The Logic Off Topic list

I am 30 Arvid :(

> Note that a lot of the people who claims that analog is best is old enough
> to have memories about them.

Last bondage session I had was with an arp 2600 patch cabels, today I use
apple USB cables, they are more shiny!
> ...or they have a romantic feeling about old times (Check out things like
> clothing and the music they listen to etc...) People say that the strongest
> emotions you can relate to is "nostalgia".
And does there opinion count when it comes to synth?! maybe I should check
out Nsync 
> If you talk to the averege teenager about digital vs. analog they really
> don't care.
Off course, they sound different and its a matter of choice, taste:-)

> PS. I'm not saying that one is better than the other...

> Can you, or anyone else who can, explain to me what that 'analogue' 
> sound is? I don't seem to understand it, :-).


An explanation is a bit difficult I think. Especially because you've 
heard many good analog synths.
Imo analog's sound more "natural". They fit more easily in the mix too.

> I spent a couple of hours noodling with a Juno 60, Jx8-P and a Poly 
> Six and decided to go for a Yamaha CS1x(digital) syth after all, :-).

Taste is something that can't be discussed. :-)
And of course a CS1x is often more convenient.


-- 
Joeri Vankeirsbilck
joeri@...

Belway Productions      -     http://www.belway.com
List-admin   Logic-users/SoundD*ver-users/Logic-TDM

Note that a lot of the people who claims that analog is best is old enough
to have memories about them.
...or they have a romantic feeling about old times (Check out things like
clothing and the music they listen to etc...) People say that the strongest
emotions you can relate to is "nostalgia".

If you talk to the averege teenager about digital vs. analog they really
don't care.

PS. I'm not saying that one is better than the other...

--
Arvid Solvang
http://www.viagram.no/

>The sound really is a league apart from the digitally
>modelled subtractive synth.  The filter for a start is really dangerous -
>proper resonance that feedsback on itself.

Hey, why don't you guys try the Moogerfooger stomp box filter on a 
digital modeled analog synth - then you might get the best of both worlds 
- analog filters but stable sound source.

Or you could try my moogfilter if you're on a mac -
http://bowery.com/bea/free

it models the moog filter, and feeds back nicely.

> From: GAmoore@...
> Reply-To: logic-ot@yahoogroups.com
> Date: Mon, 5 Nov 2001 16:14:30 EST
> To: logic-ot@yahoogroups.com, basharar@...
> Subject: Re:  Re: [L-OT] Re: Analog synth is still better
> 
>> The sound really is a league apart from the digitally
>> modelled subtractive synth.  The filter for a start is really dangerous -
>> proper resonance that feedsback on itself.
> 
> Hey, why don't you guys try the Moogerfooger stomp box filter on a
> digital modeled analog synth - then you might get the best of both worlds
> - analog filters but stable sound source.
> 
> 
> 
> 
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>

Thoughts from the mind of Arvid Solvang, 05-11-2001:

>Note that a lot of the people who claims that analog is best is old enough
>to have memories about them.

And so...?  What is your point exactly?

>If you talk to the averege teenager about digital vs. analog they really
>don't care.

If you talk to the average teenager about almost _anything_, they 
really don't care.  Again: what's your point?

>PS. I'm not saying that one is better than the other...

You mean the old people vs. the teenagers?  Or analog v.s. digital. 
See whatyou did?  You got me all confused here...


cheers,
HJ, 39 springs young, and fondly remembering true hardware analog synths

-- 
     Hendrik Jan Veenstra
     email: mailto:h@...
     www:   http://www.ision.nl/users/h/index.html

on 6/11/01 7:15 PM, Dennis Gunn at dennisg@... wrote:

>>> The sound really is a league apart from the digitally
>>> modelled subtractive synth.  The filter for a start is really dangerous -
>>> proper resonance that feedsback on itself.
>> 
>> Hey, why don't you guys try the Moogerfooger stomp box filter on a
>> digital modeled analog synth - then you might get the best of both worlds
>> - analog filters but stable sound source.
> 
> yeah the TDM one kicks ass too. :-)
> 
> Back to the subject.  Personally I love soft synths but acknowledge
> there is a difference between them and the real thing.   I get the
> feeling it has to do with aliasing distortion.

i used the moogerfooger TDM plug-in in a session last week last week and it
was really good.  we used it to generate some sub-bass frequencies to fatten
up a nasty acoustic kick drum - worked a treat.    I love digital /plug-ins
too!

all the best,

pk

The OSCILATOR in a virtual analog is modeled after an analog VCO, so are the
other component in the signal chain.... So far virtual analog can sound FAT,
and the filters are getting there, but the interaction of electronics is
still too perfect to make the instrument more human like, ie more imperfect.

Yeah, but analog-modeled synths aren't samples. A basic square wave or 
saw wave is created mathematically (that means with numbers to start 
with), then run through some sort of d/a converter to create sound. It 
never started out as a sample. 

I have samples of analog synths on my Kurzweil K2vx - prophets, 
memorymoog, etc. And they don't sound anywhere near as alive as my Nova 
with its digitally recreated analog.]]]



>>> As Logic baby implied above, I'd say that the biggest difference between
>>> analogue and analogue modelling in digital synths is largely due to the 
'non

>>> linearities' that creep into the system in the various stages of a real
>>> analogue
>>> synth.
>
>
>No, the biggest difference is that digital is a sampled-time system, which
>means that it's subject to time aliasing (usually just called aliasing).
>It's truly a challenge to  make something sound close to it's analog
>counterpart while sampling at just above the hearing threshold (<24KHz
>bandwidth in a 48KHz system).  I was just reading an AES paper on how a
>decent peak limiter would need a 6MHz sample rate to keep aliasing noise to
>inaudible levels....  these days I guess that's a practicality, what with
>2GHz Pentiums....

> From: GAmoore@...
> 
> Yeah, but analog-modeled synths aren't samples. A basic square wave or
> saw wave is created mathematically (that means with numbers to start
> with), then run through some sort of d/a converter to create sound. It
> never started out as a sample.

Anything 'digital' is sampled.  Just because the samples are computed
instead of recorded and stored, that is immaterial.

>  >The sound really is a league apart from the digitally
>>modelled subtractive synth.  The filter for a start is really dangerous -
>>proper resonance that feedsback on itself.
>
>Hey, why don't you guys try the Moogerfooger stomp box filter on a
>digital modeled analog synth - then you might get the best of both worlds
>- analog filters but stable sound source.

yeah the TDM one kicks ass too. :-)

Back to the subject.  Personally I love soft synths but acknowledge 
there is a difference between them and the real thing.   I get the 
feeling it has to do with aliasing distortion.

>> Yeah, but analog-modeled synths aren't samples. A basic square wave or
>> saw wave is created mathematically (that means with numbers to start
>> with), then run through some sort of d/a converter to create sound. It
>> never started out as a sample.
>
>Anything 'digital' is sampled.  Just because the samples are computed
>instead of recorded and stored, that is immaterial.

Sampling involves taking measurements and recording them for 
reproduction. The whole idea of 'sampling' is to take sample frequency 
readings 44,100 times per second of anotherwise continuous source of 
sound. 

I think the analog modeled softsynths are creating a file which is of the 
same form as one which comes from samples of something, but they are 
concieved and created in a test tube so to speak. A square wave for 
example is very simple to create mathematically. These sounds never heard 
the 'light of day' until they are converted to audio. 

Because of that, they may not have many of the ill affects of sampling 
process - which happen during the analog to digital conversion - which 
these sounds never go through. For one thing there should be no 
distortion - at least if the algorhthms are well written.

> From: GAmoore@...
> 
> Sampling involves taking measurements and recording them for
> reproduction. The whole idea of 'sampling' is to take sample frequency
> readings 44,100 times per second of anotherwise continuous source of
> sound. 

You're using the popular meaning of the word sampling, whereas I'm referring
to the mathematical meaning.

> I think the analog modeled softsynths are creating a file which is of the
> same form as one which comes from samples of something, but they are
> concieved and created in a test tube so to speak.

The concept of a file is irrelevant, but in your example, the samples are
computed 'on the fly'...  this has no effect on the sound, whether they're
computed ahead of time (in this sense, recording the input of an A/D is
precomputing), or on the fly, unless there is a processing power limitation
that affects the ability to process, or you have a process that depends on
time reversal, in which case you can't do it in real time.

> A square wave for
> example is very simple to create mathematically.

Actually, it's not that simple to create an alias-free square wave.  Some
folks at CCRMA have some nice papers on the subject.


> Because of that, they may not have many of the ill affects of sampling
> process - which happen during the analog to digital conversion - which
> these sounds never go through. For one thing there should be no
> distortion - at least if the algorhthms are well written.

There should be no distortion in the digitizing (A/D) process either, if the
converters are well designed.  Any 16 bit or better sampler you have is
going to have negligible artifacts, unless you purposely distort the signal
while digitizing, or digitize at an unreasonably low level.

> From: LogicBaby <basharar@...>
>
>but the interaction of electronics is
> still too perfect to make the instrument more human like, ie more imperfect.

I don't think it's perfection as much as lack of understanding of what the
analog counterpart is really doing.  For example, hard/soft oscillator sync.
There are many ways to do this in the analog world, and they yield different
(but entirely predictable) results - which is being emulated?  Or
intentional sources of distortion - are the circuits being understood to
that level?  Most of the analog designers aren't sharing their secrets...

The other thing has to do with instrument design.  I mean, in the same way a
violin is designed to be an instrument, so is a micromoog or a minimoog (two
very different playing experiences, by the way).  Something simple such as
the range of a knob has a big effect on the palette of sounds.  Are the
virtual instruments spending time with such considerations?

>The OSCILATOR in a virtual analog is modeled after an analog VCO, so are the
>other component in the signal chain.... So far virtual analog can sound FAT,
>and the filters are getting there, but the interaction of electronics is
>still too perfect to make the instrument more human like, ie more imperfect. 

I suspect that the digital recreations of the analog circuits are more 
precise and accurate and less noisy. However for greater variety, that 
can eventually be programmed into the software too. If you can identify 
exactly what the affect you want, it can be programmed. 

The word 'model' can be used two different ways. In a theoretical sense, 
we have digital models of the analog, but things like the Microphone 
Modeller and the POD use modeling in a different sense - which is more 
like samplying what happens when various signals are fed in and looking 
what happens. They even take into account how adjusting the controls 
interact with each other on various models of amps.

But thats not what happens with digital analog synths.

>Many might have but I have read interviews with Bob Moog and it is 
>pretty clear that he wouldn't.  I have seen him say that his favorite 
>processor is a clipping transistor.

Sure, now he says that - after he has seen 30 years of people using his 
stuff and how they use it. But when he was an electrical engineer at 
Cornell, I don't think he imagined how great clipping transitors would 
sound on Buddy Holly records.

>Personally I love soft synths but acknowledge 
>there is a difference between them and the real thing.   I get the 
>feeling it has to do with aliasing distortion.

Do you notice a huge difference with vsti's and analog digital analogs 
(hardware devices that digitally recreate analog)? They sound differnet 
to me but I wonder how much of that is the affects, and how much is 
corners that are cut to make it work on a native cpu rather than a 
dedicated hardware cpu.

