Doepfer

Off the top of my head, I don't think the set of square waves forms 
an orthogonal basis, so that a decomposition in terms of square 
waves is not unique. In other words, in the square wave basis, 
the "overtones" present are not unique. Not sure how you could apply 
a filter in this case, since the idea of a filter is to strip out 
members of the basis independently of the others. 

Beyond this I think our senses confirm the decomposition of 
vibrations in terms of sine waves, and this is simply a matter of 
experience agreeing with theory. I think if the ear were to 
experience a sound and we were expected to think about it in terms 
of the various contributions of square waves it would be difficult, 
because the contribution of each square wave in a particular sound 
is not unique. You could think about a sound being composed of two 
(or more) different sets of square waves, and the answer to the 
question would become ambiguous. Two or more, or many answers would 
be correct. In the case of sine waves, there is only one answer.

Hopefully I am correct in this and not muddying the waters.


Thanks,
Doug

--- In Doepfer_a100@yahoogroups.com, "Monroe Eskew" 
<monroe.eskew@...> wrote:
>
> I'm curious about harmonics.  I've been looking for an explanation 
of why
> different waveforms have different overtones.  One explanation 
offered is in
> terms of Fourier series.  Every periodic function can be expressed 
as an
> infinite sum of sine waves of increasing frequency and decreasing 
amplitude.
>  If we look at the Fourier series for a given curve (like a 
sawtooth or
> square wave), then we can find the overtones by looking at the 
terms in the
> sum.
> 
> Now I like mathematics, but I'm not satisfied by this 
explanation.  We can
> express a function as a Fourier series, but we can also express it 
in other
> ways.  Perhaps a sine wave can be expressed as an infinite series 
of square
> waves.  Then a sine wave should have a lot of overtones.
> 
> Here's my guess--  Qualitatively, different waveforms have 
different sounds,
> and this does not necessarily need to be interpreted as having 
overtones.
>  However FILTERS are what truly reveal overtones.  But the 
function of a
> filter is determined by the fact that its resonant frequency is 
always a
> sine wave.  If we had square wave resonance, then we'd have totally
> different filters, with the square wave being the least affected 
by the
> filter.
> 
> Is that more or less correct?
> 
> Also, does the Fourier expression make the most sense to the human 
ear?
>  (i.e. Does the human ear have something akin to sine wave 
resonance?)

> 
> Thanks,
> Monroe
> 
> 
> [Non-text portions of this message have been removed]
>

I'm curious about harmonics.  I've been looking for an explanation of why
different waveforms have different overtones.  One explanation offered is in
terms of Fourier series.  Every periodic function can be expressed as an
infinite sum of sine waves of increasing frequency and decreasing amplitude.
 If we look at the Fourier series for a given curve (like a sawtooth or
square wave), then we can find the overtones by looking at the terms in the
sum.

Now I like mathematics, but I'm not satisfied by this explanation.  We can
express a function as a Fourier series, but we can also express it in other
ways.  Perhaps a sine wave can be expressed as an infinite series of square
waves.  Then a sine wave should have a lot of overtones.

Here's my guess--  Qualitatively, different waveforms have different sounds,
and this does not necessarily need to be interpreted as having overtones.
 However FILTERS are what truly reveal overtones.  But the function of a
filter is determined by the fact that its resonant frequency is always a
sine wave.  If we had square wave resonance, then we'd have totally
different filters, with the square wave being the least affected by the
filter.

Is that more or less correct?

Also, does the Fourier expression make the most sense to the human ear?
 (i.e. Does the human ear have something akin to sine wave resonance?)

Thanks,
Monroe


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

On Wed, 2 Jul 2008, Monroe Eskew wrote:

> Here's my guess--  Qualitatively, different waveforms have different sounds,
> and this does not necessarily need to be interpreted as having overtones.
> However FILTERS are what truly reveal overtones.  But the function of a
> filter is determined by the fact that its resonant frequency is always a
> sine wave.  If we had square wave resonance, then we'd have totally
> different filters, with the square wave being the least affected by the
> filter.
>
> Is that more or less correct?
>
> Also, does the Fourier expression make the most sense to the human ear?
> (i.e. Does the human ear have something akin to sine wave resonance?)

