Intermodulation difference frequencies and symmetrical distortion

[Malcolm Irving]

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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 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.
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I think you are right about room resonances. I've experienced that as well. I think in my earlier post I was just thinking of relatively quiet music in a
room with a hi-fi. Not sure how it happens, but I've thought of two possible explanations. One is that music contains transients as well as steady frequencies. A transient (such as a step-change or an impulse) actually contains all frequencies and so this could stimulate the room resonance even though the frequency is not otherwise present in the music. Secondly, if the room resonance is in a some way part of a nonlinear system then new frequencies can be created.

I also agree with your examples of the 'musicality' or otherwise of the harmonic series (both even and odd). Sequences of notes, or simultaneous notes (chords) are judged by the ear-brain system as harmonious, or otherwise, according to whether there is a simple whole number ratio between the notes. Any frequencies from the harmonic series are in simple whole number ratios and therefore form the basis of scales and harmony. The great conundrum for music is how to map the harmonic series into a scale of notes within one octave. For example, taking the 3rd harmonic, 3f, dividing it by two, 3f/2, to bring it within the octave from f to 2f. This is a 'perfect 5th'. For reasons which I haven't space to go into here, a compromise is necessary to create a reasonable number of notes within an octave and for the intervals to be consistent. For example, we would like C to G to be 3f/2 but would also like E to B to be a perfect 5th as well. However, it turns out to be impossible mathematically to get all the 5ths perfect. The solution proposed by Bach et al. is called equal temperament and means that all the notes except the octave itself have to be compromised. The 5th has to be flattened slightly, but some other intervals particularly the major 3rds are way off from what the harmonic series would 'recommend'. I suppose what I am trying to say is that it is not the odd harmonics that are 'out of tune', it is the equal tempered scale which is out of tune!

[carlg]

Ah yes I'd forgotten about temperament there, it's been more than a decade now since I last cracked an acoustics textbook. Thinking back on it I remember there are many instruments that produce non-musically related harmonics naturally. Anything with bars, bells, or plates would fall into this category so glocks, vibes, drums etc. Funny then that "chimey bell-like highs" is a common phrases you hear in this hobby when the distinctive thing about bells is exactly that they produce frequencies that are not musically or mathematically related. Bells seem to be a real favourite amongst authors of acoustics texts because they illustrate that how we perceive pitch and what's actually happening are not necessarily the same thing.

[Malcolm Irving]

... I remember there are many instruments that produce non-musically related harmonics naturally. Anything with bars, bells, or plates would fall into this category so glocks, vibes, drums etc. Funny then that "chimey bell-like highs" is a common phrases you hear in this hobby when the distinctive thing about bells is exactly that they produce frequencies that are not musically or mathematically related. Bells seem to be a real favourite amongst authors of acoustics texts because they illustrate that how we perceive pitch and what's actually happening are not necessarily the same thing.

Yes, that's a fascinating point. Strings that are 'thick enough' and 'hit hard enough' start to fall into that category. Pianos are well known to have overtones that are significantly 'out' from the mathematical harmonic series.

I remember that FM synthesis was a 'breakthrough' that first allowed Marimbas, Bells, etc. to be reasonably synthesised.

Generally, I'm very 'anti' digital modelling for guitars. (Sorry I just like tube amps!) But maybe some digital modelling could introduce this type of (not exactly harmonic) overtone to get that sought-after 'bell-like' tone!

Edit: It would have to remove the natural harmonics first though and then replace them with the new overtones - I guess it would just sound like a guitar-synth - Yuck!

[nickb]

Yes, that's a fascinating point. Strings that are 'thick enough' and 'hit hard enough' start to fall into that category. Pianos are well known to have overtones that are significantly 'out' from the mathematical harmonic series.

I remember that FM synthesis was a 'breakthrough' that first allowed Marimbas, Bells, etc. to be reasonably synthesised.

Generally, I'm very 'anti' digital modelling for guitars. (Sorry I just like tube amps!) But maybe some digital modelling could introduce this type of (not exactly harmonic) overtone to get that sought-after 'bell-like' tone!

