re;re for Tom

Thanks Mike. That's exactly the type of thing I'm bringing into question. Just to add to what you already said so clearly, mathematically speaking there seems to be an infinite amount of rational numbers which in principle approximate the tempered intervals. Given 2 power of n/12, n = 0, 1, 2,...12, then taking 2 power of (N + n/12), N = whole, rounding off to nearest odd (if available), and dividing by 2 power of N seems to supply us with an inexhaustible set. Further, the larger N the more digits we include after the decimal point and the smaller the difference between the two irrational and rational numbers. So that if N approaches infinity, then theoretically speaking at least the two numbers might become exact. (But since the resultant period is infinitely large in comparison to the component periods, then this is tantamount to saying that no period exists after all).

However, nature might step in here and provide some realistic limit to N. This might be Planck's constant, or it might be limitations in computational process, or our hearing etc...

But what is of most importance to me in this question (and is not lost on Mike) is the possibility that all waves are periodic and that it is the irrationals which are the approximation. Tom, you said something to the effect that sine waves are simplistic idealizations and therefore could see no reason why we should be discussing such matters in this forum. But isn't it true that all Fourier transforms are built of component sine waves, which applies to both optics and QM? In fact what interested me in wave theory to begin with was when I saw how well wave addition modelled musical tonality. When the wave is periodic, the tonic is simply octave equivalent to the resultant fundamental frequency (not only is 6/5 a worse approximation to the minor third than 19/16, the 5 in the denominator gives the wrong tonic, whereas 16 gives the fifth octave. We need this in order to be in a minor key. Thanks everybody for confirming this for me). So if all waves are
harmonic and periodic to the very core, and these waves are a fixture of nature and by definition can be described by Fourier analysis, then the distinction between science and music is not as clear cut as we have been led to believe. If this is correct, then Pythagoras was right after all, but in a new way. In fact, it's my view that it is the sine wave that is real and observable, not the geometrical lines that physicists draw on a page and call 'space and time', nor the rods and clocks that these are meant to represent and which they had to invent in order to compensate for this observational deficiency. As I said in the last email, this question of tuning reaches right to the very heart of Western culture with Pythagoras. Harmonia means the fitting of things together and is the root of our concept of cosmos and universal order i.e. science itself. Fact is stranger than science-fiction and waves are more interesting than any dreams about time travel.

Regards

Rick

Get the name you always wanted with the new y7mail email address.
www.yahoo7.com.au/mail

--- In tuning@yahoogroups.com, rick ballan <rick_ballan@...> wrote:
>
> Thanks Mike. That's exactly the type of thing I'm bringing into
> question. Just to add to what you already said so clearly,
> mathematically speaking there seems to be an infinite amount of
> rational numbers which in principle approximate the tempered
> intervals.

Mathematically speaking, there does not seem to be, there IS an
infinite amount of rational numbers, which not only in principle
approximate the tempered intervals. For every irrational number, there
is an infinite amount of rational numbers arbitrary close to it, and
for every rational number, there is an infinite amount of irrational
numbers arbitrary close to it. From a mathematician's point of view.
this is nearly trivial.

>
> However, nature might step in here and provide some realistic limit
> to N. This might be Planck's constant, or it might be limitations
> in computational process, or our hearing etc...

As far as music is concerned, the limitations of our hearing are the
point that matters. Absolutely no use to go down to Plancks's constant.
Physics is another matter, of course.

>
> But what is of most importance to me in this question (and is not
> lost on Mike) is the possibility that all waves are periodic and
> that it is the irrationals which are the approximation. Tom, you
> said something to the effect that sine waves are simplistic
> idealizations and therefore could see no reason why we should be
> discussing such matters in this forum. But isn't it true that all
> Fourier transforms are built of component sine waves, which applies
> to both optics and QM?

But both Fourier transforms and sine waves are still abstractions. I
got to say that I, too, do not really get what you are after.

> In fact what interested me in wave theory to begin with was when I
> saw how well wave addition modelled musical tonality. When the wave
> is periodic, the tonic is simply octave equivalent to the resultant
> fundamental frequency (not only is 6/5 a worse approximation to the
> minor third than 19/16, the 5 in the denominator gives the wrong
> tonic, whereas 16 gives the fifth octave. We need this in order to
> be in a minor key. Thanks everybody for confirming this for me). So
> if all waves are harmonic and periodic to the very core, and these
> waves are a fixture of nature and by definition can be described by
> Fourier analysis,

Whether the waves ARE harmonic (rational) or only very close to
harmonic (irratonal approximating a rational) is irrelevant if the
difference is below the limitations of human hearing, isn't it?

> then the distinction between science and music is
> not as clear cut as we have been led to believe.

About that, I certainly agree!
--
Hans Straub

Hans wrote...

> > Thanks Mike. That's exactly the type of thing I'm bringing into
> > question. Just to add to what you already said so clearly,
> > mathematically speaking there seems to be an infinite amount of
> > rational numbers which in principle approximate the tempered
> > intervals.
>
> Mathematically speaking, there does not seem to be, there IS an
> infinite amount of rational numbers, which not only in principle
> approximate the tempered intervals. For every irrational number,
> there is an infinite amount of rational numbers arbitrary close
> to it,

Actually I don't think that's true.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hans wrote...
>
> >
> > Mathematically speaking, there does not seem to be, there IS an
> > infinite amount of rational numbers, which not only in principle
> > approximate the tempered intervals. For every irrational number,
> > there is an infinite amount of rational numbers arbitrary close
> > to it,
>
> Actually I don't think that's true.
>

Why not?
Give me a counter-example. then.
--
Hans Straub

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:

> > > Mathematically speaking, there does not seem to be, there IS an
> > > infinite amount of rational numbers, which not only in principle
> > > approximate the tempered intervals. For every irrational number,
> > > there is an infinite amount of rational numbers arbitrary close
> > > to it,
> >
> > Actually I don't think that's true.
>
> Why not?
> Give me a counter-example. then.
> --
> Hans Straub

There isn't even a single rational for every irrational
number -- how could there be an infinity of them?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "hstraub64" <straub@> wrote:
>
> > > > Mathematically speaking, there does not seem to be, there IS an
> > > > infinite amount of rational numbers, which not only in principle
> > > > approximate the tempered intervals. For every irrational number,
> > > > there is an infinite amount of rational numbers arbitrary close
> > > > to it,
> > >
> > > Actually I don't think that's true.
> >
> > Why not?
> > Give me a counter-example. then.
> > --
> > Hans Straub
>
> There isn't even a single rational for every irrational
> number -- how could there be an infinity of them?

Hi Carl,

"for every irrational number there is an infinite amount of rational
numbers arbitrary close to it" just means that

for every irrational number r and for every positive real number e > 0
there exists a rational number q so that |r - q| < e. This is true!

Consider pi: there is a sequence of rational numbers where successive
members get closer and closer to pi, for example the sequence

3 / 1
31 / 10
314 / 100
3141 / 1000
31415 / 10000
314159 / 100000
and so on...

Kalle

Work with me here:

For any two given integers, including consecutive, there is an infinite
number of non integers between them.
For any given ratio of two integers, there are an infinite number of non
integers between that ratio and another, even adjacent, ratio of two
integers.
Each number representing the ratio of two integers is one rational
number.
Thus there are many irrational numbers for each rational number.

Notwithstanding the unending numbers of integers and spaces between
them.

Howard Cornell

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Carl Lumma
Sent: Tuesday, June 03, 2008 11:20 AM
To: tuning@yahoogroups.com
Subject: [tuning] Re: re;re for Tom

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ,
"hstraub64" <straub@...> wrote:

> > > Mathematically speaking, there does not seem to be, there IS an
> > > infinite amount of rational numbers, which not only in principle
> > > approximate the tempered intervals. For every irrational number,
> > > there is an infinite amount of rational numbers arbitrary close
> > > to it,
> >
> > Actually I don't think that's true.
>
> Why not?
> Give me a counter-example. then.
> --
> Hans Straub

There isn't even a single rational for every irrational
number -- how could there be an infinity of them?

-Carl

Hans:
> > > For every irrational number, there is an infinite amount
> > > of rational numbers arbitrary close to it,

Me:
> > There isn't even a single rational for every irrational
> > number -- how could there be an infinity of them?

Kalle:
> "for every irrational number there is an infinite amount of
> rational numbers arbitrary close to it" just means that
>
> for every irrational number r and for every positive real
> number e > 0 there exists a rational number q so
> that |r - q| < e. This is true!

What Hans said would be true if he'd said "for any" instead
of "for every". But what you say here isn't true even then.
You can get arbitrarily close but you can't get closer as
many times as there are reals closer.

-Carl

the whole number series is infinite, isn't it?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, > "hstraub64" <straub@...> wrote:
>
> > > > Mathematically speaking, there does not seem to be, there IS an
> > > > infinite amount of rational numbers, which not only in principle
> > > > approximate the tempered intervals. For every irrational number,
> > > > there is an infinite amount of rational numbers arbitrary close
> > > > to it,
> > >
> > > Actually I don't think that's true.
> >
> > Why not?
> > Give me a counter-example. then.
> > --
> > Hans Straub
>
> There isn't even a single rational for every irrational
> number -- how could there be an infinity of them?
>
> -Carl
>
>

If the whole number series is infinite, just imagine how big the real
number system is! We have just been "discussing" how many of one kind
of number can be placed between other kinds of numbers. We can do that
if we only look at the relative preponderance of one kind of number,
which helps as long as we do not worry about the total number of either
kind being uncountable.

Howard Cornell

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Kraig Grady
Sent: Tuesday, June 03, 2008 2:20 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

the whole number series is infinite, isn't it?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/
<http://anaphoria.com/> >

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/
<http://anaphoriasouth.blogspot.com/> >

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
<mailto:tuning%40yahoogroups.com>,
> "hstraub64" <straub@...> wrote:
>
> > > > Mathematically speaking, there does not seem to be, there IS an
> > > > infinite amount of rational numbers, which not only in principle
> > > > approximate the tempered intervals. For every irrational number,
> > > > there is an infinite amount of rational numbers arbitrary close
> > > > to it,
> > >
> > > Actually I don't think that's true.
> >
> > Why not?
> > Give me a counter-example. then.
> > --
> > Hans Straub
>
> There isn't even a single rational for every irrational
> number -- how could there be an infinity of them?
>
> -Carl
>
>

It is as big as the whole number series... infinite. There are not many sizes of infinity, just one.

Oz.

On Jun 3, 2008, at 10:59 PM, Cornell III, Howard M wrote:

> If the whole number series is infinite, just imagine how big the > real number system is! We have just been "discussing" how many of > one kind of number can be placed between other kinds of numbers. We > can do that if we only look at the relative preponderance of one > kind of number, which helps as long as we do not worry about the > total number of either kind being uncountable.
>
> Howard Cornell
,

That's the problem. Infinity is merely a concept. An infinitely long
straight line has an infinite number of points. AND there are an
infinite number of points between any two adjacent points of the line.
Cantor devised a hierarchy of infinities, and identified several. The
appropriate reasoning for our discussion on numbers is as follows.

Consider the interval from 0 to 1. The reasoning can be repeated for
all other intervals between integers.

Assign each point that are rational, that is, can be expressed as the
ratio of two integers, to a counting number in order of their quotient
from smallest, 0+ to 1. Clearly this will be an infinite number of
discrete points, with a lot of space between them. That space is full
of the numbers that are not ratios of two integers, that is, irrational
numbers, and there are infinitely many numbers between each of them; but
there are clearly more of them than rational numbers, and this is just
between 0 and 1. To complete the "proof" just repeat infinite number of
times. If you need to.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Tuesday, June 03, 2008 3:09 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

It is as big as the whole number series... infinite. There are not many
sizes of infinity, just one.

Oz.

On Jun 3, 2008, at 10:59 PM, Cornell III, Howard M wrote:

If the whole number series is infinite, just imagine how big the
real number system is! We have just been "discussing" how many of one
kind of number can be placed between other kinds of numbers. We can do
that if we only look at the relative preponderance of one kind of
number, which helps as long as we do not worry about the total number of
either kind being uncountable.

Howard Cornell

,

I certainly have a hard time grasping how there can be more irrational numbers between any two whole numbes as opposed to rational numbers. They are both infinite, hence, the same "amount". Infinity is a value without a conclusion. If all the irrational numbers are counted along with rational numbers they will amount to the same value, that is to say, there will be no end to them. There can be nothing more to a continuous increase of numbers. Mayhap you are thinking that the pace of counting is different? Or perhaps, you are assuming that there should be more than a single irrational number that corresponds to a rational number at a given pace of counting? That would not change the fact that there are as many rational numbers as there are irrational numbers.

Oz.

On Jun 3, 2008, at 11:34 PM, Cornell III, Howard M wrote:

> That's the problem. Infinity is merely a concept. An infinitely > long straight line has an infinite number of points. AND there are > an infinite number of points between any two adjacent points of the > line. Cantor devised a hierarchy of infinities, and identified > several. The appropriate reasoning for our discussion on numbers is > as follows.
>
> Consider the interval from 0 to 1. The reasoning can be repeated > for all other intervals between integers.
>
> Assign each point that are rational, that is, can be expressed as > the ratio of two integers, to a counting number in order of their > quotient from smallest, 0+ to 1. Clearly this will be an infinite > number of discrete points, with a lot of space between them. That > space is full of the numbers that are not ratios of two integers, > that is, irrational numbers, and there are infinitely many numbers > between each of them; but there are clearly more of them than > rational numbers, and this is just between 0 and 1. To complete the > "proof" just repeat infinite number of times. If you need to.
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On > Behalf Of Ozan Yarman
> Sent: Tuesday, June 03, 2008 3:09 PM
> To: tuning@yahoogroups.com
> Subject: Re: [tuning] Re: re;re for Tom
>
>
> It is as big as the whole number series... infinite. There are not > many sizes of infinity, just one.
>
> Oz.
>
> On Jun 3, 2008, at 10:59 PM, Cornell III, Howard M wrote:
>
>> If the whole number series is infinite, just imagine how big the >> real number system is! We have just been "discussing" how many of >> one kind of number can be placed between other kinds of numbers. >> We can do that if we only look at the relative preponderance of one >> kind of number, which helps as long as we do not worry about the >> total number of either kind being uncountable.
>>
>> Howard Cornell
> ,
>

[ Attachment content not displayed ]

Oz,

I'm not assuming. Try not to think of an infinity that has "an equal
number of everything" in it. Try to think of just a reasonable number
of two kinds of things and expand it later. The point is that by
definition there are fewer numbers that are rational than there are
irrational numbers: because the definition of irrational numbers ALLOWS
more numbers to qualify. Don't worry about infinities yet. Just
consider, as in my previous email, how many rational numbers you can
CONCEPTUALLY put in the interval 0 to 1 and compare that to the number
of irrational numbers you can also assign to that interval. No matter
how many rational numbers you put in, you can BY DEFINITION put in more
irrational numbers, no matter how you calculate them! Still, don't
worry about counting them yet. Now, consider what you have found--you
can always put in more irrational numbers than rational numbers, even
between 0 and 1. No matter how many rational numbers you put in, there
is ALWAYS room for more irrational numbers between them. Start with
just a few rationals: you can insert as many irrationals as you have
time for between them. That's the point. It's true you can always put
in more rationals--and you can always put in EVEN MORE irrationals.
This is why infinity is not just an uncountable number of things; there
are properties associated with the kinds of things being counted.

It's just that in an infinite number of arbitrary numbers there will be
more irrationals because there will always be fewer rationals, based on
how they are defined.

See mathforum.org/isaac/problems/cantor1.html for Cantor's attempt to
make infinity less paradoxical. That page has a link to cantor2 and
beyond for further explanation of the hierarchy of infinities.

I hope that this helps.

Howard

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Tuesday, June 03, 2008 3:59 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

I certainly have a hard time grasping how there can be more irrational
numbers between any two whole numbes as opposed to rational numbers.
They are both infinite, hence, the same "amount". Infinity is a value
without a conclusion. If all the irrational numbers are counted along
with rational numbers they will amount to the same value, that is to
say, there will be no end to them. There can be nothing more to a
continuous increase of numbers. Mayhap you are thinking that the pace of
counting is different? Or perhaps, you are assuming that there should be
more than a single irrational number that corresponds to a rational
number at a given pace of counting? That would not change the fact that
there are as many rational numbers as there are irrational numbers.

Oz.

On Jun 3, 2008, at 11:34 PM, Cornell III, Howard M wrote:

That's the problem. Infinity is merely a concept. An
infinitely long straight line has an infinite number of points. AND
there are an infinite number of points between any two adjacent points
of the line. Cantor devised a hierarchy of infinities, and identified
several. The appropriate reasoning for our discussion on numbers is as
follows.

Consider the interval from 0 to 1. The reasoning can be
repeated for all other intervals between integers.

Assign each point that are rational, that is, can be expressed
as the ratio of two integers, to a counting number in order of their
quotient from smallest, 0+ to 1. Clearly this will be an infinite
number of discrete points, with a lot of space between them. That space
is full of the numbers that are not ratios of two integers, that is,
irrational numbers, and there are infinitely many numbers between each
of them; but there are clearly more of them than rational numbers, and
this is just between 0 and 1. To complete the "proof" just repeat
infinite number of times. If you need to.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of! Ozan Yarman
Sent: Tuesday, June 03, 2008 3:09 PM
To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] Re: re;re for Tom

It is as big as the whole number series... infinite. There are
not many sizes of infinity, just one.

Oz.

On Jun 3, 2008, at 10:59 PM, Cornell III, Howard M wrote:

If the whole number series is infinite, just imagine how
big the real number system is! We have just been "discussing" how many
of one kind of number can be placed between other kinds of numbers. We
can do that if we only look at the r! elative preponderance of one kind
of number, which helps as lon g as we do not worry about the total
number of either kind being uncountable.

Howard Cornell

,

[ Attachment content not displayed ]

http://www.jcu.edu/math/Vignettes/infinity.htm

It seems I am stuck in pre-1870s math.

Oz.

On Jun 4, 2008, at 12:54 AM, Mike Battaglia wrote:

> There are different degrees of infinity. In set theory, such degrees > are referred to as CARDINALITIES. One can see whether one set has a > higher cardinality (is "more infinite") than another set by seeing > if the two can be put into a ONE-TO-ONE CORRESPONDENCE with one > another. If the two sets have that property - anything in one set > can be matched by something in the other - then they are of the same > cardinality.
>
> Take the set of natural numbers: it goes on forever in one > direction. How about the set of just the positive even numbers? Is > that "less" infinite than the natural numbers?
>
> 0 1 2 3 4 ...
> 0 2 4 6 8 ...
>
> They both go on forever in one direction, and a direct one-to-one > correspondence can be made between the two. For every element in the > even numbers, there is an element in the natural number set that > will match up for it. There aren't more natural numbers than even > numbers because we will never "run out" of numbers. For each natural > number generated, there is a matching even number as well.
>
> In fact, even this works:
>
> 0 1 2 3 dog 4 5 6 ...
> 0 1 2 3 4 5 6 7 ...
>
> The set of all natural numbers and the word "dog" has the same > cardinality as the set of all natural numbers without it. This is > again because they can be put into that one-to-one correspondence I > keep taling about.
>
> How about the set of all integers? After all, it goes on forever in > two directions: negatively and positively. Maybe we'll reach some > "bigger" infinity in that way.
>
> If we take all of the negative numbers and pair them with the odd > numbers...
> Integers: -1 -2 -3 ...
> Naturals: 1 3 5 ...
>
> and the positive numbers and pair them with the even numbers...
> Integers: 1 2 3 ...
> Naturals: 2 4 6 ...
>
> Then we get this:
>
> Integers: 0 -1 1 -2 2 -3 3 ...
> Naturals: 0 1 2 3 4 5 6 ...
>
> You see that the two sets can be matched up that way. The integers > can be expressed as a list of numbers increasing forever in one > direction. There is that one to one correspondence. There is a > "bijection" between the two sets. Therefore they are of the same > degree of infinity - they have the same cardinality.
>
> What about the set of all rational numbers? Those might be of a > greater degree of infinity than the natural numbers and the integers.
>
> Any rational number can be expressed as an integer fraction. This > means we can graph set of all rational numbers can be as a plane -- > the horizontal axis would represent an increase in the denominator > and the vertical axis would represent an increase in the numerator. > So is the degree of infinity of a plane more than that of a line?
>
> For a long, long time, mathematicians thought the answer was "yes." > And then someone came up with a pattern so obvious that it was > almost stupid: If you just draw diagonal lines going from (0,1) to > (1,0), and then from (0,2) to (2,0), and etc, you hit everything. So > you can list all of the numbers and pair them up with the natural > numbers.
>
> Well, if you look at it this way:
>
> 1: 1/1
>
> 2: 1/2
> 3: 2/1
>
> 4: 1/3
> 5: 2/2
> 6: 3/1
>
> 7: 1/4
> 8: 2/3
> 9: 3/2
> 10: 4/1
>
> etc.
>
> Then every positive rational number in existence will be reached - > some multiple times (which is okay). To include the negative > rationals as well:
>
> 1: 1/1
> 2: -1/1
>
> 3: 1/2
> 4: -1/2
> 5: 2/1
> 6: -2/1
>
> 7: 1/3
> 8: -1/3
> 9: 2/2
> 10: -2/2
> 11: 3/1
> 12: -3/1
>
> etc.
>
> So we can pair up all of the rational numbers with the natural > numbers. A bijection exists between the two sets, they are of the > same degree of infinity, etc.
>
> Another way to say this is that the set of all rational numbers, > natural numbers, integers can all be ENUMERATED. You can get every > single one of these sets and find an "enumerator" function that will > generate every member of this set if we just let it run forever.
>
> So this seems to be in accordance with your hypothesis, right? That > every infinite set has the same size of infinity. BUT, what about > all of the real numbers?
>
> Well let's put it this way: let's just find all of the real numbers > between 0 and 1. If THOSE have a higher degree of infinity than the > natural numbers, then obviously ALL of the real numbers will have a > higher cardinality than the natural numbers as well.
>
> Now, if we can enumerate the real numbers (if there is any function > that will produce every single one of the reals), then we can just > write that list down, obviously.
>
> I claim that I have some incredibly complicated way that will do > just that. Here's my enumeration:
>
> 0.131972...
> 0.574659...
> 0.213331...
> 0.567893...
> 0.324324...
> 0.211323...
>
> etc. Don't worry about the pattern. Now here's the problem:
>
> 0.131972...
> 0.574659...
> 0.213331...
> 0.567893...
> 0.324324...
> 0.211323...
> .
> .
> .
> 0.284934...
>
> Do you see the pattern? For each of the numbers in bold in the > enumerator (the diagonals), I have put the number that is one more > than that and generated a new number at the bottom in red as well. > So where is THAT number in the enumerator?
>
> Wherever in the enumerator that number is, it will eventually > intersect the diagonal. When it does, at the precise digit that it > intersects - that number will have to be the number OF the diagonal > and that number PLUS ONE AT THE SAME TIME. This is impossible.
>
> So you laugh at me and prove me wrong. My enumerator isn't an > enumerator after all.
>
> The key here is: ANY enumerator for the reals will fall prey to this > same problem. The numbers can be listed and "diagonalized" in the > same way.
>
> So anything that claims to be an enumerator for the reals, isn't.
>
> This means that the reals can't be listed, so they can't be put into > a one-to-one correspondence with the natural numbers. They are of a > higher degree of infinity. If you try this approach with just the > irrationals, you will see that they can't be enumerated as well.
>
> They are all "UNCOUNTABLE."
>
> And that's the story -Mike
>
> On Tue, Jun 3, 2008 at 4:59 PM, Ozan Yarman > <ozanyarman@...> wrote:
> > I certainly have a hard time grasping how there can be more > irrational
> > numbers between any two whole numbes as opposed to rational > numbers. They
> > are both infinite, hence, the same "amount". Infinity is a value > without a
> > conclusion. If all the irrational numbers are counted along with > rational
> > numbers they will amount to the same value, that is to say, there > will be no
> > end to them. There can be nothing more to a continuous increase of > numbers.
> > Mayhap you are thinking that the pace of counting is different? Or > perhaps,
> > you are assuming that there should be more than a single > irrational number
> > that corresponds to a rational number at a given pace of counting? > That
> > would not change the fact that there are as many rational numbers > as there
> > are irrational numbers.
> > Oz.
> > On Jun 3, 2008, at 11:34 PM, Cornell III, Howard M wrote:
> >
> > That's the problem. Infinity is merely a concept. An infinitely > long
> > straight line has an infinite number of points. AND there are an > infinite
> > number of points between any two adjacent points of the line. > Cantor
> > devised a hierarchy of infinities, and identified several. The > appropriate
> > reasoning for our discussion on numbers is as follows.
> >
> > Consider the interval from 0 to 1. The reasoning can be repeated > for all
> > other intervals between integers.
> >
> > Assign each point that are rational, that is, can be expressed as > the ratio
> > of two integers, to a counting number in order of their quotient > from
> > smallest, 0+ to 1. Clearly this will be an infinite number of > discrete
> > points, with a lot of space between them. That space is full of > the numbers
> > that are not ratios of two integers, that is, irrational numbers, > and there
> > are infinitely many numbers between each of them; but there are > clearly more
> > of them than rational numbers, and this is just between 0 and 1. To
> > complete the "proof" just repeat infinite number of times. If you > need to.
> > ________________________________
> > From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On > Behalf Of
> > Ozan Yarman
> > Sent: Tuesday, June 03, 2008 3:09 PM
> > To: tuning@yahoogroups.com
> > Subject: Re: [tuning] Re: re;re for Tom
> >
> >
> > It is as big as the whole number series... infinite. There are not > many
> > sizes of infinity, just one.
> > Oz.
> > On Jun 3, 2008, at 10:59 PM, Cornell III, Howard M wrote:
> >
> > If the whole number series is infinite, just imagine how big the > real number
> > system is! We have just been "discussing" how many of one kind of > number
> > can be placed between other kinds of numbers. We can do that if > we only
> > look at the relative preponderance of one kind of number, which > helps as
> > long as we do not worry about the total number of either kind being
> > uncountable.
> >
> > Howard Cornell
> >
> > ,
> >
> >
>

funny that the diagrams are of a lambdoma!