'Not exactly' and 'to a fair degree' are the answers.  It happens
that the Fourier is roughly analogous to the physiological process
of sound detection.  The cochlea discretizes sound in both
frequency and time and so perceptual overtones correspond to
components in the Fourier expansion.  If hearing worked on a
different principle like zero crossing or peak detection, music
theory would be radically different.

m

--
Monty Brandenberg

Hi Monroe

> ways.  Perhaps a sine wave can be expressed as an infinite series of square
> waves.  Then a sine wave should have a lot of overtones.
I never tried to analyse this mathematically or experimentally, but I 
can imagine, that the sum of overtones of all the squares would be null 
(extiction by elimination due to inverted overtones of same amplitude).

Florian

I don't believe you can create a sine by adding signals with overtones. You can transfer the 
energy around in the spectrum with specific phase cancellations. But every time you 
cancel an overtone you create or reinforce another. 

In nature, sine functions are prevalent everywhere. On the other hand, square waves are 
non-existent and must always be synthesized. It would very atypical for nature to miss 
something as mathematically elegant as sines in favor of something more convoluted.


--- In Doepfer_a100@yahoogroups.com, "Monroe Eskew" <monroe.eskew@...> wrote:

>
> I'm curious about harmonics.  I've been looking for an explanation of why
> different waveforms have different overtones.  One explanation offered is in
> terms of Fourier series.  Every periodic function can be expressed as an
> infinite sum of sine waves of increasing frequency and decreasing amplitude.
>  If we look at the Fourier series for a given curve (like a sawtooth or
> square wave), then we can find the overtones by looking at the terms in the
> sum.
> 
> Now I like mathematics, but I'm not satisfied by this explanation.  We can
> express a function as a Fourier series, but we can also express it in other
> ways.  Perhaps a sine wave can be expressed as an infinite series of square
> waves.  Then a sine wave should have a lot of overtones.
> 
> Here's my guess--  Qualitatively, different waveforms have different sounds,
> and this does not necessarily need to be interpreted as having overtones.
>  However FILTERS are what truly reveal overtones.  But the function of a
> filter is determined by the fact that its resonant frequency is always a
> sine wave.  If we had square wave resonance, then we'd have totally
> different filters, with the square wave being the least affected by the
> filter.
> 
> Is that more or less correct?
> 
> Also, does the Fourier expression make the most sense to the human ear?
>  (i.e. Does the human ear have something akin to sine wave resonance?)
> 
> Thanks,
> Monroe
> 
> 
> [Non-text portions of this message have been removed]
>

Pretty sure my original post on this subject is weak (or worse), but 
I think this article might help. One should be able to construct an 
arbitrary periodic function using a non-trigonmetric basis. In other 
words, you *can* "create a sine by adding signals."

http://ieeexplore.ieee.org/iel5/31/23557/01085543.pdf?
arnumber=1085543

Further, seems to me the ear can be trained to listen for non-sine 
basis functions (aka "overtones"). Take for example, the sound of 
two or more oboes. Pretty sure one could separate the parts by 
pitch, although the parts aren't themselves pure sines. Muting one 
of the oboe parts would be filtering that "frequency", wouldn't it?

I guess I am agreeing with the original poster, and the answer as to 
why we analyse signals in terms of trigonometric basis functions is 
just mathematical (and design?) convenience. 

Doug


--- In Doepfer_a100@yahoogroups.com, "laryn91" <caymus91@...> wrote:
>
> I don't believe you can create a sine by adding signals with 
overtones. You can transfer the 
> energy around in the spectrum with specific phase cancellations. 
But every time you 
> cancel an overtone you create or reinforce another. 
> 
> In nature, sine functions are prevalent everywhere. On the other 
hand, square waves are 
> non-existent and must always be synthesized. It would very 
atypical for nature to miss 
> something as mathematically elegant as sines in favor of something 
more convoluted.
> 
> 
> --- In Doepfer_a100@yahoogroups.com, "Monroe Eskew" 
<monroe.eskew@> wrote:
> >
> > I'm curious about harmonics.  I've been looking for an 
explanation of why
> > different waveforms have different overtones.  One explanation 
offered is in
> > terms of Fourier series.  Every periodic function can be 
expressed as an
> > infinite sum of sine waves of increasing frequency and 
decreasing amplitude.
> >  If we look at the Fourier series for a given curve (like a 
sawtooth or
> > square wave), then we can find the overtones by looking at the 
terms in the
> > sum.
> > 
> > Now I like mathematics, but I'm not satisfied by this 
explanation.  We can
> > express a function as a Fourier series, but we can also express 
it in other
> > ways.  Perhaps a sine wave can be expressed as an infinite 
series of square
> > waves.  Then a sine wave should have a lot of overtones.
> > 
> > Here's my guess--  Qualitatively, different waveforms have 
different sounds,
> > and this does not necessarily need to be interpreted as having 
overtones.
> >  However FILTERS are what truly reveal overtones.  But the 
function of a
> > filter is determined by the fact that its resonant frequency is 
always a
> > sine wave.  If we had square wave resonance, then we'd have 
totally
> > different filters, with the square wave being the least affected 
by the
> > filter.
> > 
> > Is that more or less correct?
> > 
> > Also, does the Fourier expression make the most sense to the 
human ear?
> >  (i.e. Does the human ear have something akin to sine wave 
resonance?)