> From: Dennis Gunn <dennisg@...>
> 
> Once about 9 years ago on the DAW I remember writing about how the
> digital wave form for a sine wave up around say 19.k looks like a
> broken spiral staircase and how I find it hard to believe that any
> amount of filtering is going to turn that into a clean sine wave.  A
> lot of peaple called me ignorant for being decieved by the look of
> data but Nothing I have seen or heard since has ever convinced me I
> was wrong.

You're not hearing the data - you hear the data after it's been timed and
output through a D/A converter, which includes a reconstruction filter.  To
compare apples to apples you need to hook an oscilloscope up to that analog
output and compare the two.  You'll see that the digital sinewave looks the
same.

There have also been studies done showing that different looking waveforms
(due to the relative phase of thier harmonics) sound the same - so don't be
deceived by how a waveform looks.  It's useful information, to be sure, like
when editing, but it's not perfectly correlated to the sound.

> From: GAmoore@...
> 
> The word 'model' can be used two different ways. In a theoretical sense,
> we have digital models of the analog, but things like the Microphone
> Modeller and the POD use modeling in a different sense - which is more
> like samplying what happens when various signals are fed in and looking
> what happens. They even take into account how adjusting the controls
> interact with each other on various models of amps.
> 
> But thats not what happens with digital analog synths.

The way the POD uses the word modelling is the same way as the synths -
modelling the *behaviour* of the analog circuitry (as opposed to modelling
the circuitry itself).

The mic modeller abuses the term, because what it's really doing is
determining a static impulse response of a mic (already a simplifying
assumption) and the convolving some arbitrary input with the impulse
response (in other words 'running the input through' the impulse response).

Actually, as I recall the Memorymoog had 18 oscillators - as many as 3 
per voice. However, even under normal conditions you had to let it warm 
up 10 minutes then hit a 'tune' button. I suspect the oscillators were 
always drifting out of tune a tiny amount. Maybe that is part of their 
charm.

> From: GAmoore@...
> 
> Actually, as I recall the Memorymoog had 18 oscillators - as many as 3
> per voice. However, even under normal conditions you had to let it warm
> up 10 minutes then hit a 'tune' button. I suspect the oscillators were
> always drifting out of tune a tiny amount. Maybe that is part of their
> charm.

And the Polymoog had, I think it was 3 or 4 oscillators per key, times 88!

Are you sure? The Polymoog came out a few years before the Memorymoog as 
I recall and was pretty cheesy. Anyway, that same pitch wandering could 
be programmed into to a softsynth, couldn't it?

>> Actually, as I recall the Memorymoog had 18 oscillators - as many as 3
>> per voice. However, even under normal conditions you had to let it warm
>> up 10 minutes then hit a 'tune' button. I suspect the oscillators were
>> always drifting out of tune a tiny amount. Maybe that is part of their
>> charm.
>
>And the Polymoog had, I think it was 3 or 4 oscillators per key, times 88!

Thoughts from the mind of GAmoore@..., 07-11-2001:

>  >> Actually, as I recall the Memorymoog had 18 oscillators - as many as 3
>>>  per voice. However, even under normal conditions you had to let it warm
>>>  up 10 minutes then hit a 'tune' button. I suspect the oscillators were
>>>  always drifting out of tune a tiny amount. Maybe that is part of their
>>>  charm.
>>
>  >And the Polymoog had, I think it was 3 or 4 oscillators per key, times 88! 
>
>Are you sure? The Polymoog came out a few years before the Memorymoog as
>I recall and was pretty cheesy. Anyway, that same pitch wandering could
>be programmed into to a softsynth, couldn't it?

The polymoog had a custom-made chip per key.  Each chip contained 
(more or less probably) an entire monophonic synthesizer.  That's the 
only way they were able to build polyphonic synths at the time.

-- 
     Hendrik Jan Veenstra
     email: mailto:h@...
     www:   http://www.ision.nl/users/h/index.html

marc lindahl <marc@...> wrote:

>> From: Dennis Gunn <dennisg@...>
>>
>> Once about 9 years ago on the DAW I remember writing about how the
>> digital wave form for a sine wave up around say 19.k looks like a
>> broken spiral staircase and how I find it hard to believe that any
>> amount of filtering is going to turn that into a clean sine wave.  A
>> lot of peaple called me ignorant for being decieved by the look of
>> data but Nothing I have seen or heard since has ever convinced me I
>> was wrong.
>
>You're not hearing the data - you hear the data after it's been timed and
>output through a D/A converter, which includes a reconstruction filter.  To
>compare apples to apples you need to hook an oscilloscope up to that analog
>output and compare the two.  You'll see that the digital sinewave looks the
>same.
>
>There have also been studies done showing that different looking waveforms
>(due to the relative phase of thier harmonics) sound the same - so don't be
>deceived by how a waveform looks.  It's useful information, to be sure, like
>when editing, but it's not perfectly correlated to the sound.

This observation is not relevant to a sine wave as any change in appearance
at the *audio*  output outside of a strict shift in phase, can only be
distortion.

S.

Dennis Gunn <dennisg@...> wrote:

>Once about 9 years ago on the DAW I remember writing about how the
>digital wave form for a sine wave up around say 19.k looks like a
>broken spiral staircase and how I find it hard to believe that any
>amount of filtering is going to turn that into a clean sine wave.

Nine years is a long time in the history of DAWs...

It is entirely feasible to digitally filter a square wave to produce a
sinewave within the practical limits of a digital systems parameters,
which is effectively what is going on in what you describe above.
In any case, why is it hard to believe? Isn't filtering one of
the cornerstones of sound processing? Was Fourier wrong??

The problem lies in allowing >> nyquist frequencies to get to the
the output, as for example would happen in an unfiltered saw or
square oscillator at even modest frequencies with a less than ideal
filter at the converter stage. Better filter implementaions,
Greater sample rates and higher bit depths allwork to minimise this
problem...

>A
>lot of peaple called me ignorant for being decieved by the look of
>data but Nothing I have seen or heard since has ever convinced me I
>was wrong.

Though I don't entirely disagree with you, from what I have learned of
you on this list I don't find that statement a complete surprise.

S.

GA Moore wrote:

>I suspect that the digital recreations of the analog circuits are more
>precise and accurate and less noisy.
Does 'suspect' mean you're not sure?

>However for greater variety, that
>can eventually be programmed into the software too. If you can identify
>exactly what the affect you want, it can be programmed.
Can't argue with an 'if'.

>The word 'model' can be used two different ways.
??
The only two ways you can use this word that I can see, in this present 
context anyway, would be what we could perhaps call 'first-order' and 
'second-order' modelling. In first'order modelling you would possibly model 
the actual behaviour of a real something. In second-order modelling you 
would perhaps model the behaviour (or method) by which some other 
first-order gadget models something else?

But even then ... as far as I can see ... that could eventually be reduced 
to a simple case of first-order modelling because what you would actually 
be modelling is the behaviour of a very real something ... whose behaviour 
just happened to include the ability of that something to model something 
else. You do not actually have to know that you are modelling something 
that can model something else when you are trying to model that something. 
What are your two different ways of modelling something. I'm genuinely 
curious ... for as far as I can see the objects under discussion are all 
trying to model something and so I'm a little puzzled what the difference is.

>In a theoretical sense,
>we have digital models of the analog, but things like the Microphone
>Modeller and the POD use modeling in a different sense - which is more
>like samplying what happens when various signals are fed in and looking
>what happens. They even take into account how adjusting the controls
>interact with each other on various models of amps.
I do not understand how either the Antares Microphone Modeller or the POD 
are modelling 'in a different sense'.

Also ... does the modus operandi of Microphone Modeller truly have 
something to do with sampling? I thought the mathematical basis of all 
filtering, which as far as I know is basically what the MM does, was 
convolution? Or am I mistaken here? I thought, but I could be wrong, that 
the output of any digital filter, say a function y(k), was related to its 
input, say a different function x(k), through the relevant impulse 
response, another function, h(k)? Not that this is my area of expertise, or 
anything of course. I am happy to bow down to superior wisdom ... but I 
always thought that what related the two was convolution? So ... what this 
boils down to is how we come by the various functions, no? Well, that's a 
whole other issue in itself, but as far as I can see an input response 
FUNCTION is not a sample. One can criticize the MM for making some pretty 
unrealistic assumptions about the nature of its input functions and for the 
arbitrariness with which it selects the impulse responses, but I don't see 
how one criticize it for not being a sample when it isn't??!  Anyway, as 
far as I know the MM has anything remotely to do with sampling but uses 
some pretty nifty Fourier techniques to achieve its purposes.

If the impulse response has a finite length, and the input also has a 
finite length then -- at least that's what I always thought -- it is only 
necessary to store x(k) in a suitable vector and then convolve the two. 
You'v got blind source separation and all kinds of other stuff. You need 
source separation algorithms as well. You do have to deal with the reverb 
and absorption characteristics of your room. But this only needs finite 
impulse responses or FIR filters. You still only need to convolve, though, 
with whatever FUNCTION you devise to represent the original sound source. 
That's what I thought, anyway. Where's the need for sampling in the 
operation of such a device? I could be wrong about this, though. It's a 
well-known fact that I'm wrong about many things. But usually I'm fortunate 
in that when I do parade my ignorance -- way too often I admit -- some kind 
soul somewhere graciously corrects me and my knowledge increases.

Kool Musick
Keep Musick Kool


_________________________________________________________
Do You Yahoo!?
Get your free @... address at http://mail.yahoo.com

>The polymoog had a custom-made chip per key.  Each chip contained
>(more or less probably) an entire monophonic synthesizer.  That's the
>only way they were able to build polyphonic synths at the time.

Not quite. The Polymoog used  two master oscillators routed through
divide-down chips, like electronic organs, string synths, and a few early
Korg poly synths. This is why it sounded so thin and crappy. So it did not
have a complete mono synth on each key -- but it did have a filter and amp
for each key... of course, the filter here is not the famed Moog filter
either. It did have the advantage of being 71-note polyphonic though.

The Polymoog was a costly failure for Moog. It suffered numerous production
delays and was a major disappointment in terms of sound (although Klaus
Schulze and Gary Numan were among the few who could put it to good use). The
Polymoog was quickly forgotten when real polyphonic synths like the Prophet
5 came along.

> From: GAmoore@...
> 
> Are you sure? The Polymoog came out a few years before the Memorymoog as
> I recall and was pretty cheesy. Anyway, that same pitch wandering could
> be programmed into to a softsynth, couldn't it?

Yeah, I'm sure.  It had a synth with I think it was 4 different sounds, one
for each key!  Insanely great! :)

Sure, pitch wandering, temperature drift, could be programmed into a soft
synth.  The simplest would be a very low frequency LFO modulating the
oscillator frequency.  But the thing about the moog's is, if you're doing a
live gig on stage, as the stage heats up it gets more and more "out" in one
direction (mine goes sharp) - I suppose you could build a temperature - to -
MIDI controller and feed it to your soft synth :)

> From: Spectro <spectro@...>

> This observation is not relevant to a sine wave as any change in appearance
> at the *audio*  output outside of a strict shift in phase, can only be
> distortion.

Which observation?  The first one or the second one?

I like the soft synths but love the old analog synths better. 

Love all those knobs etc..  