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Ozan Yarman wrote:
> http://www.jcu.edu/math/Vignettes/infinity.htm > <http://www.jcu.edu/math/Vignettes/infinity.htm>
>
> It seems I am stuck in pre-1870s math.
>
> Oz.
>
> On Jun 4, 2008, at 12:54 AM, Mike Battaglia wrote:
>
>> There are different degrees of infinity. In set theory, such degrees >> are referred to as CARDINALITIES. One can see whether one set has a >> higher cardinality (is "more infinite") than another set by seeing if >> the two can be put into a ONE-TO-ONE CORRESPONDENCE with one another. >> If the two sets have that property - anything in one set can be >> matched by something in the other - then they are of the same >> cardinality.
>> >> Take the set of natural numbers: it goes on forever in one direction. >> How about the set of just the positive even numbers? Is that "less" >> infinite than the natural numbers?
>> >> 0 1 2 3 4 ...
>> 0 2 4 6 8 ...
>> >> They both go on forever in one direction, and a direct one-to-one >> correspondence can be made between the two. For every element in the >> even numbers, there is an element in the natural number set that will >> match up for it. There aren't more natural numbers than even numbers >> because we will never "run out" of numbers. For each natural number >> ge! nerated, there is a matching even number as well. >> >> In fact, even this works:
>> >> 0 1 2 3 dog 4 5 6 ...
>> 0 1 2 3 4 5 6 7 ...
>> >> The set of all natural numbers and the word "dog" has the same >> cardinality as the set of all natural numbers without it. This is >> again because they can be put into that one-to-one correspondence I >> keep taling about.
>> >> How about the set of all integers? After all, it goes on forever in >> two directions: negatively and positively. Maybe we'll reach some >> "bigger" infinity in that way.
>> >> If we take all of the negative numbers and pair them with the odd >> numbers...
>> Integers: -1 -2 -3 ...
>> Naturals: 1 3 5 ...
>> >> and the positive numbers and pair them with the even numbers...
>> Integers: 1 2 3 ...
>> Naturals: 2 4 6 ...
>> >> Then we get this:
>> >> Integers: 0 -1 1 -2 2 -3 3 ...Naturals: 0 1 2 3 4 5 6 ...
>> >> You see that the two sets can be matched up that way. The integers >> can be expressed as a list of numbers increasing forever in one >> direction. There is that one to one correspondence. There is a >> "bijection" between the two sets. Therefore they are of the same >> degree of infinity - they have the same cardinality.
>> >> What about the set of all rational numbers? Those might be of a >> greater degree of infinity than the natural numbers and the integers.
>> >> Any rational number can be expressed as an integer fraction. This >> means we can graph set of all rational numbers can be as a plane -- >> the horizontal axis would represent an increase in the denominator >> and the vertical axis would represent an increase in the numerator. >> So is the degree of infinity of a plane more than that of a line?
>> >> For a long, long time, mathematicians thought the answer was "yes." >> And then someone came up with a pattern so obvious that it was almost >> stupid: If you just draw diagonal lines going from (0,1) to (1,0), >> and then from (0,2) to (2,0), and etc, you hit everything. So you can >> list all of the numbers and pair them up with the natural numbers.
>> >> Well, if you look at it this way:
>> >> 1: 1/1
>> >> 2: 1/2
>> 3: 2/1
>> >> 4: 1/3
>> 5: 2/2
>> 6: 3/1
>> >> 7: 1/4
>> 8: 2/3
>> 9: 3/2
>> 10: 4/1
>> >> etc.
>> >> Then every positive rational number in existence will be reached - >> some multiple times (which is okay). To include the negative >> rationals as well:
>> >> 1: 1/1
>> 2: -1/1
>> >> 3: 1/2
>> 4: -1/2
>> 5: 2/1
>> 6: -2/1
>> >> 7: 1/3
>> 8: -1/3
>> 9: 2/2
>> 10: -2/2
>> 11: 3/1
>> 12: -3/1
>> >> etc.
>> >> So we can pair up all of the rational numbers with the natural >> numbers. A bijection exists between the two sets, they are of the >> same degree of infinity, etc.
>> >> Another way to say this is that the set of all rational numbers, >> natural numbers, integers can all be ENUMERATED. You can get every >> single one of these sets and find an "enumerator" function that will >> generate every member of this set if we just let it run forever.
>> >> So this seems to be in accordance with your hypothesis, right? That >> every infinite set has the ! same siz e of infinity. BUT, what about >> all of the real numbers?
>> >> Well let's put it this way: let's just find all of the real numbers >> between 0 and 1. If THOSE have a higher degree of infinity than the >> natural numbers, then obviously ALL of the real numbers will have a >> higher cardinality than the natural numbers as well.
>> >> Now, if we can enumerate the real numbers (if there is any function >> that will produce every single one of the reals), then we can just >> write that list down, obviously.
>> >> I claim that I have some incredibly complicated way that will do just >> that. Here's my enumeration:
>> >> 0.131972...
>> 0.574659...
>> 0.213331...
>> 0.567893...
>> 0.324324...
>> 0.211323...
>> >> etc. Don't worry about the pattern. Now here's the problem:
>> >> 0.*_1_*31972...
>> 0.5*_7_*4659...
>> 0.21*_3_*331...
>> 0.567*_8_*93...
>> 0.3243*_2_*4...
>> 0.21132*_3_*...
>> *.*
>> *.*
>> *.*
>> *0.284934...*
>> >> Do you see the pattern? For each of the numbers in bold in the >> enumerator (the diagonals), I have put the number that is one more >> than that and generated a new number at the bottom in red as well. So >> where is THAT number in the enumerator?
>> >> Wherever in the enumerator that number is, it will eventually >> intersect the diagonal. When it does, at the precise digit that it >> intersects - that number will have to be the number OF the diagonal >> and that number PLUS ONE AT THE SAME TIME. This is impossible.
>> >> So you laugh at me and prove me wrong. My enumerator isn't an >> enumerator after all.
>> >> The key here is: ANY enumerator for the reals will fall prey to this >> same problem. The numbers can be listed and "diagonalized" in the >> same way.
>> >> So anything that claims to be an enumerator for the reals, isn't.
>> >> This means that the reals can't be listed, so they can't be put into >> a one-to-one correspondence with the natural numbers. They are of a >> higher degree of infinity. If you try this approach with just the >> irrationals, you will see that they can't be enumerated as well.
>> >> They are all "UNCOUNTABLE."
>> >> And that's the story -Mike
>> >> On Tue, Jun 3, 2008 at 4:59 PM, Ozan Yarman >> <ozanyarman@... <mailto:ozanyarman@...>> wrote:
>> > I certainly have a hard time grasping how there can be more irrational
>> > numbers between any two whole numbes as opposed to rational numbers. >> They
>> > are both infinite, hence, the same "amount". Infinity is a value >> without a
>> > conclusion. If all the irrational numbers are counted along with >> rational
>> > numbers they will amount to the same value, that is to say, there >> will be no
>> > end to them. There can be nothing more to a continuous increase of >> numbers.
>> > Mayhap you are thinking that the pace of counting is different? Or >> perhaps,
>> > you are assuming that there should be more than a single irrational >> number
>> > that corresponds to a rational number at a given pace of counting? That
>> > would not change the fact that there are as many rational numbers as >> there
> > are irrational numbers.
> > Oz.
> > On Jun 3, 2008, at 11:34 PM, Cornell III, Howard M wrote:
> >
> > That's the problem. Infinity is merely a concept. An infinitely long
> > straight line has an infinite number of points. AND there are an > infinite
> > number of points between any two adjacent points of the line. Cantor
> > devised a hierarchy of infinities, and identified several. The > appropriate
> > reasoning for our discussion on numbers is as follows.
> > > > Consider the interval from 0 to 1. The reasoning can be repeated > for all
> > other intervals between integers.
> > > > Assign each point that are rational, that is, can be expressed as > the ratio
> > of two integers, to a counting number in order of their quotient from
> > smallest, 0+ to 1. Clearly this will be an infinite number of discrete
> > points, with a lot of space between them. That space is full of the > numbers
> > that are not ratios of two integers, that is, i! rrationa l numbers, > and there
> > are infinitely many numbers between each of them; but there are > clearly more
> > of them than rational numbers, and this is just between 0 and 1. To
> > complete the "proof" just repeat infinite number of times. If you > need to.
> > ________________________________
> > From: tuning@yahoogroups.com > <mailto:tuning@yahoogroups.com> [mailto:tuning@yahoogroups.com > <mailto:tuning@yahoogroups.com>] On Behalf Of
> > Ozan Yarman
> > Sent: Tuesday, June 03, 2008 3:09 PM
> > To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
> > Subject: Re: [tuning] Re: re;re for Tom
> >
> >
> > It is as big as the whole number series... infinite. There are not many
> > sizes of infinity, just one.
> > Oz.
> > On Jun 3, 2008, at 10:59 PM, Cornell III, Howard M wrote:
> >
> > If the whole number series is infinite, just imagine how big the > real number
> > system is! We have just been "disc! ussing" how many of one kind of > number
> > can be placed between other kinds of numbers. We can do that if we only
> > look at the relative preponderance of one kind of number, which helps as
> > long as we do not worry about the total number of either kind being
> > uncountable.
> > > > Howard Cornell
> >
> > ,
> >
> >
>
>

Not necessarily. By the logic there will always be FEWER rationals,
because they need to be ratios of integers, and these will never exceed
the number of numbers that do not need to be ratios of integers.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Mike Battaglia
Sent: Tuesday, June 03, 2008 6:02 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

By that same logic, you can always put an infinite number of rationals
between any two rationals. So no matter how many rationals you have, you
will always have EVEN MORE rationals.

In order to prove the uncountability of the irrationals, you need to
assume their countability and find a contradiction in the statements
proceeding from it.

-Mike

On Tue, Jun 3, 2008 at 6:37 PM, Cornell III, Howard M
<howard.m.cornell.iii@... <mailto:howard.m.cornell.iii@...> >
wrote:

Oz,

I'm not assuming. Try not to think of an infinity that has "an
equal number of everything" in it. Try to think of just a reasonable
number of two kinds of things and expand it later. The point is that
by definition there are fewer numbers that are rational than there are
irrational numbers: because the definition of irrational numbers ALLOWS
more numbers to qualify. Don't worry about infinities yet. Just
consider, as in my previous email, how many rational numbers you can
CONCEPTUALLY put in the interval 0 to 1 and compare that to the number
of irrational numbers you can also assign to that interval. No matter
how many rational numbers you put in, you can BY DEFINITION put in more
irrational numbers, no matter how you calculate them! Still, don't
worry about counting them yet. Now, consider ! ;what you have found--you
can always put in more irrational numbers than rational numbers, even
between 0 and 1. No matter how many rational numbers you put in, there
is ALWAYS room for more irrational numbers between them. Start with
just a few rationals: you can insert as many irrationals as you have
time for between them. That's the point. It's true you can always put
in more rationals--and you can always put in EVEN MORE irrationals.
This is why infinity is not just an uncountable number of things; there
are properties associated with the kinds of things being counted.

It's just that in an infinite number of arbitrary numbers there
will be more irrationals because there will always be fewer rationals,
based on how they are defined.

See mathforum.org/isaac/problems/cantor1.html
<http://mathforum.org/isaac/problems/cantor1.html> for Cantor's attempt
to make infinity less paradoxical. That page has a link to cantor2 and
beyond for further explanation of the hierarchy of infinities.

I hope that this helps.

Howard

________________________________

From: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning@yahoogroups.com> ] On
Behalf Of Ozan Yarman

Sent: Tuesday, June 03, 2008 3:59 PM

To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] Re: re;re for Tom

I certainly have a hard time grasping how there can be more
irrational numbers between any two whole numbes as opposed to rational
numbers. They are both infinite, hence, the same "amount". Infinity is a
value without a conclusion. If all the irrational numbers are counted
along with rational numbers they will amount to the same value, that is
to say, there will be no end to them. There can be nothing more to a
continuous increase of numbers. Mayhap you are thinking that the pace of
counting is different? Or perhaps, you are assuming that there should be
more than a single irrational number that corresponds to a rational
number at a given pace of counting? That would not change the fact that
there are as many rational numbers as there are irrational numbers.

Oz.

On Jun 3, 2008, at 11:34 PM, Cornell III, Howard M wrote:

That's the problem. Infinity is merely a concept. An
infinitely long straight line has an infinite number of points. AND
there are an infinite number of points between any two adjacent points
of the line. Cantor devised a hierarchy of infinities, and identified
several. The appropriate reasoning for our discussion on numbers is as
follows.

Consider the interval from 0 to 1. The reasoning can be
repeated for all other intervals between integers.

Assign each point that are rational, that is, can be
expressed as the ratio of two integers, to a counting number in order of
their quotient from smallest, 0+ to 1. Clearly this will be an infinite
number of discrete points, with a lot of space between them. That space
is full of the numbers that are not ratios of two integers, that is,
irrational numbers, and there are infinitely many numbers between each
of them; but there are clearly more of them than rational numbers, and
this is just between 0 and 1. To complete the "proof" just repeat
infinite number of times. If you need to.

________________________________

From: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of! Ozan Yarman
Sent: Tuesday, June 03, 2008 3:09 PM
To: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] Re: re;re for Tom

It is as big as the whole number series... infinite.
There are not many sizes of infinity, just one.

Oz.

On Jun 3, 2008, at 10:59 PM, Cornell III, Howard M
wrote:

If the whole number series is infinite, just
imagine how big the real number system is! We have just been
"discussing" how many of one kind of number can be placed between other
kinds of numbers. We can do that if we only look at the r! elative
preponderance of one kind of number, which helps as lon g as we do not
worry about the total number of either kind being uncountable.

Howard Cornell

,

Mike! What a well written explanation. "Transfinites 101".

Ozan, you are not alone in insisting that the idea that some
infinities are bigger than others is just silly.
http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory

But you have to admire Cantor's ingenuity. It's a beautiful argument.
As explained in the above Wikipedia article, it is the consequence of
adopting certain axioms of set theory, any one of which you are free
to deny.

I was first taught to consider it Cantor's _paradox_, rather than
Cantor's _theorem_, by Nathaniel Hellerstein, who I met in the early
1990's at Interval Research in Palo Alto, where there were a number of
George Spencer-Brown (Laws of Form) groupies like myself. I read an
unpublished paper of Hellerstein's back then, entitled "Contra
Cantor". It referred to "Cantor's tottering tower" (of infinities).
And I think he conversely said the Burali-Forti paradox should be
called the Burali-Forti theorem. I forget the details, but I remember
he did the diagonalisation in binary, and something about the
anti-diagonal number corresponding to a 2-adic.

I just googled him and found he has a book out called "Diamond: A
paradox logic", which I just ordered from Amazon. Should be fun.

Beats me what this has to do with tuning. :-)

-- Dave Keenan

Whew, what a relief... thanks Dave! I was almost about to consider myself a complete buffoon for not agreeing with something as trivial as different sizes of infinities.

"As many rationals as there are irrationals" should be our motto!

Oz.

On Jun 5, 2008, at 4:46 PM, Dave Keenan wrote:

> Mike! What a well written explanation. "Transfinites 101".
>
> Ozan, you are not alone in insisting that the idea that some
> infinities are bigger than others is just silly.
> http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory
>
> But you have to admire Cantor's ingenuity. It's a beautiful argument.
> As explained in the above Wikipedia article, it is the consequence of
> adopting certain axioms of set theory, any one of which you are free
> to deny.
>
> I was first taught to consider it Cantor's _paradox_, rather than
> Cantor's _theorem_, by Nathaniel Hellerstein, who I met in the early
> 1990's at Interval Research in Palo Alto, where there were a number of
> George Spencer-Brown (Laws of Form) groupies like myself. I read an
> unpublished paper of Hellerstein's back then, entitled "Contra
> Cantor". It referred to "Cantor's tottering tower" (of infinities).
> And I think he conversely said the Burali-Forti paradox should be
> called the Burali-Forti theorem. I forget the details, but I remember
> he did the diagonalisation in binary, and something about the
> anti-diagonal number corresponding to a 2-adic.
>
> I just googled him and found he has a book out called "Diamond: A
> paradox logic", which I just ordered from Amazon. Should be fun.
>
> Beats me what this has to do with tuning. :-)
>
> -- Dave Keenan
>

The only problem you really had was thinking of "infinite" as a size!

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Thursday, June 05, 2008 9:35 AM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

Whew, what a relief... thanks Dave! I was almost about to consider
myself a complete buffoon for not agreeing with something as trivial
as different sizes of infinities.

"As many rationals as there are irrationals" should be our motto!

Oz.

On Jun 5, 2008, at 4:46 PM, Dave Keenan wrote:

> Mike! What a well written explanation. "Transfinites 101".
>
> Ozan, you are not alone in insisting that the idea that some
> infinities are bigger than others is just silly.
> http://en.wikipedia.org/wiki/Controversy_over_Cantor
<http://en.wikipedia.org/wiki/Controversy_over_Cantor> 's_theory
>
> But you have to admire Cantor's ingenuity. It's a beautiful argument.
> As explained in the above Wikipedia article, it is the consequence of
> adopting certain axioms of set theory, any one of which you are free
> to deny.
>
> I was first taught to consider it Cantor's _paradox_, rather than
> Cantor's _theorem_, by Nathaniel Hellerstein, who I met in the early
> 1990's at Interval Research in Palo Alto, where there were a number of
> George Spencer-Brown (Laws of Form) groupies like myself. I read an
> unpublished paper of Hellerstein's back then, entitled "Contra
> Cantor". It referred to "Cantor's tottering tower" (of infinities).
> And I think he conversely said the Burali-Forti paradox should be
> called the Burali-Forti theorem. I forget the details, but I remember
> he did the diagonalisation in binary, and something about the
> anti-diagonal number corresponding to a 2-adic.
>
> I just googled him and found he has a book out called "Diamond: A
> paradox logic", which I just ordered from Amazon. Should be fun.
>
> Beats me what this has to do with tuning. :-)
>
> -- Dave Keenan
>

I never thought so at all. In fact, I imagined you were considering lesser and greater infinities.

Oz.

On Jun 5, 2008, at 6:25 PM, Cornell III, Howard M wrote:

>
> The only problem you really had was thinking of "infinite" as a size!
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On > Behalf Of Ozan Yarman
> Sent: Thursday, June 05, 2008 9:35 AM
> To: tuning@yahoogroups.com
> Subject: Re: [tuning] Re: re;re for Tom
>
> Whew, what a relief... thanks Dave! I was almost about to consider
> myself a complete buffoon for not agreeing with something as trivial
> as different sizes of infinities.
>
> "As many rationals as there are irrationals" should be our motto!
>
> Oz.
>
> On Jun 5, 2008, at 4:46 PM, Dave Keenan wrote:
>
> > Mike! What a well written explanation. "Transfinites 101".
> >
> > Ozan, you are not alone in insisting that the idea that some
> > infinities are bigger than others is just silly.
> > http://en.wikipedia.org/wiki/Controversy_over_Cantor's_theory
> >
> > But you have to admire Cantor's ingenuity. It's a beautiful > argument.
> > As explained in the above Wikipedia article, it is the consequence > of
> > adopting certain axioms of set theory, any one of which you are free
> > to deny.
> >
> > I was first taught to consider it Cantor's _paradox_, rather than
> > Cantor's _theorem_, by Nathaniel Hellerstein, who I met in the early
> > 1990's at Interval Research in Palo Alto, where there were a > number of
> > George Spencer-Brown (Laws of Form) groupies like myself. I read an
> > unpublished paper of Hellerstein's back then, entitled "Contra
> > Cantor". It referred to "Cantor's tottering tower" (of infinities).
> > And I think he conversely said the Burali-Forti paradox should be
> > called the Burali-Forti theorem. I forget the details, but I > remember
> > he did the diagonalisation in binary, and something about the
> > anti-diagonal number corresponding to a 2-adic.
> >
> > I just googled him and found he has a book out called "Diamond: A
> > paradox logic", which I just ordered from Amazon. Should be fun.
> >
> > Beats me what this has to do with tuning. :-)
> >
> > -- Dave Keenan
> >
>

Ah, so. I was actually saying the proportion of irrationals will exceed
the proportion of rationals based on their definitions.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Thursday, June 05, 2008 12:41 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

I never thought so at all. In fact, I imagined you were considering
lesser and greater infinities.