> > 
> > Thanks,
> > Monroe
> > 
> > 
> > [Non-text portions of this message have been removed]
> >
>

> Sine functions can be represented as Taylor series. 
 
Perfect example of a sine being constructed. 
 
>I'm guessing, the
> human ear naturally decomposes a wave into its Fourier components, 
absorbing
> the energy from the lower frequencies in sine form, and then 
passing off the
> rest down the tube.

I'm sticking with sines and cosines as a convenient analytical 
representation (that includes a mathematical analysis of vibrations 
in the ear too). I don't think the ear knows diddly about Fourier ;) 
I would go back to the idea that your ear/mind can separate the 
parts of a musical sound based on the timbres of the constituent 
instruments, not only in the case that they are pipes or flutes, or 
whatever particular timbre is closest to a sine. I think the 
ear/mind is really good at this, actually. If there is a bird 
chirping and a lion roaring at the same time, I bet some of the 
Fourier terms are overlapping, but there would be no doubt in 
mentally separating the sounds according to timbre. Should I go 
further and say that spectrally rich tones are easier for the mind 
to categorize than "pure" ones?

> 
> This leaves open the question of a synthesizer filter based on a 
different
> resonance waveform.  Any thoughts on whether that's possible, what 
it would
> sound like?

Not a designer, but I bet there are contexts in which using tri or 
square is more convenient than sines. Especially in digital 
synthesis. 

Doug

In your example, what are the "pitch parts" you're separating when analyzing the two 
oboes? SINE waves - right?  Not square or some other arbitrary function. 

Since I can only read the fragmented summary in the referenced paper, so maybe I'm 
understanding it totally incorrectly. But it may not be relevant since the harmonic phase 
relations don't appear to be held constant. In other words yes, you can get a sine if you 
add two signals of different shape. Which is not what the poster asked.

I believe the poster was wondering if any function can be transformed from a non-circular 
plane - like a square or rectangular one. Maybe it's just my limited knowledge, but I 
haven't heard of such a thing. I can try to *approximate" a sine from a square plane...


--- In Doepfer_a100@yahoogroups.com, "Doug" <dougc356@...> wrote:

>
> Pretty sure my original post on this subject is weak (or worse), but 
> I think this article might help. One should be able to construct an 
> arbitrary periodic function using a non-trigonmetric basis. In other 
> words, you *can* "create a sine by adding signals."
> 
> http://ieeexplore.ieee.org/iel5/31/23557/01085543.pdf?
> arnumber=1085543
> 
> Further, seems to me the ear can be trained to listen for non-sine 
> basis functions (aka "overtones"). Take for example, the sound of 
> two or more oboes. Pretty sure one could separate the parts by 
> pitch, although the parts aren't themselves pure sines. Muting one 
> of the oboe parts would be filtering that "frequency", wouldn't it?
> 
> I guess I am agreeing with the original poster, and the answer as to 
> why we analyse signals in terms of trigonometric basis functions is 
> just mathematical (and design?) convenience. 
> 
> Doug
> 
> 
> --- In Doepfer_a100@yahoogroups.com, "laryn91" <caymus91@> wrote:
> >
> > I don't believe you can create a sine by adding signals with 
> overtones. You can transfer the 
> > energy around in the spectrum with specific phase cancellations. 
> But every time you 
> > cancel an overtone you create or reinforce another. 
> > 
> > In nature, sine functions are prevalent everywhere. On the other 
> hand, square waves are 
> > non-existent and must always be synthesized. It would very 
> atypical for nature to miss 
> > something as mathematically elegant as sines in favor of something 
> more convoluted.
> > 
> > 
> > --- In Doepfer_a100@yahoogroups.com, "Monroe Eskew" 
> <monroe.eskew@> wrote:
> > >
> > > I'm curious about harmonics.  I've been looking for an 
> explanation of why
> > > different waveforms have different overtones.  One explanation 
> offered is in
> > > terms of Fourier series.  Every periodic function can be 
> expressed as an
> > > infinite sum of sine waves of increasing frequency and 
> decreasing amplitude.
> > >  If we look at the Fourier series for a given curve (like a 
> sawtooth or
> > > square wave), then we can find the overtones by looking at the 
> terms in the
> > > sum.
> > > 
> > > Now I like mathematics, but I'm not satisfied by this 
> explanation.  We can
> > > express a function as a Fourier series, but we can also express 
> it in other
> > > ways.  Perhaps a sine wave can be expressed as an infinite 
> series of square
> > > waves.  Then a sine wave should have a lot of overtones.
> > > 
> > > Here's my guess--  Qualitatively, different waveforms have 
> different sounds,
> > > and this does not necessarily need to be interpreted as having 
> overtones.
> > >  However FILTERS are what truly reveal overtones.  But the 
> function of a
> > > filter is determined by the fact that its resonant frequency is 
> always a
> > > sine wave.  If we had square wave resonance, then we'd have 
> totally
> > > different filters, with the square wave being the least affected 
> by the
> > > filter.
> > > 
> > > Is that more or less correct?
> > > 
> > > Also, does the Fourier expression make the most sense to the 
> human ear?
> > >  (i.e. Does the human ear have something akin to sine wave 
> resonance?)
> > > 
> > > Thanks,
> > > Monroe
> > > 
> > > 
> > > [Non-text portions of this message have been removed]
> > >
> >
>