[Non-text portions of this message have been removed]

> From: Spectro <spectro@...>
> 
> The problem lies in allowing >> nyquist frequencies to get to the
> the output, as for example would happen in an unfiltered saw or
> square oscillator at even modest frequencies with a less than ideal
> filter at the converter stage.

Actually, the problem is in how you generate synthetic waveforms.  In the
digital (discrete time) world, you could programmatically generate a
waveform in such a way that it's alreay aliased, so that no amount of
post-filtering could save it.  There's no rule that says that just because
you're in the digital realm you can't make aliased tones.  Once you realize
that, then you realize that it actually is not that easy to generate
non-aliased waveforms, and that you have to be very careful not to
accidentally generate aliasing with some process (such as clipping).

> From: Kool Musick <koolmusick@...>

> Also ... does the modus operandi of Microphone Modeller truly have
> something to do with sampling? I thought the mathematical basis of all
> filtering, which as far as I know is basically what the MM does, was
> convolution? Or am I mistaken here? I thought, but I could be wrong, that
> the output of any digital filter, say a function y(k), was related to its
> input, say a different function x(k), through the relevant impulse
> response, another function, h(k)? Not that this is my area of expertise, or
> anything of course. I am happy to bow down to superior wisdom ... but I
> always thought that what related the two was convolution?

You're right, for the limited (but widely applicable case) of what's called
a Linear Time-Invariant function.  That means the mic being modelled
responds the same from moment to moment.  Most things work pretty much like
this.  An example of something that doesn't is a compressor, which therefore
can't be modelled as an impulse response.  But most other audio things
(filters, rooms, mics, amps, instruments, most acoustic things, reverbs) can
be.




> So ... what this 
> boils down to is how we come by the various functions, no? Well, that's a
> whole other issue in itself, but as far as I can see an input response
> FUNCTION is not a sample. One can criticize the MM for making some pretty
> unrealistic assumptions about the nature of its input functions and for the
> arbitrariness with which it selects the impulse responses, but I don't see
> how one criticize it for not being a sample when it isn't??!

Sonic Foundry's Acoustic Modeller does the same thing but allows the user to
measure anything and apply it as an impulse response (filter).  So in a way,
sampling is subtly involved, since you excite whatever your mic or whatever
is, and sample that, then feed it to the program which processes the sample
into an impulse response.

GA Moore wrote:

>Sampling involves taking measurements and recording them for
>reproduction.
Depends what you mean by 'recording', I would guess.

>I think the analog modeled softsynths are creating a file which is of the
>same form as one which comes from samples of something,
I am curious. What file are they creating?

>A square wave for example is very simple to create mathematically.
I am curious again. How would you, personally, do this? I had always 
thought it was actually very hard to do.

>These sounds never heard
>the 'light of day' until they are converted to audio.
Well, what I always thought was that if you look carefully at the output 
end of a sampler ... then it really doesn't care two hoots what it's 
original data source was and where it came from. The sampler just computes. 
What it computes is what 'noise' it's going to output given the data stream 
passing through it at that moment. The source of that data is pretty 
inconsequential to the sampler at that point -- although it's obviously 
pretty important to us humans because we tend to gauge the effectiveness of 
the sampler by comparing that output to some original input that we have in 
mind. Can't see that that's the sampler's problem, though, to be honest!!

What I mean is, I'm not exactly clear about this 'light of day' bit.



Strictly speaking, the word 'sampler' comes from the Latin word 'exemplum' 
and simply therefore means 'the example to be followed'. Therefore, when 
you 'store' or 'record' the original data, all you are doing is storing an 
example that is to be followed ... i.e. a string of data that you will 
later use to recreate that 'something' of which you have taken an example. 
The output stage of a sampler does not 'know' that, originally, you got 
that data by taking measurements at a given bit rate. How could it 'know' 
such a thing? All you have is data. Not sound. So ... when it gets to the 
output stage, all the sampler is doing is mathematics and working with its 
data. At least ... that's my impression of things.

It's true that what you are giving the sampler is an example data stream -- 
modelled after a 'real world' exemplum -- that you would really rather like 
it to follow and recreate. In practical terms it's also true, a sampler is 
only really much use to us humans if it follows pretty strictly those 
examples taken from 'the real world'. However, that is because that is what 
we choose to do with it. Once it comes to the output stage, however, in 
every single case the sampler is simply working with the data and 
ultimately spitting out a sound ... but there is no de facto reason WHY -- 
looking strictly at that sampler -- any given sound to be created should in 
fact ever have previously existed 'in the real world'. It's questionable, 
in fact, if it ever did because what the sampler has is data and not a 
sound. All it did was take an example and it's surely a moot point whether 
or not that example is 'the real thing'. I was originally taught that it's 
only in fact a model of the real thing meaning that a sampler is really 
just a modelling device.

It is very easy, in fact, to make a sampler spit out an unearthly sound 
that has never before existed. All one has to do is mess around with data 
-- even if that data is some other sample. This is because all those newly 
designed sounds coming out of some sound designer's mind have never before 
seen the light of day. Peter Gabriel, for example, was famously able to 
transpose some very basic samples up and down a couple of octaves and so 
produce some long and complex timbres ... which had never before seen the 
light of day. Nor had they existed anywhere before ... except in the mind 
of one Peter Gabriel.

Roland for example, when they took to making samplers, used a method of 
differential interpolation to 'recreate' noises. It's hard to see how any 
of that had ever seen the light of day before either.

Far as I know, historically, a sampler is just a common or garden 
synthesizer. I always thought that that's why they always featured in books 
on synthesis techniques. The thing about a sampler -- or so I always 
understood -- is just that it so happens to have an infinite array of 
waveforms available to it simply because at the input stage it is possible 
to throw some data at it that starts off life by being an example of a 
waveform that we humans would immediately associate (when suitably 
reconverted) with some kind of 'natural sound' in 'the real world'. But 
from the point of example-taking onwards a sampler is just another 
synthesizer. The only difference between a sampler and any other kind of 
synthesizer is the source of them waveforms. Samplers do have to deal with 
some pretty complex waveforms -- i.e. mathematical data. But then again ... 
a square wave is pretty complex too! The sampler-as-synthesizer simply does 
not know -- how could it? -- that it is dealing with a waveform that 
originated as an example of something 'in the real world'.

I could well be mistaken, but I was taught when I learned about these 
things (a long time ago, admittedly and age is dimming my memory) that this 
'fact' is actually 100% irrelevant to the mode of operation of a sampler 
when it comes to the output stage. It's simply dealing with a waveform 
represented in numbers and so is just another synthesizer -- meaning that 
they're all equal in their digitalness-ness. They differ only in the source 
of wave forms whether this be physical modelling, granular synthesis or 
anything else.

Come to that there's getting to be very little difference these days 
between sampling and hard disk recording. They both deal with digitizing. 
All we do is use the HD as a real-time playback device. And, just like 
seems to have happened with sampling per se, the 'recording' and 
'manipulation' and 'performance' of data is pretty much all of a muchness, 
really. You can lift get things off your computer -- plug-ins, stimulation 
of MIDI instruments -- and then produce a final audio output that never in 
fact existed 'in the real world'. Your basic song in really just one long 
and complex and manipulable waveform.

But ... then again ... maybe I'm totally mistaken in my understanding of 
samplers and synthesizers ... I am mistaken about a lot of things, I guess, 
although I do like to keep learning.

>Because of that, they may not have many of the ill affects of sampling
>process - which happen during the analog to digital conversion - which
>these sounds never go through.
Can't argue with a 'may'.

>For one thing there should be no
>distortion - at least if the algorhthms are well written.
Can't argue with a 'should'. And ... I guess ... all we really need is a 
well-written algorithm. Lotta highly intelligent and creative people 
working on that one. Maybe they'll do it one day soon. Here's hoping, anyway.

Kool Musick
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marc lindahl wrote:
(in response to Kool Musick)

>You're right, for the limited (but widely applicable case) of what's called
>a Linear Time-Invariant function.  ... An example of something that 
>doesn't is a compressor, which therefore
>can't be modelled as an impulse response.  But most other audio things
>(filters, rooms, mics, amps, instruments, most acoustic things, reverbs) can
>be.
Thank you.

>Sonic Foundry's Acoustic Modeller does the same thing but allows the user to
>measure anything and apply it as an impulse response (filter).  So in a way,
>sampling is subtly involved, since you excite whatever your mic or whatever
>is, and sample that, then feed it to the program which processes the sample
>into an impulse response.

And thanks again.

Kool Musick
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>Sure, pitch wandering, temperature drift, could be programmed into a soft
>synth.  The simplest would be a very low frequency LFO modulating the
>oscillator frequency.  But the thing about the moog's is, if you're doing a
>live gig on stage, as the stage heats up it gets more and more "out" in one
>direction (mine goes sharp) - I suppose you could build a temperature - to -
>MIDI controller and feed it to your soft synth :)

Forget the temp gage. Just put the drift control on a midi CC, and have a 
Phatboy controller and turn the knob yourself. You probably need a VLFO 
and some randomness  to the various pitch distinctions.

>>In any case, why is it hard to believe? Isn't filtering one of
>>the cornerstones of sound processing? Was Fourier wrong??
>
>I don't know if he was wrong or right but I am pretty sure that his 
>mathematics describe something that is difficult to manifest 
>perfectly in an audio circuit in the real world and it is my opinion 
>that plays a significant factor in the character that we all hear in 
>digital audio that makes it a little less than the "perfect" 
>recording medium we all dreamed it would be way back when it was new.


Just a side note - but Fourier was not a mathematician. I think he was a 
physcist. He played fast and loose with the symbols to derive his 
results. The mathematicians were shocked by the lack of rigor. He was 
making all sorts of assumptions that were not on solid ground. However, 
over many decades there were able rigorously proove and further derive 
his ideas.

>  >Personally I love soft synths but acknowledge
>>there is a difference between them and the real thing.   I get the
>>feeling it has to do with aliasing distortion.
>
>Do you notice a huge difference with vsti's and analog digital analogs
>(hardware devices that digitally recreate analog)? They sound differnet
>to me but I wonder how much of that is the affects, and how much is
>corners that are cut to make it work on a native cpu rather than a
>dedicated hardware cpu.
>

I think I am talking about the difference between digital and analog 
rather than virtual and hard.

Once about 9 years ago on the DAW I remember writing about how the 
digital wave form for a sine wave up around say 19.k looks like a 
broken spiral staircase and how I find it hard to believe that any 
amount of filtering is going to turn that into a clean sine wave.  A 
lot of peaple called me ignorant for being decieved by the look of 
data but Nothing I have seen or heard since has ever convinced me I 
was wrong.

>  >Many might have but I have read interviews with Bob Moog and it is
>>pretty clear that he wouldn't.  I have seen him say that his favorite
>>processor is a clipping transistor.
>
>Sure, now he says that - after he has seen 30 years of people using his
>stuff and how they use it. But when he was an electrical engineer at
>Cornell, I don't think he imagined how great clipping transitors would
>sound on Buddy Holly records.

You may be right, but that's not the way *he* tells it.