Oz.

On Jun 5, 2008, at 6:25 PM, Cornell III, Howard M wrote:

The only problem you really had was thinking of "infinite" as a
size!

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of Ozan Yarman
Sent: Thursday, June 05, 2008 9:35 AM
To: tuning@yahoogroups.com <mailto:tu! ning@yahoogroups.com>
Subject: Re: [tuning] Re: re;re for Tom

Whew, what a relief... thanks Dave! I was almost about to
consider
myself a complete buffoon for not agreeing with something as
trivial
as different sizes of infinities.

"As many rationals as there are irrationals" should be our
motto!

Oz.

On Jun 5, 2008, at 4:46 PM, Dave Keenan wrote:

> Mike! What a well written explanation. "Transfinites 101".
>
> Ozan, you are not alone in insisting that the idea that some
> infinities are bigger than others is just silly.
> http://en.wikipedia.org/wiki/Controversy_over_Cantor
<http://en.wikipedia.org/wiki/Controversy_over_Cantor> 's_theory
>
> But you have to admire Cantor's ingenuity. It's a beautiful
argument.
> As explained in the above Wikipedia article, it is the
consequence of
> adopting c! ertain axioms of set theory, any one of which you
are free
> to den y.
>
> I was first taught to consider it Cantor's _paradox_, rather
than
> Cantor's _theorem_, by Nathaniel Hellerstein, who I met in the
early
> 1990's at Interval Research in Palo Alto, where there were a
number of
> George Spencer-Brown (Laws of Form) groupies like myself. I
read an
> unpublished paper of Hellerstein's back then, entitled "Contra
> Cantor". It referred to "Cantor's tottering tower" (of
infinities).
> And I think he conversely said the Burali-Forti paradox should
be
> called the Burali-Forti theorem. I forget the details, but I
remember
> he did the diagonalisation in binary, and something about the
> anti-diagonal number corresponding to a 2-adic.
>
> I just googled him and found he has a book out called
"Diamond: A
> paradox logic", which I just ordered from Amazon. Should be
fun.
>
> Beats me what this has to do with tuning. :-)
>
> -- Dave Keenan
>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> But you have to admire Cantor's ingenuity. It's a beautiful
> argument.

It's also a core element in at least one of Godel's famous
incompleteness theorems, and in related work by Turing.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> asked:

> There isn't even a single rational for every irrational
> number -- how could there be an infinity of them?
>
Hi Carl,

...because there exist even already an
http://en.wikipedia.org/wiki/Infinite_set
of
http://en.wikipedia.org/wiki/Natural_number
s.
"This set is countably infinite:
it is infinite but countable by definition."

Hence integers and rationals turn out to be
http://en.wikipedia.org/wiki/Equinumerous
due to the vice-versa
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
correspondence in an
http://en.wikipedia.org/wiki/Bijection
http://en.wikipedia.org/wiki/Bijective_numeration

Hope that helps to answer yours question.
Yours Sincerely
A.S.

[ Attachment content not displayed ]

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> It is as big as the whole number series... infinite. There are not
> many sizes of infinity, just one.
>
Hi Oz,

that can be expressed by
http://en.wikipedia.org/wiki/Cardinal_number
s in terms of
http://en.wikipedia.org/wiki/Aleph_number
s.
Hence there are much more 'infinties', than "just one".

Even beyond that, there exist also
http://en.wikipedia.org/wiki/Transfinite
numbers too, an
http://en.wikipedia.org/wiki/Uncountable_set
which..
"is an infinite set which is too big to be countable."
Proof:
http://www.apronus.com/math/uncountable.htm

At leasr attend there the:
http://www.apronus.com/music/flashpiano.htm

Yours Sincerely
A.S.

> > There isn't even a single rational for every irrational
> > number -- how could there be an infinity of them?
>
> Hi Carl,
>
> ...because there exist even already an
> http://en.wikipedia.org/wiki/Infinite_set
> of
> http://en.wikipedia.org/wiki/Natural_number
> s.
> "This set is countably infinite:
> it is infinite but countable by definition."
>
> Hence integers and rationals turn out to be
> http://en.wikipedia.org/wiki/Equinumerous
> due to the vice-versa
> http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
> correspondence in an
> http://en.wikipedia.org/wiki/Bijection
> http://en.wikipedia.org/wiki/Bijective_numeration
>
> Hope that helps to answer yours question.
> Yours Sincerely
> A.S.

Hi Andreas,

I don't see how any of this answers my question.

-Carl

Carl,

Because rational numbers are the ratios of integers, you can assign a
counting number to each rational number.
Granted you can do this forever without finishing; but for each rational
number there is a counting number that can be assigned to it. The thing
is that you cannot even count the irrational numbers between two
rational numbers! No matter how many you place there, there will always
be more. Because there are demonstrably MORE irrational numbers than
rational, the set of all numbers will be MOSTLY irrational by a LARGE
margin.

No matter that the supply of numbers is endless, it is the union of
rational and irrational numbers. And for each rational number you find,
there are an uncountable number of irrational numbers.

It's not possible to take an infinite number of numbers and try to count
which are rational and which are irrational; for the reason given
above--you may count all day and never even come across a rational
number. But you KNOW there are some. And that there just aren't "as
many" (relatively) as the irrationals.

Howard

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Carl Lumma
Sent: Thursday, June 05, 2008 3:38 PM
To: tuning@yahoogroups.com
Subject: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

> > There isn't even a single rational for every irrational
> > number -- how could there be an infinity of them?
>
> Hi Carl,
>
> ...because there exist even already an
> http://en.wikipedia.org/wiki/Infinite_set
<http://en.wikipedia.org/wiki/Infinite_set>
> of
> http://en.wikipedia.org/wiki/Natural_number
<http://en.wikipedia.org/wiki/Natural_number>
> s.
> "This set is countably infinite:
> it is infinite but countable by definition."
>
> Hence integers and rationals turn out to be
> http://en.wikipedia.org/wiki/Equinumerous
<http://en.wikipedia.org/wiki/Equinumerous>
> due to the vice-versa
> http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
<http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument>
> correspondence in an
> http://en.wikipedia.org/wiki/Bijection
<http://en.wikipedia.org/wiki/Bijection>
> http://en.wikipedia.org/wiki/Bijective_numeration
<http://en.wikipedia.org/wiki/Bijective_numeration>
>
> Hope that helps to answer yours question.
> Yours Sincerely
> A.S.

Hi Andreas,

I don't see how any of this answers my question.

-Carl

[ Attachment content not displayed ]

Here is a test: for every irrational number you come up with, I can come up with a rational number and we can carry this process to infinity, ending up with an equal amount of irrational and rational numbers.

Oz.

On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:

> Carl,
>
> Because rational numbers are the ratios of integers, you can assign > a counting number to each rational number.
> Granted you can do this forever without finishing; but for each > rational number there is a counting number that can be assigned to > it. The thing is that you cannot even count the irrational numbers > between two rational numbers! No matter how many you place there, > there will always be more. Because there are demonstrably MORE > irrational numbers than rational, the set of all numbers will be > MOSTLY irrational by a LARGE margin.
>
> No matter that the supply of numbers is endless, it is the union of > rational and irrational numbers. And for each rational number you > find, there are an uncountable number of irrational numbers.
>
> It's not possible to take an infinite number of numbers and try to > count which are rational and which are irrational; for the reason > given above--you may count all day and never even come across a > rational number. But you KNOW there are some. And that there just > aren't "as many" (relatively) as the irrationals.
>
> Howard
>
>

[ Attachment content not displayed ]

Howard wrote:

> Carl,
>
> Because rational numbers are the ratios of integers, you can
> assign a counting number to each rational number.
> Granted you can do this forever without finishing; but for each
> rational number there is a counting number that can be assigned
> to it. The thing is that you cannot even count the irrational
> numbers between two rational numbers!

Why are you telling me this? I'm the one who pointed it out.
This thread is hopelessly off-topic and I can't see it getting
better unless people bother to read the messages they're
replying to.

-Carl

For every irrational number that can be produced, a natural number can be made to correspond with it, ad infinitum.

Oz.

On Jun 6, 2008, at 1:41 AM, Mike Battaglia wrote:

> How about we take it a step further: For every irrational number in > existence, can you come up with a natural number?
>
> On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman > <ozanyarman@...> wrote:
>
> Here is a test: for every irrational number you come up with, I can > come up with a rational number and we can carry this process to > infinity, ending up with an equal amount of irrational and rational > numbers.
>
> Oz.
>
> On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:
>
>> Carl,
>>
>> Because rational numbers are the ratios of integers, you can assign >> a counting number to each rational number.
>> Granted you can do this forever without finishing; but for each >> rational number there is a counting number that can be assigned to >> it. The thing is that you cannot even count the irrational numbers >> between two rational numbers! No matter how many you place there, >> there will always be more. Because there are demonstrably MORE >> irrational numbers than rational, the set of all numbers will be >> MOSTLY irrational by a LARGE margin.
>>
>> No matter that the supply of numbers is endless, it is the union of >> rational and irrational numbers. And for each rational number you >> find, there are an uncountable number of irrational numbers.
>>
>> It's not possible to take an infinite number of numbers and try to >> count which are rational and which are irrational; for the reason >> given above--you may count all day and never even come across a >> rational number. But you KNOW there are some. And that there just >> aren't "as many" (relatively) as the irrationals.
>>
>> Howard
>>
>>
>
>

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Carl:
> Howard's got it right.
>
> In addition to his statements, you might want to consider

You might want to consider reading the messages you're replying
to. I'm the one who pointed out that Hans and Kalle were wrong.
Meanwhile, this is not the place to transcribe your 'math for
liberal arts majors' textbooks.

-Carl

[ Attachment content not displayed ]

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
> >>>> Mathematically speaking, there does not seem to be, there IS an
>>>> infinite amount of rational numbers, which not only in principle
>>>> approximate the tempered intervals. For every irrational number,
>>>> there is an infinite amount of rational numbers arbitrary close
>>>> to it,

<snip>

> There isn't even a single rational for every irrational
> number -- how could there be an infinity of them?

Let's try another answer here. I think the point is that the set of rational numbers that approximates a given irrational number will also approximate some other irrational number to the same or infinitesimally different tolerance. So there's a mapping from an irrational number and a tolerance to a set of rational numbers but it isn't one-to-one.

I can't say if the set of sets of rationals is countable... But maybe it isn't and that would also answer your question. There's certainly a big family of irrational numbers, like square roots and logarithms, that can be uniquely represented by continued fraction sequences.

Somewhere Hans's original statement got interpreted as meaning there are more rationals than irrationals. That doesn't follow. Certainly his statement is correct -- think about continued fractions (or the Stern-Brocot tree as Kraig pointed out). But it's also true that there's an infinite number of irrational numbers that approximate any given rational number. I think Gene said this means both sets are infinite and dense.

Graham

> Somewhere Hans's original statement got interpreted as
> meaning there are more rationals than irrationals.

That follows from "for every", as I already said.

-Carl

> I can't say if the set of sets of rationals is countable...
> But maybe it isn't and that would also answer your
> question.

???

Is there really any doubt about the rhetorical nature of
my "question", or are you just trolling?

-Carl

Mike wrote...

> I think a thread about why there are more irrational numbers than
> rational numbers is a perfectly good place to discuss why there
> are more irrational numbers than rational numbers.

Sorry Mike, but math threads are off-topic for this list.

-Carl

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Yeah. Uncountability. The set of all truths is uncountable. Learning
about
> Godel's theorem changed my life for a little bit, as I went on a huge
> philosophical kick about the whole thing - you should have seen me,
walking
> around the lake at my school at 6 AM raving to myself like a
madman... >:)
>
> My question is, if the set of all natural numbers has cardinality
> aleph-zero, and the set of all reals has cardinality aleph-one
(let's assume
> for right now there is nothing between them), what about the next
level up,
> aleph-two? The power set of the set of all reals will have this
cardinality,
> but what "property" does it share?
>
> My question is, aleph-one is distinguished from aleph-zero by its
> "uncountability." What property do the aleph-two cardinals have?
Even more
> uncountability? :P I think it has something to do with the concept of a
> "choice function," but I'm not sure.
>
> The debate on whether or not the set of all reals is of cardinality
> aleph-one or not doesn't need to be brought into this - my question is
> specifically what the properties of sets that has the same
cardinality as
> the power set of R are.
>
> On Thu, Jun 5, 2008 at 2:18 PM, Carl Lumma <carl@...> wrote:
>
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Dave
Keenan"
> > <d.keenan@> wrote:
> >
> > > But you have to admire Cantor's ingenuity. It's a beautiful
> > > argument.
> >
> > It's also a core element in at least one of Godel's famous
> > incompleteness theorems, and in related work by Turing.
> >
> > -Carl
> >
> > Since the sine wave clearly states that time is proportional to
period, t = nT where n is number of cycles (i.e. t= T+T+...+nfactors),
do you think that the length of the time line would be countably
infinite, aleph-naught N(0), since each cycle is a whole number? In
other words, n = N(0) for ANY value of T.
> >
>

> Since the sine wave clearly states that time is proportional to
> period, t = nT where n is number of cycles (i.e. t= T+T+...+nfactors),
> do you think that the length of the time line would be countably
> infinite, aleph-naught N(0), since each cycle is a whole number? In
> other words, n = N(0) for ANY value of T.

I'd love to know what this has to do with microtonal music.

-Carl

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> Since the sine wave clearly states that time is proportional to
> period, t = nT where n is number of cycles (i.e. t= T+T+...+nfactors),
> do you think that the length of the time line would be countably
> infinite, aleph-naught N(0), since each cycle is a whole number? In
> other words, n = N(0) for ANY value of T.
>

Lengths are not cardinal numbers. The question whether countable or
incountable is meaningless in the case of lengths, isn't it?
--
Hans Straub

Carl Lumma wrote:
>> I can't say if the set of sets of rationals is countable... >> But maybe it isn't and that would also answer your >> question.

I've concluded that, yes, the set of sets of rationals is uncountable. After all an irrational number written as a decimal expansion is essentially a set of rationals. So it's easier to assign a single, unique set of rationals to each irrational than to assign a single rational to each irrational. Whatever the point of that was.

> Is there really any doubt about the rhetorical nature of
> my "question", or are you just trolling?

It's perfectly clear that your question is rhetorical, and that the argument it's supporting is false. Either that or your argument isn't clear. There's nothing wrong with what Hans originally said (bar some typos).

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> >> I can't say if the set of sets of rationals is countable...
> >> But maybe it isn't and that would also answer your
> >> question.
>
> I've concluded that, yes, the set of sets of rationals is
> uncountable.

But the set of rationals is definitely countable!

Proof:

For every integer n, the set of all rationals with both numerator and
denominator smaller than n is finite. So we can arrange all of these
sets one after another and this way get a numbering of all the
rationals.
--
Hans Straub

No we cannot. If we don't get tired you WILL run out of rational
numbers and I NEVER will run out of irrational numbers. In fact I only
have to give you TWO rational numbers and you can give me irrational
numbers between them until you change your mind.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Thursday, June 05, 2008 5:39 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

Here is a test: for every irrational number you come up with, I can come
up with a rational number and we can carry this process to infinity,
ending up with an equal amount of irrational and rational numbers.

Oz.

On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:

Carl,

Because rational numbers are the ratios of integers, you can
assign a counting number to each rational number.
Granted you can do this forever without finishing; but for each
rational number there is a counting number that can be assigned to it.
The thing is that you cannot even count the irrational numbers between
two rational numbers! No matter how many you place there, there will
always be more. Because there are demonstrably MORE irrational numbers
than rational, the set of all numbers will be MOSTLY irrational by a
LARGE margin.

No matter that the supply of numbers is endless, it is the union
of rational and irrational numbers. And for each rational number you
find, there are an uncountable number of irrational numbe! rs.

It's not possible to take an infinite number of numbers and try
to count which are rational and which are irrational; for the reason
given above--you may count all day and never even come across a rational
number. But you KNOW there are some. And that there just aren't "as
many" (relatively) as the irrationals.

Howard

Sure. But we already agree there will be no end to that argument. Try
to understand the difference between kinds of numbers that are naturally
limited to identify with natural numbers and numbers that are only
arbitrarily identified with natural numbers by trying to count them.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Mike Battaglia
Sent: Thursday, June 05, 2008 5:42 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

How about we take it a step further: For every irrational number in
existence, can you come up with a natural number?

On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman <ozanyarman@...
<mailto:ozanyarman@...> > wrote:

Here is a test: for every irrational number you come up with, I
can come up with a rational number and we can carry this process to
infinity, ending up with an equal amount of irrational and rational
numbers.

Oz.

On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:

Carl,

Because rational numbers are the ratios of integers, you
can assign a counting number to each rational number.
Granted you can do this forever without finishing; but
for each rational number there is a counting number that can be assigned
to it. The thing is that you cannot even count the irrational numbers
between two rational numbers! No matter how many you place there, there
will always be more. Because there are demonstrably MORE irrational
numbers than rational, the set of all numbers will be MOSTLY irrational
by a LARGE margin.

No matter that the supply of numbers is endless, it is
the union of rational and irrational numbers. And for each rational
number you find, there are an uncountable number of irrational numbers.

It's not possible to take an infinite number of numbers
and try to count which are rational and which are irrational; for the
reason given above--you may count all day and never even come across a
rational number. But you KNOW there are some. And that there just
aren't "as many" (relatively) as the irrationals.

Howard

Yes. But not a RATIONAL number, which is "more unique".

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Thursday, June 05, 2008 6:43 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

For every irrational number that can be produced, a natural number can
be made to correspond with it, ad infinitum.

Oz.

On Jun 6, 2008, at 1:41 AM, Mike Battaglia wrote:

How about we take it a step further: For every irrational number
in existence, can you come up with a natural number?


On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman
<ozanyarman@... <mailto:ozanyarman@...> > wrote:

Here is a test: for every irrational number you come up
with, I can come up with a rational number and we can carry this process
to infinity, ending up with an equal amount of irrational and rational
numbers.

Oz.

On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:

Carl,

Because rational numbers are the ratios of
integers, you can assign a counting number to each rational number.
Granted you can do this forever without
finishing; but for each rational number there is a counting number that
can be assigned to it. The thing is that you cannot even count the
irrational numbers between two rational numbers! No matter how many you
place there, there will always be more. Because there are demonstrably
MORE irrational numbers than rational, the set of all numbers will be
MOSTLY irrational by a LARGE margin.

No matter that the supply of numbers is endless,
it is the union of rational and irrational numbers. And for each
rational number you find, there are an uncountable number of irrational
numbers.

It's not possible to take an infinite number of
numbers and try to count which are rational and which are irrational;
for the reason given above--you may count all day and never even come
across a rational number. But you KNOW there are some. And that there
just aren't "as many" (relatively) as the irrationals.

Howard

Carl,

I wrote that because you "didn't see how [what Andreas wrote] answers"
your question. It would be nice to resolve this so that we all
understand the nature of rational numbers.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Carl Lumma
Sent: Thursday, June 05, 2008 6:42 PM
To: tuning@yahoogroups.com
Subject: [tuning] for Carl's quest: Why there are infinite many
rationals?

Howard wrote:

> Carl,
>
> Because rational numbers are the ratios of integers, you can
> assign a counting number to each rational number.
> Granted you can do this forever without finishing; but for each
> rational number there is a counting number that can be assigned
> to it. The thing is that you cannot even count the irrational
> numbers between two rational numbers!

Why are you telling me this? I'm the one who pointed it out.
This thread is hopelessly off-topic and I can't see it getting
better unless people bother to read the messages they're
replying to.

-Carl

Graham,

Continued fractions do not necessarily indicate irrationality. Consider
1/3.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Graham Breed
Sent: Thursday, June 05, 2008 10:53 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

Carl Lumma wrote:
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ,
"hstraub64" <straub@...> wrote:
>
>>>> Mathematically speaking, there does not seem to be, there IS an
>>>> infinite amount of rational numbers, which not only in principle
>>>> approximate the tempered intervals. For every irrational number,
>>>> there is an infinite amount of rational numbers arbitrary close
>>>> to it,

<snip>

> There isn't even a single rational for every irrational
> number -- how could there be an infinity of them?

Let's try another answer here. I think the point is that
the set of rational numbers that approximates a given
irrational number will also approximate some other
irrational number to the same or infinitesimally different
tolerance. So there's a mapping from an irrational number
and a tolerance to a set of rational numbers but it isn't
one-to-one.

I can't say if the set of sets of rationals is countable...
But maybe it isn't and that would also answer your
question. There's certainly a big family of irrational
numbers, like square roots and logarithms, that can be
uniquely represented by continued fraction sequences.

Somewhere Hans's original statement got interpreted as
meaning there are more rationals than irrationals. That
doesn't follow. Certainly his statement is correct -- think
about continued fractions (or the Stern-Brocot tree as Kraig
pointed out). But it's also true that there's an infinite
number of irrational numbers that approximate any given
rational number. I think Gene said this means both sets are
infinite and dense.

Graham

Graham,

An irrational number might approximate a rational number; but if it is
not the ratio of two integers it is not rational.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Graham Breed
Sent: Friday, June 06, 2008 4:54 AM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: re;re for Tom

Carl Lumma wrote:
>> I can't say if the set of sets of rationals is countable...
>> But maybe it isn't and that would also answer your
>> question.

I've concluded that, yes, the set of sets of rationals is
uncountable. After all an irrational number written as a
decimal expansion is essentially a set of rationals. So
it's easier to assign a single, unique set of rationals to
each irrational than to assign a single rational to each
irrational. Whatever the point of that was.

> Is there really any doubt about the rhetorical nature of
> my "question", or are you just trolling?

It's perfectly clear that your question is rhetorical, and
that the argument it's supporting is false. Either that or
your argument isn't clear. There's nothing wrong with what
Hans originally said (bar some typos).

Graham

You can assign each added wavelength a new countable number, the current
"time". But why?

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of hstraub64
Sent: Friday, June 06, 2008 3:12 AM
To: tuning@yahoogroups.com
Subject: [tuning] Re: re;re for Tom

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ,
"rick_ballan" <rick_ballan@...> wrote:
>
> Since the sine wave clearly states that time is proportional to
> period, t = nT where n is number of cycles (i.e. t= T+T+...+nfactors),
> do you think that the length of the time line would be countably
> infinite, aleph-naught N(0), since each cycle is a whole number? In
> other words, n = N(0) for ANY value of T.
>

Lengths are not cardinal numbers. The question whether countable or
incountable is meaningless in the case of lengths, isn't it?
--
Hans Straub

Irrational numbers and rational numbers are numerically infinite in
number.

However, an approximation of irrational numbers by rational numbers,
and vice versa, is a different topic.

For example, in 12-TET, the "major third," or the "twelfth root of
two to the fourth power," has a value of 400.00 cents. One possible
rational approximation is ratio 223/177, which has a value of 399.95
cents; another approximation is ratio 63/50, which has a value of
400.11 cents; etc.!

By definition, all approximations are conceptually infinite in
number.