Edit: It would have to remove the natural harmonics first though and then replace them with the new overtones - I guess it would just sound like a guitar-synth - Yuck!

I'm pro-digital because of it's potential. I'm not a fan of digital modelling per se as to my way of thinking that is failing to realise it's true potential. Not that it doesn't have it's place. I see the potential as being able to synthesise sounds that would be practically impossible in the analog domain. In the context of the current conversation, this means being able to generate that perfect over-driven sound, if only it can be defined. I have a suspicion that it is elusive simply because it depends on the musical context, the ambiance, the ability of the the player and the mood of the listener i.e. subjective factors. Nevertheless, it should be possible to make a generalized approximation that is better than we are doing now.

Like it or not, but it is the wave of the future. We might cling to to our warm tubes but if we don't get on the boat soon we'll be left in their dying glow on the shoreline

[Mike Sulzer]

Room resonances do not create new frequencies. They are just resonances which emphasize certain frequencies.

[carlg]

That doesn't match my experience but what do I know. I'm apparently pretty rusty at this stuff. I know that what you describe is certainly one thing they can do, make some notes sound louder than others, but it's my understanding that resonances (of any type) are able to be energized by impulses (drums, note attacks etc), different but harmonically related frequencies, or (depending on their q) different frequencies that are within a range that's able to excite them. I've always thought that once energized they ring at their own frequency, which could be different than the exciting one, but I could be wrong.

[Mike Sulzer]

That doesn't match my experience but what do I know. I'm apparently pretty rusty at this stuff. I know that what you describe is certainly one thing they can do, make some notes sound louder than others, but it's my understanding that resonances (of any type) are able to be energized by impulses (drums, note attacks etc), different but harmonically related frequencies, or (depending on their q) different frequencies that are within a range that's able to excite them. I've always thought that once energized they ring at their own frequency, which could be different than the exciting one, but I could be wrong.

A true impulse contains all frequencies and thus can excite any resonance.

[loudthud]

I remember a thread from years ago where a novice was wanting to "fix" a low frequency wavering he heard from his amp as you let a note slowly decay. Thinking it was some kind of beating against the power supply ripple, I tried playing my 5E3 with a regulated DC supply. It didn't make any difference. Could it be the low frequency IM product?

[Gingertube]

FWIW (probably not much)
This is a copy of something I posted to AudioAsylum HIFI forum yonks back:

Intermodulation Distortion (IMD) and Harmonic Distortion (HD) are produced by the same mechanism. You can't have one without the other.
To get to IMD discussion we need to look at Harmonic Distortion first.

Back in 1997 Lynn Olsen wrote an excellent article on amplifier distortion (Glass Audio Vol9 No 4)

In his introduction he wrote:
"....the subjective correlation between the total harmonic distortion (THD) measurement and what you actually hear is close to zero".

He then went on to give an excellent analogy:
"The fault is not with the subjective perception of the listener, but with the measurement itself. There is nothing very new in this; you can measure all you want, but a mass spectrometer is not going to find a lot of difference between lunch at a high school cafeteria and the best dinner at a four-star restaurant. To foolishly assert that the mass-spec. machine is right,.....,is an example of simple ignorance trying to cover its nakedness with a fig leaf of science."

The following summarises, in my own words, what he had to say.

To attempt to make some correlations between harmonic distortion measurements and perceived sound you need to first separate Even harmonics and Odd harmonics and view these as separate sets of data.

Even harmonics are generated by asymmetrical distortion mechanisms and odd harmonics are generated by symmetrical distortion mechanisms.

As an example, look at the long tail pair. To reduce the Even Harmonics you must balance the currents in the 2 devices such that there is little or no asymmetry. Once this is done the residual distortion will be odd harmonics due to the symmetrical nonlinearities in the 2 devices.

This fundamental concept is important, and we get some big clues when we extend are consideration of distortion to include Intermodulation Distortion (IM).

Let us, for example, consider 2 frequencies in the important 1 to 5 kHz region.

For example use 3kHz and 4kHz:
Most of us understand that superimposing these (mixing in a non- linear system) will produce new frequencies and we often refer to these as sidebands. This is too simplistic.

