posted 25 March 2002 21:08            

Start with two equal amplitude tones, say 300 Hz and 500 Hz. The second IMD products are at 200 Hz and 800 Hz. Both of these should have the same amplitude, and for every increase of 1 dB in the parent tones, these two tones should increase by 2 dB.

The third order IMD products are located at 700 Hz and 100 Hz. These should both be the same amplitude and for every increase of 1 dB in the parent tones, these two should increase by 3 dB.

Now our scale of reference is unknown to us by some offset in dB, but the difference between where the 2nd order intercept occurs and where the 3rd order intercept occurs should be independent of that unknown offset.

According to my calculations, this difference should equal

4.75dB + 10*log10[2/3(2-a)/(1-a)] where the Benade exponent is a.

For a = 0.5 this should be 7.76 dB, a = 0.6 => 8.43 dB, and a = 0.7 => 9.36 db.

So all of this hinges on being able to detect differences of 1 dB in the probe tone levels.

...after dinner I will have a go at this...

- DM

[Well... the answer is a resounding ???. I can't determine the amplitude of a probe tone to better than about 3 dB, and the measurements of the 3rd order IMD products is simply a ridiculous outcome. This one will have to await a better approach, like OAE Tomography...

Oh, there is absolutely no problem detecting beats at these various IMD product frequencies. None at all... The difficulty is trying to determine what level of probe tone produces the strongest possible beat notes. That turns out to be a very broad function of probe amplitude. So broad in fact that I get an answer to this question of a = 0 +/- 100, which means a totally useless answer...]

[This message has been edited by David McClain (edited 25 March 2002).]

posted 26 March 2002 03:07            

Since the IMD tones decay at rates of 2x and 3x of the parent tones, I setup the Kyma to forcibly decay the probe tone at these multiples, while the parent tones were stepped in increments of 3 dB. Hence, if I tune up on the strong signals then at maximum attenuation I should still hear the beatnotes. If I am off a little then that error gets magnified at the successive attenuations.

I also tried numerous detunings above and below the expected IMD tones, to avoid masking artifacts, and to compare both slow beatnotes and fast beatnotes all the way up to the onset of harsh tones, at around 10-15 Hz difference.

This is a very tough measurement to make with any degree of accuracy, but my results are now showing a spread of 7-8 dB between the 2nd and 3rd IMD intercept points. That implies a Benade exponent somewhere in the range slightly below 0.4 to slightly above 0.6. These in turn suggest that one's perception (mine) of "twice as loud" (whatever that really means...) should be between 10 dB and 15 dB of signal amplitude increase.

After numerous trials I fail to show any separation beyond 8 dB and a fraction between the two IMD intercept points. Note that this test is not affected by apparent hearing loss. All that is necessary is to balance the probe tone level such that maximum beat strength is observed, however strong that appears to be.

So at the very least I have bounded his exponent on the bright side for corrective compression gain, and we are well away from 1.0 which is the condition for linearity. There really is a verifiable nonlinearity, and if it follows the Sones model above 40 dBSPL then our exponent lies between 0.4 and 0.6. But even if this model is grossly incorrect, the mere fact that I so readily perceive these IMD tones with a probe tone shows that substantial nonlinearity exists in my ears. (Yours too!).

- DM

posted 26 March 2002 06:45            

Many times I have needed to have the log of the spectrol envelopes when morphing. What I do is, do an analysis first, and then do the convertion from linear to log. What I don't do is destort the signal before it goes into the analysis to avoid having to do the lin to log convertion later, and then try to undo the intermodulation caused by the destortion.

It seems to me that you are saying that, because we perceive signal level in a log form, that the log must be added before our ears and brain derive the spectrol info.

Have I misunderstood this?

also if you make a mix of 300Hz and 500Hz and then start to destort it, the first two new signals that will appear , will be 400Hz and 100hz and not the sum and the differance.

posted 26 March 2002 12:48            

No I was careful to avoid distorting the parent signals. Typically I operated at -12 to -24 dBFS, generally at -20 dBFS. The IMD product tones were quite apparent with the probe tone technique, no problem hearing them at all.