Cris

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> For every irrational number that can be produced, a natural number
can
> be made to correspond with it, ad infinitum.
>
> Oz.
>
> On Jun 6, 2008, at 1:41 AM, Mike Battaglia wrote:
>
> > How about we take it a step further: For every irrational number
in
> > existence, can you come up with a natural number?
> >
> > On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman
> > <ozanyarman@...> wrote:
> >
> > Here is a test: for every irrational number you come up with, I
can
> > come up with a rational number and we can carry this process to
> > infinity, ending up with an equal amount of irrational and
rational
> > numbers.
> >
> > Oz.
> >
> > On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:
> >
> >> Carl,
> >>
> >> Because rational numbers are the ratios of integers, you can
assign
> >> a counting number to each rational number.
> >> Granted you can do this forever without finishing; but for
each
> >> rational number there is a counting number that can be assigned
to
> >> it. The thing is that you cannot even count the irrational
numbers
> >> between two rational numbers! No matter how many you place
there,
> >> there will always be more. Because there are demonstrably
MORE
> >> irrational numbers than rational, the set of all numbers will
be
> >> MOSTLY irrational by a LARGE margin.
> >>
> >> No matter that the supply of numbers is endless, it is the
union of
> >> rational and irrational numbers. And for each rational number
you
> >> find, there are an uncountable number of irrational numbers.
> >>
> >> It's not possible to take an infinite number of numbers and try
to
> >> count which are rational and which are irrational; for the
reason
> >> given above--you may count all day and never even come across
a
> >> rational number. But you KNOW there are some. And that there
just
> >> aren't "as many" (relatively) as the irrationals.
> >>
> >> Howard
> >>
> >>
> >
> >
>

Interesting....but wouldn't this kind of be assuming the rational estimations would end up getting infinitely many results, but at the same time results that merged around the actual "correct note"...almost as if approaching, but never quite touching, a limit?

--- On Fri, 6/6/08, Cris Forster <cris.forster@...> wrote:
From: Cris Forster <cris.forster@...>
Subject: [tuning] for Carl's quest: Why there are infinite many rationals? was Re: re;re for Tom
To: tuning@yahoogroups.com
Date: Friday, June 6, 2008, 8:35 AM

Irrational numbers and rational numbers are numerically infinite in

number.

However, an approximation of irrational numbers by rational numbers,

and vice versa, is a different topic.

For example, in 12-TET, the "major third," or the "twelfth root of

two to the fourth power," has a value of 400.00 cents. One possible

rational approximation is ratio 223/177, which has a value of 399.95

cents; another approximation is ratio 63/50, which has a value of

400.11 cents; etc.!

By definition, all approximations are conceptually infinite in

number.

Cris

--- In tuning@yahoogroups. com, Ozan Yarman <ozanyarman@ ...> wrote:

>

> For every irrational number that can be produced, a natural number

can

> be made to correspond with it, ad infinitum.

>

> Oz.

>

> On Jun 6, 2008, at 1:41 AM, Mike Battaglia wrote:

>

> > How about we take it a step further: For every irrational number

in

> > existence, can you come up with a natural number?

> >

> > On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman

> > <ozanyarman@ ...> wrote:

> >

> > Here is a test: for every irrational number you come up with, I

can

> > come up with a rational number and we can carry this process to

> > infinity, ending up with an equal amount of irrational and

rational

> > numbers.

> >

> > Oz.

> >

> > On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:

> >

> >> Carl,

> >>

> >> Because rational numbers are the ratios of integers, you can

assign

> >> a counting number to each rational number.

> >> Granted you can do this forever without finishing; but for

each

> >> rational number there is a counting number that can be assigned

to

> >> it. The thing is that you cannot even count the irrational

numbers

> >> between two rational numbers! No matter how many you place

there,

> >> there will always be more. Because there are demonstrably

MORE

> >> irrational numbers than rational, the set of all numbers will

be

> >> MOSTLY irrational by a LARGE margin.

> >>

> >> No matter that the supply of numbers is endless, it is the

union of

> >> rational and irrational numbers. And for each rational number

you

> >> find, there are an uncountable number of irrational numbers.

> >>

> >> It's not possible to take an infinite number of numbers and try

to

> >> count which are rational and which are irrational; for the

reason

> >> given above--you may count all day and never even come across

a

> >> rational number. But you KNOW there are some. And that there

just

> >> aren't "as many" (relatively) as the irrationals.

> >>

> >> Howard

> >>

> >>

> >

> >

>

Conceptual infinity, as opposed to numerical infinity, is subject to
tolerances -- or rounding to n decimal places -- determined by the
mathematician.

Cris

--- In tuning@yahoogroups.com, Michael Sheiman <djtrancendance@...>
wrote:
>
> Interesting....but wouldn't this kind of be assuming the rational
estimations would end up getting infinitely many results, but at the
same time results that merged around the actual "correct
note"...almost as if approaching, but never quite touching, a limit?
>
> --- On Fri, 6/6/08, Cris Forster cris.forster@... wrote:
> From: Cris Forster cris.forster@...
> Subject: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom
> To: tuning@yahoogroups.com
> Date: Friday, June 6, 2008, 8:35 AM
>
>
>
>
>
>
>
>
>
>
>
> Irrational numbers and rational numbers are
numerically infinite in
>
> number.
>
>
>
> However, an approximation of irrational numbers by rational
numbers,
>
> and vice versa, is a different topic.
>
>
>
> For example, in 12-TET, the "major third," or the "twelfth root of
>
> two to the fourth power," has a value of 400.00 cents. One
possible
>
> rational approximation is ratio 223/177, which has a value of
399.95
>
> cents; another approximation is ratio 63/50, which has a value of
>
> 400.11 cents; etc.!
>
>
>
> By definition, all approximations are conceptually infinite in
>
> number.
>
>
>
> Cris
>
>
>
> --- In tuning@yahoogroups. com, Ozan Yarman
<ozanyarman@ ...> wrote:
>
> >
>
> > For every irrational number that can be produced, a natural
number
>
> can
>
> > be made to correspond with it, ad infinitum.
>
> >
>
> &gt; Oz.
>
> >
>
> > On Jun 6, 2008, at 1:41 AM, Mike Battaglia wrote:
>
> >
>
> > > How about we take it a step further: For every
irrational number
>
> in
>
> > > existence, can you come up with a natural number?
>
> > >
>
> > > On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman
>
> > > <ozanyarman@ ...> wrote:
>
> &gt; >
>
> > > Here is a test: for every irrational number you come up
with, I
>
> can
>
> > > come up with a rational number and we can carry this
process to
>
> > > infinity, ending up with an equal amount of irrational
and
>
> rational
>
> > > numbers.
>
> > >
>
> > > Oz.
>
> > >
>
> > > On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:
>
> > >
>
> > >&gt; Carl,
>
> > >>
>
> > >> Because rational numbers are the ratios of integers,
you can
>
> assign
>
> > >> a counting number to each rational number.
>
> > >> Granted you can do this forever without finishing;
but for
>
> each
>
> > >> rational number there is a counting number that can
be assigned
>
> to
>
> > >> it. The thing is that you cannot even count the
irrational
>
> numbers
>
> > >> between two rational numbers! No matter how many
you place
>
> there,
>
> > >> there will always be more. Because there are
demonstrably
>
> MORE
>
> > >> irrational numbers than rational, the set of all
numbers will
>
> be
>
> > >> MOSTLY irrational by a LARGE margin.
>
> > >>
>
> > >> No matter that the supply of numbers is endless, it
is the
>
> union of
>
> > >> rational and irrational numbers. And for each
rational number
>
> you
>
> > >> find, there are an uncountable number of irrational
numbers.
>
> > >>
>
> >; >> It's not possible to take an infinite number of
numbers and try
>
> to
>
> > >> count which are rational and which are irrational;
for the
>
> reason
>
> > >> given above--you may count all day and never even
come across
>
> a
>
> > >> rational number. But you KNOW there are some. And
that there
>
> just
>
> &gt; >> aren't "as many" (relatively) as the irrationals.
>
> > &gt;>
>
> > >> Howard
>
> > >>
>
> > >>
>
> > >
>
> > >
>
> >
>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Carl Lumma wrote:
> >> I can't say if the set of sets of rationals is countable...
> >> But maybe it isn't and that would also answer your
> >> question.
>
> I've concluded that, yes, the set of sets of rationals is
> uncountable.

The rationals are countable.

> > Is there really any doubt about the rhetorical nature of
> > my "question", or are you just trolling?
>
> It's perfectly clear that your question is rhetorical, and
> that the argument it's supporting is false.

Needless to say I disagree.

-Carl

I think it's quite funny that the guy behind all this was called
Cantor...

Now tell me this: I am sitting at a rectangular table. How can I tell
whether the ratio of its width to its length is rational or irrational?

~~~T~~~

Any ratio of two integers is rational.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Tom Dent
Sent: Friday, June 06, 2008 12:54 PM
To: tuning@yahoogroups.com
Subject: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

I think it's quite funny that the guy behind all this was called
Cantor...

Now tell me this: I am sitting at a rectangular table. How can I tell
whether the ratio of its width to its length is rational or irrational?

~~~T~~~

--- In tuning@yahoogroups.com, "Cornell III, Howard
M" <howard.m.cornell.iii@...> wrote:
>
> Any ratio of two integers is rational.
>
> ________________________________
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
Behalf
> Of Tom Dent
> Sent: Friday, June 06, 2008 12:54 PM
> To: tuning@yahoogroups.com
> Subject: [tuning] for Carl's quest: Why there are infinite many
> rationals? was Re: re;re for Tom
>
>
>
>
> I think it's quite funny that the guy behind all this was called
> Cantor...

And how about "Aleph", LOL.
>
> Now tell me this: I am sitting at a rectangular table. How can I
tell
> whether the ratio of its width to its length is rational or
irrational?
>
> ~~~T~~~
>

A. it's a rectangle and B. it doesn't fall on your lap.

[ Attachment content not displayed ]

> > I think a thread about why there are more irrational numbers than
> > rational numbers is a perfectly good place to discuss why there
> > are more irrational numbers than rational numbers.
>
> Sorry Mike, but math threads are off-topic for this list.
>
> -Carl

Then go bother the person who started the thread. I'm just enjoying a
discussion, as is everyone else.

Let me remind you that this is a spinoff thread from a thread that was
originally about tuning. The topic we are discussing here is necessary
to fully understand the other thread. If it really bothers you that
much that for a second it looked like you believed that the set of all
reals was countable, then feel free to push for moderation and rain on
everyone's parade.

-Mike

Cameron,

Even rectangles can have integer dimensions. Perhaps the table is 60
inches long and 30 inches wide. The ratio is 30/60 or 1/2 which is
rational. If it were pi feet long and the square root of 2 feet wide
that would be a different matter.

It doesn't matter that it doesn't fall on your lap; but that is
nevertheless a good quality for a table to have.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Cameron Bobro
Sent: Friday, June 06, 2008 2:05 PM
To: tuning@yahoogroups.com
Subject: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ,
"Cornell III, Howard
M" <howard.m.cornell.iii@...> wrote:
>
> Any ratio of two integers is rational.
>
> ________________________________
>
> From: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ] On
Behalf
> Of Tom Dent
> Sent: Friday, June 06, 2008 12:54 PM
> To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
> Subject: [tuning] for Carl's quest: Why there are infinite many
> rationals? was Re: re;re for Tom
>
>
>
>
> I think it's quite funny that the guy behind all this was called
> Cantor...

And how about "Aleph", LOL.
>
> Now tell me this: I am sitting at a rectangular table. How can I
tell
> whether the ratio of its width to its length is rational or
irrational?
>
> ~~~T~~~
>

A. it's a rectangle and B. it doesn't fall on your lap.

Exactly. I think the fact that you can access rational numbers makes
them more unique, not just that there are fewer of them, which is also a
quality of uniqueness.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Mike Battaglia
Sent: Friday, June 06, 2008 2:23 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

Nononono. The rational numbers are just as unique as the natural
numbers. The irrational numbers are more unique.

The idea is that if I give out irrational numbers and you give rational
numbers, that doesn't actually prove anything. I will never be able to
come up with a way to systematically access EVERY irrational number.
There is, however, a way to systematically access every natural and
rational number. This is the difference between the rational and
irrational numbers.

For example, let's say I have a list of every irrational number, paired
up with some corresponding natural number.

1: 0.123239...
2: 0.423242...
3: 0.123463...
4: 0.132948...
5: 0.534854...
6: 0.348275...
.
.
.

So let's say that this just goes on forever and ends up hitting EVERY
irrational number. Let's say there is some "pattern" in the irrational
numbers I've hit on that will systematically hit every one. We can also
construct a number by taking the first digit of the first number, and
the second of the second number, etc:

1: 0.127439...
2: 0.423942...
3: 0.623763...
4: 0.142948...
5: 0.534854...
6: 0.348275...
.
.
.

D: 0.123955...

It's easy to see that this number will appear in the list somewhere,
since the list supposedly has *every* irrational number in it. The place
where this number meets the diagonal in the list will have the digits
simply "align," as the number formed by the diagonal will simply be the
same as this number here.

BUT... what about the number formed by taking the diagonal and
incrementing each digit by 1?

1: 0.127439...
2: 0.423942...
3: 0.623763...
4: 0.142948...
5: 0.534854...
6: 0.348275...
.
.
.

D: 0.234066...

This number, as this list is supposed to contain "every" irrational
number, will also have to be in the list somewhere. What about where
this number meets the diagonal? What will the digit be? The number will,
by definition, be one digit higher than the diagonal - therefore, where
the number meets the diagonal, that digit will have to be the digit of
the diagonal AND that digit plus one *at the same time*. So it breaks
down.

Interesting, but what does it mean? It means that any supposed "list"
containing all of the irrational numbers won't contain all of the
irrational numbers. The concept of a list containing every irrational
number is contradictory. No list can be made that will contain all of
the irrational numbers, even if we let that list go on forever.

On the other hand, a list can be made that WILL contain all of the
natural numbers, and all of the rational numbers, if we let them run
forever. By starting at 0 and incrementing by 1, we'll eventually get
every natural number, and check the other thread for a way to have every
rational number be eventually hit in the same way.

This means that you can't pair up every natural number with every
irrational number. If that were possible, then you'd be able to write
them out in such a list, which would then prove itself to not be a list
at all.

So in other words, any way you have to compute systematically every
irrational number will fail. There will always be a number you will
miss. Which is kind of worrying for certain fields of mathematics and
computer science, that there exist "uncomputable" numbers.

Pi, e, etc. are all computable. We have formulas to compute these
numbers to as much precision as we'd like. If we left these formulas
running forever, then we'd have pi and e and such exactly. Uncomputable
numbers, however, are of a different species entirely.

For the record, the set of all "computable" numbers is on par with the
natural and rational numbers.

On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...
<mailto:howard.m.cornell.iii@...> > wrote:
>
>
>
>
>
>
>
> Yes. But not a RATIONAL number, which is "more unique".
>
> ________________________________
> From: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning@yahoogroups.com> ] On
Behalf Of
> Ozan Yarman
> Sent: Thursday, June 05, 2008 6:43 PM
> To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
> Subject: Re: [tuning] for Carl's quest: Why there are infinite many
> rationals? was Re: re;re for Tom
>
>
>
>
>
>
> For every irrational number that can be produced, a natural number can
be
> made to correspond with it, ad infinitum.
>
>
> Oz.
>
>
> On Jun 6, 2008, at 1:41 AM, Mike Battaglia wrote:
>
>
> How about we take it a step further: For every irrational number in
> existence, can you come up with a natural number?
>
>
> On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman <ozanyarman@...
<mailto:ozanyarman@...> >
> wrote:
>
> >
> >
> >
> >
> >
> >
> >
> > Here is a test: for every irrational number you come up with, I can
come
> up with a rational number and we can carry this process to infinity,
ending
> up with an equal amount of irrational and rational numbers.
> >
> >
> > Oz.
> >
> >
> >
> > On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:
> >
> >
> >
> > Carl,
> >
> > Because rational numbers are the ratios of integers, you can assign
a
> counting number to each rational number.
> > Granted you can do this forever without finishing; but for each
rational
> number there is a counting number that can be assigned to it. The
thing is
> that you cannot even count the irrational numbers between two rational
> numbers! No matter how many you place there, there will always be
more.
> Because there are demonstrably MORE irrational numbers than rational,
the
> set of all numbers will be MOSTLY irrational by a LARGE margin.
> >
> > No matter that the supply of numbers is endless, it is the union of
> rational and irrational numbers. And for each rational number you
find,
> there are an uncountable number of irrational numbers.
> >
> > It's not possible to take an infinite number of numbers and try to
count
> which are rational and which are irrational; for the reason given
above--you
> may count all day and never even come across a rational number. But
you
> KNOW there are some. And that there just aren't "as many"
(relatively) as
> the irrationals.
> >
> > Howard
> >
> >
>
> /! div>
>
>
>
>
>
>
>

Mike wrote:
> Then go bother the person who started the thread.

That sounds a lot like "he started it".

> I'm just enjoying a discussion, as is everyone else.

You've polled all 1300 members, I take it?

> Let me remind you that this is a spinoff thread from a thread
> that was originally about tuning.

So it's off-topic. There's actually a list for that:

/metatuning

> The topic we are discussing here is necessary
> to fully understand the other thread.

It isn't. Rationals and irrationals approximate one
another to far beyond the limits of human perception.
That's all that's relevant for microtonal music.

-Carl

Howard:

I think he's saying, given an actual real life table, will the sides
ever have perfectly rational dimensions? That is, is the ratio of each
side to an "inch" going to be rational, as well as the ratio of the
sides to each other?

-Mike

On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...> wrote:
>
>
>
>
>
>
>
> Cameron,
>
> Even rectangles can have integer dimensions. Perhaps the table is 60 inches
> long and 30 inches wide. The ratio is 30/60 or 1/2 which is rational. If
> it were pi feet long and the square root of 2 feet wide that would be a
> different matter.
>
> It doesn't matter that it doesn't fall on your lap; but that is nevertheless
> a good quality for a table to have.
>
> ________________________________
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
> Cameron Bobro
> Sent: Friday, June 06, 2008 2:05 PM
> To: tuning@yahoogroups.com
> Subject: [tuning] for Carl's quest: Why there are infinite many rationals?
> was Re: re;re for Tom
>
>
>
>
>
> --- In tuning@yahoogroups.com, "Cornell III, Howard
> M" <howard.m.cornell.iii@...> wrote:
> >
> > Any ratio of two integers is rational.
> >
> > ________________________________
> >
> > From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
> Behalf
> > Of Tom Dent
> > Sent: Friday, June 06, 2008 12:54 PM
> > To: tuning@yahoogroups.com
> > Subject: [tuning] for Carl's quest: Why there are infinite many
> > rationals? was Re: re;re for Tom
> >
> >
> >
> >
> > I think it's quite funny that the guy behind all this was called
> > Cantor...
>
> And how about "Aleph", LOL.
> >
> > Now tell me this: I am sitting at a rectangular table. How can I
> tell
> > whether the ratio of its width to its length is rational or
> irrational?
> >
> > ~~~T~~~
> >
>
> A. it's a rectangle and B. it doesn't fall on your lap.
>
>
>
>

Ah. I thought you meant by "unique" that they had less of a pattern
amongst them - there is no pattern amongst the irrationals, but there
is among the rationals.

Nice to see miscommunications resolved, I think we're all on the same page :)

-Mike

On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...> wrote:
>
>
>
>
>
>
>
> Exactly. I think the fact that you can access rational numbers makes them
> more unique, not just that there are fewer of them, which is also a quality
> of uniqueness.
>
> ________________________________
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
> Mike Battaglia
> Sent: Friday, June 06, 2008 2:23 PM
>
> To: tuning@yahoogroups.com
> Subject: Re: [tuning] for Carl's quest: Why there are infinite many
> rationals? was Re: re;re for Tom
>
>
>
>
>
> Nononono. The rational numbers are just as unique as the natural numbers.
> The irrational numbers are more unique.
>
> The idea is that if I give out irrational numbers and you give rational
> numbers, that doesn't actually prove anything. I will never be able to come
> up with a way to systematically access EVERY irrational number. There is,
> however, a way to systematically access every natural and rational number.
> This is the difference between the rational and irrational numbers.
>
> For example, let's say I have a list of every irrational number, paired up
> with some corresponding natural number.
>
> 1: 0.123239...
> 2: 0.423242...
> 3: 0.123463...
> 4: 0.132948...
> 5: 0.534854...
> 6: 0.348275...
> .
> .
> .
>
> So let's say that this just goes on forever and ends up hitting EVERY
> irrational number. Let's say there is some "pattern" in the irrational
> numbers I've hit on that will systematically hit every one. We can also
> construct a number by taking the first digit of the first number, and the
> second of the second number, etc:
>
> 1: 0.127439...
> 2: 0.423942...
> 3: 0.623763...
> 4: 0.142948...
> 5: 0.534854...
> 6: 0.348275...
> .
> .
> .
>
> D: 0.123955...
>
> It's easy to see that this number will appear in the list somewhere, since
> the list supposedly has *every* irrational number in it. The place where
> this number meets the diagonal in the list will have the digits simply
> "align," as the number formed by the diagonal will simply be the same as
> this number here.
>
> BUT... what about the number formed by taking the diagonal and incrementing
> each digit by 1?
>
>
> 1: 0.127439...
> 2: 0.423942...
> 3: 0.623763...
> 4: 0.142948...
> 5: 0.534854...
> 6: 0.348275...
> .
> .
> .
>
> D: 0.234066...
>
> This number, as this list is supposed to contain "every" irrational number,
> will also have to be in the list somewhere. What about where this number
> meets the diagonal? What will the digit be? The number will, by definition,
> be one digit higher than the diagonal - therefore, where the number meets
> the diagonal, that digit will have to be the digit of the diagonal AND that
> digit plus one *at the same time*. So it breaks down.
>
> Interesting, but what does it mean? It means that any supposed "list"
> containing all of the irrational numbers won't contain all of the irrational
> numbers. The concept of a list containing every irrational number is
> contradictory. No list can be made that will contain all of the irrational
> numbers, even if we let that list go on forever.
>
> On the other hand, a list can be made that WILL contain all of the natural
> numbers, and all of the rational numbers, if we let them run forever. By
> starting at 0 and incrementing by 1, we'll eventually get every natural
> number, and check the other thread for a way to have every rational number
> be eventually hit in the same way.
>
> This means that you can't pair up every natural number with every irrational
> number. If that were possible, then you'd be able to write them out in such
> a list, which would then prove itself to not be a list at all.
>
> So in other words, any way you have to compute systematically every
> irrational number will fail. There will always be a number you will miss.
> Which is kind of worrying for certain fields of mathematics and computer
> science, that there exist "uncomputable" numbers.
>
> Pi, e, etc. are all computable. We have formulas to compute these numbers to
> as much precision as we'd like. If we left these formulas running forever,
> then we'd have pi and e and such exactly. Uncomputable numbers, however, are
> of a different species entirely.
>
> For the record, the set of all "computable" numbers is on par with the
> natural and rational numbers.
>
> On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...> wrote:
> >
> >
> >
> >
> >
> >
> >
> > Yes. But not a RATIONAL number, which is "more unique".
> >
> > ________________________________
> > From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
> > Ozan Yarman
> > Sent: Thursday, June 05, 2008 6:43 PM
> > To: tuning@yahoogroups.com
> > Subject: Re: [tuning] for Carl's quest: Why there are infinite many
> > rationals? was Re: re;re for Tom
> >
> >
> >
> >
> >
> >
> > For every irrational number that can be produced, a natural number can be
> > made to correspond with it, ad infinitum.
> >
> >
> > Oz.
> >
> >
> > On Jun 6, 2008, at 1:41 AM, Mike Battaglia wrote:
> >
> >
> > How about we take it a step further: For every irrational number in
> > existence, can you come up with a natural number?
> >
> >
> > On Thu, Jun 5, 2008 at 6:38 PM, Ozan Yarman <ozanyarman@...>
> > wrote:
> >
> > >
> > >
> > >
> > >
> > >
> > >
> > >
> > > Here is a test: for every irrational number you come up with, I can come
> > up with a rational number and we can carry this process to infinity,
> ending
> > up with an equal amount of irrational and rational numbers.
> > >
> > >
> > > Oz.
> > >
> > >
> > >
> > > On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:
> > >
> > >
> > >
> > > Carl,
> > >
> > > Because rational numbers are the ratios of integers, you can assign a
> > counting number to each rational number.
> > > Granted you can do this forever without finishing; but for each rational
> > number there is a counting number that can be assigned to it. The thing
> is
> > that you cannot even count the irrational numbers between two rational
> > numbers! No matter how many you place there, there will always be more.
> > Because there are demonstrably MORE irrational numbers than rational, the
> > set of all numbers will be MOSTLY irrational by a LARGE margin.
> > >
> > > No matter that the supply of numbers is endless, it is the union of
> > rational and irrational numbers. And for each rational number you find,
> > there are an uncountable number of irrational numbers.
> > >
> > > It's not possible to take an infinite number of numbers and try to count
> > which are rational and which are irrational; for the reason given
> above--you
> > may count all day and never even come across a rational number. But you
> > KNOW there are some. And that there just aren't "as many" (relatively) as
> > the irrationals.
> > >
> > > Howard
> > >
> > >
> >
> > /! div>
> >
> >
> >
> >
> >
> >
> >
>
>
>
>

Oh, why of course. Let's instead say the table is 15 and 3/4 inches by
19 and 7/8 inches. If we convert to units of 1/8 inch, that would be
126 by 159 so the ratio would be 126/159 or 159/126. This would work of
course to any accuracy of measurement.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Mike Battaglia
Sent: Friday, June 06, 2008 3:46 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

Howard:

I think he's saying, given an actual real life table, will the sides
ever have perfectly rational dimensions? That is, is the ratio of each
side to an "inch" going to be rational, as well as the ratio of the
sides to each other?