>
> In your example, what are the "pitch parts" you're separating when 
analyzing the two 
> oboes? SINE waves - right?  Not square or some other arbitrary 
function. 

The "oboe function" (steady state). I picked oboe, because it's not 
a sine. Each part is not a sine, yet the mind is easily able to 
separate them, and not in terms of equivalent sums of bell sounds. 
Just an example of how the ear is not hobbled by only being able to 
separate sounds into sines.


>Which is not what the poster asked.

Yeah, I went from the specific case he mentioned... using the set of 
square waves as a basis... to a set of *any* signals as a basis. And 
this is exactly what the paper addresses (well there are some 
assumptions about the candidate basis functions). It even goes so 
far to use tri waves as an example. There are even oboe graphs in 
there! It's a math paper in IEEE with music! Ha ha. 

> 
> I believe the poster was wondering if any function can be 
transformed from a non-circular 
> plane - like a square or rectangular one. Maybe it's just my 
limited knowledge, but I 
> haven't heard of such a thing. I can try to *approximate" a sine 
from a square plane...

This is exactly what the paper is addressing.

Doug

The reason the ear can separate the two oboes is because an expected harmonic 
relationship is known as a reference point for analysis. So the two "parts" can be 
reconstructed by the brain to identify two oboes. So it looks like the ear-brain maybe does 
know (pre-wired?) Fourier series ;-)

It's even more complex than that - the brain also keeps maintains a timbre map (vocal 
tract resonances) of every speaker so you can hear individual speakers in a crowded room 
(I do speech research work). Within the noisy signal, very weak individual speaker signals 
can be easily correlated, extracted and decoded.

The brain is mostly tracking resonances rather then individual harmonics, but there is a lot 
of evidence the human input transducers are spectral + time types. Experiments with 
critical banding and masking indicate sine harmonics are being detected by the ear.

I'll have to see if I can find your reference in full somewhere. Sounds very interesting!



--- In Doepfer_a100@yahoogroups.com, "Doug" <dougc356@...> wrote:

>
> >
> > In your example, what are the "pitch parts" you're separating when 
> analyzing the two 
> > oboes? SINE waves - right?  Not square or some other arbitrary 
> function. 
> 
> The "oboe function" (steady state). I picked oboe, because it's not 
> a sine. Each part is not a sine, yet the mind is easily able to 
> separate them, and not in terms of equivalent sums of bell sounds. 
> Just an example of how the ear is not hobbled by only being able to 
> separate sounds into sines.
> 
> 
> >Which is not what the poster asked.
> 
> Yeah, I went from the specific case he mentioned... using the set of 
> square waves as a basis... to a set of *any* signals as a basis. And 
> this is exactly what the paper addresses (well there are some 
> assumptions about the candidate basis functions). It even goes so 
> far to use tri waves as an example. There are even oboe graphs in 
> there! It's a math paper in IEEE with music! Ha ha. 
> 
> > 
> > I believe the poster was wondering if any function can be 
> transformed from a non-circular 
> > plane - like a square or rectangular one. Maybe it's just my 
> limited knowledge, but I 
> > haven't heard of such a thing. I can try to *approximate" a sine 
> from a square plane...
> 
> This is exactly what the paper is addressing.
> 
> Doug
>

Yes, but not vary precise and lots of hi-freq artifacts. 

