David Sheirer's MIT PhD
thesis "Music-Listening Systems." You should download the
pdf of the thesis and read the entire text at

http://web.media.mit.edu/~eds/thesis/

Of course, the comments on my citations show that no one ever
bothers any of the references I cite, since that would require
people to shut off their computers and get a life. Thus, internet
Alzheimer's...the fantastic delusion that "if it ain't on the net, it
isn't knowledge."

>Ah, but how do we CREATE music? Any darn well way we please, these
days.
> We write in 13/8 against 5/4 time if we want. We use tunings
dependent on
>astrological signs if we want. We make the piece a palindrome if we
want.
> All of this is acceptable. Regardless of whether the math is audible
in the
>music, *it has a connection with music because it is used to create
it.*

>CAN WE AGREE ON THIS?

No, since the opposite seems to be the case in virtually every one of
your
conclusions and assertions.

You have here combined four different causal fallacies in one
assertion:

(1) The fallacy of insignificance -- X is taken as a cause of Y
when in fact X is a negligible contributing factor.

(2) The fallacy of reversing the directin of cause and effect.

(3) The fallacy of joint effect -- mistaking X as a cause for Y,
when in fact
both X and Y are both caused by some third factor W.

(4) The fallacy of complex cause, in which X is incorrect
identified as
the entire cause for Y when in fact X is only a partial
cause. The
best example of this is genetic risk factors for disease.

If the mathematics proves inaudible, then you are falling into
the fallacy of insignificance. This is the same fallacy Aristotle fell
into
when he claimed that sheets of ice do not sink in water because
they are broad and flat and cannot divide the water, while a piece
of iron is sharp and blocky and divides the water and so the
iron sinks.

Galileo refuted Aristotle by taking a piece of ice and pushing it
to the bottom of a bowl of water. The ice divided the water's
surface, but nonetheless rose again. Likewise, Galileo took
an iron needle and placed it on a piece of paper, then pulled
the piece of paper from under the needle. The iron needle
floated on the water.

The causes Aritstotle alleged were in fact unimportant.
Galileo showed what the true causes were -- density and
surface tension.

You cannot so easily brush off the significance of whether
the mathematics is audible. Experiments conducted
by Carol Krumhansl that trained college musicians
cannot hear mathematical structures in music even
after having the structuers explained to them. Krumhansl
concluded that deep mathematical structures are
inaudible, and the features to which listeners react
are primarily surface features. In another experiment
in which lsiteners identified snippets of music only
1/4 second long, the main mode of identification was
found to be timbre. Thus, timbre seem to be the
primary audible property that listeners glom onto.
Mathematical structures at a deep level appear to be
wholly inaudible. It is therefore unwise to assert
math as a cause for how a composition sounds,
as opposed to timbre, or other aspects of a composition
which listening experiments have shown to be
far more important for listeners.

"The relatively recent concern of music theorists with
the detailed cnalaysis of hierarchic structure has led to
an incongruous coupling -- a coupling of nineteenth-
centry ideologya dn twentieth-century science. What
seems to have happened is this: the Romantic esteem
for the mysters of the `profound' -- in Wordsworth's words,
of `thoughts that do often lie too deep for tears' -- let
to the valuing of what lay beneath the surface of things.
Miscontrstruing this belief, theorists in more than one
field have been beguiled into believing that `deepest
level of structure' [for isntance, mathematical structures
whicha re not direclty audible] -- is `profound' and hence
more significant than the patent patterning of the
phenomenal foreground.

"But there is a `profound' difference between valuing a theory
for its explanatory power and elegance, and valuing teh
experience of the phenomenon that is explained. To call a
work of art `profound' is to character the experience of
that work, _not_ to comprehend the general principle upon
which it is based. The `thoughts which do often lie too
deep for tears' result form the personal experience of `the
meanest flower that blows.' What is profound is nto the deep
structure of a flower (or of a piece of muisc), but the experience
of a particular work -- its power to move us.

"Clearly depth of explanatory theory and depth of aesthetic
experience got mixed up. Theories, which are propositional
constructs _about_ phenomena, are valued for their generality,
clarity, and verifiabliity. The more general a theory, the deeper
it is said to be. But works or art are not propositions _about_
phenomena, they _are_ phenomena -- phenomena which are
valued for the psceific experiences they provide. And because
such experiences are difficult to describe and fully explain, they
are felt to be profound and often characterized as `ineffable.'"
[[Meyer, Leonard B., "A Universe Of Universals," The
Journal of Musicology, Winter 1998, Vol. XVI, No. 1,
pg. 13]

Next, there is the fallacy of reversal of cause and effect.

