Harmonics in oscillator-
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From: David Bernier on 30 Jan 2010 14:34 I.N. Galidakis wrote: > David Bernier wrote: >> I.N. Galidakis wrote: >>> JEMebius wrote: >>>> I.N. Galidakis wrote: >>>>> I am playing around with a specific oscillator. I have identified 5 modes >>>>> of oscillation, which correspond to 7 frequencies N_j, therefore the system >>>>> phasor is: >>>>> >>>>> p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) >>>>> >>>>> Do I need to also include all the harmonics >>>>> >>>>> exp(2*Pi*i*k*N_j*t), k=2,3,4,... >>>>> >>>>> in the phasor, in order to analyze it in the Fourier sense? >>>>> >>>>> In other words, is the system phasor p(t) or q(t), with: >>>>> >>>>> q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) >>>>> >>>>> Thanks, >>> Hi Johan, >>> >>>> I guess you have identified 7 modes, not 5 modes of oscillation? >>> No, it's 5 modes, but two modes vibrate at two frequencies each: Mode 1 >>> vibrates at N_1 and 1/N_1 and Mode 2 vibrates at N_2 and 1/N_2. The rest >>> vibrate at one mode, total 7 frequencies. >>> >>>> Anyhow, one does not need the extra terms to obtain a decent Fourier >>>> analysis; >>> The reason I asked is because everywhere I asked, they told me that any >>> oscillator which broadcasts at N_j, also broadcasts at k*N_j, all of the >>> harmonics. For my case, this complicates matters considerably, because >>> according to Wiki, the harmonics fall as ~1/k^2 and I don't know how to >>> model them using an appropriate function. >> [...] >> >> From what I remember about the vibrating string (such as one idealized >> guitar string), the string end-points could be say at 0 centimeters >> and 48 centimeters. In the fundamental mode, the string goes >> up and down say 200 times a second. (pure 200 Hz tone). > > Hi David, > > Although vibrating strings are a good example of oscillators, I believe that the > case of music is only a particular example of general oscillator theory. > > In any case, I have some experience with musical instruments and if the A0 piano > chord vibrates at 440 Hz, it will also generate all the higher harmonics, at > 880, 1760, ... n*440, etc., n=2,3,4,... > > This means that if I apply the damper ONLY on the A0 piano chord after it is > struck, there will be some residual resonance at all the higher A's, like A1, > A2, A3, up to the right limit of the piano. > > In other words, if I silence A0 with the damper, the rest of the A's will > resonate for a short time. > > In THAT sense I am interested in what sense to include all the harmonics in my > spectrum. [...] I think I misunderstood. The amplitude (or maybe power) of the harmonics falls as 1/(k^2), perhaps ... According to what Wikipedia says about the audio CD standard [so-called "red book"] , Wikipedia: "An audio CD can represent frequencies up to 22.05 kHz, the Nyquist frequency [...] " Reference: < http://en.wikipedia.org/wiki/Red_Book_(audio_CD_standard) > . I don't know if this helps, but harmonics with frequencies above about ~= 20,000 Hertz practically don't matter for human perception of sound. regards, David Bernier
From: I.N. Galidakis on 4 Feb 2010 23:24 David Bernier wrote: > I.N. Galidakis wrote: >> David Bernier wrote: >>> I.N. Galidakis wrote: >>>> JEMebius wrote: >>>>> I.N. Galidakis wrote: >>>>>> I am playing around with a specific oscillator. I have identified 5 modes >>>>>> of oscillation, which correspond to 7 frequencies N_j, therefore the >>>>>> system phasor is: >>>>>> >>>>>> p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) >>>>>> >>>>>> Do I need to also include all the harmonics >>>>>> >>>>>> exp(2*Pi*i*k*N_j*t), k=2,3,4,... >>>>>> >>>>>> in the phasor, in order to analyze it in the Fourier sense? >>>>>> >>>>>> In other words, is the system phasor p(t) or q(t), with: >>>>>> >>>>>> q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) >>>>>> >>>>>> Thanks, >>>> Hi Johan, >>>> >>>>> I guess you have identified 7 modes, not 5 modes of oscillation? >>>> No, it's 5 modes, but two modes vibrate at two frequencies each: Mode 1 >>>> vibrates at N_1 and 1/N_1 and Mode 2 vibrates at N_2 and 1/N_2. The rest >>>> vibrate at one mode, total 7 frequencies. >>>> >>>>> Anyhow, one does not need the extra terms to obtain a decent Fourier >>>>> analysis; >>>> The reason I asked is because everywhere I asked, they told me that any >>>> oscillator which broadcasts at N_j, also broadcasts at k*N_j, all of the >>>> harmonics. For my case, this complicates matters considerably, because >>>> according to Wiki, the harmonics fall as ~1/k^2 and I don't know how to >>>> model them using an appropriate function. >>> [...] >>> >>> From what I remember about the vibrating string (such as one idealized >>> guitar string), the string end-points could be say at 0 centimeters >>> and 48 centimeters. In the fundamental mode, the string goes >>> up and down say 200 times a second. (pure 200 Hz tone). >> >> Hi David, >> >> Although vibrating strings