GA Moore wrote:

>Just a side note - but Fourier was not a mathematician.
Yes he was.
For example, he taught mathematics at L'Ecole Polytechnique -- and other 
prestigious places. Why would they hire him as a mathematics lecturer if he 
wasn't a mathematician? To be honest, I can't find a single Internet site 
that says he wasn't.

>I think he was a physcist.
Does that mean you are not sure? Fourier could more strictly be described 
as a mathematical physicist. Not the same thing at all.

>He played fast and loose with the symbols to derive his
>results.
Not exactly, no.

>The mathematicians were shocked by the lack of rigor.
Not exactly, no. Fourier's work was later shown to be valid in all 
important respects.

>He was
>making all sorts of assumptions that were not on solid ground.
Not exactly, no. Depends who you're asking. Depends what you mean by 'solid 
ground'.

>However,
>over many decades there were able rigorously proove and further derive
>his ideas.
Not exactly, no.

Fourier claimed: "... but herein we have dealt with a single case only of a 
more general problem, which consists in developing any function whatever in 
an infinite series of sines of cosines of multiple arcs". He was extending 
the work of Lagrange. It was the basis of his investigation into heat 
flows. This necessarily involved the boundary value problem ... and 
therefore made it a problem within the calculus. The theory of boundary 
value problems is that of making solutions of differential equations fit a 
set of prescribed initial conditions. One way or another, most problems in 
mathematical physics reduce to boundary value problems. Usually, you solve 
for those, and you've solved your problem.

As to whether or not Fourier did actually solve his boundary value 
problems, this pretty much depends on your attitude to the distinction 
between pure and applied mathematics. Depends, in other words, what you are 
prepared to regard as 'a valid solution'. Very few pure mathematicians make 
a viable contribution to mathematical physics. The aims and standards of 
proof are somewhat different. Lagrange, Gauss, Archimedes, Euler, Riemann 
... some manage to straddle both camps. Fourier was like that. Far as I 
know, Jean Baptiste Fourier is generally regarded as a consummate 
mathematician.

A major difference between Fourier and others was the attitude to the word 
'function'. In that sense Fourier was of the lineage of Daniel Bernoulli 
who is generally regarded as the first really definable mathematical 
physicist. To Fourier, 'function' included not just those functions given 
by formulae, but also arbitrary representations -- i.e. sets of seemingly 
meaningless and disconnected points. This was what the stricter school of 
pure mathematicians did not seem to like.

Fourier initiated an enquiry into the nature of a function and what it took 
to sum them 'infinitesimally' given that not all functions, in Fourier's 
view, would be 'continuous' in what later became the 'accepted' sense. 
Roughly speaking, what Fourier provided was a method by which given 
functions with only a finite number of discontinuities in a given interval, 
and that therefore contained only a finite number of turning points within 
that interval, could nevertheless be summed by regarding them as an 
infinite sum of sines, cosines, or both. What Fourier pointed out was that 
sines and cosines are periodic, and that periodicity was a way to treat a 
function sufficiently well enough to allow outstanding and intractable 
physical problems -- such as the flow of heat through metal -- to be 
solved. Not only that, but any solution would be accurate enough for any 
reasonable purpose ... even though it might not be 'pin-point' accurate in 
the kind of way beloved by pure mathematicians. That was only ever really 
the issue. At least, that's what I always understood.

On the pure mathematicians' side, about 25 years later Dirichlet conducted 
the first rigorous (i.e. pure mathematical) study into Fourier series. He 
was the first to present a set of properly defined sufficiency conditions 
-- i.e. conditions that satisfied pure mathematicians. However, the 
Dirichlet definitions simply confirmed Fourier's original view about the 
generality of functions. But, against what Fourier had claimed, Dirichlet 
demonstrated that some of the 'inclusivenesses' and 'generalities' that 
Fourier proclaimed for his method, and thus for fully generalized sets of 
points, were false. This is probably what you are referring to. I do not 
really know, though.

Dirichlet achieved his aim with a classic and elegant reductio ab asurdum 
argument. One of the best in the canons of mathematical reasoning. But 
overall, what Dirichlet nonetheless did, in spite of this, was extend the 
function concept, within pure mathematics, in the way Fourier had 
originally suggested. The word 'function' therefore came to include very 
general and much broader correspondences of the Fourier kind.

Riemann later extended Fourier and Dirichlet's work yet further into the 
concept of the definite integral. Thus brought in yet more kinds of 
functions that failed to meet the 'piecewise continuous' requirements 
established principally by Cauchy. Riemann's demonstration that infinite 
sets of discontinuities did not necessarily remove the integrability 
property from the given set of points and/or functions was a big step 
forwards in helping to classify such sets. Riemann, though, limited his 
investigations to Fourier series. He did not go any further into the set 
theory itself. Later researchers, however, were able to establish the 
uniqueness of Fourier series. That's my understanding of the matter, anyway.

Therefore, what Fourier demonstrated was that there were indeed functions 
that although legally and properly expressible as functions, and that could 
therefore be integrated and differentiated, nevertheless did not have 
graphs that could be sketched. This was what the pure mathematicians found 
difficult to comprehend, what they initially had doubts about, and what 
they set about investigating. After all, they were forced to investigate 
it. They could hardly just dismiss it. Much like Heaviside did to later 
pure mathematicians with his somewhat eccentric method of solving the 
differential equations produced by wireless telegraphy, they simply had to 
check what Fourier did out ... simply because it worked. Same with 
Heaviside. They both produced results. If the pure mathematicians hadn't 
checked either of these guys out properly and then bent their theories to 
fit, they would have looked very stupid indeed.

Seems to me that the important point is that Fourier was quite justified in 
ignoring his (pure mathematical) critics. His, after all, was the first 
great step forward in boundary phenomena. As far as mathematical physicists 
were concerned the Fourier method worked just fine. It was plenty rigorous 
enough to produce the results they wanted, and to the kind of accuracy they 
desired in order to make problems in physics tractable. Fourier had made 
life a lot easier. Just find the appropriate Fourier coefficients.

While true there are theoretical restrictions, which Fourier admittedly 
bypassed, to the full and theoretical generality of his approach, those 
exceptions are of very little -- of almost no -- consequence to the fields 
of practice in which Fourier series are mostly used: the study of periodic 
phenomena in nature. Within that field, the exceptions to the Fourier 
method, although important to pure mathematicians, can on a physical level 
be regarded as anomalous cases with no real physical significance. And ... 
it is the physical significance that mathematical physicists and applied 
mathematicians are interested in. Mathematical physicists and applied 
mathematicians are both really mathematicians, though. As was Fourier.

In Fourier's defence, many say that he was quite right to ignore his 
critics ... but that he was probably mistaken in not conceding that those 
critics did maybe have a point when it came to discussing all possible 
curves in all possible mathematical universes .... even though the vast 
majority of those curves or points that might be dredged can make no 
conceivable appearance in the reality that mathematicians try to describe 
through their equations.

To be honest, Fourier published copious numbers of papers in pure 
mathematics. E.g. his work on determinate equations. Lagrange had shown how 
the roots of such an equation could be extracted by using a second equation 
whose roots were the squares of the differences of the roots of the 
original equation. Sturm gave the final solution. But ... this seems about 
as pure mathematical as you can get to me. Surely, anyone who can do that 
is a mathematician?

I am overall intrigued -- and a bit surprised -- by your statement that: 
'as a side note -- Fourier was not a mathematician'. As I said earlier, 
every book and Internet site I look at says that he was; and is. I have 
done my best to indicate why I am so puzzled by your claim, although I 
completely accept that everything I know is erroneous and therefore in need 
of correction.

Kool Musick
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Bravo, Kool!  I feel like I just sat thru an episode on the Discovery
channel!

One thing - it's "reductio ad absurdum" - reducing to absurdity.

> From: Kool Musick <koolmusick@...>
> 
> Dirichlet achieved his aim with a classic and elegant reductio ab asurdum
> argument.

marc lindahl wrote:

>One thing - it's "reductio ad absurdum" - reducing to absurdity.

Whoops.
I miswrote.
'Reductio ad absurdum' is exactly what I meant to write! But ... if that is 
my only error, and if nothing else I wrote can be 'reduced to an 
absurdity', then I am gratified. I do not know many things; but I had some 
truly excellent teachers to whom I am deeply grateful. The few things they 
managed to drill into my rather obtuse skull I like to know well. I owe 
them at least that much for they really were very good to me. Thank you 
most sincerely for your kind words. They are much appreciated.

Kool Musick
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> Lagrange, Gauss, Archimedes, Euler, Riemann 


These guys have caused me an awful lot of headaches in the last couple 
of years!!! :-))

> E.g. his work on determinate equations. Lagrange had shown how
> the roots of such an equation could be extracted by using a second 
> equation
> whose roots were the squares of the differences of the roots of the
> original equation. Sturm gave the final solution. But ... this seems 
> about
> as pure mathematical as you can get to me. Surely, anyone who can do that
> is a mathematician?

Question: is this the kind of stuff one would have to study e.g. in a 
bachelor business degree?
Reason why I ask this:  I had a math professor who made us study all 
these things and I argued that this was a bit "over the top" for 
business studies.
(although it's not a real bachelor, it's called "licence" over here, but 
in the US it's regarded as a bachelor)

I didn't do very well on that exam.

> I am overall intrigued -- and a bit surprised -- by your statement that:
> 'as a side note -- Fourier was not a mathematician'.

I don't think anyone will ever argue with you again on mathematics. :-) 
Not after reading this mail. :-)))

I didn't follow the thread and I'm a bit surprised to see what the 
question "is analog still better" can do to people! :-)))))))

>
>
>

-- 
Joeri Vankeirsbilck
joeri@...

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List-admin   Logic-users/SoundD*ver-users/Logic-TDM

>  > From: Dennis Gunn <dennisg@...>
>>
>>  Once about 9 years ago on the DAW I remember writing about how the
>>  digital wave form for a sine wave up around say 19.k looks like a
>>  broken spiral staircase and how I find it hard to believe that any
>>  amount of filtering is going to turn that into a clean sine wave.  A
>>  lot of peaple called me ignorant for being decieved by the look of
>>  data but Nothing I have seen or heard since has ever convinced me I
>>  was wrong.
>
>You're not hearing the data - you hear the data after it's been timed and
>output through a D/A converter, which includes a reconstruction filter.  To
>compare apples to apples you need to hook an oscilloscope up to that analog
>output and compare the two.  You'll see that the digital sinewave looks the
>same.
>
>There have also been studies done showing that different looking waveforms
>(due to the relative phase of thier harmonics) sound the same - so don't be
>deceived by how a waveform looks.  It's useful information, to be sure, like
>when editing, but it's not perfectly correlated to the sound.
>

I am aware of all that but looking at the data gives you an idea of 
the kind of contortions it takes to get it back into some semblance 
of the intended wave form at the intended pitch and it seems to be 
pretty hard to do because I can often hear aliasing distortion in 
high frequencies on softsynths and I think so can just about everyone 
else whether they realise that is what they are hearing or not.