As noted in this thread, there are many mathematical functions that will  *approximate* a 
sine. But only a circle can make a circle. An infinite number of squares may converge to 
circle - but never hit it (I think...).


--- In Doepfer_a100@yahoogroups.com, Chris Muir <cbm@...> wrote:

>
> 
> On Jul 2, 2008, at 9:20 AM, Monroe Eskew wrote:
> 
> > Perhaps a sine wave can be expressed as an infinite series of square
> > waves.
> 
> 
> Aren't Walsh transforms squarewave-based?
> 
> -C
> 
> Chris Muir
> cbm@...	
> http://www.xfade.com
>

Expressing overtones in squarewaves is not correct.
Because the squarewave has a large number of overtones.

The (perfect) sine is the purest tone you can achive. It has no overtones.
that is the way mathematics looks at it.
If you would look in with a spectrum analyzer the (perfect) sine shows up as
a needle sticking out at one place.
In truth most sine oscillators are not entirely perfect so they do have
slight tendencies to have a few tiny overtones.
If you would analyze the squarewave this way you see a whole blur over the
with of the spectrum

The only way to create a sine out of a squarewave is by using a lowpass
filter.
Mathematically seen you could consider the lowpass filter  an integrator.


If you are interested doing research in these fields I would advise you to
take a look at Cycling 74's Max/MSP software.







On Thu, Jul 3, 2008 at 8:50 PM, laryn91 <caymus91@mac.com> wrote:

>   I don't believe you can create a sine by adding signals with overtones.
> You can transfer the
> energy around in the spectrum with specific phase cancellations. But every
> time you
> cancel an overtone you create or reinforce another.
>
> In nature, sine functions are prevalent everywhere. On the other hand,
> square waves are
> non-existent and must always be synthesized. It would very atypical for
> nature to miss
> something as mathematically elegant as sines in favor of something more
> convoluted.
>
>
> --- In Doepfer_a100@yahoogroups.com <Doepfer_a100%40yahoogroups.com>,
> "Monroe Eskew" <monroe.eskew@...> wrote:
> >
> > I'm curious about harmonics. I've been looking for an explanation of why
> > different waveforms have different overtones. One explanation offered is
> in
> > terms of Fourier series. Every periodic function can be expressed as an
> > infinite sum of sine waves of increasing frequency and decreasing
> amplitude.
> > If we look at the Fourier series for a given curve (like a sawtooth or
> > square wave), then we can find the overtones by looking at the terms in
> the
> > sum.
> >
> > Now I like mathematics, but I'm not satisfied by this explanation. We can
> > express a function as a Fourier series, but we can also express it in
> other
> > ways. Perhaps a sine wave can be expressed as an infinite series of
> square
> > waves. Then a sine wave should have a lot of overtones.
> >
> > Here's my guess-- Qualitatively, different waveforms have different
> sounds,
> > and this does not necessarily need to be interpreted as having overtones.
> > However FILTERS are what truly reveal overtones. But the function of a
> > filter is determined by the fact that its resonant frequency is always a
> > sine wave. If we had square wave resonance, then we'd have totally
> > different filters, with the square wave being the least affected by the
> > filter.
> >
> > Is that more or less correct?
> >
> > Also, does the Fourier expression make the most sense to the human ear?
> > (i.e. Does the human ear have something akin to sine wave resonance?)
> >
> > Thanks,
> > Monroe
> >
> >
> > [Non-text portions of this message have been removed]
> >
>
>  
>


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

I agree completely. The true nature of the analogue world is that  
there are very few if any abrupt transitions to any event. Nothing  
naturally goes instantly from one state to another like a square wave  
(I guess you would have to go sub-atomic first but even that is  
theory). In fact it is very hard to make a true square wave with no  
overshoot or bounce in the real world - air pressure waves can't be  
made that abrupt because of the compressible nature of air for  
example. One must also acknowledge that pure sine wave sound doesn't  
exists in nature either - even the purest pipe tone has overtones.

On Jul 3, 2008, at 11:50 AM, laryn91 wrote:
>
> In nature, sine functions are prevalent everywhere. On the other  
> hand, square waves are
> non-existent and must always be synthesized. It would very atypical  
> for nature to miss
> something as mathematically elegant as sines in favor of something  
> more convoluted.
>

Sine functions can be represented as Taylor series.  The terms of the Taylor
series are not periodic functions however.  But we can redo the Taylor
series, taking advantage of the periodic nature of the sine wave, doing it
normally on the interval [0,2pi) and then "re-centering" at each multiple of
2pi, just copying the function from the [0,2pi) interval.  The result is the
sine wave expressed as a sum of increasingly curvy sawtooths, albeit all of
the same frequency.

