For Tom

Well Tom, you are reading the statement "potential to be periodic" out of context. I was talking to Mike about a paper I have written in which the principle of wave addition is conserved under a relativistic transformation, and which incidentally the current view of "dispersive" waves does not achieve (e.g. Since quantum light and matter waves rely on the emission and absorption of frequency-energy, then these are concerned with the difference and sum and are familiar as a type of beat frequencies. In other words, they can still be periodic since this is independent of the question of rationality and so forth).

But my actual question to the group was whether or not all irrational intervals could in fact be approximations to rational intervals in the higher octaves of the harmonic series. Just as one among many examples, the ratio 181/128 approximates the flat fifth (sq root of 2) to 3 decimal places. If we cannot make a distinction past a certain number of decimal places, then it might be just as valid to argue that the tempered system is an approximation to harmonics after all. Or it might be undecidable and depend upon harmonic context?

As to your statement that periodic functions and Fourier analysis have been known for centuries, well I'll admit they have been used uncritically for centuries. For despite widespread "scientific" belief, these are not primordial concepts at all but rely on a number system dating as far back as Pythagoras. If you look at the basic sine wave, then space and time are quantities that are defined in proportion to wavelength and period, the factor of proportionality being the number of cycles. In other words, the very idea of proportion and ratio was written into the sine wave from the very beginning. Mechanics seems to have forgotten that it was in fact Pythagoras (circa 570 B. C.) who invented maths-science when he discovered that the world of harmonia, seen by the Greeks to the very source of natural mystery and wonder, corresponded to the series of natural numbers. Scientists then (albeit unwittingly) rewrote history to make it appear that it was wholly
deducible from mechanics. Therefore, when the matter wave was discovered in 1928, instead of this being hailed as a remarkable and uncanny confirmation of Pythagoras, they continued to reduce it to mechanics, hence, the self-contradictory view of dispersive waves which do not conserve periodicity under transformation and therefore contravene the principle of relativity (i.e. the condition that the math'l form of a law be the same for all observers. Apparently, the laws of musical harmony are not considered worthy. But in fact they apply to all waves and therefore have relevance for all physics).

So how could scientists miss such a basic oversight for so long? Well you answered this yourself; "I find it unlikely that you have discovered anything new". Since you are merely making a premature guess, then I would say that this type of attitude is the true source of ill-defined terminology. Mathematics is a big field my friend and wave theory is really in its infancy.

Get the name you always wanted with the new y7mail email address.
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rick ballan wrote:

> But my actual question to the group was whether or not all irrational > intervals could in fact be approximations to rational intervals in the > higher octaves of the harmonic series. Just as one among many examples, > the ratio 181/128 approximates the flat fifth (sq root of 2) to 3 > decimal places. If we cannot make a distinction past a certain number of > decimal places, then it might be just as valid to argue that the > tempered system is an approximation to harmonics after all. Or it might > be undecidable and depend upon harmonic context?

They *could* be, for all we could tell. And for some instruments they are (synthesizers, Hammond organs). But there's no privileged rational approximation.

If there's something to this for acoustic instruments I'd guess it's more to do with the room acoustics than the harmonic context.

Graham

you could look at periodic motion as the set of harmonics or you see it in term of the Farey series or the stern brocot tree. But then again you could see the whole continuum in terms of golden/noble numbers. that such a series converge on

Sorry for the delay but JI is short for Just intonation. the basic idea there is they might approach any interval in the terms you describe. With those working with equal temperament they will approximate it by those means.

I am confused how one can differentiate one continuum from another since in the end they are indistinguishable.
yes i would like to see your paper and a big fan of Hume

/^_,',',',_ //^ /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/>

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

rick ballan wrote:
> Well Tom, you are reading the statement "potential to be periodic" out > of context. I was talking to Mike about a paper I have written in > which the principle of wave addition is conserved under a relativistic > transformation, and which incidentally the current view of > "dispersive" waves does not achieve (e.g. Since quantum light and > matter waves rely on the emission and absorption of frequency-energy, > then these are concerned with the difference and sum and are familiar > as a type of beat frequencies. In other words, they can still be > periodic since this is independent of the question of rationality and > so forth).
>
> But my actual question to the group was whether or not all irrational > intervals could in fact be approximations to rational intervals in the > higher octaves of the harmonic series. Just as one among many > examples, the ratio 181/128 approximates the flat fifth (sq root of 2) > to 3 decimal places. If we cannot make a distinction past a certain > number of decimal places, then it might be just as valid to argue that > the tempered system is an approximation to harmonics after all. Or it > might be undecidable and depend upon harmonic context?
>
> As to your statement that periodic functions and Fourier analysis have > been known for centuries, well I'll admit they have been used > uncritically for centuries. For despite widespread "scientific" > belief, these are not primordial concepts at all but rely on a number > system dating as far back as Pythagoras. If you look at the basic sine > wave, then space and time are quantities that are defined in > proportion to wavelength and period, the factor of proportionality > being the number of cycles. In other words, the very idea of > proportion and ratio was written into the sine wave from the very > beginning. Mechanics seems to have forgotten that it was in fact > Pythagoras (circa 570 B. C.) who invented maths-science when he > discovered that the world of harmonia, seen by the Greeks to the very > source of natural mystery and wonder, corresponded to the series of > natural numbers. Scientists then (albeit unwittingly) rewrote history > to make it appear that it was wholly deducible from mechanics. > Therefore, when the matter wave was discovered in 1928, instead of > this being hailed as a remarkable and uncanny confirmation of > Pythagoras, they continued to reduce it to mechanics, hence, the > self-contradictory view of dispersive waves which do not conserve > periodicity under transformation and therefore contravene the > principle of relativity (i.e. the condition that the math'l form of a > law be the same for all observers. Apparently, the laws of musical > harmony are not considered worthy. But in fact they apply to all waves > and therefore have relevance for all physics).
>
> So how could scientists miss such a basic oversight for so long? Well > you answered this yourself; "I find it unlikely that you have > discovered anything new". Since you are merely making a premature > guess, then I would say that this type of attitude is the true source > of ill-defined terminology. Mathematics is a big field my friend and > wave theory is really in its infancy.
>
> ------------------------------------------------------------------------
> Get the name you always wanted with the new y7mail email address > <http://au.rd.yahoo.com/mail/taglines/au/y7mail/default/*http://au.mail.yahoo.com/?p1=ni&p2=general&p3=tagline&p4=other>. >
>