-Mike

On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...
<mailto:howard.m.cornell.iii%40lmco.com> > wrote:
>
>
>
>
>
>
>
> Cameron,
>
> Even rectangles can have integer dimensions. Perhaps the table is 60
inches
> long and 30 inches wide. The ratio is 30/60 or 1/2 which is rational.
If
> it were pi feet long and the square root of 2 feet wide that would be
a
> different matter.
>
> It doesn't matter that it doesn't fall on your lap; but that is
nevertheless
> a good quality for a table to have.
>
> ________________________________
> From: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ] On
Behalf Of
> Cameron Bobro
> Sent: Friday, June 06, 2008 2:05 PM
> To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
> Subject: [tuning] for Carl's quest: Why there are infinite many
rationals?
> was Re: re;re for Tom
>
>
>
>
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ,
"Cornell III, Howard
> M" <howard.m.cornell.iii@...> wrote:
> >
> > Any ratio of two integers is rational.
> >
> > ________________________________
> >
> > From: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ] On
> Behalf
> > Of Tom Dent
> > Sent: Friday, June 06, 2008 12:54 PM
> > To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
> > Subject: [tuning] for Carl's quest: Why there are infinite many
> > rationals? was Re: re;re for Tom
> >
> >
> >
> >
> > I think it's quite funny that the guy behind all this was called
> > Cantor...
>
> And how about "Aleph", LOL.
> >
> > Now tell me this: I am sitting at a rectangular table. How can I
> tell
> > whether the ratio of its width to its length is rational or
> irrational?
> >
> > ~~~T~~~
> >
>
> A. it's a rectangle and B. it doesn't fall on your lap.
>
>
>
>

- And how about the ratio of an inch to a centimetre, or to an atomic
radius, or to any other measuring unit you can think of?

But putting that aside (if we can), how could I possibly tell the
difference between a table with 'perfectly rational' dimensions and
one with, I guess, perfectly irrational ones? Please give exact
instructions, you may consider the use of tape measures, micrometers,
interferometers, and any other kind of equipment.

~~~T~~~

[Carl, please don't give away the punchline yet...]

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> Howard:
>
> I think he's saying, given an actual real life table, will the sides
> ever have perfectly rational dimensions? That is, is the ratio of each
> side to an "inch" going to be rational, as well as the ratio of the
> sides to each other?
>
> -Mike
>
> On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...> wrote:
> >
> > Cameron,
> >
> > Even rectangles can have integer dimensions. Perhaps the table is
60 inches
> > long and 30 inches wide. The ratio is 30/60 or 1/2 which is
rational. If
> > it were pi feet long and the square root of 2 feet wide that would
be a
> > different matter.
> >
> > It doesn't matter that it doesn't fall on your lap; but that is
nevertheless
> > a good quality for a table to have.
> >
> > ________________________________
> > From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
Behalf Of
> > Cameron Bobro
> > Sent: Friday, June 06, 2008 2:05 PM
> > To: tuning@yahoogroups.com
> > Subject: [tuning] for Carl's quest: Why there are infinite many
rationals?
> > was Re: re;re for Tom
> >
> >
> > --- In tuning@yahoogroups.com, "Cornell III, Howard
> > M" <howard.m.cornell.iii@> wrote:
> > >
> > > Any ratio of two integers is rational.
> > >
> > > ________________________________
> > >
> > > From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
> > Behalf
> > > Of Tom Dent
> > > Sent: Friday, June 06, 2008 12:54 PM
> > > To: tuning@yahoogroups.com
> > > Subject: [tuning] for Carl's quest: Why there are infinite many
> > > rationals? was Re: re;re for Tom
> > >
> > > I think it's quite funny that the guy behind all this was called
> > > Cantor...
> >
> > And how about "Aleph", LOL.
> > >
> > > Now tell me this: I am sitting at a rectangular table. How can I
> > tell
> > > whether the ratio of its width to its length is rational or
> > irrational?
> > >
> > > ~~~T~~~
> > >
> >
> > A. it's a rectangle and B. it doesn't fall on your lap.

> > Let me remind you that this is a spinoff thread from a thread
> > that was originally about tuning.
>
> So it's off-topic. There's actually a list for that:
>
> /metatuning

I'm looking at metatuning now. The first three topics I found involved
a joke of the agreement between FEMA and NIST, someone selling their
CD's on ebay, and a link to a news article about Albert Hoffman dying.

Doesn't really seem like it would really fit in over there.

Especially when the discussion over rational and irrational numbers is
necessary to understand the entire point of Rick Ballan's original
tuning-related thread. People shouldn't have to go to another yahoo
group to understand this. Believe it or not, not everyone understands
math well enough to already know this stuff. Therefore, given the
discussion at hand, it's "related" to tuning.

It's not too much of a stretch for things related to tuning to be
discussed on a tuning forum.

> > The topic we are discussing here is necessary
> > to fully understand the other thread.
>
> It isn't. Rationals and irrationals approximate one
> another to far beyond the limits of human perception.
> That's all that's relevant for microtonal music.

Who are you to decide what's relevant to microtonal music? The point
of Rick's original thread is whether the 12 tone equal tempered scale
is actually a JI scale involving "higher" overtones due to various
quantum mechanics-related effects.

So to understand that, he has to first explain various aspects of
quantum mechanics to the people who don't already understand.
Alternatively, people have different views of quantum mechanics and so
a discussion of it is formed, the results of which will reflect
directly back on the original tuning-related idea. Regardless of this,
we then hear cries about it being "off-topic."

Then, we eventually get past the quantum mechanics side of things and
we end up in a discussion about whether the numbers involved are
actually rational or irrational numbers. The argument slowly builds
that they ARE actually irrational because there are "more" irrational
numbers than rational numbers. Some people disagree, and so a
discussion about that starts in another side-thread. This side-thread
is obviously necessary because it once again ties directly back to
Rick Ballan's original argument.

Quite a few microtonally related ideas have already come directly out
of this, first and foremost the notion that 16:19:24 might actually be
the minor chord that we're all used to rather than 10:12:15. From that
there are a lot more questions, such as what are we actually hearing
when we listen to temperament? Out of tune JI chords, or "in tune"
higher-harmonic JI chords - how does it all work?

Of course, a discussion of this means we would get into gestalt
psychology, which would likely for organizational purposes be best
served with a side-thread of its own. This, I am sure, would be met
with cries of it being "off-topic," and so no progress will ever be
made on anything. Especially if the realizations being made are
cross-applicational, and in order to discuss them we are always going
to be met with this "off-topic" nonsense from people who are either
bent on dominating the discussion or haven't bothered to read the
original threads to see how these seemingly oblique discussions relate
back to tuning.

I, on the other hand, find this discussion - what it is we actually
perceive when we hear tempered chords - extremely interesting and at
the heart of the very reasons that I have joined the group at all. I
would also say that it's most likely related to the reasons why many
people have gotten interested in JI or microtonal music in the first
place. It seems counter-productive to force everyone to move
discussions such as these to the "joke around" group you posted above.
These are the kinds of discussions where progress can get made. If
we're not allowed to discuss these things because they would require
loose discussions about related topics, then what are we going to
discuss?

I don't think this group should be limited to just discussions about
whether 53tet or 72tet is superior or whether miracle temperaments are
more useful than diaschismatic ones.

-Mike

If you measure it to any accuracy you require, and are confident you
have obtained the correct width and length, you will find that the
proportion is rational. That is what makes irrational numbers
irrational--they do not fit into a counting number system exactly. That
is, you cannot build a table that is pi feet wide.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Tom Dent
Sent: Friday, June 06, 2008 4:23 PM
To: tuning@yahoogroups.com
Subject: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

- And how about the ratio of an inch to a centimetre, or to an atomic
radius, or to any other measuring unit you can think of?

But putting that aside (if we can), how could I possibly tell the
difference between a table with 'perfectly rational' dimensions and
one with, I guess, perfectly irrational ones? Please give exact
instructions, you may consider the use of tape measures, micrometers,
interferometers, and any other kind of equipment.

~~~T~~~

[Carl, please don't give away the punchline yet...]

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> , "Mike
Battaglia" <battaglia01@...> wrote:
>
> Howard:
>
> I think he's saying, given an actual real life table, will the sides
> ever have perfectly rational dimensions? That is, is the ratio of each
> side to an "inch" going to be rational, as well as the ratio of the
> sides to each other?
>
> -Mike
>
> On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...> wrote:
> >
> > Cameron,
> >
> > Even rectangles can have integer dimensions. Perhaps the table is
60 inches
> > long and 30 inches wide. The ratio is 30/60 or 1/2 which is
rational. If
> > it were pi feet long and the square root of 2 feet wide that would
be a
> > different matter.
> >
> > It doesn't matter that it doesn't fall on your lap; but that is
nevertheless
> > a good quality for a table to have.
> >
> > ________________________________
> > From: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ] On
Behalf Of
> > Cameron Bobro
> > Sent: Friday, June 06, 2008 2:05 PM
> > To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
> > Subject: [tuning] for Carl's quest: Why there are infinite many
rationals?
> > was Re: re;re for Tom
> >
> >
> > --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ,
"Cornell III, Howard
> > M" <howard.m.cornell.iii@> wrote:
> > >
> > > Any ratio of two integers is rational.
> > >
> > > ________________________________
> > >
> > > From: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com> ] On
> > Behalf
> > > Of Tom Dent
> > > Sent: Friday, June 06, 2008 12:54 PM
> > > To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
> > > Subject: [tuning] for Carl's quest: Why there are infinite many
> > > rationals? was Re: re;re for Tom
> > >
> > > I think it's quite funny that the guy behind all this was called
> > > Cantor...
> >
> > And how about "Aleph", LOL.
> > >
> > > Now tell me this: I am sitting at a rectangular table. How can I
> > tell
> > > whether the ratio of its width to its length is rational or
> > irrational?
> > >
> > > ~~~T~~~
> > >
> >
> > A. it's a rectangle and B. it doesn't fall on your lap.

One never runs out of rational numbers when there is an inexhaustable supply of them. They are infinite in number. Between any two irrational numbers, an infinite amount of rational numbers exist. It is possible to carry the numerators and denominators to infinite precision.

Oz.

On Jun 6, 2008, at 4:38 PM, Cornell III, Howard M wrote:

> No we cannot. If we don't get tired you WILL run out of rational > numbers and I NEVER will run out of irrational numbers. In fact I > only have to give you TWO rational numbers and you can give me > irrational numbers between them until you change your mind.
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On > Behalf Of Ozan Yarman
> Sent: Thursday, June 05, 2008 5:39 PM
> To: tuning@yahoogroups.com
> Subject: Re: [tuning] for Carl's quest: Why there are infinite many > rationals? was Re: re;re for Tom
>
>
> Here is a test: for every irrational number you come up with, I can > come up with a rational number and we can carry this process to > infinity, ending up with an equal amount of irrational and rational > numbers.
>
> Oz.
>
> On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:
>
>> Carl,
>>
>> Because rational numbers are the ratios of integers, you can assign >> a counting number to each rational number.
>> Granted you can do this forever without finishing; but for each >> rational number there is a counting number that can be assigned to >> it. The thing is that you cannot even count the irrational numbers >> between two rational numbers! No matter how many you place there, >> there will always be more. Because there are demonstrably MORE >> irrational numbers than rational, the set of all numbers will be >> MOSTLY irrational by a LARGE margin.
>>
>> No matter that the supply of numbers is endless, it is the union of >> rational and irrational numbers. And for each rational number you >> find, there are an uncountable number of irrational numbe! rs.
>>
>> It's not possible to take an infinite number of numbers and try to >> count which are rational and which are irrational; for the reason >> given above--you may count all day and never even come across a >> rational number. But you KNOW there are some. And that there just >> aren't "as many" (relatively) as the irrationals.
>>
>> Howard
>>
>>
>
>
>

[ Attachment content not displayed ]

I guess you just don't believe that if we take one irrational number out
of the counting system for each rational number, which will take a long
time, we WILL have many irrational numbers left. This is because you
CANNOT come up with a rational number for EVERY irrational number even
if you can come up with a rational number for any particular irrational
number. A rational number generator has one RULE, an irrational number
generator does not. It makes a difference.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Friday, June 06, 2008 4:53 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

One never runs out of rational numbers when there is an inexhaustable
supply of them. They are infinite in number. Between any two irrational
numbers, an infinite amount of rational numbers exist. It is possible to
carry the numerators and denominators to infinite precision.

Oz.

On Jun 6, 2008, at 4:38 PM, Cornell III, Howard M wrote:

No we cannot. If we don't get tired you WILL run out of
rational numbers and I NEVER will run out of irrational numbers. In
fact I only have to give you TWO rational numbers and you can give me
irrational numbers between them until you change your mind.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of Ozan Yarman
Sent: Thursday, June 05, 2008 5:39 PM
To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] for Carl's quest: Why there are infinite
many rationals? was Re: re;re for Tom

Here is a test: for every irrational number you come up with, I
can come up with a rational number and we can carry this process to
infinity, ending up with an equal amount of irrational and rational
numbers.

Oz.

On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:

Carl,

Because rational numbers are the ratios of integers, you
can assign a counting number to each rational number.
Granted you can do this forever without finishing; but
for each rational number there is a counting number that can be assigned
to it. The thing is that you cannot even count the irrational numbers
between two rational numbers! No matter how many you place there, there
will always be more. Because there are demonstrably MORE irrational
numbers than rational, the set of all numbers will be MOSTLY irrational
by a LARGE margin.

No matter that the supply of numbers is endless, it is
the union of rational and irrational numbers. And for each rational
number you find, there are an uncountable number of irra! tional numbe!
rs.

It's not possible to take an infinite number of numbers
and try to count which are rational and which are irrational; for the
reason given above--you may count all day and never even come across a
rational number. But you KNOW there are some. And that there just
aren't "as many" (relatively) as the irrationals.

Howard

Not at all. The true and actual dimension of the table could be
measured in atomic diameters, say. That would be an integer so the
ratio of two dimensions would always be rational. But you don't have to
be that accurate. If you think that you have measured the dimensions
with accuracy to within a certain proportion of an inch, you can
convert to that unit and also get a rational number. The point is that
you cannot make the table pi, or e [base of natural logarithms], feet
or meters in dimension. Therefore you could not measure an irrational
number! But, as we all know, you can get arbitrarily close to it. The
smallest unit you want to use to measure the dimension is what makes it
measurable. Then you can count in terms of that. Irrational numbers do
not obey that rule. They will lie between the smaller number of units
and the next larger number of units. Doesn't that make sense?

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Mike Battaglia
Sent: Friday, June 06, 2008 5:02 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

There is a conflict of ideas here:

That "almost all" numbers are irrational and uncomputable, and so the
length of an arbitrary side of a table will most likely be an irrational
number;

and the idea that there is a "smallest unit" of space which would
theoretically make all proportions of length rational.

-Mike

On Fri, Jun 6, 2008 at 5:36 PM, Cornell III, Howard M
<howard.m.cornell.iii@... <mailto:howard.m.cornell.iii@...> >
wrote:

If you measure it to any accuracy you require, and are confident
you have obtained the correct width and length, you will find that the
proportion is rational. That is what makes irrational numbers
irrational--they do not fit into a counting number system exactly. That
is, you cannot build a table that is pi feet wide.

________________________________

From: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning@yahoogroups.com> ] On
Behalf Of Tom Dent

Sent: Friday, June 06, 2008 4:23 PM

To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
Subject: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

- And how about the ratio of an inch to a centimetre, or to an
atomic
radius, or to any other measuring unit you can think of?