I think the explanation has more to do with the cochlea.  As I understand
it, It has different sensors for different frequencies, and the frequency
sensed increases as you travel further into the tube.  The sensor hairs
probably vibrate naturally in a sine waveform.  Thus, I'm guessing, the
human ear naturally decomposes a wave into its Fourier components, absorbing
the energy from the lower frequencies in sine form, and then passing off the
rest down the tube.

This leaves open the question of a synthesizer filter based on a different
resonance waveform.  Any thoughts on whether that's possible, what it would
sound like?

Monroe


On Thu, Jul 3, 2008 at 1:50 PM, laryn91 <caymus91@mac.com> wrote:

>   I don't believe you can create a sine by adding signals with overtones.
> You can transfer the
> energy around in the spectrum with specific phase cancellations. But every
> time you
> cancel an overtone you create or reinforce another.
>
> In nature, sine functions are prevalent everywhere. On the other hand,
> square waves are
> non-existent and must always be synthesized. It would very atypical for
> nature to miss
> something as mathematically elegant as sines in favor of something more
> convoluted.
>
>
> --- In Doepfer_a100@yahoogroups.com <Doepfer_a100%40yahoogroups.com>,
> "Monroe Eskew" <monroe.eskew@...> wrote:
> >
> > I'm curious about harmonics. I've been looking for an explanation of why
> > different waveforms have different overtones. One explanation offered is
> in
> > terms of Fourier series. Every periodic function can be expressed as an
> > infinite sum of sine waves of increasing frequency and decreasing
> amplitude.
> > If we look at the Fourier series for a given curve (like a sawtooth or
> > square wave), then we can find the overtones by looking at the terms in
> the
> > sum.
> >
> > Now I like mathematics, but I'm not satisfied by this explanation. We can
> > express a function as a Fourier series, but we can also express it in
> other
> > ways. Perhaps a sine wave can be expressed as an infinite series of
> square
> > waves. Then a sine wave should have a lot of overtones.
> >
> > Here's my guess-- Qualitatively, different waveforms have different
> sounds,
> > and this does not necessarily need to be interpreted as having overtones.
> > However FILTERS are what truly reveal overtones. But the function of a
> > filter is determined by the fact that its resonant frequency is always a
> > sine wave. If we had square wave resonance, then we'd have totally
> > different filters, with the square wave being the least affected by the
> > filter.
> >
> > Is that more or less correct?
> >
> > Also, does the Fourier expression make the most sense to the human ear?
> > (i.e. Does the human ear have something akin to sine wave resonance?)
> >
> > Thanks,
> > Monroe
> >
> >
> > [Non-text portions of this message have been removed]
> >
>
>  
>


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

On Jul 2, 2008, at 9:20 AM, Monroe Eskew wrote:

> Perhaps a sine wave can be expressed as an infinite series of square
> waves.


Aren't Walsh transforms squarewave-based?

-C

Chris Muir
cbm@well.com	
http://www.xfade.com

From: Chris Muir <cbm@well.com>

Subject: Re: [Doepfer_a100] hard science question
Date: Thu, 3 Jul 2008 14:40:12 -0700
Message-ID: <AD7B588A-D181-4E0B-9557-3C1ADAA7CBAF@well.com>

> 
> On Jul 2, 2008, at 9:20 AM, Monroe Eskew wrote:
> 
> > Perhaps a sine wave can be expressed as an infinite series of square
> > waves.
> 
> Aren't Walsh transforms squarewave-based?

They are. Sine generation is possible but not *THAT*  neat really.

Google the web for Walsh functions and sine and you should find several
relevant pages. Wikipedia should help you on the way as usual.

In short will all waveforms exist in sine and cosine form, i.e. 0 and 90
degrees. The simple 4 sample case is illustrative. These are the 4 base
vectors (the third line can be inversed, don't recall from the top of my head,
but literature will correct that mistake):

+1 +1 +1 +1
+1 +1 -1 -1
-1 +1 +1 -1
+1 -1 +1 -1

Looks simple and powerfull. Ah well. :)

Cheers,
Magnus

>why is one decomposition favored?

In analog electronics, sine waves are everywhere due to the way
current and voltage behave in analog circuits. Building a filter in
this environment using a trig decomposition is easier than switching
to another decomposition.

