Intermodulation difference frequencies and symmetrical distortion

[Mike Sulzer]

I have attempted to show with a bit of math that the difference frequency is not present with symmetrical distortion. It is a bit rough around the edges and not complete. It looks as though it works for the more general odd order versus even order result as well.
imFromSymFun.pdf

Thanks to a very knowledgeable person (DF96) on another forum (diyAudio) I have found out that symmetric distortion produces only odd-order intermodulation products, such as (2f1 + f2), (2f2 - f1) etc. In particular, it cannot create the difference frequency (f1-f2) as that is an even-order intermodulation product.

I think this is quite interesting, because I find the difference frequency below a distorted power chord to be a big nuisance!

Could be another reason why folks like overloaded push-pull power stages.

OK, I know a lot of other folks prefer overloaded single-ended power stages.

[Malcolm Irving]

I have attempted to show with a bit of math that the difference frequency is not present with symmetrical distortion....

Thanks Mike. That's just what I was hoping for. I'm very impressed!

[Mike Sulzer]

Thanks Mike. That's just what I was hoping for. I'm very impressed!

Thanks. You might ask how good is this power series representation of a distortion function? The answer is that it can be very good with just a few terms within a set range for x, as the attached plot shows. However, it goes bad quickly outside the interval, and so it is good enough for proving a point, but not what you would actually use if you were simulating a guitar amp in software.

[jazbo8]

Rod Elliot presents some very interesting experiments which demonstrate that symmetrical distortion creates far less intermodulation sum and difference frequencies than asymmetrical distortion:
IMD - Something New

So are we in agreement that Elliot did not discover anything new? It's a simple matter of "less in, less out" - when symmetrical distortion starts off with less harmonic components than asymmetrical distortion to begin with...

[Malcolm Irving]

I think a direct quote from Rod Elliot’s article clarifies this:

“I searched many different sources and found no reference to this most interesting behaviour, however a reader did find one other text that mentions the effect, "Audio Measurements" by Norman H Crowhurst (1958, pp98-102) [1]. The effects described are not 'new', but this is one of very few articles you will find that describe the difference between symmetrical and asymmetrical distortion and how it affects intermodulation products.”

As they say ‘There is nothing new under the sun’!

However, I think the analysis that Mike Sulzer has presented is new.

[Mike Sulzer]

So are we in agreement that Elliot did not discover anything new? It's a simple matter of "less in, less out" - when symmetrical distortion starts off with less harmonic components than asymmetrical distortion to begin with...

I do not see the direct connection between fewer harmonic components and the lack of certain IM components. Perhaps you could explain it?

[jazbo8]

I do not see the direct connection between fewer harmonic components and the lack of certain IM components. Perhaps you could explain it?

IMD consists of all the sum and the difference frequencies, since symmetrical distortion ideally has no even harmonic, then there should be no f_odd + or - f_even (and the multiples), does that make sense?

[carlg]

Are we sure we want fewer imd frequencies? Even if we assume "IMD is bad" (which I'm not sure you can in all tones/styles), the logical conclusion would be that we want lower level imd products. That's not the same thing as fewer. When undesired frequencies are inevitable it can sometimes be the case that having a great many of them tends to average out into a nonspecific background noise, but having only a few makes each one more recognizable and objectionable. That's the situation you face with room modes (resonances) in sound reinforcement. In the low frequencies the resonances are sparse and produce very noticeable errors and problems, but as you go up in frequency more and more resonances are present until you reach a point where there's so many they blur together into a smooth level. Above that point (the shcroeder frequency) we call it reverb and actually like the effect, below that where we can pick out individual frequencies that are more prominent than others we don't like it anymore.

[Malcolm Irving]

Yes, interesting point. I think harmonic distortion and intermodulation are inextricably linked. If you want a certain amount of harmonic distortion to get that ‘nice’ overdriven guitar sound, then you get a certain amount of IM with it whether you like it or not!
The problem with the intermodulation frequencies is that they are not (generally) musically related to the notes you are playing, whereas harmonics are. (They are the basis of musical harmony and scales.)
I think some people have experimented with separate small pickups for each guitar string, followed by 6 independent overdriven channels to eliminate the IM between strings! There are also stories about one of the heavier bands recording their distorted chords ‘one string at a time’ – not sure if that’s really a fact though. Anyway, you still get the IM among the harmonics of the one string you are playing.
Room resonances only emphasize or de-emphasize frequencies that are already there in the music. Although something that has a strong mechanical resonance could be stimulated to resonate at its ‘own frequency’ unrelated to the music (but that would usually have to be pretty loud to do it).
Reverb is just the superposition of a lot of variable time-delays. This can have a comb-filtering effect on the frequencies that are present in the music, but can’t introduce any new frequencies.

Edit: Not sure if that last bit is strictly correct. If you have a high amplitude sound wave hitting a reflecting surface, the surface could move in sympathy. I think that would introduce a Doppler frequency shift to higher frequency waves reflecting from that same surface.

[jazbo8]

Are we sure we want fewer imd frequencies? Even if we assume "IMD is bad" (which I'm not sure you can in all tones/styles), the logical conclusion would be that we want lower level imd products.

You are correct, for "good sounding" distortion, we like to have a mixture of odd and even distortion and their IMD. If all we had was odd order distortion, i.e., strictly symmmetrical clipping, then we end up with a rather poor sounding "buzz" box.