Kool Musick wrote:
 > > Lagrange, Gauss, Archimedes, Euler, Riemann

Joeri Vankeirsbilck wrote:
 > These guys have caused me an awful lot of headaches in the last couple
 > of years!!! :-))
Me too. Let us be very clear. There's a big difference between those who 
can actually DO mathematics, and those who are just interested in it. I 
always enjoyed mathematics, but gradually I became less and less able to 
actually do it. I can manage some basic calculus but beyond that .... Since 
I quite enjoyed the subject, however, I became very fascinated by the 
history of mathematics, and by its philosophy. So I have an interest in 
people like Peano and Godel ... and I am also very interested in how 
mathematical ideas have developed over the centuries as new ideas have 
sprung up and the like.

When it comes to actually DOING it, however, they give me as big a headache 
as you do. I can no more actually follow say Lagrange's Celestial Mechanics 
than you probably could. But I do know what he was trying to achieve, and I 
know its place in the history of mathematics etc. It's the same with the 
history of music, I guess. I am very well aware of Mozart and The Beatles, 
and the periods in their lives at which they wrote certain things and why 
and so on and so forth ... but I am no more capable of writing the stuff 
they did than I am at solving a barrel of differential equations in my head.

 > E.g. his work on determinate equations. Lagrange had shown how
 > the roots of such an equation could be extracted by using a second
 > equation
etc etc etc.


 > Question: is this the kind of stuff one would have to study e.g. in a
 > bachelor business degree?
 > Reason why I ask this:  I had a math professor who made us study all
 > these things and I argued that this was a bit "over the top" for
 > business studies.

Interesting question. My immediate thoughts are:

A person who has a very good idea for a business does not in himself or 
herself have to know very much about organizing the details of his or her 
business or of making it run on a day to day basis. He or she simply has to 
come up with a very good idea and then learn to be a suitable magnet for 
those who believe that it will work ... and who are therefore willing to 
finance it. And, of course, financing a business is a big key to making it 
successful.

Once the business starts getting established things become very different. 
It becomes important to make 'correct' and 'wise' decisions and to see that 
the particular business is well integrated into its niche, into the broader 
community, and so on and so forth. Necessarily this means -- at least in my 
view -- benefitting from the ability to reason. It means importing into 
economic management the methods and techniques that a study of mathematics 
can teach so well.

It's hard to say if Lagrange, Gauss, Archimedes, Euler, and Riemann are 
truly 'beneficial' to the running of any one business ... but since those 
are theirs are the kinds of ideas that bandied around amongst those who, 
for example, want to decide on economic policies, discover new 
technologies, and make new products, then in general I think the financiers 
and controllers of business are being very foolish indeed, on a social 
scale, if they do not know and understand what others in the economy are 
talking about, and if they cannot keep pace with them intellectually at the 
very least.

Yes ... it is expensive, economically, to teach these things and to have 
them available in society, but on the whole I think they are indispensable 
things that should be made freely available and disseminated in any society 
that can afford it. I think, on the whole, that the business community 
cannot afford NOT to learn about them ... although I do accept that when 
you look at any individual business, or at any particular business person 
within any one organization, then it is easy to see that in that particular 
circumstance it is probably 'over the top' to have to do it. So ... I 
sympathise with you and with everyone else who does not want to study it 
because, honestly, I don't very much myself. However, I think it would be a 
big mistake to take it completely out of business schools altogether. Yes, 
I think they should be in business schools, but to set against that I 
really don't think people should be penalized either for avoiding them, or 
else for not being very good at them.

I hope this answers your question adequately, albeit in a very personal 
kind of way.

 > I didn't do very well on that exam.
I sympathise!!!

Kool Musick
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Oops, I thought I had sent this mail to Kool privately. My apologies.

-- 
Joeri Vankeirsbilck
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Kool
You are obviously a clever guy, and you wrote a great article, but 
frankly, I don't think you fully know what you are talking about. I mean 
no disrespect, but your comments are misleading. I hope you won't mind if 
I set them straight - especially since everyone seems to take these 
comments as valid and insightful.

>>Just a side note - but Fourier was not a mathematician.
>Yes he was.
>For example, he taught mathematics at L'Ecole Polytechnique -- and other 
>prestigious places. Why would they hire him as a mathematics lecturer if he 
>wasn't a mathematician? ..... Fourier could more strictly be described 
>as a mathematical physicist. Not the same thing at all.

Well a mathematical physicist is not a mathematician. Is a jazz pianist 
who plays something that sounds a bit classical, and actual classical 
pianist/composer? This is not a matter of snobbishness. Real 
mathematicians learn things in detail that scientists gloss over. 

>>He was
>>making all sorts of assumptions that were not on solid ground.
>Not exactly, no. Depends who you're asking. Depends what you mean by 'solid 
>ground'.


Let me give you an example of the sort of fundamental mistake that 
Fourier made. 

If you have sequence of functions {  f_n(x) : n = 1, 2, .....} then it is 
>obivously< true that
the you can interchange the integral and the limit :

        integral( limit ( f_n(x), n -> infinty) , x = 0 to x = infinity) =
            limit ( integral(   f_n(x) , x = 0 to x = infinity)  , n -> 
infinty)

Right? Wrong. Consider this simple example :
   let  f_n(x) = 1/n for 0 <= x <= n, and f_n(x) = 0 elsewhere.

  Then the 
       integral(   f_n(x) , x = 0 to x = infinity)  = 1 for all n, 
and therefore
       limit ( integral(   f_n(x) , x = 0 to x = infinity)  , n -> 
infinty) = 1 

On the other hand, 
   limit ( f_n(x), n -> infinty)  = 0 (the zero function)
   integral( 0, n -> infinty) , x = 0 to x = infinity) = 0

Clearly 1 does not equal 0.

If you want a stronger example, consider the Heaviside function
   H_n(x) = 1 for x >= n, and 0 for x < n.

This function has a limit of the zero function also, but each of the 
integrals are infinity, so your equation above would look like
    infinity = 0
Clearly this is false.

This is the kind of thing Fourier did. He assumed these kinds of things 
could be done by 'pushing symbols around' as we say in mathematics.

>One way or another, most problems in 
>mathematical physics reduce to boundary value problems. 

I'm not sure if all of your readers will recognize that these are 
differential equations - which can be notoriously difficult to solve and 
are still studied to this day.


>As to whether or not Fourier did actually solve his boundary value 
>problems, this pretty much depends on your attitude to the distinction 
>between pure and applied mathematics. Depends, in other words, what you are 
>prepared to regard as 'a valid solution'. Very few pure mathematicians make 
>a viable contribution to mathematical physics. The aims and standards of 
>proof are somewhat different.

Exactly my point!!! However 'different' is a euphemism. Its like when 
children are asked what is 2 + 2 and someone says 4 but someone else says 
5 and we say that second answer is simply 'different' that the first. No! 
Its wrong.

Mathematical proof developed from the time of Gauss (in the early to mid 
1800's) because so many crap math was done leading to erroneous results 
.... which Fourier carried on the tradition of. 


> Lagrange, Gauss, Archimedes, Euler, Riemann 
>... some manage to straddle both camps. Fourier was like that. 

Excuse me while spit my coffee out at that statement. Guass, Archimedes, 
and Newton are considered the three greatest mathematical minds that 
lived, although Galois was clearly a genius of enormous stature who died 
prematurely. Riemann, Euler, Lagrange, along with Cauchy, Hilbert and 
others were clearly first rate also. But Fourier is not even a 
mathematician and in fact committed errors on the level of a C student.

>Far as I 
>know, Jean Baptiste Fourier is generally regarded as a consummate 
>mathematician.

Nonsense.

>A major difference between Fourier and others was the attitude to the word 
>'function'. In that sense Fourier was of the lineage of Daniel Bernoulli 
>who is generally regarded as the first really definable mathematical 
>physicist. To Fourier, 'function' included not just those functions given 
>by formulae, but also arbitrary representations -- i.e. sets of seemingly 
>meaningless and disconnected points. This was what the stricter school of 
>pure mathematicians did not seem to like.

This is completely and totally untrue. It actually the reverse. 
Weirstrauss and others in the late 1800's lead the revolution in 
mathematics that considered functions to be defined on any metric space - 
very abstract indeed - I suspect this is too abstract for manuy 
physicists to  comprehend because they like to deal with real world 
things. (I mean to no disrespect to physicists, in fact my younger 
brother has a PhD in this.) Many functions were devised around that time 
which are can not be expressed as formula - some as limits or other 
things. Mathematics is nothing if not abstract, and it was not the 
mathematicians that didn't like the greater generality.


>Fourier initiated an enquiry into the nature of a function and what it took 
>to sum them 'infinitesimally' given that not all functions, in Fourier's 
>view, would be 'continuous' in what later became the 'accepted' sense. 
>Roughly speaking, what Fourier provided was a method by which given 
>functions with only a finite number of discontinuities in a given interval, 
>and that therefore contained only a finite number of turning points within 
>that interval, could nevertheless be summed by regarding them as an 
>infinite sum of sines, cosines, or both. What Fourier pointed out was that 
>sines and cosines are periodic, and that periodicity was a way to treat a 
>function sufficiently well enough to allow outstanding and intractable 
>physical problems -- such as the flow of heat through metal -- to be 
>solved. Not only that, but any solution would be accurate enough for any 
>reasonable purpose ... even though it might not be 'pin-point' accurate in 
>the kind of way beloved by pure mathematicians. That was only ever really 
>the issue. At least, that's what I always understood.
>
>On the pure mathematicians' side, about 25 years later Dirichlet conducted 
>the first rigorous (i.e. pure mathematical) study into Fourier series. 

i.e. the first valid study... Dirichlet was first rate.

>However, the 
>Dirichlet definitions simply confirmed Fourier's original view about the 
>generality of functions. 

You mean Dirichlet actually proved Fourier's wild conjectures happened to 
work out.

>But, against what Fourier had claimed, Dirichlet 
>demonstrated that some of the 'inclusivenesses' and 'generalities' that 
>Fourier proclaimed for his method, and thus for fully generalized sets of 
>points, were false. 

uh huh!

>Riemann later extended Fourier and Dirichlet's work yet further into the 
>concept of the definite integral. 

That may be news to Sir Isaac Newton and Liebnitz who lived about 200 
years earlier, and who actually invented the definite integral.


>Thus brought in yet more kinds of 
>functions that failed to meet the 'piecewise continuous' requirements 
>established principally by Cauchy. Riemann's demonstration that infinite 
>sets of discontinuities did not necessarily remove the integrability 
>property from the given set of points and/or functions was a big step 
>forwards in helping to classify such sets. Riemann, though, limited his 
>investigations to Fourier series. He did not go any further into the set 
>theory itself. Later researchers, however, were able to establish the 
>uniqueness of Fourier series. That's my understanding of the matter, anyway.

You could be right, but it was my understanding that another French 
mathematician Lebesgue was the one who discovered the most general form 
of integration which is called  the "Lebesgue Integral" and is based on 
measure theory - which is standard material for first year grad students 
in math, or advanced senior math majors.


>Therefore, what Fourier demonstrated was that there were indeed functions 
>that although legally and properly expressible as functions, and that could 
>therefore be integrated and differentiated, nevertheless did not have 
>graphs that could be sketched. 

Again, I think this is wrong. Fourier is being credited with the 
discoveries of dozens of mathematicians who lived a generation or two 
before him.