In math, sine waves form a nice orthogonal basis, which makes the
decomposition easier to calculate.

In the ear, natural vibrations are probably best modeled with sines
and cosines. In other words, fewer basis functions are required to
describe the vibration when the trigonometric basis is used. 

It seems that we want to believe that the mechanism of hearing
involves a decomposition into a trigonometric basis followed by a
cognitive synthesis. I'm not sure why this would be, and what this
model accomplishes, other than satisfying our urge for reductive
analysis armed with the most convenient mathematical language known to
us. 

Doug

The question is basically whether overtones are objective or just one
interpretation of a wave.  Given that there are multiple ways to decompose a
given waveform, why is one decomposition favored as a way of understanding
the overtones?  If the decomposition corresponds to how the cochlea
physically works, or how the brain processes, or a combination of both, this
may help explain it.  Obviously overtones have a functional relationship to
filters, and since filters are sine-wave-based, this leads to the hypothesis
that the ear (and/or brain) shares some characteristics with those filters.

The ear doesn't have to "know" anything; it may physically function in a way
as to break the spectrum into sine components.  The cochlea is a complex
thing.

On Thu, Jul 3, 2008 at 3:42 PM, laryn91 <caymus91@mac.com> wrote:

>   In your example, what are the "pitch parts" you're separating when
> analyzing the two
> oboes? SINE waves - right? Not square or some other arbitrary function.
>
> Since I can only read the fragmented summary in the referenced paper, so
> maybe I'm
> understanding it totally incorrectly. But it may not be relevant since the
> harmonic phase
> relations don't appear to be held constant. In other words yes, you can get
> a sine if you
> add two signals of different shape. Which is not what the poster asked.
>
> I believe the poster was wondering if any function can be transformed from
> a non-circular
> plane - like a square or rectangular one. Maybe it's just my limited
> knowledge, but I
> haven't heard of such a thing. I can try to *approximate" a sine from a
> square plane...
>
> --- In Doepfer_a100@yahoogroups.com <Doepfer_a100%40yahoogroups.com>,
> "Doug" <dougc356@...> wrote:
> >
> > Pretty sure my original post on this subject is weak (or worse), but
> > I think this article might help. One should be able to construct an
> > arbitrary periodic function using a non-trigonmetric basis. In other
> > words, you *can* "create a sine by adding signals."
> >
> > http://ieeexplore.ieee.org/iel5/31/23557/01085543.pdf?
> > arnumber=1085543
> >
> > Further, seems to me the ear can be trained to listen for non-sine
> > basis functions (aka "overtones"). Take for example, the sound of
> > two or more oboes. Pretty sure one could separate the parts by
> > pitch, although the parts aren't themselves pure sines. Muting one
> > of the oboe parts would be filtering that "frequency", wouldn't it?
> >
> > I guess I am agreeing with the original poster, and the answer as to
> > why we analyse signals in terms of trigonometric basis functions is
> > just mathematical (and design?) convenience.
> >
> > Doug
>
> >
> >
> > --- In Doepfer_a100@yahoogroups.com <Doepfer_a100%40yahoogroups.com>,
> "laryn91" <caymus91@> wrote:
> > >
> > > I don't believe you can create a sine by adding signals with
> > overtones. You can transfer the
> > > energy around in the spectrum with specific phase cancellations.
> > But every time you
> > > cancel an overtone you create or reinforce another.
> > >
> > > In nature, sine functions are prevalent everywhere. On the other
> > hand, square waves are
> > > non-existent and must always be synthesized. It would very
> > atypical for nature to miss
> > > something as mathematically elegant as sines in favor of something
> > more convoluted.
> > >
> > >
> > > --- In Doepfer_a100@yahoogroups.com <Doepfer_a100%40yahoogroups.com>,
> "Monroe Eskew"
> > <monroe.eskew@> wrote:
> > > >
> > > > I'm curious about harmonics. I've been looking for an
> > explanation of why
> > > > different waveforms have different overtones. One explanation
> > offered is in
> > > > terms of Fourier series. Every periodic function can be
> > expressed as an
> > > > infinite sum of sine waves of increasing frequency and
> > decreasing amplitude.
> > > > If we look at the Fourier series for a given curve (like a
> > sawtooth or
> > > > square wave), then we can find the overtones by looking at the
> > terms in the
> > > > sum.
> > > >
> > > > Now I like mathematics, but I'm not satisfied by this
> > explanation. We can
> > > > express a function as a Fourier series, but we can also express
> > it in other
> > > > ways. Perhaps a sine wave can be expressed as an infinite
> > series of square
> > > > waves. Then a sine wave should have a lot of overtones.
> > > >
> > > > Here's my guess-- Qualitatively, different waveforms have
> > different sounds,
> > > > and this does not necessarily need to be interpreted as having
> > overtones.
> > > > However FILTERS are what truly reveal overtones. But the
> > function of a
> > > > filter is determined by the fact that its resonant frequency is
> > always a
> > > > sine wave. If we had square wave resonance, then we'd have
> > totally
> > > > different filters, with the square wave being the least affected
> > by the
> > > > filter.
> > > >
> > > > Is that more or less correct?
> > > >
> > > > Also, does the Fourier expression make the most sense to the
> > human ear?
> > > > (i.e. Does the human ear have something akin to sine wave
> > resonance?)
> > > >
> > > > Thanks,
> > > > Monroe
> > > >
> > > >
> > > > [Non-text portions of this message have been removed]
> > > >
> > >
> >
>
>  
>