There is no hard objective evidence to show that
math causes music in Western culture, as opposed to the
reverse assertion that Western music causes composers to
fixate obsessively over math because of the way Western
music is taught. The proof that you have reversed
cause and effect is simple: in virtually all other musical
cultures around the world, mathematics is not considered
a part of music, and non-Western musicians typically do
not make any use of mathematics in creating music, nor
do they attempt to understand music in terms of math.
William Sethares encountered this when he travelled to
Java and talked to gamelan tuners. They rolled their eyes
and muttered about "another westerner trying to tack
math onto what they were doing" (or words to that
effect -- personal communication).

In non-Western culture, music is usually considered
a part of magic, or religion, or it's used in ceremonies
to rid young girls of possession by alleged demons,
and so on. We do not see often see non-Westerners
using physics texts to exorcise demons, nor do we
see non-Western cultures employing rocket ships
or transistors in an effort to exorcise alleged demons
from young girls. This suggests that you have got
it wrong...and, rather than causing Western music,
mathematics is caused by the way Western music
is taught (namely, saturated with superstitions of
the kind Dante Rosati has gibbered for us with
such peerless incoherence).

Third, we have the fallacy of joint effect. Even in those
cases where a Western composer does make use
of math, there is no objective verifiable vidence showing
that the math plays any more of a role in how the
composition turns out than other causal factors.
For example, the wholly irrational causal factors of
composer's state of mind, the composer's
compositional proclivities, and so on.
To put it bluntly, everything we do -- including
composition -- boils down to inexplicable
and indescribable non-rational processes in
the human brain. Composers do not
know where a melody comes from, or why
a cadenza must be inserted at point X, or
why they decided to write a rondo rather
than a fugue -- but they also don't know
why they find themselves enamoured of
composing electronic musique concrete
rather than kazoo klezmer. The decision to
use a particular bit of math most likely arises
from the same cause as all the composer's
other musical decisions -- deep emotions.

In this case, math might explain one small
part of a composition, but the _decision_ to use
that particular bit of math -- like the decision
to use a rondo form, etc. -- are both likely to
arise from a much deeper cause... Namely,
the irrational and inexplicable emotional processes
of the human mind.

So the actual third cause for both the math
used in a particuclar composition
and the other musical decisions which make
the composition peculiarly what it is, typically
boils down to emotion and imagination and
creativity.

Fourth, we have the fallacy of the complex
cause. Even in those instances where
mathematics plays a subsidiary and
trivial role in a composition, as in the
case of polyrhythms, other equally
important factors also play a role in
how the music turns out -- namely,
cultural conditioning. A composer
does not exist outside culture. It was
the great fallacy of the serial atonal
composers to imagine that they
could stand entirely oustide culture
and outside biology. In fact, because
of the properties of the human nervous
system, atonal music typically sounds
audibly disorganized. With no
discernible melodies and no fucntional
harmonies and no perceptible rhythmic
pulse, music slides into audible
incoherence and quickly becomes
unutterably boring. Here, nature
and nurture both play a part. Culture
determines some parameters of our
response to music, but so does the
strucvutre of the human cognitive sytem
and the human auditory system. It's
not a simple either-or, which leads
us to conclude that ascribing any
material part of "how a composition
sound" to math alone is the fallacy
of the complex cause. Culture as
well as the properties of the human
auditory system are far more important
in determining how a composition sounds
than any amount of math involved.

"To explain why human beings in some
actual cultural-historical context think,
respond, and choose as they do, it it
necessary to distinguish those facets
of human behavior that are learned from
those that are innate and universal. But
it is a mistake -- albeit a common one --
to conceptualize the problem as seardch
for `musical' universals. _There are none._
TRhere are only the acoustical universals
of the physical world and the bio-psych-
ological universals of the human world.
Acoustical stimuli affect the perception,
cognition and hence practice of muisc
only through the constraining action of
bio-psychological ones." [Meyer, Leonard
B., "A Universe Of Universals," The Journal
of Musicology, Winter 1998, Vol. XVI, No.
1, pp. 5-6]

Jacob Barton continues:

>Now for the tougher question. Where is the objective verifiable
>evidence that math has any causal connection with music?
>Doesn't it completely depend on the person composing or
>the person listening? If it does, is there any way to ever find
>objective verifiable evidence?

This is the fallacy of the complex question, which assumes the
answer as part of the quesiton.

Two possibilties exist:

(1) If math has an objectively verifiable causal connection with
music, there evidence for it should abound. Where is it? Let us
see the evidence.