are a good example of oscillators, I believe that >> the case of music is only a particular example of general oscillator theory. >> >> In any case, I have some experience with musical instruments and if the A0 >> piano chord vibrates at 440 Hz, it will also generate all the higher >> harmonics, at 880, 1760, ... n*440, etc., n=2,3,4,... >> >> This means that if I apply the damper ONLY on the A0 piano chord after it is >> struck, there will be some residual resonance at all the higher A's, like A1, >> A2, A3, up to the right limit of the piano. >> >> In other words, if I silence A0 with the damper, the rest of the A's will >> resonate for a short time. >> >> In THAT sense I am interested in what sense to include all the harmonics in >> my spectrum. > [...] > > I think I misunderstood. The amplitude (or maybe power) of the harmonics > falls as 1/(k^2), perhaps ... Sorry for the delay in responding. The subject is very complex and I am just *beginning* to understand it. And to think that I've been playing the piano for 40 years :-) It seems from what I understand so far that harmonics are NOT generated in all cases, so Johan's response concerns the general *simplest* case. Harmonics are generated in many cases with musical instruments, and in this case the combination of the fundamental and the harmonics determines the "pitch" which one perceives. For example, the piano and some typical string instruments generate what are called "partials" which seem to be very close to the harmonics for the fundamental freq the instrument vibrates in: http://en.wikipedia.org/wiki/Harmonic_series_(music) (last paren has fallen off the link. Put back for link to work) Whenever this is the case, it seems that the intensity of the k-th partial (harmonic) falls off (at least) as fast as 1/k^2. This means that in this case, the Fourier series coefficients a_n satisfy: |a_n| <= M/k^2, for some positive constant M. Note however that *not* all oscillators generate harmonics, hence my confusion. A piano string does, but a synthesizer which produces a simple sine tone, does not. That's why I was confused. I actually have to examine my oscillator closely and verify which harmonics are produced in each mode. To summarize: For a simple harmonic oscillator, the harmonics are given by its Fourier coefficients, so the analysis is, as Johan sez, correct, but this does not mean that the oscillator actually *produces* those harmonics when it vibrates. It depends on the *type* of oscillator. [snip] > I don't know if this helps, but harmonics with frequencies above > about ~= 20,000 Hertz practically don't matter for human perception > of sound. Yes. That's one reason our ears are not saturated with crazy pitches, because eventually the higher harmonics fall off the audible range. Thanks everyone for all responses. > regards, > > David Bernier -- Ioannis
From: David Bernier on 5 Feb 2010 01:38 I.N. Galidakis wrote: > David Bernier wrote: >> I.N. Galidakis wrote: >>> David Bernier wrote: >>>> I.N. Galidakis wrote: >>>>> JEMebius wrote: >>>>>> I.N. Galidakis wrote: >>>>>>> I am playing around with a specific oscillator. I have identified 5 modes >>>>>>> of oscillation, which correspond to 7 frequencies N_j, therefore the >>>>>>> system phasor is: >>>>>>> >>>>>>> p(t)=sum(exp(2*Pi*i*N_j*t),j=1..7) >>>>>>> >>>>>>> Do I need to also include all the harmonics >>>>>>> >>>>>>> exp(2*Pi*i*k*N_j*t), k=2,3,4,... >>>>>>> >>>>>>> in the phasor, in order to analyze it in the Fourier sense? >>>>>>> >>>>>>> In other words, is the system phasor p(t) or q(t), with: >>>>>>> >>>>>>> q(t)=sum(sum(exp(2*Pi*i*k*N_j*t),j=1..7),k=1..oo) >>>>>>> >>>>>>> Thanks, >>>>> Hi Johan, >>>>> >>>>>> I guess you have identified 7 modes, not 5 modes of oscillation? >>>>> No, it's 5 modes, but two modes vibrate at two frequencies each: Mode 1 >>>>> vibrates at N_1 and 1/N_1 and Mode 2 vibrates at N_2 and 1/N_2. The rest >>>>> vibrate at one mode, total 7 frequencies. >>>>> >>>>>> Anyhow, one does not need the extra terms to obtain a decent Fourier >>>>>> analysis; >>>>> The reason I asked is because everywhere I asked, they told me that any >>>>> oscillator which broadcasts at N_j, also broadcasts at k*N_j, all of the >>>>> harmonics. For my case, this complicates matters considerably, because >>>>> according to Wiki, the harmonics fall as ~1/k^2 and I don't know how to >>>>> model them using an appropriate function. >>>> [...] >>>> >>>> From what I remember about the vibrating string (such as one idealized >>>> guitar string), the string end-points could be say at 0 centimeters >>>> and 48 centimeters. In the fundamental mode, the string goes >>>> up and down say 200 times a second. (pure 200 Hz tone). >>> Hi David, >>> >>> Although vibrating strings are a good example of oscillators, I believe that >>> the case of music is only a particular example of general oscillator theory. >>> >>> In any case, I have some experience with musical instruments and if the A0 >>> piano chord