>This was what the pure mathematicians found 
>difficult to comprehend, what they initially had doubts about, and what 
>they set about investigating. After all, they were forced to investigate 

I think it was the pure mathematicians who discovered these things in the 
first place. Maybe I should read up more on Fourier but the reason I 
don't know much of his work, is that I never saw any reference to any 
result of the kind you speak of - in fact the only time I ever saw his 
name mentioned was attached to the well known series. There were no 
theorems, lemmas, or corallaries named after him. There were no 
historicial footnotes about his discoveries - but there were plenty on 
many others. I heard more about Sonya Kobaleski, Weirstrauss' protege 
mathematician and young lover than I heard of Fourier.

>it. They could hardly just dismiss it. Much like Heaviside did to later 
>pure mathematicians with his somewhat eccentric method of solving the 
>differential equations produced by wireless telegraphy, they simply had to 
>check what Fourier did out ... simply because it worked. 

...at least in some ideal circumstances.... but how did he know they 
would work in all cases?

>Same with 
>Heaviside. They both produced results. If the pure mathematicians hadn't 
>checked either of these guys out properly and then bent their theories to 
>fit, they would have looked very stupid indeed.

The mathematicians fleshed out the details to make these fellows look 
important while no one remembers their names.

>Seems to me that the important point is that Fourier was quite justified in 
>ignoring his (pure mathematical) critics. His, after all, was the first 
>great step forward in boundary phenomena. As far as mathematical physicists 
>were concerned the Fourier method worked just fine. 

That true! Because mathematical phsycists are not mathematicians, and 
anything that looks reaonsable must surely be true, right? (See example 
above)


>While true there are theoretical restrictions, which Fourier admittedly 
>bypassed, to the full and theoretical generality of his approach, those 

Thats what i'm talkin' bout!

>exceptions are of very little -- of almost no -- consequence to the fields 
>of practice in which Fourier series are mostly used: the study of periodic 
>phenomena in nature. 

Some of the most complicated stuff there is comes from nature. I remember 
one of my professors telling me 20 years ago that if they could simply 
solve non-linear differential equations, then the could unlock the keys 
to such things as earthquakes and weather prediction. To this day, those 
things have not been solved.

I think the proper restatement of the sentence above would be - they work 
out well in ideal conditions or of problems that you handpick to 
cooperate with these methods.

>Within that field, the exceptions to the Fourier 
>method, although important to pure mathematicians, can on a physical level 
>be regarded as anomalous cases with no real physical significance. And ... 
>it is the physical significance that mathematical physicists and applied 
>mathematicians are interested in. Mathematical physicists and applied 
>mathematicians are both really mathematicians, though. As was Fourier.
>
>In Fourier's defence, many say that he was quite right to ignore his 
>critics ... 

his 'critics' ... you mean the people actually knew what they were doing?


Fourier had a good imagination for something but didn't have the 
mathematical horsepower to actually prove his ideas. That was left for 
mathematicians to do. In the end, his name is famous for having the 
original idea, but it would have been worthless if not on solid ground.

> From: Joeri Vankeirsbilck <joeri@...>
> 
> Question: is this the kind of stuff one would have to study e.g. in a
> bachelor business degree?
> Reason why I ask this:  I had a math professor who made us study all
> these things and I argued that this was a bit "over the top" for
> business studies.
> (although it's not a real bachelor, it's called "licence" over here, but
> in the US it's regarded as a bachelor)

It's applicable to Economics, for some of the modelling you might do, but
that would be pretty advanced.  Or, it might be used in the insurance field.

>Dennis Gunn <dennisg@...> wrote:
>
>>Once about 9 years ago on the DAW I remember writing about how the
>>digital wave form for a sine wave up around say 19.k looks like a
>>broken spiral staircase and how I find it hard to believe that any
>>amount of filtering is going to turn that into a clean sine wave.
>
>Nine years is a long time in the history of DAWs...

Yes it is. But I think it was about nine years ago I bought one the 
first batch of audio media cards that came out and was using the very 
first versions on Studio Vision.  Was it less?  Maybe.


>It is entirely feasible to digitally filter a square wave to produce a
>sinewave within the practical limits of a digital systems parameters,
>which is effectively what is going on in what you describe above.

No  a square wave isn't what I am describing and if you have ever 
looked at the digital data for a 19k sine wave in a 44.1k pcm signal 
then you know what I mean.  It looks like there will be resonances at 
other frequencies and as a matter of fact with most digital 
synthesizers I have ever used especially the older ones I can hear 
those resonances as undertones in what should be pure high 
frequencies.


>In any case, why is it hard to believe? Isn't filtering one of
>the cornerstones of sound processing? Was Fourier wrong??

I don't know if he was wrong or right but I am pretty sure that his 
mathematics describe something that is difficult to manifest 
perfectly in an audio circuit in the real world and it is my opinion 
that plays a significant factor in the character that we all hear in 
digital audio that makes it a little less than the "perfect" 
recording medium we all dreamed it would be way back when it was new.

>The problem lies in allowing >> nyquist frequencies to get to the
>the output, as for example would happen in an unfiltered saw or
>square oscillator at even modest frequencies with a less than ideal
>filter at the converter stage. Better filter implementaions,
>Greater sample rates and higher bit depths allwork to minimise this
>problem...
>
>>A
>>lot of peaple called me ignorant for being decieved by the look of
>>data but Nothing I have seen or heard since has ever convinced me I
>>was wrong.
>
>Though I don't entirely disagree with you, from what I have learned of
>you on this list I don't find that statement a complete surprise.

So your saying I am not wrong but I should be?  You are pretty 
consistent yourself.

Hi Joeri
Well I have my degrees in  mathematics and even published two papers in 
math, but I'm a bit rusty in the more advanced topics since I haven't 
dealt with them much in quite a few years. I usually teaching only 
through four semesters of calculus - none of the upper division or 
graduate stuff. 

> > Question: is this the kind of stuff one would have to study e.g. in a
> > bachelor business degree?
> > Reason why I ask this:  I had a math professor who made us study all
> > these things and I argued that this was a bit "over the top" for
> > business studies.

It depends on exactly what you are learning. From what you mentioned it 
didn't seem too relevant. However, I'm teaching business calculus now, 
and I hear that same complaint all the time. However, they don't know 
what they will later need in the professional workplace. Some financial 
analysts do use calculus or DiffEQ and make $300,000 a year. Others 
don't.  We do stuff with elasticity of demand, consumer surplus, etc. It 
seems pretty relevant to me.

Part of what you learn in math, like what you learn in philosophy or 
literature, is not a direct method you use the next day, but rather an 
ability to think more precisely and work hard to achieve a goal.

Do you want to share some problems you consider unnecessary?

Thoughts from the mind of Bruce Bartlett, 07-11-2001:

>  >The polymoog had a custom-made chip per key.  Each chip contained
>  >(more or less probably) an entire monophonic synthesizer.  That's the
>>only way they were able to build polyphonic synths at the time.
>
>Not quite. The Polymoog used  two master oscillators routed through
>divide-down chips, like electronic organs, string synths, and a few early
>Korg poly synths. This is why it sounded so thin and crappy. So it did not
>have a complete mono synth on each key -- but it did have a filter and amp
>for each key...

That's why I included the "more or less probably", as I was not 
entirely sure. Thanks for setting things straight.

cheers,
HJ
-- 
     Hendrik Jan Veenstra
     email: mailto:h@...
     www:   http://www.ision.nl/users/h/index.html

ooops sorry - that was meant to be private ....

>Hi Joeri
>Well I ....

--- In logic-ot@y..., GAmoore@a... wrote:
> ooops sorry - that was meant to be private ....
> 
> >Hi Joeri
> >Well I ....

--- In logic-ot@y..., Joeri Vankeirsbilck <joeri@b...> wrote:
> Oops, I thought I had sent this mail to Kool privately. My 
apologies.
> 


Could Emagic fix this in LA5 please!!!   ;-)

--- In logic-ot@y..., GAmoore@a... wrote:
> Let me give you an example of the sort of fundamental mistake that 
> Fourier made. 
> 
> If you have sequence of functions {  f_n(x) : n = 1, 2, .....} then it is 
> >obivously< true that
> the you can interchange the integral and the limit :
> 
>         integral( limit ( f_n(x), n -> infinty) , x = 0 to x = infinity) =
>             limit ( integral(   f_n(x) , x = 0 to x = infinity)  , n -> 
> infinty)
> 
> Right? Wrong. Consider this simple example :
>    let  f_n(x) = 1/n for 0 <= x <= n, and f_n(x) = 0 elsewhere.
> 
>   Then the 
>        integral(   f_n(x) , x = 0 to x = infinity)  = 1 for all n, 

I'm sorry but wouldn't the result of this be: limit( ((1/n)*x, x = 0 to x = y), y -> infinity) which is not equal to 1 for all n?

> and therefore
>        limit ( integral(   f_n(x) , x = 0 to x = infinity)  , n -> 
> infinty) = 1 

This result is correct although the previous assumption was not correct.

> 
> On the other hand, 
>    limit ( f_n(x), n -> infinty)  = 0 (the zero function)
>    integral( 0, n -> infinty) , x = 0 to x = infinity) = 0
> 
> Clearly 1 does not equal 0.
> 
> If you want a stronger example, consider the Heaviside function
>    H_n(x) = 1 for x >= n, and 0 for x < n.
> 
> This function has a limit of the zero function also, but each of the 
> integrals are infinity, so your equation above would look like
>     infinity = 0
> Clearly this is false.

Your conclusion is right. But I thought the equation you started with was ONLY true under certain pre-conditions, the one about the limit of the integral is the same as the integral of the limit. One of the pre-conditions being that the functions must 'converge uniformly'( I hope this is the correct term). This means that the limit( abs(f_n -f), n -> infinity) = 0, where abs is the absolute function and f is a function. You introduced a function into the equation that violates that pre-condition; both functions aren't uniformly convergent.
Check your calculus books, man. At least you gave me a jolt to check my college first year calculus stuff again. Thanks, :-).

> This is the kind of thing Fourier did. He assumed these kinds of things 
> could be done by 'pushing symbols around' as we say in mathematics.

Wrong. Fourier's equations satisfy the precondition I mention above. Maybe he didn't write it in his publications this way, but they were valid. Stop dissing Fourier, man!! :-).

Thoughts from the mind of yoonchinet@..., 08-11-2001:

>--- In logic-ot@y..., GAmoore@a... wrote:
>  >    let  f_n(x) = 1/n for 0 <= x <= n, and f_n(x) = 0 elsewhere.
>>
>>    Then the
>>         integral(   f_n(x) , x = 0 to x = infinity)  = 1 for all n,
>
>I'm sorry but wouldn't the result of this be: limit( ((1/n)*x, x = 0 
>to x = y), y -> infinity) which is not equal to 1 for all n?

The graph of the functions Greg describes looks like (in Courier please)


      |
1/n -|=======================
      |
      |
      |
      +----------------------+=======================
                             n

i.e. a horizontal line at height 1/n, from x=0 to x=n, and at height 
0 for x>n.  The integral is simply the surface between the function 
and the x-axis, which is clearly n*1/n = 1.

Your
     limit( ((1/n)*x, x = 0 to x = y), y -> infinity)
is indeed not 1 for all n (in fact, it's infinity for all n>0). 
However, this limit is not the same as the previous integral. 
(1/n)*x is a primitive function for 1/n -- but 1/n is _not_ what's 
being integrated.