In every other field where math has an objective causal
connection with the phenomena studied, we have copious evidence
for the causal connection. Math has a causal connection with
the phenomena studied by physics, and it's simple and
easy to prove it -- we find that mathematical physics produces
theories which have predictive value. We can t
(Message over 64 KB, truncated)

xenharmonic wrote:

> I contend that math has no causal connection with, and no explanatory
> power for, music. There is no objective verifiable evidence that it
> does. On the contrary -- overwhelming evidence converges from many
> different fields to show shows that math has no causal connection
> with, and no explanatory power for, music.

The "causal connection" part is easy to dismiss. Any reasonable person
should count Bach's compositions as "music" and should agree that they
couldn't have been written without mathematical techniques such as
augmentation and inversion. And unreasonable person can disagree about
anything being "music" and so the statement becomes meaningless.

The "explanatory power for" is the other side of the same coin. We can
explain certain features of Bach's compositions using mathematical
descriptions.

> (Although the state legistlature of Wisconsin DID once
> pass a law mandating that pi = 4. I have a xerox of that bill
> haning on the wall of my study.) The Pythagorean Theorem, Are you sure? These people might like to hear of it:

http://www.snopes.com/religion/pi.htm

the only example given is Indiana, where a bill fixing the value of pi
to a more complex ratio passed its first reading in 1897. I know of
that from several sources, but none mention this Wisconsin bill.

> even though it is independently discovered by many > different cultures, is always the same. This suggests that > there is an objective element to math which is outside of > human culture.

You contradict this later with:

>>At the same time, however, a lot of basic math is just > > plain true.
> > Unfortunately, this is simply not so in the commonly
> understood meaning of the word "true."

> Integer ratios as representations of audible musical > intervals have basic problems.
> > One of the most basic problems is that there exist an infinite
> quantity of positive prime integers A,B,C where > 1.0 < B/C, A/(2^N) < 2.0 such that:
> > ABS[A/(2^N) - B/C] <= epsilon, where epsilon is
> an arbitrarily small positive quantity close to, but
> greater than, 0. Yes, that's where the idea of "temperament" comes from.

> Even the benighted members of the ATL can't be
> _that_ deluded. There are, indeed, members of the ATL who object to any amount of
temperament, howerver small.

> So where are your small integers now?

Small integers are small integers. Big integers are not small integers. If a small integer does the job, you don't bother to look for big integers. This is so blindingly obvious to most people that they don't go through it in the detail you seem to expect

> Here we see a drastic example of the
> radical difference twist math and
> music. Mathematically 3 is not even
> remotely equal to 100663319. They're
> both primes, but mathematically not
> even Gene "Woolly-Headed Numerology"
> Smith could possibly write down the
> equation 3 = 100663319 with a straight
> face. Mathematically that equation is
> absolutely and unequivocally false.

Mathamatics does include the concept of approximation. The
"approximately equal" sign is like an equal sign, but with the top line
a bit wavy. In ASCII, lets call that ~=. You can then write

frac(log2(3)) ~= frac(log2(100663319))

where frac(x) is the fractional part of x such that 0 <= x < 1 and log2
is the logarithm to base 2. Gene could have written that down really
easily.

> But musically, 3 = 100663319.

That's interesting, because you seem to think that two intervals
separated by an arbitrary number of octaves are "musically equal". Have
you forgotten that octave equivalence is not a universal? Or that the
perceptual octave is not exactly 2:1?

> Ratio space lattices are musically meaningless for
> the same reason. There exist an infinite number
> of points in ratio space audibly identical to
> the pitch between 1/1 and 2/1 represented by
> any given point. Distances in ratio space are
> musically meaningless for the same reason. Yes, and you can show the same pitch on more than one place on the lattice. Fokker did this. It's one of the main things that lattices are useful for.

> Ratio space diagrams are musically
> meaningless because for any given
> audible musical pitch there is no set of > coordinates in ratio space which can be > objectively verified to correspond with
> that audible musical pitch.

Yet above you showed that an infinite number of coordinates correspond to the audible musical pitch.

> Lastly, listeners cannot tell at better than
> coin-flip rates which of two radically different
> ratio space distances corresopnds to
> a given acoustic roughness between two
> musical notes with harmonic series timbres.
> This leads us once again to the analog
> with astrology. Just as astrology charts
> ahve no meaningful connection with the
> real world, ratio space distances have
> no connection with the acoustic roughness
> which occurs between two musical notes
> which use harmonic series timbres.