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

> Pretty sure my original post on this subject is weak (or worse), but
> I think this article might help. One should be able to construct an
> arbitrary periodic function using a non-trigonmetric basis. In other
> words, you *can* "create a sine by adding signals."
>
> http://ieeexplore.ieee.org/iel5/31/23557/01085543.pdf?
> arnumber=1085543
>
> Further, seems to me the ear can be trained to listen for non-sine
> basis functions (aka "overtones"). Take for example, the sound of
> two or more oboes. Pretty sure one could separate the parts by
> pitch, although the parts aren't themselves pure sines. Muting one
> of the oboe parts would be filtering that "frequency", wouldn't it?
>
> I guess I am agreeing with the original poster, and the answer as to
> why we analyse signals in terms of trigonometric basis functions is
> just mathematical (and design?) convenience.
>
> Doug

There have been a lot of approaches to replace the Fourier
synthesis/analysis by another orthogonal basis of functions/waveforms (e.g.
Walsh functions/Hadamard transform or the Haar functions) especially because
rectangle based functions (like Walsh) can be generated much easier in the
digital world. But after all the Fourier version is the most "natural" one
as it conforms with the behaviour of the human sense of hearing. And it's
much easier to understand and to handle compared to other synthesis forms.
The other synthesis forms are mathematically correct but much more difficult
to handle and to understand compared to the simple overtone principle of the
Fourier synthesis that follows the human sense of hearing.

Just my point of view ...

Dieter Doepfer

Another benefit to harmonic theory is it's intuitive and deep. Since humans find only a 
small subset of possible sounds interesting for musical application, subtractive and 
additive synthesis has made sound design relatively easy but powerful.

On the other hand, modulation type synthesizers (FM, AM, etc.) are completely non-
intuitive and creating musical sounds with them is difficult. 

I suppose someone could create a synthesizer based on something like polynomials - but 
associating coefficient values to it's sound is not going to be anywhere near as intuitive as 
direct harmonic manipulation.


--- In Doepfer_a100@yahoogroups.com, "Monroe Eskew" <monroe.eskew@...> wrote:

>
> It's not just what's convenient.  It may be found by scientific
> investigation to be how the ear actually works.  Much work has already been
> done on this.  Here is a good starting place for information:
> http://en.wikipedia.org/wiki/Cochlea
> 
> Monroe
> 
> On Thu, Jul 3, 2008 at 7:09 PM, Doug <dougc356@...> wrote:
> 
> >
> >
> > It seems that we want to believe that the mechanism of hearing
> > involves a decomposition into a trigonometric basis followed by a
> > cognitive synthesis. I'm not sure why this would be, and what this
> > model accomplishes, other than satisfying our urge for reductive
> > analysis armed with the most convenient mathematical language known to
> > us.
> >
> > Doug
> >
> >  
> >
> 
> 
> [Non-text portions of this message have been removed]
>

It's not just what's convenient.  It may be found by scientific
investigation to be how the ear actually works.  Much work has already been
done on this.  Here is a good starting place for information:
http://en.wikipedia.org/wiki/Cochlea

Monroe

On Thu, Jul 3, 2008 at 7:09 PM, Doug <dougc356@yahoo.com> wrote:

>
>
> It seems that we want to believe that the mechanism of hearing
> involves a decomposition into a trigonometric basis followed by a
> cognitive synthesis. I'm not sure why this would be, and what this
> model accomplishes, other than satisfying our urge for reductive
> analysis armed with the most convenient mathematical language known to
> us.
>
> Doug
>
>  
>


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