Saturating a miniature audio transformer in a pedal.

[R.G.]

That scope waveform looks vaugely sawtoothy, maybe a strong 2nd harmonic? I

It's symmetrical top and bottom. That means only odd harmonics. Second harmonic gives strong asymmetry, tops being different from bottoms.

[gritz]

That scope waveform looks vaugely sawtoothy, maybe a strong 2nd harmonic? I

It's symmetrical top and bottom. That means only odd harmonics. Second harmonic gives strong asymmetry, tops being different from bottoms.

Yes. And no. The waveform shape depends on the relative phases of the waveform components. The "if-it's-symmetrical-it-must-be-odd" thing only really applies to time-invariant gain mangling, like clipping. As soon as you toss the possibility of phase shift into the mix (with e.g. inductors and / or capacitors) then anything can happen and then it's time to reach for the spectrum analyser

[Jazznoise]

I can draw up a similar waveform in most DAWs. Not to be overly reductionist, but when it looks like a sawtooth and sounds like a sawtooth then it's probably a complex waveform with partials of amplitude relationship equal to 1/n of the fundamental where n is the number of the harmonic, yadig? 

Also, phase shift in this scenario is a certainty - not a possibility. I'm not overly invested in my opinion, so don't take that as defensiveness - I'll do up a table in MAX MSP over the weekend that looks similar as I'm curious now. I can do a fourier there and we'll get an aproximation of the reality, however cackhanded.

[amptramp]

A triangle wave has all the odd harmonics, but the amplitude is the reciprocal of the square of the order compared to the square wave whose amplitude is the reciprocal of the order.  The difference between the top and bottom of the waveform gives you the even order harmonics.

F + 1/3*3F +1/5*5F+... is a square wave

F + 1/9*3F +1/25*5F+... is a triangle wave

[R.G.]

Yes. And no. The waveform shape depends on the relative phases of the waveform components. The "if-it's-symmetrical-it-must-be-odd" thing only really applies to time-invariant gain mangling, like clipping. As soon as you toss the possibility of phase shift into the mix (with e.g. inductors and / or capacitors) then anything can happen and then it's time to reach for the spectrum analyser

It's been a long time since I read through the sources that led me to my rule-of-thumb that mirror images top and bottom are composed of only odd waveforms. In the interests of being sure I'm not misremembering, I went looking for some things.

Try here: http://www.elect.mrt.ac.lk/EE201_non_sinusoidal_part_1.pdf page 8, just after the second equation, where it says

Thus is is seen that in the case of half-wave symmetry, even harmonics do not exist...

The math is based on the observed wave shape, not on any relative phasing of harmonics. That's a way of saying that whatever phase shifting was done had to be done before you observed the wave shape, and the phase shifting previously affecting the waveform is already included in the description of the waveform.

It is true that waveshape as observed on a scope is dependent on the relative phases of the harmonics, and one can change the observed waveshape by changing the phase of a harmonic, possibly by doing a harmonic synthesis setup to control it directly. So one could make a square wave look much less like a square wave by phase shifting the 5th by 90 degrees, for instance. However, what the math says is that although the wave shape changes, the harmonics making it up do not, and if it has even harmonics, it will not be half-wave symmetrical. So I think that the paper seems to be supporting "if it's symmetrical, it's only odd harmonics".

N.B.
The presumption in the math is that the waveforms are not time varying. This is not ever completely true, but it's very, very close in the real world.
Also, ANY differences from top and bottom require the introduction of even order harmonics to describe the waveform in terms of sines, cosines, harmonic number and coefficients. So if in the example of the photos shown, the (presumed) leakages were different for one half of the winding, they would in fact insert different harmonics. However, the closer it looks to symmetrical, the closer the spectrum analyzer will find the even harmonics to zero.

So I think it's yes. Help me understand if I have that wrong.

[tca]

Phase shifts can alter the visual form of the signal, but not the harmonics. Look for the cos form of the Fourier series; for each term there is the amplitude of the harmonic and the phase of that harmonic.

Cheers.