But putting that aside (if we can), how could I possibly tell
the
difference between a table with 'perfectly rational' dimensions
and
one with, I guess, perfectly irrational ones? Please give exact
instructions, you may consider the use of tape measures,
micrometers,
interferometers, and any other kind of equipment.

~~~T~~~

[Carl, please don't give away the punchline yet...]

--- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>
, "Mike Battaglia" <battaglia01@...> wrote:
>
> Howard:
>
> I think he's saying, given an actual real life table, will the
sides
> ever have perfectly rational dimensions? That is, is the ratio
of each
> side to an "inch" going to be rational, as well as the ratio
of the
> sides to each other?
>
> -Mike
>
> On 6/6/08, Cornell III, Howard M <howard.m.cornell.iii@...>
wrote:
> >
> > Cameron,
> >
> > Even rectangles can have integer dimensions. Perhaps the
table is
60 inches
> > long and 30 inches wide. The ratio is 30/60 or 1/2 which is
rational. If
> > it were pi feet long and the square root of 2 feet wide that
would
be a
> > different matter.
> >
> > It doesn't matter that it doesn't fall on your lap; but that
is
nevertheless
> > a good quality for a table to have.
> >
> > ________________________________
> > From: tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> ] On
Behalf Of
> > Cameron Bobro
> > Sent: Friday, June 06, 2008 2:05 PM
> > To: tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>

> > Subject: [tuning] for Carl's quest: Why there are infinite
many
rationals?
> > was Re: re;re for Tom
> >
> >
> > --- In tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> , "Cornell III, Howard
> > M" <howard.m.cornell.iii@> wrote:
> > >
> > > Any ratio of two integers is rational.
> > >
> > > ________________________________
> > >
> > > From: tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> ] On
> > Behalf
> > > Of Tom Dent
> > > Sent: Friday, June 06, 2008 12:54 PM
> > > To: tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com>
> > > Subject: [tuning] for Carl's quest: Why there are infinite
many
> > > rationals? was Re: re;re for Tom
> > >
> > > I think it's quite funny that the guy behind all this was
called
> > > Cantor...
> >
> > And how about "Aleph", LOL.
> > >
> > > Now tell me this: I am sitting at a rectangular table. How
can I
> > tell
> > > whether the ratio of its width to its length is rational
or
> > irrational?
> > >
> > > ~~~T~~~
> > >
> >
> > A. it's a rectangle and B. it doesn't fall on your lap.

[ Attachment content not displayed ]

Yes, it baffles me that anything other than the fact that rational numbers are as inexhaustable as irrational numbers can be claimed. Maybe this is because I think of infinity as it is meant to be: Inexhaustability.

Oz.

On Jun 7, 2008, at 1:11 AM, Cornell III, Howard M wrote:

> I guess you just don't believe that if we take one irrational number > out of the counting system for each rational number, which will take > a long time, we WILL have many irrational numbers left. This is > because you CANNOT come up with a rational number for EVERY > irrational number even if you can come up with a rational number for > any particular irrational number. A rational number generator has > one RULE, an irrational number generator does not. It makes a > difference.
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On > Behalf Of Ozan Yarman
> Sent: Friday, June 06, 2008 4:53 PM
> To: tuning@yahoogroups.com
> Subject: Re: [tuning] for Carl's quest: Why there are infinite many > rationals? was Re: re;re for Tom
>
>
> One never runs out of rational numbers when there is an > inexhaustable supply of them. They are infinite in number. Between > any two irrational numbers, an infinite amount of rational numbers > exist. It is possible to carry the numerators and denominators to > infinite precision.
>
> Oz.
>
> On Jun 6, 2008, at 4:38 PM, Cornell III, Howard M wrote:
>
>> No we cannot. If we don't get tired you WILL run out of rational >> numbers and I NEVER will run out of irrational numbers. In fact I >> only have to give you TWO rational numbers and you can give me >> irrational numbers between them until you change your mind.
>>
>> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On >> Behalf Of Ozan Yarman
>> Sent: Thursday, June 05, 2008 5:39 PM
>> To: tuning@yahoogroups.com
>> Subject: Re: [tuning] for Carl's quest: Why there are infinite many >> rationals? was Re: re;re for Tom
>>
>>
>> Here is a test: for every irrational number you come up with, I can >> come up with a rational number and we can carry this process to >> infinity, ending up with an equal amount of irrational and rational >> numbers.
>>
>> Oz.
>>
>> On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:
>>
>>> Carl,
>>>
>>> Because rational numbers are the ratios of integers, you can >>> assign a counting number to each rational number.
>>> Granted you can do this forever without finishing; but for each >>> rational number there is a counting number that can be assigned to >>> it. The thing is that you cannot even count the irrational >>> numbers between two rational numbers! No matter how many you >>> place there, there will always be more. Because there are >>> demonstrably MORE irrational numbers than rational, the set of all >>> numbers will be MOSTLY irrational by a LARGE margin.
>>>
>>> No matter that the supply of numbers is endless, it is the union >>> of rational and irrational numbers. And for each rational number >>> you find, there are an uncountable number of irra! tional numbe! >>> rs.
>>>
>>> It's not possible to take an infinite number of numbers and try to >>> count which are rational and which are irrational; for the reason >>> given above--you may count all day and never even come across a >>> rational number. But you KNOW there are some. And that there >>> just aren't "as many" (relatively) as the irrationals.
>>>
>>> Howard
>>>
>>>
>>
>>

[ Attachment content not displayed ]

Mike wrote:

> Especially when the discussion over rational and irrational
> numbers is necessary to understand the entire point of Rick
> Ballan's original tuning-related thread.

Why do you think Rick's original post had anything to do with
tuning?

> > > The topic we are discussing here is necessary
> > > to fully understand the other thread.
> >
> > It isn't. Rationals and irrationals approximate one
> > another to far beyond the limits of human perception.
> > That's all that's relevant for microtonal music.
>
> Who are you to decide what's relevant to microtonal music?
> The point of Rick's original thread is whether the 12 tone
> equal tempered scale is actually a JI scale involving
> "higher" overtones due to various quantum mechanics-related
> effects.
//
> So to understand that, he has to first explain various aspects
> of quantum mechanics to the people who don't already understand.

First you have to establish it's about tuning and not just
another crackpot physics theory that should be getting
moderated off a physics newsgroup somewhere.

> Quite a few microtonally related ideas have already come directly
> out of this, first and foremost the notion that 16:19:24 might
> actually be the minor chord that we're all used to rather than
> 10:12:15.

Can you cite the text where this amazing discovery most
recently (for the 45,000th time) "came out" on this list?
I'd love to know why you're under the apprehension it has:

* anything to do with the continuum hypothesis
* anything to do with quantum mechanics

Do tell.

> Of course, a discussion of this means we would get into gestalt
> psychology, which would likely for organizational purposes be
> best served with a side-thread of its own.

*Concepts from* gestalt psychology have been discussed *in the
context of* microtonal music and *applied to* psychoacoustics
research many times on this list. I've got no problem with it.

> This, I am sure, would be met with cries of it being
> "off-topic," and so no progress will ever be made on anything.

You're right: there's definitely a problem with progress being
made around here. The lack of a FAQ is partially to blame.

> I, on the other hand, find this discussion - what it is we
> actually perceive when we hear tempered chords - extremely
> interesting

Yeah, that'd be great. I'm not seeing much of it and it isn't
what I'm complaining about.

> These are the kinds of discussions where progress can get made.

If that were true you wouldn't still be having them.

> I don't think this group should be limited to just discussions
> about whether 53tet or 72tet is superior or whether miracle
> temperaments are more useful than diaschismatic ones.

Oh, I agree. And like Don Rumsfeld, I'm also against cutting
people's heads off.

-Carl

The point you stated in your second paragraph is the point I was trying
to make: the act of measuring the dimension forces it into being
rational. I did not mean to imply that there is an accurate enough
measurement system to actually make irrational dimensions rational.
Let's look at it one more time in a different way. Let's say the table
is exactly thirty pi centimeters in one dimension. You cannot measure
it exactly because it is not a countable multiple of any length unit you
can use. But, of course, you could measure it approximately, which would
give a rational number based on the units of length, like using a laser
beam and measuring in angstroms; but even then you cannot be sure that
it is thirty pi centimeters.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Mike Battaglia
Sent: Friday, June 06, 2008 5:42 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

Right. So you're on the side that since everything can be broken up into
arbitrary measuring units, then all proportions are rational.

I think the question really breaks down to whether anything exists on a
scale smaller than the Planck length, which is something quantum physics
has differing theories on. It could be that our MEASUREMENTS will always
yield rational approximations to irrational numbers - or that our
measurements will always yield rational approximations of even finer
rational numbers.

On Fri, Jun 6, 2008 at 6:27 PM, Cornell III, Howard M
<howard.m.cornell.iii@... <mailto:howard.m.cornell.iii@...> >
wrote:

Not at all. The true and actual dimension of the table could be
measured in atomic diameters, say. That would be an integer so the
ratio of two dimensions would always be rational. But you don't have to
be that accurate. If you think that you have measured the dimensions
with accuracy to within a certain proportion of an inch, you can
convert to that unit and also get a rational number. The point is that
you cannot make the table pi, or e [base of natural logarithms], feet
or meters in dimension. Therefore you could not measure an irrational
number! But, as we all know, you can get arbitrarily close to it. The
smallest unit you want to use to measure the dimension is what makes it
measurable. Then you can count in terms of that. Irrational numbers do
not obey that rule. They will lie between the smaller number of units
and the next larger number of units. Doesn't that make sense?


________________________________

From: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
[mailto:tuning@yahoogroups.com <mailto:tuning@yahoogroups.com> ] On
Behalf Of Mike Battaglia
Sent: Friday, June 06, 2008 5:02 PM
To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] for Carl's quest: Why there are infinite
many rationals? was Re: re;re for Tom

There is a conflict of ideas here:

That "almost all" numbers are irrational and uncomputable, and
so the length of an arbitrary side of a table will most likely be an
irrational number;

and the idea that there is a "smallest unit" of space which
would theoretically make all proportions of length rational.

-Mike

On Fri, Jun 6, 2008 at 5:36 PM, Cornell III, Howard M
<howard.m.cornell.iii@... <mailto:howard.m.cornell.iii@...> >
wrote:

If you measure it to any accuracy you require, and are
confident you have obtained the correct width and length, you will find
that the proportion is rational. That is what makes irrational numbers
irrational--they do not fit into a counting number system exactly. That
is, you cannot build a table that is pi feet wide.

________________________________

From: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of Tom Dent

Sent: Friday, June 06, 2008 4:23 PM

To: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com>
Subject: [tuning] for Carl's quest: Why there are
infinite many rationals? was Re: re;re for Tom

- And how about the ratio of an inch to a centimetre, or
to an atomic
radius, or to any other measuring unit you can think of?

But putting that aside (if we can), how could I possibly
tell the
difference between a table with 'perfectly rational'
dimensions and
one with, I guess, perfectly irrational ones? Please
give exact
instructions, you may consider the use of tape measures,
micrometers,
interferometers, and any other kind of equipment.

~~~T~~~

[Carl, please don't give away the punchline yet...]

--- In tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> , "Mike Battaglia" <battaglia01@...>
wrote:
>
> Howard:
>
> I think he's saying, given an actual real life table,
will the sides
> ever have perfectly rational dimensions? That is, is
the ratio of each
> side to an "inch" going to be rational, as well as the
ratio of the
> sides to each other?
>
> -Mike
>
> On 6/6/08, Cornell III, Howard M
<howard.m.cornell.iii@...> wrote:
> >
> > Cameron,
> >
> > Even rectangles can have integer dimensions. Perhaps
the table is
60 inches
> > long and 30 inches wide. The ratio is 30/60 or 1/2
which is
rational. If
> > it were pi feet long and the square root of 2 feet
wide that would
be a
> > different matter.
> >
> > It doesn't matter that it doesn't fall on your lap;
but that is
nevertheless
> > a good quality for a table to have.
> >
> > ________________________________
> > From: tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> ] On
Behalf Of
> > Cameron Bobro
> > Sent: Friday, June 06, 2008 2:05 PM
> > To: tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com>
> > Subject: [tuning] for Carl's quest: Why there are
infinite many
rationals?
> > was Re: re;re for Tom
> >
> >
> > --- In tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> , "Cornell III, Howard
> > M" <howard.m.cornell.iii@> wrote:
> > >
> > > Any ratio of two integers is rational.
> > >
> > > ________________________________
> > >
> > > From: tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com> ] On
> > Behalf
> > > Of Tom Dent
> > > Sent: Friday, June 06, 2008 12:54 PM
> > > To: tuning@yahoogroups.com
<mailto:tuning%40yahoogroups.com>
> > > Subject: [tuning] for Carl's quest: Why there are
infinite many
> > > rationals? was Re: re;re for Tom
> > >
> > > I think it's quite funny that the guy behind all
this was called
> > > Cantor...
> >
> > And how about "Aleph", LOL.
> > >
> > > Now tell me this: I am sitting at a rectangular
table. How can I
> > tell
> > > whether the ratio of its width to its length is
rational or
> > irrational?
> > >
> > > ~~~T~~~
> > >
> >
> > A. it's a rectangle and B. it doesn't fall on your
lap.

Exactly. That is why we get headaches if we go straight to making
assumptions about how many of what kinds of things are actually in an
infinitude of things. We have to understand from the start what makes
the things different and whether and how that affects the mix.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Ozan Yarman
Sent: Friday, June 06, 2008 5:43 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

Yes, it baffles me that anything other than the fact that rational
numbers are as inexhaustable as irrational numbers can be claimed. Maybe
this is because I think of infinity as it is meant to be:
Inexhaustability.

Oz.

On Jun 7, 2008, at 1:11 AM, Cornell III, Howard M wrote:

I guess you just don't believe that if we take one irrational
number out of the counting system for each rational number, which will
take a long time, we WILL have many irrational numbers left. This is
because you CANNOT come up with a rational number for EVERY irrational
number even if you can come up with a rational number for any particular
irrational number. A rational number generator has one RULE, an ir!
rational number generator does not. It makes a difference.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of Ozan Yarman
Sent: Friday, June 06, 2008 4:53 PM
To: tuning@yahoogroups.com <mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] for Carl's quest: Why there are infinite
many rationals? was Re: re;re for Tom

One never runs out of rational numbers when there is an
inexhaustable supply of them. They are infinite in number. Between any
two irrational numbers, an infinite amount of rational numbers exist. It
is possible to carry the numerators and denominators to infinite !
precision.

Oz.

On Jun 6, 2008, at 4:38 PM, Cornell III, Howard M wrote:

No we cannot. If we don't get tired you WILL run out of
rational numbers and I NEVER will run out of irrational numbers. In
fact I only have to give you TWO rational numbers and you can give me
irrational numbers between them until you change your mind.

________________________________

From: tuning@yahoogroups.com
[mailto:tuning@yahoogroups.com <mailto:tuning@yahoogroups.com> ] On
Behalf Of Ozan Yarman
Sent: Thursday, June 05, 2008 5:39 PM
To: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com>
! Subject: Re: [tuning] for Carl's quest: Why there are
infinite many rationals? was Re: re;re for Tom

Here is a test: for every irrational number you come up
with, I can come up with a rational number and we can carry this process
to infinity, ending up with an equal amount of irrational and rational
numbers.

Oz.

On Jun 6, 2008, at 1:06 AM, Cornell III, Howard M wrote:

Carl,

Because rational numbers are the ratios of
integers, you can assign a counting number to each rational number.
Granted you can do this forever without
finishing; but for each rational number there is a counting number that
can be assigned to it. The thing is that you cannot even count the
irrational numbers between two rational numbers! No matter how many you
place there, there will always be more. Because there are demonstrably
MORE irrational numbers than rational, the set of all numbers will be
MOSTLY irrational by a LARGE margin.

No matter that the supply of numbers is endless,
it is the union of rational and irrational numbers. And for each
rational number you find, there are an uncoun! table number of irra!
tional numbe! rs.

It's not possible to take an infinite number of
numbers and try to count which are rational and which are irrational;
for the reason given above--you may count all day and never even come
across a rational number. But you KNOW there are some. And that there
just aren't "as many" (relatively) as the irrationals.

Howard

That's right. When you limit the combinations for some of the things,
but not others, you just won't get as many of them.

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Mike Battaglia
Sent: Friday, June 06, 2008 5:55 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] for Carl's quest: Why there are infinite many
rationals? was Re: re;re for Tom

The irrational numbers and the rational numbers are both inexhaustible.
But the rational numbers are the inexhaustible results of following one
specific pattern, and the irrational numbers have one further degree of
freedom as they have no specific pattern that causes them.

Here's another way of putting it: there are always more combinations of
atoms than atoms themselves. There are always more combinations of
things than the things themselves. This applies to infinite sets as
well. For the natural numbers, there will always be more combinations of
natural numbers than the natural numbers as well.

If you laid out all of the natural numbers, they would be infinite in
actuality. However, possible combinations and things to do with those
numbers are infinite in another way - they are infinite in potential.
The "infinite in potential" is greater than the "infinite in actuality."
Or rather, it's another shade of infinity. This is another way of
thinking about it.

-Mike

On Fri, Jun 6, 2008 at 6:42 PM, Ozan Yarman <ozanyarman@...
<mailto:ozanyarman@...> > wrote:

Yes, it baffles me that anything other than the fact that
rational numbers are as inexhaustable as irrational numbers can be
claimed. Maybe this is because I think of infinity as it is meant to be:
Inexhaustability.

Oz.

On Jun 7, 2008, at 1:11 AM, Cornell III, Howard M wrote:

I guess you just don't believe that if we take one
irrational number out of the counting system for each rational number,
which will take a long time, we WILL have many irrational numbers left.
This is because you CANNOT come up with a rational number for EVERY
irrational number even if you can come up with a rational number for any
particular irrational number. A rational number generator has one RULE,
an irrational number generator does not. It makes a difference.

________________________________

From: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of Ozan Yarman
Sent: Friday, June 06, 2008 4:53 PM
To: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] for Carl's quest: Why there are
infinite many rationals? was Re: re;re for Tom

One never runs out of rational numbers when there is an
inexhaustable supply of them. They are infinite in number. Between any
two irrational numbers, an infinite amount of rational numbers exist. It
is possible to carry the numerators and denominators to infinite
precision.

Oz.

On Jun 6, 2008, at 4:38 PM, Cornell III, Howard M wrote:

No we cannot. If we don't get tired you WILL
run out of rational numbers and I NEVER will run out of irrational
numbers. In fact I only have to give you TWO rational numbers and you
can give me irrational numbers between them until you change your mind.

________________________________

From: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> [mailto:tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com> ] On Behalf Of Ozan Yarman
Sent: Thursday, June 05, 2008 5:39 PM
To: tuning@yahoogroups.com
<mailto:tuning@yahoogroups.com>
Subject: Re: [tuning] for Carl's quest: Why
there are infinite many rationals? was Re: re;re for Tom

Here is a test: for every irrational number you
come up with, I can come up with a rational number and we can carry this
process to infinity, ending up with an equal amount of irrational and
rational numbers.

Oz.

On Jun 6, 2008, at 1:06 AM, Cornell III, Howard
M wrote:

Carl,

Because rational numbers are the ratios
of integers, you can assign a counting number to each rational number.
Granted you can do this forever without
finishing; but for each rational number there is a counting number that
can be assigned to it. The thing is that you cannot even count the
irrational numbers between two rational numbers! No matter how many you
place there, there will always be more. Because there are demonstrably
MORE irrational numbers than rational, the set of all numbers will be
MOSTLY irrational by a LARGE margin.

No matter that the supply of numbers is
endless, it is the union of rational and irrational numbers. And for
each rational number you find, there are an uncountable number of irra!
tional numbe! rs.

It's not possible to take an infinite
number of numbers and try to count which are rational and which are
irrational; for the reason given above--you may count all day and never
even come across a rational number. But you KNOW there are some. And
that there just aren't "as many" (relatively) as the irrationals.

Howard

Hey, we're just waterboarding for now....

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf
Of Carl Lumma
Sent: Friday, June 06, 2008 5:58 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: How to keep discussions on-topic

Mike wrote:

> Especially when the discussion over rational and irrational
> numbers is necessary to understand the entire point of Rick
> Ballan's original tuning-related thread.

Why do you think Rick's original post had anything to do with
tuning?

> > > The topic we are discussing here is necessary
> > > to fully understand the other thread.
> >
> > It isn't. Rationals and irrationals approximate one
> > another to far beyond the limits of human perception.
> > That's all that's relevant for microtonal music.
>
> Who are you to decide what's relevant to microtonal music?
> The point of Rick's original thread is whether the 12 tone
> equal tempered scale is actually a JI scale involving
> "higher" overtones due to various quantum mechanics-related
> effects.
//
> So to understand that, he has to first explain various aspects
> of quantum mechanics to the people who don't already understand.

First you have to establish it's about tuning and not just
another crackpot physics theory that should be getting
moderated off a physics newsgroup somewhere.

> Quite a few microtonally related ideas have already come directly
> out of this, first and foremost the notion that 16:19:24 might
> actually be the minor chord that we're all used to rather than
> 10:12:15.

Can you cite the text where this amazing discovery most
recently (for the 45,000th time) "came out" on this list?
I'd love to know why you're under the apprehension it has:

* anything to do with the continuum hypothesis
* anything to do with quantum mechanics

Do tell.

> Of course, a discussion of this means we would get into gestalt
> psychology, which would likely for organizational purposes be
> best served with a side-thread of its own.

*Concepts from* gestalt psychology have been discussed *in the
context of* microtonal music and *applied to* psychoacoustics
research many times on this list. I've got no problem with it.

> This, I am sure, would be met with cries of it being
> "off-topic," and so no progress will ever be made on anything.

You're right: there's definitely a problem with progress being
made around here. The lack of a FAQ is partially to blame.

> I, on the other hand, find this discussion - what it is we
> actually perceive when we hear tempered chords - extremely
> interesting

Yeah, that'd be great. I'm not seeing much of it and it isn't
what I'm complaining about.

> These are the kinds of discussions where progress can get made.

If that were true you wouldn't still be having them.

> I don't think this group should be limited to just discussions
> about whether 53tet or 72tet is superior or whether miracle
> temperaments are more useful than diaschismatic ones.

Oh, I agree. And like Don Rumsfeld, I'm also against cutting
people's heads off.

-Carl

> Why do you think Rick's original post had anything to do with
> tuning?

Because it did. Go reread it. As I stated like 3 sentences down,

> > The point of Rick's original thread is whether the 12 tone
> > equal tempered scale is actually a JI scale involving
> > "higher" overtones due to various quantum mechanics-related
> > effects.

> First you have to establish it's about tuning and not just
> another crackpot physics theory that should be getting
> moderated off a physics newsgroup somewhere.

It's pretty hard to establish that it's a valid physics theory without
talking about physics. I suppose we should first coordinate a
mass-move between all of the participants in the discussion to another
yahoo group. Then we could come back here and try to tell everyone
what we figured out. But then we'd have the same discussion again with
the people who didn't make the move. And then I assume that discussion
would be met with cries of it being off topic.

Bureaucracy impedes progress.

> > Quite a few microtonally related ideas have already come directly
> > out of this, first and foremost the notion that 16:19:24 might
> > actually be the minor chord that we're all used to rather than
> > 10:12:15.
>
> Can you cite the text where this amazing discovery most
> recently (for the 45,000th time) "came out" on this list?
> I'd love to know why you're under the apprehension it has:

Aaron Wolf in Rick's original thread mentioned about how 16:19 has an
entirely different character for a minor third than 5:6. I compared
the two chords and he is right. We're definitely used to the 16:19
"sound" from 12tet. I was under the impression that 16:19 was too high
up in the harmonic series to be "useful," and that I'd hear 16:19 as
an out of tune 5:6. Clearly I was wrong.

The idea is that 16:19 isn't an RI approximation to the equal tempered
minor third as much as the equal tempered minor third is an
approximation to 16:19. Or is it? This is basically the discussion
we're trying to have, amidst the cries of it being off topic.
Obviously at the heart of this discussion is the relationship between
irrational and rational numbers.

> * anything to do with the continuum hypothesis

I never said anything about it, except maybe for a sentence in a
thread where I was explaining above said relationship between
irrational and rational numbers

> * anything to do with quantum mechanics

Because the realizations being sparked here came out of Rick's quantum
mechanical ideas. Cross-field realizations happen all the time. Most
people think rational intonation is best served as an approximation to
temperament which is best served as an approximation to lower-harmonic
JI. Rick thinks otherwise, and it's an idea worth looking into.

> > Of course, a discussion of this means we would get into gestalt
> > psychology, which would likely for organizational purposes be
> > best served with a side-thread of its own.
>
> *Concepts from* gestalt psychology have been discussed *in the
> context of* microtonal music and *applied to* psychoacoustics
> research many times on this list. I've got no problem with it.

Once again: Go back and reread Rick's original thread if you want to
see how the current discussion evolved out of a tuning-related
argument. Especially when you accuse others of "not reading the posts"
they're responding to. It's kind of irritating to have someone walk
midway into a discussion and then argue about how it's off-topic.

> > This, I am sure, would be met with cries of it being
> > "off-topic," and so no progress will ever be made on anything.
>
> You're right: there's definitely a problem with progress being
> made around here. The lack of a FAQ is partially to blame.

I'm also thinking that an actual online forum would be useful as well,
with different sub-forums for different topics. I don't know how
people are getting these messages, but I have every single one emailed
to my inbox, and every time there's a topic change, gmail puts it into
another "thread," which is responsible for some of the clutter. The
same applies to yahoo's web-based view of the group - we have to
subscribe to different groups such as tuning-math and meta-tuning to
get to different related topics.

There would be no argument about what threads are off topic or not -
we wouldn't have to join different groups for related topics and post
to them, when said groups might feasibly have a completely different
subscriber base, and thus we'd have to get all participants in the
discussion to sign up for yet another yahoo group which might be
initially moderated and so on.

Of course, getting everyone to use a new forum would be difficult -
although if we could somehow "link it" to the current yahoo group that
would make things easy. I don't know how feasible that is, though.

> > These are the kinds of discussions where progress can get made.
>
> If that were true you wouldn't still be having them.

What?

> > I don't think this group should be limited to just discussions
> > about whether 53tet or 72tet is superior or whether miracle
> > temperaments are more useful than diaschismatic ones.
>
> Oh, I agree. And like Don Rumsfeld, I'm also against cutting
> people's heads off.

How clever.

-Mike

> The point you stated in your second paragraph is the point I was trying to
> make: the act of measuring the dimension forces it into being rational. I
> did not mean to imply that there is an accurate enough measurement system to
> actually make irrational dimensions rational. Let's look at it one more
> time in a different way. Let's say the table is exactly thirty pi
> centimeters in one dimension. You cannot measure it exactly because it is
> not a countable multiple of any length unit you can use. But, of course, you
> could measure it approximately, which would give a rational number based on
> the units of length, like using a laser beam and measuring in angstroms;
> but even then you cannot be sure that it is thirty pi centimeters.

Haha, NOW I feel like we're all starting to get pretty far off-topic...

You can get a piece of wood "pi" feet long by doing this:

1. Construct a circle whose radius is 1/2 of a foot using a compass.
2. The diameter will be one foot wide, and so the circumference will
be pi feet around.
3. Get a piece of rope, lie it around the circumference of the circle,
cut the rope so that it fits.
4. Measure a piece of wood to be as long as the rope. Cut it. Done.

You will have a piece of wood that is pi feet long and subject to only
the same inaccuracies in measurement that you would get by trying to
measure something that is a rational number of feet long. An inch
might not be an inch either.

Maybe this should be moved to meta-tuning after all, as we are
starting to drift away from the microtonal aspects of it that I was
using to defend all of this stuff in the other thread.