[nickb]

You are correct, for "good sounding" distortion, we like to have a mixture of odd and even distortion and their IMD. If all we had was odd order distortion, i.e., strictly symmmetrical clipping, then we end up with a rather poor sounding "buzz" box.

I think we are searching for the Holy Grail here. Some ratio of even to odd plus a certain shape of filter perhaps in some feng-shui arrangement? Just look at the plethora of stomp boxes out there. It seems that perhaps no one knows the answer. My concern is we are getting away from objective measures. I found the very low beat frequency suppression effect for asymmetrical very compelling and is easy to demonstrate.

Can you provide some authoritative sources that define exactly what is 'good'?

[Mike Sulzer]

I think some people have experimented with separate small pickups for each guitar string, followed by 6 independent overdriven channels to eliminate the IM between strings!

Yes, and I for one find the six separate distortions useless: too boring. Although classic power chords minimize unpleasant IM, I think good composition and playing uses some level of additional dissonance to create different musical effects that were not used before electric guitars.

[carlg]

Malcolm,

You're right, there are many differences between the room mode situation and IMD. I didn't mean to suggest they're perfect analogies, it was just the first example that popped into my head of a situation where having many errors averaging together could be preferable to having fewer, but more individually identifiable ones. I just wanted to play devil's advocate a bit and point out that those situations do exist, so maybe fewer imd frequencies = better wasn't the given everyone seemed to be taking it for. It could be, but it seems worth thinking about.

Room resonances only emphasize or de-emphasize frequencies that are already there in the music. Although something that has a strong mechanical resonance could be stimulated to resonate at its ‘own frequency’ unrelated to the music (but that would usually have to be pretty loud to do it).

I could be wrong but I'm under the impression that room resonances can definitely add frequencies that aren't present in the original music. In the most extreme examples I've been at clubs where once the band starts playing all I can hear in the bass is a single, endlessly droning frequency. I can see that the bass player is changing notes and shredding away, but pretty much all I can hear where I'm standing is this one frequency that I've always assumed is being set by the dimensions of the room. Every now and then the bass player will hit a note that for some reason suppresses that mode and the droning suddenly stops, which unfortunately only makes it's more noticeable since now that error frequency is switching on and off. Live sound can be a real pain.

The problem with the intermodulation frequencies is that they are not (generally) musically related to the notes you are playing, whereas harmonics are. (They are the basis of musical harmony and scales.)

The even harmonics are octaves at least, but the odd harmonics don't seem to be musically related. The first couple do happen to fall close to recognizable notes though. The 3rd harmonic is very close to a P5, but the 5th harmonic is about 14 cents flat from a M3. I don't know how musical that would sound, particularly if you're playing in a minor key. Odd order harmonics beyond that are adding musically unrelated frequencies, and odd harmonics would be the only type generated by the symmetrical clipping being talked up in this thread.

To go a little deeper, this whole idea about "musically related" distortion ( that "even order = desirable, odd order = bad because it's not musically related" or "IMD = bad because it's not musically related" etc) is one of those common wisdoms you do hear a lot, but I think it's very worth taking some time to question whether you agree with it. There are so many different tones and musical styles that it's very hard for any generalization like this to be true.

Edit: I made a big error in this next paragraph. Please skip this part
If you want to hear what this supposedly ideal tone sounds like, made of the root note with nothing but these "desirable" even harmonics added, just plug into a perfectly clean amp and switch to bridge pickup or turn up the treble. There you go. The raw sound the string makes already is the root plus every even harmonic. The harmonics are normally at a lower level than the fundamental, but switching to the bridge pickup and/or turning up a treble knob is all it takes to add as much "even order harmonics" as your heart desires.

I don't think a bridge pickup into a perfectly clean amp with the treble cranked is really the sound most people are looking for, although the Bakersfield twang crowd seems to like it. With the crunchy and overdriven tones most people want though, I believe adding a certain amount of NON-musically related distortion is exactly the goal, not the thing to be avoided, and it seems very plausible to me that IMD could be a desired ingredient in that recipe.

In more heavily distorted sounds I don't think there's any question that non-musically related distortion has always been an iconic part of the sound. Reminds me of a little moment in a video where John Petrucci is over at mesa boogie beta testing one of their amps. He's really stoked on the heavy rhythm tone and remarks it's got that kind of attack where "you can't quite tell what pitch is being played". I don't think musically related distortion frequencies could ever get you there. Link to that video

[Mike Sulzer]

If you want to hear what this supposedly ideal tone sounds like, made of the root note with nothing but these "desirable" even harmonics added, just plug into a perfectly clean amp and switch to bridge pickup or turn up the treble. There you go. The raw sound the string makes already is the root plus every even harmonic. The harmonics are normally at a lower level than the fundamental, but switching to the bridge pickup and/or turning up a treble knob is all it takes to add as much "even order harmonics" as your heart desires.

The odd harmonics are just as present. Certainly the fundamental can be much less than the level of the first several harmonics, but the pitch of the note is perceived correctly. That has to do with the spacing of the harmonics.

But I think that interned is really something very different.

[carlg]

Yeah I realized I made a boner there as soon as I pressed submit and read it back. Although it's already 10am out here I still haven't had coffee

If anything that's an even better argument against the old "even harmonics = good, odd harmonics = bad" story though. If odd harmonics sounded so bad and unmusical, why do we like the sound of acoustic guitars?