{GAmoore again]
>  > and therefore
>>         limit ( integral(   f_n(x) , x = 0 to x = infinity)  , n ->
>>  infinty) = 1

Correct.  This limit is simply (well, loosely speaking) an infinite 
line at height 1/infinite, which still encloses a surface of 1.


>  > On the other hand,
>>     limit ( f_n(x), n -> infinty)  = 0 (the zero function)
>>     integral( 0, n -> infinty) , x = 0 to x = infinity) = 0

Also correct.


cheers,
HJ
-- 
     Hendrik Jan Veenstra
     email: mailto:h@...
     www:   http://www.ision.nl/users/h/index.html

--- In logic-ot@y..., Hendrik Jan Veenstra <h@k...> wrote:
> Thoughts from the mind of yoonchinet@y..., 08-11-2001:
> 
> >--- In logic-ot@y..., GAmoore@a... wrote:
> >  >    let  f_n(x) = 1/n for 0 <= x <= n, and f_n(x) = 0 elsewhere.
> >>
> >>    Then the
> >>         integral(   f_n(x) , x = 0 to x = infinity)  = 1 for all n,
> >
> >I'm sorry but wouldn't the result of this be: limit( ((1/n)*x, x = 0 
> >to x = y), y -> infinity) which is not equal to 1 for all n?
> 
> The graph of the functions Greg describes looks like (in Courier please)
> 
> 
>       |
> 1/n -|=======================
>       |
>       |
>       |
>       +----------------------+=======================
>                              n
> 
> i.e. a horizontal line at height 1/n, from x=0 to x=n, and at height 
> 0 for x>n.  The integral is simply the surface between the function 
> and the x-axis, which is clearly n*1/n = 1.
> 
> Your
>      limit( ((1/n)*x, x = 0 to x = y), y -> infinity)
> is indeed not 1 for all n (in fact, it's infinity for all n>0). 
> However, this limit is not the same as the previous integral. 
> (1/n)*x is a primitive function for 1/n -- but 1/n is _not_ what's 
> being integrated.


You are right! My mistake; the integration should've been done piece-wise. Thanks for correcting me.

Yoonchi.

I am sorry for the delay in responding properly but I have been very busy 
all day.

GA Moore wrote:

>Kool ... wrote a great article,
Heaven's sake ... it was just an email!!!

>  but
>frankly, I don't think you fully know what you are talking about.
Probably not.
I am happy to bow to your superior wisdom.

>  I mean
>no disrespect, but your comments are misleading.
OK.

>I hope you won't mind if
>I set them straight -
Not at all.

>especially since everyone seems to take these
>comments as valid and insightful.
Really? Then it really is best if you set them straight pretty smartish.

>Well a mathematical physicist is not a mathematician.
Depends who you ask, to be honest.

>Real mathematicians learn things in detail that scientists gloss over.
Depends on the historical epoch in which they are judged. This distinction 
between 'mathematician' and 'scientist', as also between 'pure' and 
'applied' mathematics is largely a modern contrivance and convention. 
Historically, they were not originally distinct. By your argument the 
Egyptians and Babylonians were not 'real' mathematicians either and 
therefore should not figure in books on the history of mathematics. Seems 
absurd to me.

>Let me give you an example of the sort of fundamental mistake that
>Fourier made.
Since these were not the 'errors' that Fourier made I have zero interest in 
your examples. The errors that Fourier ACTUALLY made I am very interested 
in. As was Dirichlet, actually.


>If you want a stronger example,
Not particularly, no thank you, because (a) I really don't see what point 
they are making; and (b) contrary to your assertion, they are simply NOT 
the kinds of errors that Fourier made

>This is the kind of thing Fourier did.
No.

>  He assumed these kinds of things
>could be done by 'pushing symbols around' as we say in mathematics.
Since you are not even close to what Fourier actually asserted and did, I 
really do not understand where this judgement comes from.

Anyway ... here's the kicker ... what exactly is wrong with that?
To put it another way ... if mathematics is going to increase, improve, and 
extend its boundaries, someone or other has to come up with exciting new 
ideas, things with new possibilities, new skeletons on which flesh and 
clothing can be put. So ... the people who come up with the skeletons ... 
by your criteria they are 'not really mathematicians'? Seems absurd to me. 
Nonetheless, you are entitled to your opinion. I am entitled to mine.

> >One way or another, most problems in
> >mathematical physics reduce to boundary value problems.
>I'm not sure if all of your readers will recognize that these are
>differential equations
So what? I gave an accurate description of the nature of most problems in 
mathematical physics. They reduce to boundary value problems.

Kool said:
> >As to whether or not Fourier did actually solve his boundary value
> >problems, this pretty much depends on your attitude to the distinction
> >between pure and applied mathematics. Depends, in other words, what you are
> >prepared to regard as 'a valid solution'. Very few pure mathematicians make
> >a viable contribution to mathematical physics. The aims and standards of
> >proof are somewhat different.
>
>Exactly my point!!!
My point also.

>However 'different' is a euphemism.
No it isn't.

>Its like when
>children are asked what is 2 + 2 and someone says 4 but someone else says
>5 and we say that second answer is simply 'different' that the first. No!
>Its wrong.
This is plain ridiculous. 5 is known to be wrong at the moment that it is 
offered because we have a valid method AT THE PRESENT for coming by the 
correct answer. At the time that Fourier was working, there was no method 
for judging whether answers were right or wrong. That is what everybody was 
struggling with. It is all very well to sit in a comfy armchair now, some 
400 years later, and smirk and say 'blah blah was wrong'. At the time, 
there was no method.

The problem of the day and of the time was that, probably because there was 
no general method, mathematicians and scientists were nonetheless 
consistently able to get correct answers using Fourier's methods, although 
they had not the slightest idea why or how. You may know NOW, sitting in 
the 21st century, but they didn't know back then. So ... they set about 
investigating how and why they could sometimes get correct results, and 
they set about trying to find out when and how it wouldn't work either. 
They knew ALL ABOUT 2+2=5, so your example is just fatuous and plain 
ridiculous . And what they managed to do was come up with proper methods.

By the same token, the Ancient Egyptians did really know how to produce a 
right-angle triangle whenever they wanted to by using what we now call the 
Pythagorean Triangle. They did not, however, have what we would now regard 
as a valid proof. The Greeks caused a revolution by coming up with such 
things. So ... the Ancient Egyptians were not real mathematicians? Is that 
what we are to conclude? So ... when someone writes a book on the history 
of mathematics they should go straight to Ancient Greece and not bother at 
all with Ahmes and people like that? Because Ahmes and people like that 
could not possibly be real mathematicians because they were not Greeks? 
Strange book on the history of mathematics that would be IMHO.

>Mathematical proof developed from the time of Gauss (in the early to mid
>1800's) because so many crap math was done leading to erroneous results
>.... which Fourier carried on the tradition of.

Let's see ... Carl Friedrich Gauss, 1777-1855, Jean Baptiste Fourier, 
1768-1830. Pretty contemporaneous.

Gauss radically changed the nature of mathematics. It just became a 
completely different subject. He helped change it because of, admittedly, 
the problems caused by the 'intuitive' approach of people like Fourier. And 
... let's not forget that the 'Fourier approach' was pretty much the same 
approach adopted by Euler, 1707-1783, Lagrange, 1736-1813, and, really 
every other mathematician in history up to that point. Mathematics in that 
era was certainly in a mess. That mess was caused by calculus. And ... 
Gauss more than anyone showed the way out of it. Speaking generally, the 
Gauss method was WAY better than the Fourier method, and you've got no 
quarrel from me if that's all you're saying. However, your proposition that 
Fourier was not 'really a mathematician' is denied. It is denied not just 
by me but by every single book ever written on the history of mathematics, 
and every single Internet site I have examined. I invite anyone reading 
this to check out that fact.

Point is, Fourier lived at the very time that mathematics was changing to 
become what it is today; and changing in such a way as to make it clear 
that the real future of mathematics lay in the methods of Gauss and those 
who followed him and not those who worked like Fourier. It was, however, a 
transitional era. And ... Fourier belonged to that era. A member of the 
theoretical old guard of his day maybe, but nevertheless a real and working 
mathematician of that day, and one of the ones involved in demonstrating 
perhaps why there ought to be a change ... as well as demonstrating what 
was valuable about the old methods. Because what Fourier did was most 
valuable.

>Excuse me while spit my coffee out
Go ahead.

>But Fourier is not even a mathematician
You are entitled to your opinion. I do not share it.
And, to be honest, nor does anyone else I have read who has printed a book 
on the history of mathematics share that opinion. You cannot, in fact, 
write a history of mathematics and the calculus without mentioning Fourier. 
What he did was that important.

>and in fact committed errors on the level of a C student.
Depends where you're standing to judge him. At the time, it was not 
possible to give A's B's and C's because there were no standards and nobody 
to set an exam, and nobody to mark it. Nobody would even have known what to 
PUT in the exam, come to that. This was fringe and trend-setting stuff.

Anyway, Fourier was plenty smart enough to set people a challenge. A man 
must surely be pretty smart to set people a challenge that it took people 
like Dirichlet and Riemann and Lebsgue to resolve. Smart as Fermat, I'd 
say, who also set a challenge that it took aeons to resolve. Or are you 
going to say that Fermat wasn't a real mathematician either?

> >Far as I
> >know, Jean Baptiste Fourier is generally regarded as a consummate
> >mathematician.
>
>Nonsense.
Once again, you are entitled to your opinion. I do not share it. The 
majority opinion sides with me. I rest content with that.

>Mathematics is nothing if not abstract, and it was not the
>mathematicians that didn't like the greater generality.
This is not even remotely close to what I said.

>You mean Dirichlet actually proved Fourier's wild conjectures happened to
>work out.
If they worked out ... then they could not have been that wild, could they? 
Seems as though they were spot on to me.

> >Riemann later extended Fourier and Dirichlet's work yet further into the
> >concept of the definite integral.
>
>That may be news to Sir Isaac Newton and Liebnitz who lived about 200
>years earlier, and who actually invented the definite integral.
Sorry, but you are again misrepresenting what I said. I clearly said 
'extended ... yet further into'. I did NOT say 'invent'.

> >Riemann, though, limited his
> >investigations to Fourier series. He did not go any further into the set
> >theory itself. Later researchers, however, were able to establish the
> >uniqueness of Fourier series. That's my understanding of the matter, anyway.

>You could be right, but it was my understanding that another French
>mathematician Lebesgue was the one who discovered the most general form
>of integration ...
Let's see ... Riemann, 1826-1866, Lebesgue, 1875-1941.
Strange .... I could SWEAR that my original sentence has a 'Later' in it. 
And ... what do I see when I check those dates? I see that Lebesgue was 
born 9 years after Riemann had died. Seems like a 'Later' to me.

>which is called  the "Lebesgue Integral" and is based on
>measure theory - which is standard material for first year grad students
>in math, or advanced senior math majors.
Your point being?