So what "radically different" coordinates are you choosing here? 4:3 compared to 11:8, or 3:2 compared to some impossibly complex ratio that's almost the same pitch?

> 13 is musically equal to 17^3/(2^5*3^3*7) > > In cents, 17^3/(2^5*3^3*7) = 840.175319435
> while the just 13/8 = 840.5276617693.

This is thoroughly bizarre, because not only do you make the factors of 2 explicit, but you get them wrong. The unison vector here is 4914:4013.

4914 = 13 * 2 * 3^3 * 7
4913 = 17^3

In cents, 17^2/(2^5 * 3^3 * 7) = -359.8 cents. To get 840.2, you need 17^2/(2^4 * 3^3 * 7).

> Since an infinite number of musically
> nearby points in ratio space audibly
> counterfeit every other coordinate,
> this means that in ratio space
> [1,0,0,0] musically = [0,-2,4,0]

Making that equivalence gives you a planar temperament. One which Gene has named after me, and written music in. It's exactly the supposed numerology you object to.

> This makes points in ratio space
> musically meaninglesss since it
> is obviously impossible for any
> listener to tell which coordinates
> in ratio space correspond to a
> given just interval.

In the absence of context.

> The
> coordinates for the 13/8 correspond
> to both [0,0,0,0,0,1,0] and
> [-3,0,-1,0,0,3], and unless the
> listener can reliably and verifiably
> hear a difference of 0.4 cents > in real microtonal music in the
> real world (which no one can)
> there is absolutely no way for
> any human being to tell the > difference.

Not that Gene corrected this as the comma 4914:4913. It doesn't matter if anybody can tell the difference, because there's no evidence that four digit frequency ratios have any audible meaning. The 13-limit is borderline, but if this interval can be perceived as a ratio it'll be unquivocally 13/8.

> Moreover, there exists
> not just one duplicate of the 13/8
> in ratio space, there are infinitely
> many, most of them quite close to
> the origin -- which is to say,
> whithin no more than 20 axis
> clicks out. What's a lattice click? What does "most" mean in the context of an infinity?

Don't worry, it doesn't matter. 40 steps is huge anyway. Assuming a measly 7-limit octave-equivalent lattice, that's 10 million possible intervals within the octave. How big is your keyboard?

(Sorry Gene, yes, not everybody uses keyboards)

> Accordingly, it becomes reasonable
> to ask the question -- What musical
> information does ratio space convey?
> > Ratio sxpace does not convey infomration
> about audible pitch height.
>
> Ratio space does not convey information
> about audible interval width.

That depends on how you draw the lattice.

> Ratio space does not convey information
> about audible roughness twixt two sustained
> picthes which both use harmonic series timbres.
> (For conveience let's agree that we're talking
> about equiamplitude steady-state harmonics
> 1 through 6. But it could be any buzzy harmonic
> series tone made up non-gapped low order
> overtones.) It gives you a good idea about the roughness. The roughness may be less than you predict, which means the point was badly chosen. Often, the rule is "one step consonance, everything else dissonance".

> And the proof is clear. If ratio space conveyed
> information about any of these aspects of
> music, listeners would be able to quickly and
> easily be able to distinguish twixt two radically
> different ratio-space lattices by ear. They cannot.

No, the "quickly and easily" is a hurdle you've added. As long as the points are distinguishable, then the lattice carries meaning.

> So if listeners cannot audibly distinguish twixt
> two raidcally sets of points and distances in
> ratio space by ear, then what musical information
> does ratio space convey?

Are these deliberately chosen "radically different" points that give the same intervals? How are they musically arrived at? Lattices are primarily used to show progressions that move by small steps on the lattice. In that case, you won't have some absurdly complex ratio arriving in the middle of the piece. If a complex ratio happens to get modulated to, then it can be tempered out.

In the simplest case, a 5-limit octave-equivalent lattice illustrates funcional harmony. It's been used for this purpose without any connection to frequency ratios.

> Pitch, by contrast with loudness, is mapped logarithmically and > tonotopically to a semicircular arc of the auditory cortex. (When
> Gabor denies this documented fact, I shall provided references
> raising serious questions about his credibility.)

That looks mathematical. Does it have anything to do with music?

> In non-Western culture, music is usually considered > a part of magic, or religion, or it's used in ceremonies
> to rid young girls of possession by alleged demons,
> and so on.

Is this what punk bands do in Tokyo?

> of the complex cause. Culture as
> well as the properties of the human
> auditory system are far more important
> in determining how a composition sounds
> than any amount of math involved.

No shit!

Graham

In the bottomless chum bucket of the ATL, a rational discussion

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