[sdiy] Generating acyclic waveforms?

[Magnus Danielson]

Ian Fritz wrote:
> At 07:06 PM 3/21/2010, Magnus Danielson wrote:
>
>>> You wouldn't see a significant effect like this in winds because the 
>>> waves are longitudinal waves in air and therefore have negligible 
>>> dispersion.  Additionally, driven systems such as winds excite 
>>> coherent waves, so there can be no speeding up of specific harmonics 
>>> over many cycles.
>> However, the end-impedance would allow for a similar effect, such 
>> that the resonance frequency of the overtones would not match up 
>> exactly with the integer multiples frequency-wise form the base 
>> frequency. To illustrate the impedance effect on a wind instrument...
>
>
> No. The played notes have harmonic spectra, because they are 
> periodic.  (In mathematical terms this is known as "Fourier's 
> theorem").  You are perhaps thinking of the resonances of the 
> impedance curves, which are affected by a number of parameters, 
> including the shape of the bore and end effects.  The art of making 
> instruments is to try to get the two spectra to match reasonably 
> well.  This is *extensively* discussed in Benade's book "Fundamentals 
> of Musical Acoustics", chapters 20-23.
>
> Ian
>
I think you entirely missed my knowledge of both Fourier transforms and 
acoustical impedances and related resonances. I also think you might 
have missed the finer details that occurs in acoustical resonators such 
as used in crystal oscillators, where fundamental frequency and 
overtones is known to be non-harmonic due to among other things 
impedances, and that also many modes of oscillation occurs. It is not 
entirely impossible, that similar effects do haunt air-resonant 
open-ended resonators such as wind instruments, which is what I proposed 
passingly. If you do not match the impedance carefully, the resonances 
will not match up harmonically and the produces waveform will be 
periodic at a frequency much lower than the fundamental, but not have 
the usual relationships to the fundamental tone, as you then point out.

Best Regards,
Magnus