-Mike

On Fri, Jun 6, 2008 at 6:35 PM, Mike Battaglia <battaglia01@...> wrote:
> Haha, NOW I feel like we're all starting to get pretty far off-topic...
>
> You can get a piece of wood "pi" feet long by doing this:
>
> 1. Construct a circle whose radius is 1/2 of a foot using a compass.
> 2. The diameter will be one foot wide, and so the circumference will
> be pi feet around.
> 3. Get a piece of rope, lie it around the circumference of the circle,
> cut the rope so that it fits.
> 4. Measure a piece of wood to be as long as the rope. Cut it. Done.
>
> You will have a piece of wood that is pi feet long and subject to only
> the same inaccuracies in measurement that you would get by trying to
> measure something that is a rational number of feet long. An inch
> might not be an inch either.
>
> Maybe this should be moved to meta-tuning after all, as we are
> starting to drift away from the microtonal aspects of it that I was
> using to defend all of this stuff in the other thread.
>
> -Mike

Mike,

A much better approach would be to use your compass to make the
length of your table sqrt(2) times the width. No messing around with
the rope, just use the compass to mark off a length equal to the width
and then use the compass again to mark off a length equal to the
diagonal of the square. By the Pythagorean theorem the diagonal of the
square is sqrt(2) times the side and the sqrt(2) is not a rational
number. Thus it is possible to have the two sides of your table not
being in rational proportions.

David Bowen

> On Fri, Jun 6, 2008 at 6:35 PM, Mike Battaglia <battaglia01@...>
> wrote:
> > Haha, NOW I feel like we're all starting to get pretty far off-topic...
> >
> > You can get a piece of wood "pi" feet long by doing this:
> >
> > 1. Construct a circle whose radius is 1/2 of a foot using a compass.
> > 2. The diameter will be one foot wide, and so the circumference will
> > be pi feet around.
> > 3. Get a piece of rope, lie it around the circumference of the circle,
> > cut the rope so that it fits.
> > 4. Measure a piece of wood to be as long as the rope. Cut it. Done.
> >
> > You will have a piece of wood that is pi feet long and subject to only
> > the same inaccuracies in measurement that you would get by trying to
> > measure something that is a rational number of feet long. An inch
> > might not be an inch either.
> >
> > Maybe this should be moved to meta-tuning after all, as we are
> > starting to drift away from the microtonal aspects of it that I was
> > using to defend all of this stuff in the other thread.
> >
> > -Mike
>
> Mike,
>
> A much better approach would be to use your compass to make the
> length of your table sqrt(2) times the width. No messing around with
> the rope, just use the compass to mark off a length equal to the width
> and then use the compass again to mark off a length equal to the
> diagonal of the square. By the Pythagorean theorem the diagonal of the
> square is sqrt(2) times the side and the sqrt(2) is not a rational
> number. Thus it is possible to have the two sides of your table not
> being in rational proportions.
>
> David Bowen

Haha! I didn't even think of that. Indeed.

-Mike

Mike wrote...

> > * anything to do with quantum mechanics
>
> Because the realizations being sparked here came out of Rick's
> quantum mechanical ideas.

Has Rick posted something about quantum mechanics?
(Answer: no.)

> Cross-field realizations happen all the time.

Yup... but not this time.

> Most people think rational intonation is best served as an
> approximation to temperament

"Most people" according to...? Rational intonation is an
approximation to temperament??

> which is best served as an approximation to lower-harmonic
> JI. Rick thinks otherwise, and it's an idea worth looking into.
>
> > > Of course, a discussion of this means we would get into
> > > gestalt psychology, which would likely for organizational
> > > purposes be best served with a side-thread of its own.
> >
> > *Concepts from* gestalt psychology have been discussed *in the
> > context of* microtonal music and *applied to* psychoacoustics
> > research many times on this list. I've got no problem with it.
>
> Once again: Go back and reread Rick's original thread if you
> want to see how the current discussion evolved out of a
> tuning-related argument. Especially when you accuse others of
> "not reading the posts" they're responding to. It's kind of
> irritating to have someone walk midway into a discussion and
> then argue about how it's off-topic.

I've read every post. I asked for cites in hopes you
might reconsider how you've interpreted them.

> > > This, I am sure, would be met with cries of it being
> > > "off-topic," and so no progress will ever be made on anything.
> >
> > You're right: there's definitely a problem with progress being
> > made around here. The lack of a FAQ is partially to blame.
>
> I'm also thinking that an actual online forum would be useful
> as well, with different sub-forums for different topics. I
> don't know how people are getting these messages, but I have
> every single one emailed to my inbox, and every time there's a
> topic change, gmail puts it into another "thread," which is
> responsible for some of the clutter. The same applies to yahoo's
> web-based view of the group - we have to subscribe to different
> groups such as tuning-math and meta-tuning to get to different
> related topics.

I get most of the groups by e-mail, but I use a mail client
that's somewhat less obnoxious than gmail. But I happen to
read this list on Yahoo's web site, which is how most people
do it (or did when I was a moderator). The website is annoying
but provides access (on a good day) to a decade worth of
archives. Which is a definite value-add for noobs such as
yourself.

Lately Yahoo also provides RSS feeds. I haven't tried it,
but you could subscribe to all the lists and read them in
something like Google Reader.

> Of course, getting everyone to use a new forum would be
> difficult

Aaron Johnson tried to move MMM to Google Groups and failed.
I tried to move tuning-math to freelists and failed. There
have been many threads about moving this list and in all of
them some people were against it.

> > > These are the kinds of discussions where progress can
> > > get made.
> >
> > If that were true you wouldn't still be having them.
>
> What?

If these conversations were so progressive they wouldn't
recur twice every year for a decade as if they were original.
A big problem that mailing lists have is a lack of
permanence, a lack of focal points. Wikis address that
problem, but often don't have enough in the chronology
dept. This community uses both technologies but it would
be great if there were some magic hybrid of the two. Many
wiki engines feature blog integration but it's usually
pretty hokey IMO. There is apparently no substitute for
manual meta-analysis, FAQ-writing, etc. And to date nobody
has taken the time.

> > > I don't think this group should be limited to just
> > > discussions about whether 53tet or 72tet is superior
> > > or whether miracle temperaments are more useful than
> > > diaschismatic ones.
> >
> > Oh, I agree. And like Don Rumsfeld, I'm also against cutting
> > people's heads off.
>
> How clever.

Jon Stewart deserves the credit there. At any rate, please
leave the straw men at home.

-Carl

--- In tuning@yahoogroups.com, "Cornell III, Howard M"
<howard.m.cornell.iii@...> wrote:
>
> Not at all. The true and actual dimension of the table could be
> measured in atomic diameters, say.

As you look closer at the table, you'll find the edges harder
and harder to make out. Even if you refrain from lighting it on
fire with your microscope, you'll find that it won't even sit
still while you're measuring. All measurements are thus an
abstraction of some sort. You can use whatever numbers you like
to represent them. You'll be more successful if your choice of
number system makes sense in terms of the model you're using,
and you'll eventually outbreed people who choose poorly. And
that's about it for the truth of weights and measures.

And that's the danger of off-topic posts. Even I reply to them.

-Carl

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--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> >
> > Since the sine wave clearly states that time is proportional to
> > period, t = nT where n is number of cycles (i.e. t= T+T+...+nfactors),
> > do you think that the length of the time line would be countably
> > infinite, aleph-naught N(0), since each cycle is a whole number? In
> > other words, n = N(0) for ANY value of T.
> >
>
> Lengths are not cardinal numbers. The question whether countable or
> incountable is meaningless in the case of lengths, isn't it?
> --
> Hans Straub
>
Well not exactly. If we apply Cantors proofs we reach an interesting
situation. 1.If we set n = N(0), then t = nT = N(0)T = N(0) because
the product of countable infinity by a number is itself.
2. If we set T = N(0) we have t = nN(0)=N(0) again.
3. If both n and T = N(0) we get t =N(0)N(0)=N(0) yet again because
N(0)squared also equals itself.
Now as to the thesis of continuity of a line which involves
uncountable infinity N(1), we can actually map all real numbers back
inside a single period. Given any real number, it can be written in
the form N + r where N is whole and r is the remainder i.e. the
numbers after the decimal. It's very easy to prove that for any
periodic function Y (say a sine wave),Y(N + r) = Y(r). In other words,
we can create a modulus out of the period.
So we reach this interesting and anti-intuitive circumstance; there
are more numbers inside a single period than there are number of periods.

If the recent exchanges about the edges of tables etc. are off -topic, then let's be off-topic.

I expect that the ir/rational postings and exchanges are prolly now completed, so are we ready to move on to something a little more adventurous/exciting?

I, for one, can see the connection, even if Carl is still floundering about in his counting box, enumerating the cogs on his ancient Greek wheels.

BTW I realise that the maths people will tell me that sine waves by definition are two dimensional, yet my guitar strings certainly move in three (or more) dimensions; so could sine waves and Fourier transforms be a rather limited way of

"looking at the maths, mechanics and physics of things "musical and harmonious"?

Maybe this "sinewave-simplification" in the traditional model, i.e. that musical harmonics (are only to found at small number integer frequency ratios) is another reason why the traditional model/paradigm doesn't work very well.

(Don't worry, Rick. Carl is used to me winding him up, and regularly threatens to censor me, but Carl has resisted the temptation for the past 15 odd years, so I hope his inner curiosity will again prevail.)

On 7 Jun 2008, at 05:03, rick_ballan wrote:

> --- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
>
> >
> > --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > >
> > > Since the sine wave clearly states that time is proportional to
> > > period, t = nT where n is number of cycles (i.e. t= T+T+...> +nfactors),
> > > do you think that the length of the time line would be countably
> > > infinite, aleph-naught N(0), since each cycle is a whole number? > In
> > > other words, n = N(0) for ANY value of T.
> > >
> >
> > Lengths are not cardinal numbers. The question whether countable or
> > incountable is meaningless in the case of lengths, isn't it?
> > --
> > Hans Straub
> >
> Well not exactly. If we apply Cantors proofs we reach an interesting
> situation. 1.If we set n = N(0), then t = nT = N(0)T = N(0) because
> the product of countable infinity by a number is itself.
> 2. If we set T = N(0) we have t = nN(0)=N(0) again.
> 3. If both n and T = N(0) we get t =N(0)N(0)=N(0) yet again because
> N(0)squared also equals itself.
> Now as to the thesis of continuity of a line which involves
> uncountable infinity N(1), we can actually map all real numbers back
> inside a single period. Given any real number, it can be written in
> the form N + r where N is whole and r is the remainder i.e. the
> numbers after the decimal. It's very easy to prove that for any
> periodic function Y (say a sine wave),Y(N + r) = Y(r). In other words,
> we can create a modulus out of the period.
> So we reach this interesting and anti-intuitive circumstance; there
> are more numbers inside a single period than there are number of > periods.
>
>
>>

Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

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Mike wrote...
> > Has Rick posted something about quantum mechanics?
> > (Answer: no.)
>
> From the "Hi!" thread:
>
> > Sorry, forgot your actual question. In fact I never said that
> > ALL waves are periodic but only that they must have the
> > potential to be. What I have proved is that superposition and
> > harmony is a universal law of nature, perhaps the most
> > fundamental. Thus, a perfect fifth will transform to a perfect
> > fifth, a flat-fifth (irrational) to a flat-fifth, but not a
> > perfect fifth to a flat-fifth. Logical and obvious I know, but
> > that is not what is currently being maintained in relativity
> > and quantum mechanics.

There's nothing about quantum mechanics here. It's just
a vacuous assertion that "perfect fifths" can't "transform"
into "irrational flat fifths" because of "relativity and
quantum mechanics". Yeah baby.

> Just because this isn't a "new realization" to you doesn't
> mean that we shouldn't be allowed to discuss it.

Generally on academic mailing lists the goal is to improve
the S/N ratio by banning crackpot theories and discouraging
repetitive conversation. We haven't done that here yet but
I definitely didn't invent the art and I'm not a Bad Man
for suggesting its use.

> RI, especially in the Hammond Organ tuning for
> example, is usually pitched as a 'cost-effective' way
> to approximate equal temperament.

That's just one example. The real wheelers and dealers
of RI are rational number mystics like La Monte Young and
his disciples.

> > I've read every post. I asked for cites in hopes you
> > might reconsider how you've interpreted them.
>
> one by Charles Lucy:

Charles Lucy is infamous for being of a proponent of a
theory nobody other than himself can understand. He
claims variously that:
* He's discovered the long-lost tuning of John Harrison,
which he's named after... himself.
* He sometimes claims it's significantly different from
other meantones even though nobody else agrees and he
almost certainly couldn't tell the difference himself in
a blind listening test. He sometimes backpedals on this
only to re-assert it later.
* That these special qualities are do to one or both of
two facts:
* The beat rates of irrational intervals based on pi
entrain the brain in alpha states.
* The irrational intervals based on pi capture something
about the "spherical wavefronts" of sound.

We tolerate him though because he's a resident eccentric.
And he outranks me in microtonal seniority by a couple of
decades.

> From Charles Lucy:
>
> > I'm really glad to see that Rick is looking at this from a
> > scientific POV.
//
> I suppose you should respond to Charles and tell him that
> scientific POV's are "off-topic in this forum".

LOL

> Furthermore, why do you care? This is the current topic of
> discussion on a tuning forum. You started pouting about it
> being off topic as a reaction to -of all reasons- people
> thinking that you thought the set of all irrationals was
> countable. Who cares?

Plenty of fine musicians and theorists have been driven off
this list over the years due to the high message volume.
And that was back when the messages did frequently contain
exciting new discoveries. A tremendous amount of knowledge
has been aggregated here, and it's worth caring about.
There's a tremendous amount of potential here too, but as
it stands I don't think this forum is ready for the rapid
growth the topic of microtonal music is about to experience.
Maybe it loses its central position, and maybe that's overdue
anyway, I don't know.

-Carl

Actually the theory has a problem since we started with perfect 3/2 and our society decided to go with irrational ones. All because of a madman called Plato!!!
they didn't give him hemlock soon enough (being the undemocratic he was.)

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
> Mike wrote...
> > > Has Rick posted something about quantum mechanics?
> > > (Answer: no.)
> >
> > From the "Hi!" thread:
> >
> > > Sorry, forgot your actual question. In fact I never said that
> > > ALL waves are periodic but only that they must have the
> > > potential to be. What I have proved is that superposition and
> > > harmony is a universal law of nature, perhaps the most
> > > fundamental. Thus, a perfect fifth will transform to a perfect
> > > fifth, a flat-fifth (irrational) to a flat-fifth, but not a
> > > perfect fifth to a flat-fifth. Logical and obvious I know, but
> > > that is not what is currently being maintained in relativity
> > > and quantum mechanics.
>
> There's nothing about quantum mechanics here. It's just
> a vacuous assertion that "perfect fifths" can't "transform"
> into "irrational flat fifths" because of "relativity and
> quantum mechanics". Yeah baby.
>
> > Just because this isn't a "new realization" to you doesn't
> > mean that we shouldn't be allowed to discuss it.
>
> Generally on academic mailing lists the goal is to improve
> the S/N ratio by banning crackpot theories and discouraging
> repetitive conversation. We haven't done that here yet but
> I definitely didn't invent the art and I'm not a Bad Man
> for suggesting its use.
>
> > RI, especially in the Hammond Organ tuning for
> > example, is usually pitched as a 'cost-effective' way
> > to approximate equal temperament.
>
> That's just one example. The real wheelers and dealers
> of RI are rational number mystics like La Monte Young and
> his disciples.
>
> > > I've read every post. I asked for cites in hopes you
> > > might reconsider how you've interpreted them.
> >
> > one by Charles Lucy:
>
> Charles Lucy is infamous for being of a proponent of a
> theory nobody other than himself can understand. He
> claims variously that:
> * He's discovered the long-lost tuning of John Harrison,
> which he's named after... himself.
> * He sometimes claims it's significantly different from
> other meantones even though nobody else agrees and he
> almost certainly couldn't tell the difference himself in
> a blind listening test. He sometimes backpedals on this
> only to re-assert it later.
> * That these special qualities are do to one or both of
> two facts:
> * The beat rates of irrational intervals based on pi
> entrain the brain in alpha states.
> * The irrational intervals based on pi capture something
> about the "spherical wavefronts" of sound.
>
> We tolerate him though because he's a resident eccentric.
> And he outranks me in microtonal seniority by a couple of
> decades.
>
> > From Charles Lucy:
> >
> > > I'm really glad to see that Rick is looking at this from a
> > > scientific POV.
> //
> > I suppose you should respond to Charles and tell him that
> > scientific POV's are "off-topic in this forum".
>
> LOL
>
> > Furthermore, why do you care? This is the current topic of
> > discussion on a tuning forum. You started pouting about it
> > being off topic as a reaction to -of all reasons- people
> > thinking that you thought the set of all irrationals was
> > countable. Who cares?
>
> Plenty of fine musicians and theorists have been driven off
> this list over the years due to the high message volume.
> And that was back when the messages did frequently contain
> exciting new discoveries. A tremendous amount of knowledge
> has been aggregated here, and it's worth caring about.
> There's a tremendous amount of potential here too, but as
> it stands I don't think this forum is ready for the rapid
> growth the topic of microtonal music is about to experience.
> Maybe it loses its central position, and maybe that's overdue
> anyway, I don't know.
>
> -Carl
>
>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote/asked:
>
> Now tell me this: I am sitting at a rectangular table. How can I tell
> whether the ratio of its width to its length is rational or >irrational?
>
hi Tom,

all 3 sides of a rectangular are rational
if and only if the sides are in ratios of:
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html
http://mathworld.wolfram.com/PythagoreanTriple.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html#uadgen
traces them back to the initial proportion of:
3:4:5
http://de.wikipedia.org/wiki/Quartsextakkord
http://fr.wikipedia.org/wiki/Accord_de_quarte_et_sixte
enlish entry still lacking

all other cases, except the above one, the
http://en.wikipedia.org/wiki/Hypotenuse
s
do have irrational lenngths
http://mathworld.wolfram.com/RightTriangle.html
due to the square-root symbol.

http://www.emis.de/journals/JIS/VOL4/DUBNER/pyth.pdf

Yours Sincerely
A.S.

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Well, since Planck time is about 5.39121 * 10^-44 s, then Planck frequency is 5.39121 * 10^44 Hz, which is 140 octaves higher than 386.8 Hz, or 976.9 cents higher than A3 = 220 Hz. Unless I got my math wrong anyway.

So shouldn't we be tuning everything to G4 = 386.8 Hz?

~D.

Chris Vaisvil wrote:
> Is everyone talking about notes or intervals?
> > This seems to be a silly conversation to me.
> > A note can be -any- frequency.
> > The counting here is arbitrary
> > A second is not the only measure of time.
> > What if you counted time in.... 1/3 of a second?
> > Or years?
> > Or hours?
> > All of those are equally valid.... the really are.
>
> A particular interval comes out irrational only from one point of view.
> > change the way you relate to frequency - woosh! the irrational vanishes.
> > frequencies also correspond to energy levels so I do imagine Planck's > constant applies
> > But for heavens sake - h =\,\,\, 6.626\ 070\ 95(44) \times 10^{-34}\ > \mbox{J}\cdot\mbox{s} \,\,\,
> > is so small as to be zero and frequancy totally analog for the > purposes of music.
>

--- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:
>
> Well, since Planck time is about 5.39121 * 10^-44 s, then Planck
> frequency is 5.39121 * 10^44 Hz, which is 140 octaves higher than
> 386.8 Hz, or 976.9 cents higher than A3 = 220 Hz. Unless I got my
> math wrong anyway.
>
> So shouldn't we be tuning everything to G4 = 386.8 Hz?
>
> ~D.

Here's something vaguely to do with quantum mechanics. Just to
reaffirm my status as a jaded old codger, I'll let everyone know
that the Planck frequency thing has come up on this list, MMM,
and tuning-math, probably at least half a dozen times in total
so far.

Danny, I think you need to multiply the starting number by 2 to
get the period before inverting. Since the Planck time
represents the least time it takes to traverse the Planck
distance, you need to come back to complete an oscillation.
You then wind up at 340Hz (after taking out an obscene number
of octaves), which is close to F 349 in the 12-ET A=440 system.

-Carl

> Just to reaffirm my status as a jaded old codger

Hahaha

> MMM,

I've seen this referenced a few times in your post, and I found the
MakeMicroMusic group from it. Now I feel like it's christmas, so I
have to thank you for that.

-Mike

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> > BTW I realise that the maths people will tell me that sine waves by
> > definition are two dimensional, yet my guitar strings certainly
move in
> > three (or more) dimensions; so could sine waves and Fourier
transforms be
> a
> > rather limited way of
> > "looking at the maths, mechanics and physics of things "musical and
> > harmonious"?
>
> Interesting... But what about computer speakers, where the waves
produced
> from each are almost completely two-dimensional? (that is, there is
little
> to no lateral motion involved)
>
> Perhaps if we could finally figure out THAT model... :D then we could
> "explode" it outward into 3 dimensions. Also I think that phase is a
huge
> factor in everything, and it seems to be often overlooked.
>
> > Maybe this "sinewave-simplification" in the traditional model,
i.e. that
> > musical harmonics (are only to found at small number integer
frequency
> > ratios) is another reason why the traditional model/paradigm
doesn't work
> > very well.
>
> Interesting.
>
> I am well aware of your theories of how there are special irrational
points
> of resonance involving multiples of pi... it all ties into a
question that I
> would have liked to discuss over the past few days:
>
> What is it that we hear when we hear an equal-tempered major chord?
Often
> this chord is cited as having a relatively in-tune 3/2 and a slightly
> out-of-tune 5/4... But is there some other way that we are really
hearing
> it?
>
> Rick has been talking recently about how maybe we're hearing 3 steps of
> 12tet as being a slightly out-of-tune 19/16 ratio rather than a
really out
> of tune 6/5 ratio. Which begs the question as to what are we
hearing? Do we
> psychologically "move" these ratios to the nearest JI one (except
that there
> are an infinite amount of them)? If I play a JI minor chord with
the minor
> third as 19/16 am I hearing THAT as an out of tune 6/5?
>
> Or, if imposing the JI structure on these doesn't work, is there any
sort of
> structure or way of understanding equal temperaments?
>
> I think the context of it might matter. There is definitely some
difference
> between 11-tet and 12-tet. I think that if you play this chord:
>
> 1:2:3:4:5:6
>
> but you sharpen the "5" slightly (perhaps even to a different rational
> multiple/overtone), you will still most likely hear the resemblence
to the
> overtone series and hear it that way. The sharpened "5" will cause some
> beating that you may or may not like, depending on the context. On
the other
> hand, 3 steps of 12tet does sound an awful lot like 16:19, which has an
> entirely different *character* than 5:6. What is the nature of this
change
> in character, and what does this character mean? Why do we sometimes
hear
> intervals "guided" to their nearest match in a harmonic series, and
why do
> we sometimes hear them as new entities in their own right with different
> feelings and characters to them?
>
> I suppose the question is one of psychology: what causes someone to
hear a
> certain interval as a mistuned form of another interval?
>
> Abstractly put, what causes someone, when experiencing a phenomenon,
to view
> that phenomenon as an altered version of another phenomenon?
>
> An offshoot of this question is musically, how can we "guide" a
listener to
> hear different intervals as "new" phenomena, rather than "out-of-tune"
> versions of ones they already know? Many people upon hearing
JI-tuned music
> for the first time often feel that it is simply an "out-of-tune"
version of
> 12tet. Why do they hear it that way, rather than simply as having a
> different *character* (as 16:19 and 5:6 had different characters)?
>
> I suspect it has something to do with psychological things having
nothing to
> do with music causing one to cling to things one is already familiar
with -
> and yet the question is inextricably tied in with music theory,
especially
> microtonal theory.
>
> Just some thoughts.
>
> -Mke
>
> > (Don't worry, Rick. Carl is used to me winding him up, and regularly
> > threatens to censor me, but Carl has resisted the temptation for
the past
> 15
> > odd years, so I hope his inner curiosity will again prevail.)
>
Not at all Mike, I'm having a great time. You guys have confirmed for
me what I always suspected about the minor 3, which gives me enough
experimental scope to argue reasonable doubt about the other
intervals. I get so frustrated at the anti-intellectual/artistic
attitude in Australia where academics are mostly resting on other
peoples laurels, and heated debate about an idea is usually a sign
that there is something to it.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
>
> > But you have to admire Cantor's ingenuity. It's a beautiful
> > argument.
>
> It's also a core element in at least one of Godel's famous
> incompleteness theorems, and in related work by Turing.

