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  • Monroe Eskew
    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
    Message 1 of 21 , Jul 2, 2008
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      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]
    • Doug
      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
      Message 2 of 21 , Jul 2, 2008
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        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]
        >
      • mcb, inc.
        ... 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.
        Message 3 of 21 , Jul 2, 2008
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          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
        • Florian Anwander
          Hi Monroe ... 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
          Message 4 of 21 , Jul 3, 2008
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            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
          • laryn91
            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.
            Message 5 of 21 , Jul 3, 2008
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              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]
              >
            • Doug
              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
              Message 6 of 21 , Jul 3, 2008
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                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]
                > >
                >
              • achtung_999
                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
                Message 7 of 21 , Jul 3, 2008
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                  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@...> 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]
                • James Husted
                  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
                  Message 8 of 21 , Jul 3, 2008
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                    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.
                    >
                  • Monroe Eskew
                    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,
                    Message 9 of 21 , Jul 3, 2008
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                      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@...> 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]
                    • Doug
                      ... Perfect example of a sine being constructed. ... absorbing ... passing off the ... I m sticking with sines and cosines as a convenient analytical
                      Message 10 of 21 , Jul 3, 2008
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                        > 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
                      • laryn91
                        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.
                        Message 11 of 21 , Jul 3, 2008
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                          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]
                          > > >
                          > >
                          >
                        • Doug
                          ... analyzing the two ... 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
                          Message 12 of 21 , Jul 3, 2008
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                            >
                            > 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
                          • Chris Muir
                            ... Aren t Walsh transforms squarewave-based? -C Chris Muir cbm@well.com http://www.xfade.com
                            Message 13 of 21 , Jul 3, 2008
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                              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
                            • laryn91
                              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
                              Message 14 of 21 , Jul 3, 2008
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                                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
                                >
                              • Magnus Danielson
                                From: Chris Muir Subject: Re: [Doepfer_a100] hard science question Date: Thu, 3 Jul 2008 14:40:12 -0700 Message-ID:
                                Message 15 of 21 , Jul 3, 2008
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                                  From: Chris Muir <cbm@...>
                                  Subject: Re: [Doepfer_a100] hard science question
                                  Date: Thu, 3 Jul 2008 14:40:12 -0700
                                  Message-ID: <AD7B588A-D181-4E0B-9557-3C1ADAA7CBAF@...>

                                  >
                                  > 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
                                • laryn91
                                  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
                                  Message 16 of 21 , Jul 3, 2008
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                                    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
                                    >
                                  • Monroe Eskew
                                    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
                                    Message 17 of 21 , Jul 3, 2008
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                                      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@...> 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]
                                    • Doug
                                      ... 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
                                      Message 18 of 21 , Jul 3, 2008
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                                        >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
                                      • hardware@doepfer.de
                                        ... There have been a lot of approaches to replace the Fourier synthesis/analysis by another orthogonal basis of functions/waveforms (e.g. Walsh
                                        Message 19 of 21 , Jul 4, 2008
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                                          > 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
                                        • Monroe Eskew
                                          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
                                          Message 20 of 21 , Jul 4, 2008
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                                            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]
                                          • laryn91
                                            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,
                                            Message 21 of 21 , Jul 4, 2008
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                                              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]
                                              >
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