vibrates at 440 Hz, it will also generate all the higher >>> harmonics, at 880, 1760, ... n*440, etc., n=2,3,4,... >>> >>> This means that if I apply the damper ONLY on the A0 piano chord after it is >>> struck, there will be some residual resonance at all the higher A's, like A1, >>> A2, A3, up to the right limit of the piano. >>> >>> In other words, if I silence A0 with the damper, the rest of the A's will >>> resonate for a short time. >>> >>> In THAT sense I am interested in what sense to include all the harmonics in >>> my spectrum. >> [...] >> >> I think I misunderstood. The amplitude (or maybe power) of the harmonics >> falls as 1/(k^2), perhaps ... > > Sorry for the delay in responding. > > The subject is very complex and I am just *beginning* to understand it. And to > think that I've been playing the piano for 40 years :-) It seems from what I > understand so far that harmonics are NOT generated in all cases, so Johan's > response concerns the general *simplest* case. > > Harmonics are generated in many cases with musical instruments, and in this case > the combination of the fundamental and the harmonics determines the "pitch" > which one perceives. For example, the piano and some typical string instruments > generate what are called "partials" which seem to be very close to the harmonics > for the fundamental freq the instrument vibrates in: > > http://en.wikipedia.org/wiki/Harmonic_series_(music) > > (last paren has fallen off the link. Put back for link to work) > > Whenever this is the case, it seems that the intensity of the k-th partial > (harmonic) falls off (at least) as fast as 1/k^2. This means that in this case, > the Fourier series coefficients a_n satisfy: > > |a_n| <= M/k^2, for some positive constant M. > > Note however that *not* all oscillators generate harmonics, hence my confusion. > A piano string does, but a synthesizer which produces a simple sine tone, does > not. > > That's why I was confused. I actually have to examine my oscillator closely and > verify which harmonics are produced in each mode. > > To summarize: For a simple harmonic oscillator, the harmonics are given by its > Fourier coefficients, so the analysis is, as Johan sez, correct, but this does > not mean that the oscillator actually *produces* those harmonics when it > vibrates. It depends on the *type* of oscillator. Hello Ioannis, I'm more familiar with sound synthesis ideas (adding waves or waveforms) much more than music theory. Wikipedia on the decay over time of various harmonics: "Typically, high-frequency harmonics will die out more quickly than the lower harmonics." From: < http://en.wikipedia.org/wiki/ADSR_envelope#Imitative_synthesis > The article on timbre has a spectrogram in time and frequency, with intensity given by a color: < http://en.wikipedia.org/wiki/Timbre > To properly define a real-time FFT, one way is with a windowed Fourier transform: < http://en.wikipedia.org/wiki/Short-time_Fourier_transform > But there are many choices of windows ... David Bernier >> I don't know if this helps, but harmonics with frequencies above >> about ~= 20,000 Hertz practically don't matter for human perception >> of sound. > > Yes. That's one reason our ears are not saturated with crazy pitches, because > eventually the higher harmonics fall off the audible range. > > Thanks everyone for all responses. > >> regards, >> >> David Bernier
From: I.N. Galidakis on 5 Feb 2010 02:28
David Bernier wrote: [snip] > Hello Ioannis, > > I'm more familiar with sound synthesis ideas (adding waves or > waveforms) much more than music theory. > > > Wikipedia on the decay over time of various harmonics: > "Typically, high-frequency harmonics will die out more quickly than the lower > harmonics." > > From: > < http://en.wikipedia.org/wiki/ADSR_envelope#Imitative_synthesis > > > The article on timbre has a spectrogram in time and frequency, with > intensity given by a color: > < http://en.wikipedia.org/wiki/Timbre > Thanks. Yes. The totally hillarious thing is that I've done all that programmatically many years ago[*], but from a mathematical standpoint the story is entirely different. (Using a shameless plug) if you go to: http://www.virtualcomposer2000.com/Screenshots/index.html you can scroll down and see two FFT spectra: One for a sine tone and one for a (synthesizer) piano tone. The harmonics in the piano spectrum fall off really fast. Note also how the spectrum of the sine tone looks like a(n approximate) Dirac(x-440). But I think the issue is settled. I now have to figure out if my oscillator produces any harmonics. And I have no idea how to do that, because the frequencies are not in the audible sound range. > To properly define a real-time FFT, one way is with a windowed Fourier > transform: < http://en.wikipedia.org/wiki/Short-time_Fourier_transform > > > But there are many choices of windows ... [*] which virtually proves that programmers usually have no idea about what they are doing :-) > David Bernier -- Ioannis |