> >Therefore, what Fourier demonstrated was that there were indeed functions
> >that although legally and properly expressible as functions, and that could
> >therefore be integrated and differentiated, nevertheless did not have
> >graphs that could be sketched.
>
>Again, I think this is wrong. Fourier is being credited with the
>discoveries of dozens of mathematicians who lived a generation or two
>before him.
What Fourier is being credited with is demonstrating that there did in fact 
exist proper functions that could be expressed as functions and that worked 
as functions, but that nobody had yet invented a proper theory for. If he 
had not had that original idea, what on EARTH would there have been for 
those later 'dozens of mathematicians' to research into? Fourier stood at 
the head of a whole new branch of mathematics, and it is rightly named 
after him.

>Maybe I should read up more on Fourier
I think so, to be frank and honest. There's rather a lot about him you just 
don't seem to know. Which is probably why you have been reduced to making 
things that you then claim he simply must have done, and which he frankly 
did not do by any conceivable reading of history just to try to make your 
point. Me ... I just stick to what he actually did ... and I just also 
stick to readily admitting the things he did not do. My job is easy because 
I do not have to invent problems that Fourier never remotely tried to solve 
as examples of how stupid he was. He may have been stupid, but please stick 
to demonstrating this with problems that he did actually tackle. And ... if 
you stick to problems that he did actually tackle you will observe that he 
was right the bulk of the time ... which was the problem everyone else had 
with him. Please stick to what he actually did. Life's a lot easier that way.

>but the reason I
>don't know much of his work, is that I never saw any reference to any
>result of the kind you speak of - in fact the only time I ever saw his
>name mentioned was attached to the well known series.
I wonder why on earth the Fourier series is quite so well known? Let me see 
... could it be because it's INCREDIBLY useful, because it stimulated new 
mathematics, because it still provides wonderful research topics today ... 
stuff like that?

>  There were no
>theorems, lemmas, or corallaries named after him.

So what? He is named for what he did, and for the theoretical contribution 
he did actually make to the history of mathematical thought.

In about 1750 (please forgive me for not remembering the exact date) Euler 
defined a function as a variable quantity that is dependent on another 
quantity. This does, admittedly begin to approach today's definition. 
However, it is simply not good enough. It was FOURIER, and not anyone else, 
who stood up against that definition. He met great opposition in doing so, 
but NOT for the reasons that you personally have given. It was FOURIER, and 
not anyone else, who took the very important step of demonstrating that the 
proposed Euler definition was totally unsatisfactory. He did this -- as you 
know -- by introducing series with sines and cosines as terms. It was this, 
and only this, that led to the later concept that a given representation of 
a function might only be valid for a certain range of values. I think even 
you will surely agree that this is a vitally important part of what we call 
calculus today, no? Well ... Fourier did that. And it was based on 
FOURIER'S suggestions -- and not anybody else's suggestions for there were 
a lot doing the rounds at the time -- that Dirichlet proposed that a 
function is a correspondence that assigns a unique value of the dependent 
variable to every permitted value of an independent variable. Please not 
the importance of 'unique' and 'permitted'. Since you teach calculus you 
will appreciate how vital all of this is. That is what Fourier initiated. 
Nobody else initiated it. Everybody else wanted to trot merrily along with 
what Euler and Cauchy had proposed. Fourier just smiled and showed that it 
simply would not do. He had his idea. It worked. Dirichlet agreed with him. 
Just as well that Dirichlet did, actually.

>There were no
>historicial footnotes about his discoveries
This is because Fourier's discoveries were not footnotes. Seems plain and 
simple to me. Fourier's discoveries were absolutely vital to the theories 
and development of mathematics. He challenged the prevailing definitions. 
He showed that they were totally inadequate. Since you seem to like 
bringing in rather ridiculous analogies, then it's as if Fourier climbed in 
a boxing ring with Euler -- and many important others -- and knocked them 
down. He showed that they were all wrong and totally barking up the wrong 
tree in trying to define a function. They were all wrong. He was the one on 
the right track. He was not ALL right, but he was on a path that was much 
more fruitful than anyone else's. Therefore, he has a whole branch of 
mathematics named after him. And rightly so.

Seems to me that to deny that Fourier is a mathematician is a bit like 
denying that someone who can enter a boxing ring with Muhammad Ali and 
knock him down is not really a boxer but could only have done it by a 
fluke. Nice fluke. Maybe so ... but he'll go down in the history of boxing 
anyway ... and even though people will argue for ever afterwards if he 
really did knock down Muhammad Ali with skill, or whether Ali just kind of 
happened to fall over his own feet by accident while Fourier was 
simultaneously flinging his arm. Either way, the man fell. And ... Fourier 
was there swinging his arm. There's no denying that.

> >it. They could hardly just dismiss it. Much like Heaviside did to later
> >pure mathematicians with his somewhat eccentric method of solving the
> >differential equations produced by wireless telegraphy, they simply had to
> >check what Fourier did out ... simply because it worked.
>
>...at least in some ideal circumstances.... but how did he know they
>would work in all cases?
He didn't. He was wrong there, as others proved, and as I have accepted. 
But he was right that they worked IN SOME CASES. The 'some cases' bit was 
absolutely vital at the time. It is simply that you are suffering from 
rather a large dose of hindsight, because the only thing you know about is 
the theories within calculus that actually worked. You seem to know zip 
about the many theories that didn't work. But then ... why should you? 
They're of scant interest to a working mathematician after all. 
Nevertheless, I strongly suggest you curl up with a good book on the 
history of calculus one day and read a bit more about the host of theories 
of that time that DIDN'T work. Lots of good people studied them and lots of 
good people couldn't get anywhere with any of them. Therefore (a bit like 
the phlogiston theory) they are known only to the few people like me who 
have an arcance kind of interest in the history of the subject.

Fourier is famous because out of all the truly crackpot theories of the day 
about limits and calculus and so on and so forth, HIS was the one that 
contained the seed that actually bore fruit. I think, actually, if you knew 
a bit more about all those bogus ideas that simply didn't pan out; and if 
you found out a bit more about how even people like Cauchy were locking 
themselves up in dark rooms and screaming and going almost mad (with Cauchy 
that really and nearly happened); and if you read a bit more about how they 
were clutching at every straw they could find to try to resolve these 
issues, then I think you'd have just a tad more respect for Fourier and for 
his achievements. I repeat, if you really think that Fourier's an idiot and 
a bogus mathematician then you really should catch sight of some of them 
others. You'd probably give up the subject in order not to be associated 
with them.

Nobody is denying that Fourier was wrong in certain respects with his 
ideas. But ... there were lots of others who were totally wrong in every 
way. And that's really the point here. Out of all the people who were 
totally wrong, he was the one person who had a really rather smart idea 
that could be worked on. None of the others were in the least workable. And 
that, surely, is a notable achievement. In my boo, that makes him a 
mathematician. Quite why in your book it doesn't is a bit beyond me, 
frankly, but everyone else who studies this subject sides with me so I feel 
quite OK about it all.

In any case ... what's so shameful, and what's there to turn up your nose 
at, at the errors of mathematicians. Euler was wrong about a fair few 
things as well. So ... are you now going to propose that Euler must have 
been a bad mathematician because he made a couple of boo-boos? Just where 
would that kind of madness end?

> >Same with
> >Heaviside. They both produced results. If the pure mathematicians hadn't
> >checked either of these guys out properly and then bent their theories to
> >fit, they would have looked very stupid indeed.
>
>The mathematicians fleshed out the details to make these fellows look
>important while no one remembers their names.
This is out and out snobbery. Of the worst kind.

>That true! Because mathematical phsycists are not mathematicians, and
>anything that looks reaonsable must surely be true, right?
So also is this.

Paul Dirac is another example. A fine mathematician who found a method for 
solving equations that made a major contribution to theoretical physics. 
You, doubtless, would call him 'not really a mathematician'. Nevertheless, 
he has a whole algebra named after him on account of what he contributed to 
mathematical and algebraic theory whilst dealing with the mathematics of 
quanta. But then hay ... mathematical physicists are not mathematicians, 
right? Strange then, isn't it, that Dirac should have a whole system of 
algebra named after him. Probably, though, it's not a bona fide part of 
algebraic theory. How could it be? Dirac was 'only' a mathematical 
physicist, after all. Not really a REAL mathematician. Yet ... there his 
system of algebra stands. What ARE we going to do about that?

> >While true there are theoretical restrictions, which Fourier admittedly
> >bypassed, to the full and theoretical generality of his approach, those
>
>Thats what i'm talkin' bout!
I never denied them.

>I think the proper restatement of the sentence above would be - they work
>out well in ideal conditions or of problems that you handpick to
>cooperate with these methods.
This is what applied mathematicians, mathematical physicists, and 
physicists do. They do not try to solve problems that do not help them 
understand better how nature works. They concentrate on solving those kinds 
of problems that are relevant to the solutions of practical problems. 
What's wrong with that, exactly?

> >In Fourier's defence, many say that he was quite right to ignore his
> >critics ...
>
>his 'critics' ... you mean the people actually knew what they were doing?
Fourier knew what he was doing. It was them other guys who had a problem.

>Fourier had a good imagination for something but didn't have the
>mathematical horsepower to actually prove his ideas.
What you mean is, to prove his ideas to YOUR satisfaction, which is 
essentially to the satisfaction of a set of criteria that only entered 
mathematics through the work of Gauss and with whom he was admittedly 
contemporaneous.

As a whole, mathematics did very much better to follow the path of Gauss 
than of Fourier. I am not for one moment denying that. It would be rampant 
stupidity. Nevertheless, there is a role in mathematics for the kinds of 
creative insights that people like Fourier have. He set a challenging 
problem. He showed his methods would work in at least some circumstances. 
He set such a problem that it took generations before people could sort it 
out. He kept people like Dirichlet and Riemann well occupied. Not bad for 
"a mere hack". Congratulations, Mr. Fourier. And, in fact, this is a big 
issue in mathematical pedagogy. Why do so many mathematicians burn out at 
such a young age? Is it because the insistence on logical rigour gradually 
stifles their imagination? How are people like Ramanujan to be encouraged, 
without at the same time sacrificing the things that make mathematics so 
valuable and the subject it is. Mathematics will definitely change. It 
always has. It will change when more people of other cultures start doing 
it more intensively.

>That was left for mathematicians to do.
It was left for other mathematicians to do. Fourier was a mathematician.

>In the end, his name is famous for having the original idea,
... and ... what on earth ELSE could he be possibly be famous for??!!!

>but it would have been worthless if not on solid ground.
???!!!!!

I find this a truly astounding but ununderstandable remark. It would surely 
have been even MORE worthless if it had never been thought of. Where would 
we be today without Fourier's original demonstration that the definitions 
of an integral -- as proposed by people like Euler and Lagrange and so many 
others, and as so tamely accepted by everyone else in his day -- were 
false? This is why his name went forth into mathematics.

Fourier was a mathematician. You are entitled to your opinion. I am 
entitled to mine. Mine, though, has the singular virtue that it is accepted 
by every historian of mathematics I have ever come across and whose book I 
have read and whose Internet site I have looked at. This doesn't mean that 
you're wrong. Just means that there's more people like me than there are 
people like you. But hey ... I really am a very boring little guy. Just one 
of the crowd.

Feel free to say anything further that you wish on this subject. Me ... 
with these two emails I'm now writing, I'm done with it. I have way more 
important things in life than this.

Fourier was a mathematician.

Kool Musick
Keep Musick Kool


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