[sdiy] Harmonic content of the "sigmoid" half-sine wave

[Jerry Gray-Eskue]

<<One interesting effect is where you add together many high frequency
harmonics, but none of the lower order ones. The resulting waveforms
consist of a burst of energy with not much going on inbetween. The
period of repetition is the whole thing from one burst to another,
and this suggests a fundamental frequency which isn't present>>

I think it would be safe to say that the period of repetition is equal to
the least common division of all the wave form periods, that is to say it
would only repeat on the lowest number of full cycles for all the waves.


It is also my understanding that if you mix 2 sine waves you get four
frequencies out, the original frequencies, the sum of the frequencies, and
the difference of the frequencies. With each additional sine wave added you
get the same effect interacting with each of the other frequencies.

So is the suggested fundamental not there, or is it created by the total
interaction of the original sine waves, their least common fundamental
frequency, ( even though not present in the generation of the waveform )
and represented by the resulting wave form?


-----Original Message-----
From: Tom Wiltshire [mailto:tom at electricdruid.net]
Sent: Wednesday, June 03, 2009 9:27 AM
To: Jerry Gray-Eskue
Cc: Synth-diy at dropmix.xs4all.nl
Subject: Re: [sdiy] Harmonic content of the "sigmoid" half-sine wave



On 3 Jun 2009, at 14:12, Jerry Gray-Eskue wrote:

>
> Thanks Aaron
>
> I think I get the feel for what is going on, correct me if I am
> wrong on any
> of these points.
>
> As I understand it, A Fourier series define the theoretical
> combination of
> sine waves that produce any repeating waveform.
>
> In any case that it appears that a sine wave is being produced but
> at less
> than all 4 quadrants over the period of the waveform, the apparent
> fractional sine wave is merely the result of the content of various
> higher
> frequencies as there is no combination of sine waves that will
> result in
> sine wave short of 4 quadrants.
>
> This leads me to an assumption: for any repeating waveform the
> period of
> repeat ion is the period of the fundamental frequency - I do not
> have a
> "warm fuzzy" on this assumption, it correct?

Yes; or at least that's my understanding too. Note that the
fundamental frequency by this definition might not be the same as the
heard fundamental pitch, but that's a question of psychoacoustics
rather than physics.

As an example, imagine you add a sub-octave sine wave to a repeating
waveform. The sub-octave waveform will cause alternate wavecycles to
differ (since one will be added to the positive part of the wave and
one to the negative) so the period of repetition halves along with
the fundamental frequency.

One interesting effect is where you add together many high frequency
harmonics, but none of the lower order ones. The resulting waveforms
consist of a burst of energy with not much going on inbetween. The
period of repetition is the whole thing from one burst to another,
and this suggests a fundamental frequency which isn't present. Examples:

http://www.electricdruid.com/Lumps2.png
http://www.electricdruid.com/Lumps2_spectrum.png
http://www.electricdruid.com/Lumps4.png
http://www.electricdruid.com/Lumps4_spectrum.png

Going back to the sigmoid waveform, I once tried generating this
waveform from harmonics for use as a distortion waveshaper (it's a
soft-clip curve) but unfortunately I wasn't happy with the results,
so I never kept them...otherwise I'd have a similar graph for that
curve - Sorry!

Regards,
Tom