Carl,

That's a very good point. Ozan or I can choose not to accept one or
other of the axioms that lead to Cantor's transfinites, so that they
all collapse to a single infinity, but I suspect there will then be
some other paradox that is more or less objectionable.

I think that a paradox is like a bulge in a wall-to-wall carpet. If
you get rid of it in one place, it just pops up somewhere else. Better
to know where your paradox is. Maybe Cantor's transfinites are the
equivalent of putting the bulge under the couch where it is least
objectionable.

Herman,

Hofstadter's 'Gödel, Escher, Bach: An Eternal Golden Braid' is one of
my favourites. It got me onto Hofstadter and Dennett's 'The Mind's I',
which I like even more, and then everything I could find by Daniel
Dennett. IMHO, Daniel Dennett and Ken Wilber are the two greatest
philosophers of consciousness, although in some ways, Raymond Smullyan
is their intersection.

Paradox and consciousness go hand-in-hand. Consciousness just _is_
paradoxical. The best I can see is to take a Zen-like attitude to
paradoxes and not keep trying to eliminate them. It's futile.

-- Dave Keenan

> Carl,
>
> That's a very good point. Ozan or I can choose not to accept one or
> other of the axioms that lead to Cantor's transfinites, so that they
> all collapse to a single infinity, but I suspect there will then be
> some other paradox that is more or less objectionable.

I think the concept of transfinite numbers is different from the
concept of infinity. Infinity I think we usually just assume means
without bound, goes on forever, etc. The real numbers, the rationals,
the natural numbers are all definitely "infinite." But you have this
quality that the real numbers have, that they are continuous, and that
they can't be put "on par" with the natural numbers, and these
transfinite numbers are just a way of categorizing all of that. Maybe
it's erroneous and misleading to think that it's tied in with the
concept of "infinity."

> Hofstadter's 'Gödel, Escher, Bach: An Eternal Golden Braid' is one of
> my favourites. It got me onto Hofstadter and Dennett's 'The Mind's I',
> which I like even more, and then everything I could find by Daniel
> Dennett. IMHO, Daniel Dennett and Ken Wilber are the two greatest
> philosophers of consciousness, although in some ways, Raymond Smullyan
> is their intersection.

I've got to read this book, everyone in my entire life keeps telling
me about it. Honestly I fully expect my cat to tell me to read it
tomorrow.

> Paradox and consciousness go hand-in-hand. Consciousness just _is_
> paradoxical. The best I can see is to take a Zen-like attitude to
> paradoxes and not keep trying to eliminate them. It's futile.

Or is it? *gong*

-Mike

> Not at all Mike, I'm having a great time. You guys have confirmed for
> me what I always suspected about the minor 3, which gives me enough
> experimental scope to argue reasonable doubt about the other
> intervals. I get so frustrated at the anti-intellectual/artistic
> attitude in Australia where academics are mostly resting on other
> peoples laurels, and heated debate about an idea is usually a sign
> that there is something to it.

I usually find that if nobody else knows what you're talking about
that it's usually a good sign that you're onto something. :)

Doubly so if people yell at you for being wrong and "out of line," but
you know that they aren't really addressing the core issue at the
heart of realization that you've had.

I have always thought it interesting that 19/16 is often labeled as
some kind of "quasi-equal" minor third, as if it's merely an oddity -
a semi-useful JI-way to reach at the equal tempered minor third if for
some reason you want to do that. Never really gave it much thought
that maybe it's just its own separate, independent JI interval.

Just out of curiosity, do you have a list of each of the 12-tet
intervals and the other rational relationships approximating them
beyond the usual ones?

-Mike

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
> I think the concept of transfinite numbers is different from the
> concept of infinity. Infinity I think we usually just assume means
> without bound, goes on forever, etc. The real numbers, the rationals,
> the natural numbers are all definitely "infinite." But you have this
> quality that the real numbers have, that they are continuous, and that
> they can't be put "on par" with the natural numbers, and these
> transfinite numbers are just a way of categorizing all of that. Maybe
> it's erroneous and misleading to think that it's tied in with the
> concept of "infinity."

Maybe. Another possibility is that the whole idea of a continuum (of
infinite precision) is fundamentally paradoxical. To quote Nat
Hellerstein, "Is zero positive or negative?", "Is dawn night or day?"

I visited the catacombs today and unearthed Nat Hellerstein's "Contra
Cantor" paper from behind some decaying skulls. I must correct
something I said in an earlier message. It was not the Burali-Forti
paradox that he said should be called a theorem, it was Skolem's paradox.
http://en.wikipedia.org/wiki/Skolem%27s_paradox

> I've got to read this book [GEB: an EGB], everyone in my entire life
keeps telling
> me about it. Honestly I fully expect my cat to tell me to read it
> tomorrow.

What's this:

[getting louder] Meeeeeeee [pitch suddenly drops due to doppler
effect] owwwwww [fading away]. Sound it out loud. "Meeeeeee\owwwwww".

The answer, backwards: .setaks rellor delleporp-tekcor no tac a s'tI

> > Paradox and consciousness go hand-in-hand. Consciousness just _is_
> > paradoxical. The best I can see is to take a Zen-like attitude to
> > paradoxes and not keep trying to eliminate them. It's futile.
>
> Or is it? *gong*

It is futile for Mike Battaglia to try to prove this sentence true. :-)

-- Dave Keenan

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:
> >> Well, since Planck time is about 5.39121 * 10^-44 s, then Planck >> frequency is 5.39121 * 10^44 Hz, which is 140 octaves higher than
>> 386.8 Hz, or 976.9 cents higher than A3 = 220 Hz. Unless I got my
>> math wrong anyway.
>>
>> So shouldn't we be tuning everything to G4 = 386.8 Hz?
>>
>> ~D.
>> >
> Here's something vaguely to do with quantum mechanics. Just to
> reaffirm my status as a jaded old codger, I'll let everyone know
> that the Planck frequency thing has come up on this list, MMM,
> and tuning-math, probably at least half a dozen times in total
> so far.
> Come to think of it, I think I brought it up long ago. I can never remember.
> Danny, I think you need to multiply the starting number by 2 to
> get the period before inverting. Since the Planck time
> represents the least time it takes to traverse the Planck
> distance, you need to come back to complete an oscillation.
> You then wind up at 340Hz (after taking out an obscene number
> of octaves), which is close to F 349 in the 12-ET A=440 system.
> Okay, so "Planck frequency" is 2 * 1/tP, makes sense. Make that 139 octaves down. And I did get the math wrong: the reciprocal of 5.39121 * 10^-44 is 1.85487 * 10^43, and half of that is 9.27436 * 10^42 Hz. 134 octaves lower than that is 425.858 Hz, which is 56.559 cents flat of A-440. But that assumes that Planck time is really the shortest length of time that matters; there is doubt among physicists nowadays.

Anyway, this is getting OT, so I'll stop now. ~D.

[ Attachment content not displayed ]

I really enjoy keeping an eye and ear to this list, but traffic sometimes gets out of hand, both from volume and tenor. May I make two suggestions to improve this? They are to write with restraint and to use back channels whenever a posting to the whole list is neither necessary nor polite.

When I first began reading the original Mills list, it was well over a year before I posted an item. I waited in order to become familiar with the discussion to date. Once I began participating, I quickly found that some of the best and most valuable communication took place in the form of the back channel, that is direct emails to individuals rather than posts to the entire list. I still probably posted more to the list than was good for me (and certainly for the rest of you), but the posts to the main list were improved by the back channel discussion. Moreover, although the list then had no official moderation, the back channels carried out this function in a more polite and gentle way. I have the impression that Jon Szanto admirably continued this practice when he began the MMM list, resorting to full list scoldings (and encouragement) only when absolutely necessary and then clearly identified himself as the "list Mom".

Daniel Wolf

I really enjoy keeping an eye and ear to this list, but traffic sometimes
gets out of hand, both from volume and tenor. May I make two suggestions
to improve this? They are to write with restraint and to use back channels
whenever a posting to the whole list is neither necessary nor polite.

When I first began reading the original Mills list, it was well over a
year before I posted an item. I waited in order to become familiar with
the discussion to date. Once I began participating, I quickly found that
some of the best and most valuable communication took place in the form of
the back channel, that is direct emails to individuals rather than posts
to the entire list. I still probably posted more to the list than was good
for me (and certainly for the rest of you), but the posts to the main list
were improved by the back channel discussion. Moreover, although the list
then had no official moderation, the back channels carried out this
function in a more polite and gentle way. I have the impression that Jon
Szanto admirably continued this practice when he began the MMM list,
resorting to full list scoldings (and encouragement) only when absolutely
necessary and then clearly identified himself as the "list Mom".

Daniel Wolf

--
Using Opera's revolutionary e-mail client: http://www.opera.com/mail/

> Maybe. Another possibility is that the whole idea of a continuum (of
> infinite precision) is fundamentally paradoxical. To quote Nat
> Hellerstein, "Is zero positive or negative?", "Is dawn night or day?"

"What equal temperament is the harmonic series in?"

> I visited the catacombs today and unearthed Nat Hellerstein's "Contra
> Cantor" paper from behind some decaying skulls. I must correct
> something I said in an earlier message. It was not the Burali-Forti
> paradox that he said should be called a theorem, it was Skolem's paradox.
> http://en.wikipedia.org/wiki/Skolem%27s_paradox

Skolem's paradox seems to stem right from the axiom of choice. I never
liked the axiom of choice anyway.

Nonetheless the theorem that Skolem's paradox stems from, the
"Löwenheim Skolem-Theorem" is way over my head from the way it's
explained on Wikipedia. I need to understand that one better before I
can even evaluate how the paradox is possible.

>> I've got to read this book [GEB: an EGB], everyone in my entire life
> keeps telling
>> me about it. Honestly I fully expect my cat to tell me to read it
>> tomorrow.
>
> What's this:
>
> [getting louder] Meeeeeeee [pitch suddenly drops due to doppler
> effect] owwwwww [fading away]. Sound it out loud. "Meeeeeee\owwwwww".
>
> The answer, backwards: .setaks rellor delleporp-tekcor no tac a s'tI

That took me way too long to translate backwards, rofl

>> > Paradox and consciousness go hand-in-hand. Consciousness just _is_
>> > paradoxical. The best I can see is to take a Zen-like attitude to
>> > paradoxes and not keep trying to eliminate them. It's futile.
>>
>> Or is it? *gong*
>
> It is futile for Mike Battaglia to try to prove this sentence true. :-)

If this sentence is true, then Dave Keenan is wrong about that last
sentence of his.

-Mike

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> > Not at all Mike, I'm having a great time. You guys have confirmed for
> > me what I always suspected about the minor 3, which gives me enough
> > experimental scope to argue reasonable doubt about the other
> > intervals. I get so frustrated at the anti-intellectual/artistic
> > attitude in Australia where academics are mostly resting on other
> > peoples laurels, and heated debate about an idea is usually a sign
> > that there is something to it.
>
> I usually find that if nobody else knows what you're talking about
> that it's usually a good sign that you're onto something. :)
>
> Doubly so if people yell at you for being wrong and "out of line," but
> you know that they aren't really addressing the core issue at the
> heart of realization that you've had.
>
> I have always thought it interesting that 19/16 is often labeled as
> some kind of "quasi-equal" minor third, as if it's merely an oddity -
> a semi-useful JI-way to reach at the equal tempered minor third if for
> some reason you want to do that. Never really gave it much thought
> that maybe it's just its own separate, independent JI interval.
>
> Just out of curiosity, do you have a list of each of the 12-tet
> intervals and the other rational relationships approximating them
> beyond the usual ones?
>
> -Mike
>
It made me laugh the other day when you said that you were walking
around uni mumbling to yourself trying to figure out Godels theorem. I
did a similar thing when I was trying to figure out inverses. Freq
inverse to period, and two lots of inverse transformations made me
completely dislexic. I stripped the gears of my sisters Corolla going
from fourth to first. She doesn't know to this day.

Yeah you must be the only one who hasn't read Hofstadler's GEB (EBG,
BGE). It got me interested in Turing, Godel etc...I remember being
obsessed with that book and Nietzsche's Beyond good and evil at the
same time (phew, what a mix) which shocked and fascinated me. I think
that every one who eventually thinks for themselves have to go through
that mumbly period of confusion and doubt. I'm no longer surprised
that many academics are just what Nietzsche called schoolmasters who
regurgitate "facts" without understanding them in the slightest. GEB
is great for turning that confusion into fun paradoxes and the like.

Well I arrived at 19/16 because I knew that 5 couldn't be the tonic
since it's not 2 to the power of N (=123...). Didn't make sense that
the whole minor key didn't agree with tonic = octave equiv. of
fundamental. So I spent days looking for ratio approximations and
found it eventually, probably by accident. Fortunately, I found a much
simpler way. You'll need a calculator.
For any interval 2 power p/12, p = 0,1,2...11, take 2 power of (N +
p/12), N is whole. This will give you a number with a whole part M and
remainder r after the decimal point. Now there are two possibilities
1. we round off M to nearest odd, or 2. M is already odd and we ignore
r. But the best numbers are those where M is already odd AND rounds
off to M. The ratio we are looking for is M divided by 2 to the power
of N i.e. the N we started with.
E.g. p = 3, N = 4 gives minor 3rd 19/16. 2 power 4.25 gives 19.0273...
which rounds off to 19 numerator. 2 power of 4 gives 16 denominator.

I'm assuming that other tets are also base 2? Is that correct? For eg,
is 17-tet 2 power of p/17, p = 0,1,2,...16? If so then the above
method still holds.

P.S. No ones answered my stuff about the length of time being
aleph-naught since cycles are discreet and whole, but number of points
within a single period being aleph-one since we need continuity.
Didn't it get through or is it just a daft thought?

Rick

> It made me laugh the other day when you said that you were walking
> around uni mumbling to yourself trying to figure out Godels theorem. I
> did a similar thing when I was trying to figure out inverses. Freq
> inverse to period, and two lots of inverse transformations made me
> completely dislexic. I stripped the gears of my sisters Corolla going
> from fourth to first. She doesn't know to this day.

Hahaha, damn.

> Yeah you must be the only one who hasn't read Hofstadler's GEB (EBG,
> BGE). It got me interested in Turing, Godel etc...I remember being
> obsessed with that book and Nietzsche's Beyond good and evil at the
> same time (phew, what a mix) which shocked and fascinated me. I think
> that every one who eventually thinks for themselves have to go through
> that mumbly period of confusion and doubt. I'm no longer surprised
> that many academics are just what Nietzsche called schoolmasters who
> regurgitate "facts" without understanding them in the slightest. GEB
> is great for turning that confusion into fun paradoxes and the like.

Yeah, I have to read it. I'm about to buy it on Amazon right now.

> Well I arrived at 19/16 because I knew that 5 couldn't be the tonic
> since it's not 2 to the power of N (=123...). Didn't make sense that
> the whole minor key didn't agree with tonic = octave equiv. of
> fundamental. So I spent days looking for ratio approximations and
> found it eventually, probably by accident. Fortunately, I found a much
> simpler way. You'll need a calculator.
> For any interval 2 power p/12, p = 0,1,2...11, take 2 power of (N +
> p/12), N is whole. This will give you a number with a whole part M and
> remainder r after the decimal point. Now there are two possibilities
> 1. we round off M to nearest odd, or 2. M is already odd and we ignore
> r. But the best numbers are those where M is already odd AND rounds
> off to M. The ratio we are looking for is M divided by 2 to the power
> of N i.e. the N we started with.
> E.g. p = 3, N = 4 gives minor 3rd 19/16. 2 power 4.25 gives 19.0273...
> which rounds off to 19 numerator. 2 power of 4 gives 16 denominator.

Again, I'll have to check that out.

> I'm assuming that other tets are also base 2? Is that correct? For eg,
> is 17-tet 2 power of p/17, p = 0,1,2,...16? If so then the above
> method still holds.
>
> P.S. No ones answered my stuff about the length of time being
> aleph-naught since cycles are discreet and whole, but number of points
> within a single period being aleph-one since we need continuity.
> Didn't it get through or is it just a daft thought?

Well, that isn't how those numbers work. Aleph-null and Aleph-one are
cardinal numbers used to explain the size of infinite sets. The length
of time increases without bound, and the amount of points between a
single period are aleph one. Really, I'm not sure exactly what you're
saying here. Can you clarify?

-Mike

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
> "What equal temperament is the harmonic series in?"

8539-EDO, for all practical purposes. Of course the keyboard is hell
to get thru doorways and the fretwire is a complete bastard.

> Skolem's paradox seems to stem right from the axiom of choice. I
> never liked the axiom of choice anyway.
>
> Nonetheless the theorem that Skolem's paradox stems from, the
> "Löwenheim Skolem-Theorem" is way over my head from the way it's
> explained on Wikipedia. I need to understand that one better before
> I can even evaluate how the paradox is possible.

Hey, I can't even evaluate how _I_ am possible. ;-)

Another correction re Hellerstein's 'Contra Cantor': I said he said
the binary diagonal number was a 2-adic. In fact he said it was a
dyadic (a much simpler thing) but in this case with a finite number of
bits which are zero or one (false or true) followed by an infinite
number of bits having the truth value "paradoxical", represented as
the imaginary truth value "i" which is its own negation. This he
interprets as uncertainty, i.e. finite precision.

> > It is futile for Mike Battaglia to try to prove this sentence
true. :-)
>
> If this sentence is true, then Dave Keenan is wrong about that last
> sentence of his.

As you probably realised, mine was intended to be a kind of Godel
sentence for Mike Battaglia. Something that anyone _other_ than Mike
Battaglia could readily prove, similar to the one first directed by C
H Whitely to J R Lucas in 1962.

Yours is a "Santa Claus sentence" or Curry Paradox, by which it seems
anyone can prove anything. I think Hellerstein thinks this is what
Cantor is doing, attempting to deduce something (the existence of a
hierarchy of transfinites) from a paradox.

-- Dave Keenan

--- In tuning@yahoogroups.com, "Mike Battaglia" <battaglia01@...> wrote:
>
> > It made me laugh the other day when you said that you were walking
> > around uni mumbling to yourself trying to figure out Godels theorem. I
> > did a similar thing when I was trying to figure out inverses. Freq
> > inverse to period, and two lots of inverse transformations made me
> > completely dislexic. I stripped the gears of my sisters Corolla going
> > from fourth to first. She doesn't know to this day.
>
> Hahaha, damn.
>
> > Yeah you must be the only one who hasn't read Hofstadler's GEB (EBG,
> > BGE). It got me interested in Turing, Godel etc...I remember being
> > obsessed with that book and Nietzsche's Beyond good and evil at the
> > same time (phew, what a mix) which shocked and fascinated me. I think
> > that every one who eventually thinks for themselves have to go through
> > that mumbly period of confusion and doubt. I'm no longer surprised
> > that many academics are just what Nietzsche called schoolmasters who
> > regurgitate "facts" without understanding them in the slightest. GEB
> > is great for turning that confusion into fun paradoxes and the like.
>
> Yeah, I have to read it. I'm about to buy it on Amazon right now.
>
> > Well I arrived at 19/16 because I knew that 5 couldn't be the tonic
> > since it's not 2 to the power of N (=123...). Didn't make sense that
> > the whole minor key didn't agree with tonic = octave equiv. of
> > fundamental. So I spent days looking for ratio approximations and
> > found it eventually, probably by accident. Fortunately, I found a much
> > simpler way. You'll need a calculator.
> > For any interval 2 power p/12, p = 0,1,2...11, take 2 power of (N +
> > p/12), N is whole. This will give you a number with a whole part M and
> > remainder r after the decimal point. Now there are two possibilities
> > 1. we round off M to nearest odd, or 2. M is already odd and we ignore
> > r. But the best numbers are those where M is already odd AND rounds
> > off to M. The ratio we are looking for is M divided by 2 to the power
> > of N i.e. the N we started with.
> > E.g. p = 3, N = 4 gives minor 3rd 19/16. 2 power 4.25 gives 19.0273...
> > which rounds off to 19 numerator. 2 power of 4 gives 16 denominator.
>
> Again, I'll have to check that out.
>
> > I'm assuming that other tets are also base 2? Is that correct? For eg,
> > is 17-tet 2 power of p/17, p = 0,1,2,...16? If so then the above
> > method still holds.
> >
> > P.S. No ones answered my stuff about the length of time being
> > aleph-naught since cycles are discreet and whole, but number of points
> > within a single period being aleph-one since we need continuity.
> > Didn't it get through or is it just a daft thought?
>
> Well, that isn't how those numbers work. Aleph-null and Aleph-one are
> cardinal numbers used to explain the size of infinite sets. The length
> of time increases without bound, and the amount of points between a
> single period are aleph one. Really, I'm not sure exactly what you're
> saying here. Can you clarify?
>
> -Mike
>
Well imagine a universe where waves dominate i.e. our very own. Our
sense of space and time come from waves of all types and every one of
these can be Fourier Analysed down to component sine waves (time for
eg is represented by the CLASS of all frequencies, not by the
arbitrary choice of 1 second). Take f = 1/T, time = t, and cycles = n
= ft. We then get t = nT. e.g. f = 440Hz, then 2 seconds requires n =
880 cycles. Time is now a dependent variable, not a separate concept.

Now observe that number of cycles also means number of periods. t = nT
can also be written t = T + T +...n factors. Therefore a complete
cycle is "1" and the total number of cycles is N(0), countable
infinity. If n = N(0) then t = N(0) sec since the product of
aleph-zero by a number is itself. The length of the time line is N(0).

"But n does not have to be whole" you protest. Here's the rub. All
real numbers on the t line can be mapped back into a single period.
Let n = ft = N + r where N = 1,2,3... and 0 < r > 1. Since adding N to
the sine function Y does not affect it then Y(N + r) = Y(r). because
the sine wave has to be continuous, this means that the number of
numbers inside a single period, r, is N(1), the number of real numbers.

We reach the peculiar circumstance that the number of numbers within a
single period (the number or amount of r's) is larger than the number
of periods (number of N's), i.e. a typical Cantor-esque proof. But we
also see that the number of periods for one wave is rational and not
irrational. Interesting.

I still get the feeling you're going to ask me something.

Rick