I still have no idea what Rick is talking about. Yes, any irrational
number can be well approximated by a sequence of rational numbers, and
I suppose vice versa. I don't see why this should have any consequence
for physics, since the difference can always be made unmeasurably
small (which Kraig effectively already said). Physicists don't believe
that distances or times or frequencies can be defined or measured
infinitely precisely, whatever theory you have is going to break down
when you go beyond the Planck scale. The world is, so far as anyone
knows, not made up of perfect, simple mathematical objects like sine
waves, at least no-one has found any medium that is perfectly linear
and non-dispersive.

But yes, it is premature to start debating something when we haven't
heard the main point of it yet. Why don't you send a message with the
abstract of the paper or some equations so we can find out what you
think is fundamentally wrong with quantum mechanics. (Notwithstanding
that this is an almost totally inappropriate place to discuss it...)
My at-work signature follows :

Thomas Dent
------------------------
Theoretical Physics
University of Heidelberg

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> you could look at periodic motion as the set of harmonics or you see it
> in term of the Farey series or the stern brocot tree. But then again
you
> could see the whole continuum in terms of golden/noble numbers. that
> such a series converge on
>
> Sorry for the delay but JI is short for Just intonation. the basic idea
> there is they might approach any interval in the terms you describe.
> With those working with equal temperament they will approximate it by
> those means.
>
> I am confused how one can differentiate one continuum from another
since
> in the end they are indistinguishable.
> yes i would like to see your paper and a big fan of Hume
>
> /^_,',',',_ //^ /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/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>

I think that Rick is saying that we view 3 steps of 12-tet as being an
out of tune 6/5, but maybe we actually perceive it as a relatively
in-tune 19/16. Furthermore, just apply this concept to every step in
the scale.

Rick's theory, as far as I understand it is actually a restatement of
what you said - that any measurement will break down beyond the Planck
scale. So Rick is saying that when we say that equal temperament is
made up of waves that are in irrational proportions to one another, it
is actually so that they are not, because the measurements that we
have to determine their relationship to one another are of limited
accuracy, not only due to practical reasons, but due to this
theoretical constraint. Furthermore, I think he's saying that it's not
the measurements that break down at this level, but that there is
theoretically no smaller unit that can exist than that of the Planck
length. I think he's taking the Planck length as it applies to
measurements of space and is applying it to the frequency/periodicity
of waveforms.

So he's not saying that these irrational numbers are just APPROXIMATED
by a continued fraction sequence, but that there is a point in which
there is NO DIFFERENCE from a certain term arbitrarily far into the
sequence and the irrational number itself as it applies to the
frequency of a given wave, as the difference in the wavelength would
be small enough that it would be undetectable (or not exist). This
obviously doesn't apply to the abstract concept of numbers themselves,
but when they are instantiated in reality, then these theoretical
limits would apply.

At least that's what I think he's saying. And furthermore, he seems to
have been met with some positive response, as a lot of people have
responded that 19/16 sounds much more like the "minor" chord they are
used to than 6/5.

-Mike

On Sun, Jun 1, 2008 at 6:05 PM, Tom Dent <stringph@...> wrote:
>
> I still have no idea what Rick is talking about. Yes, any irrational
> number can be well approximated by a sequence of rational numbers, and
> I suppose vice versa. I don't see why this should have any consequence
> for physics, since the difference can always be made unmeasurably
> small (which Kraig effectively already said). Physicists don't believe
> that distances or times or frequencies can be defined or measured
> infinitely precisely, whatever theory you have is going to break down
> when you go beyond the Planck scale. The world is, so far as anyone
> knows, not made up of perfect, simple mathematical objects like sine
> waves, at least no-one has found any medium that is perfectly linear
> and non-dispersive.
>
> But yes, it is premature to start debating something when we haven't
> heard the main point of it yet. Why don't you send a message with the
> abstract of the paper or some equations so we can find out what you
> think is fundamentally wrong with quantum mechanics. (Notwithstanding
> that this is an almost totally inappropriate place to discuss it...)
> My at-work signature follows :
>
> Thomas Dent
> ------------------------
> Theoretical Physics
> University of Heidelberg
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>>
>> you could look at periodic motion as the set of harmonics or you see it
>> in term of the Farey series or the stern brocot tree. But then again
> you
>> could see the whole continuum in terms of golden/noble numbers. that
>> such a series converge on
>>
>> Sorry for the delay but JI is short for Just intonation. the basic idea
>> there is they might approach any interval in the terms you describe.
>> With those working with equal temperament they will approximate it by
>> those means.
>>
>> I am confused how one can differentiate one continuum from another
> since
>> in the end they are indistinguishable.
>> yes i would like to see your paper and a big fan of Hume
>>
>> /^_,',',',_ //^ /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/>
>>
>> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>>
>
>