The maths works like this:
Take 2 sinusoidal signal (well we use cosine rather than sine to keep the maths simple)
Signal 1 = a1(cos x)
Signal 2 = a2(cos y)

From output = a1 cos(x) + a2 cos(y)
Exapanding this and substituting in the trigonometric identity cos(x) + cos(y) = 1/2(cos(x+y) + cos(x-y)) and a lot of tedious algebra later we end up with an expression for:

1) the 2nd order term , For the superposition of 2 signals of x = 3KHz and y = 4kHz then four(4) new frequencies are created, 2 off 2nd harmonic terms and 2 off IM sidebands :
2x = 6kHz , 2nd harmonic of x
2y = 8kHz , 2nd harmonic of y
x + y = 7kHz , IM sideband
x - y = 1kHz , IM sideband

2) The 3rd order term gives six (6) new frequencies. 2 off 3rd harmonics and 4 off IM sidebands
3x= 9kHz , 3rd harmonic of x
3y = 12kHz , 3rd harmonic of y
2x + y = 10kHz , IM Sideband
2x - y = 2kHz , IM Sideband
2y + x = 11kHz , IM sideband
2y - x = 5kHz , IM Sideband

The important thing to note here is that the 2nd order distortion results in new intermodulation product frequencies (sidebands) which are remote from the original frequencies. (1kHz and 7kHz)

The 3rd order distortion results in more intermodulation products, two(2) of which are very close to the original frequencies. (2kHz and 5kHz)

To exagerate this (but still a valid example) see what happens with 14kHz and 15kHz
2nd order IM terms are 1kHz and 29kHz (very remote from the original 14 and 15kHz)
3rd order products are 13kHz and 16kHz (very close to the original 14 and 15 kHz)

Higher order terms produce even more IM Sidebands
I might be wrong (because I did'nt extend the algebra past the 2nd and 3rd order terms) but I think the 4th order will produce 6 IM sidebands and the 5th will produce 8 IM Sidebands.

Then consider other sources of a signal with which we can intermodulate.

One of the most important will be the 100Hz (120Hz in the US etc) residual power supply ripple which depending upon the quality of your power supply may well have 200, 300, 400Hz etc. harmonics as well.

At this point the "stray" IM products can run to thousands.

This is why a no feedback single ended triode amp with 2% of 2nd harmonic distortion but no 3rd, 4th, 5th etc can sound stunning and why an SS amp with 0.001% THD distortion which is primarily odd (and high) order distortion can sound awful.

It is also why some amplifiers which sound lovely with folk or jazz music which have a lower number of simultaneous tones (sparse spectra) can sound seriously rubbish when reproducing large choirs and orchestras (many closely spaced tones).

So what can we do about it?

The answer is simply to follow the established "popular wisdom".
1) Keep Power Supplies super clean
2) Don't allow high order harmonic distortion products
3) What harmonic distortion there is should be even harmonics ONLY

So here is my theory / explanation / WAG,
In short, higher order distortions produce many more Intermodulation Sidebands
Even order distortions create IM Sidebands remote from the original frequencies
Odd order distortion produce IM Sidebands close to the original frequencies.

End of copied material.

Cheers,
Ian

[Malcolm Irving]

Even harmonics are generated by asymmetrical distortion mechanisms and odd harmonics are generated by symmetrical distortion mechanisms.

I'd just like to clarify this point a little.
Asymmetric distortion produces both even and odd harmonics, in general.
To get odd harmonics only, the distortion must have ‘odd’ symmetry, i.e. f(x) = -f(-x).
To get even harmonics only, the distortion must have ‘even’ symmetry, i.e. f(x) = f(-x).

[uneumann]

Even order distortions create IM Sidebands remote from the original frequencies
Odd order distortion produce IM Sidebands close to the original frequencies.

Hmmm - I thought post 6 implied the opposite about the closeness of IM sidebands. Did I miss something here?

[Malcolm Irving]

In my posts, I seem to have overlooked the possibility of the ‘difference frequency’ being created by odd-order IM, as follows: f1=100 f2=150, f2-f1=50, but the odd-order product 2f1-f2 is also 50.