The IMD intercept point is that sound level in the parent tones such that these IMD product tones would have equal amplitudes to their parent tones. I never pushed the system that hard to actually go there... not sure my ears could even tolerate it. Instead you chart the growth of the IMD tones as a function of parent tone amplitude and then you ascertain where this IMD growth line intersects the line of unit slope which represents the parent growth. That is the IMD intercept point. In our case we have two different intercepts, one for 2nd order IMD, and another for 3rd order IMD. The dB difference between these two intercept points is what I am trying to measure.

Fine radio receivers are generally rated in terms of 3rd order IMD intercept points as an indication of how distortion free the front-end amplifiers are. (high-gain RF amps at the antenna end) On the basis of a comparison like this with a fine radio receiver, I would grade our ears as F-, but the dynamic range is incredible!

The Log in the Sones expression is merely there because we like to work in units of dB. It does nothing to distort the sounds. Whenever you specifiy an oscillator amplitude in dB measure, you are doing the same thing I amd doing here. Harmless as far as quality of sound production.

- DM

[In terms of apparent loudness, I was working primarily in the vicinity of 60 dBSPL -- comfortably loud, and well above the limit of 40 dBSPL in the range 200 Hz to 800 Hz.]

posted 26 March 2002 13:18            

I wanted to find a more objective means to measure the Benade exponent. Researchers using JND's in the UK and Germany have reported values ranging from 0.47 to 0.8. Quite a big spread. Perhaps that spread is real and it reflects the variability of humans. But then perhaps it is an artifact of this ad-hoc numeric grade assignment.

Using the dB difference between the 2nd and 3rd order IMD intercept points is a more physics-oriented approach. I have never seen any of the researchers report using such a technique, but then the researchers in these fields are often Psychologists and MD's, neither of whom would be expected as a group to have much in the way of physics knowledge.

As a kid I met Dr. Benade at Case-Western Reserve University. I remember going into his lab and seeing his horn measurements on his bench, with hypodermic needles feeding in probe sounds and microphones. He had ancient pen and ink chart recorders running all over the place. [Computers in those days occupied large rooms and were never used for running and recording experimental results except at places like Brookhaven and other national accelerator facilities.]

I was very impressed. I was a musician leaning toward a career in science, not even yet in college. I have fond memories of Dr. Benade, all the more because he used physics to understand my other love of sound. I feel that this experiment is a tribute to his memory...

posted 26 March 2002 14:20            

I was obtaining 2nd order IMD intercepts around -8 dBFS and 3rd order intercepts around -15 dBFS. In terms of VU these are +2 dBVU and -5 dBVU! That means that at normal listening levels with music, these IMD products are equal in amplitude to their parent tones. Whether we realize it or not, we are hearing their effects in the blend of musical sound. These add to the richness of music to a surprising degree. They are not subtle side effects as I had thought.

So when a compression technique like MP3 discards some sound levels on the basis of apparent masking, they also discard to possibility to produce these secondary tones, and that no doubt, is one reason why you can spot an MP3 a mile away!! The music is lacking that "certain something" and it is not entirely attributable to their 15 or 16 KHz cutoff.

- DM

[This also explains why, as I have often heard recording engineers remark, EQ applied gently at one frequency affects all other frequencies as well. Indeed!]

posted 27 March 2002 05:48            

When resynthersizing speech,If we omit the upper partials the speach sounds totaly dull, but if we listen only to those upper partials , they seem so quiet and almost imposiable to hear. It seems unbeleavable that those almost inaudable scratchy noises can make such a hugh difference to the normal sound.

Maybe those upper partials are causing more to happen at the lower frequencies (mid frequencies) when they are combined with the lower harmonics, and not just adding to the treble content.

posted 27 March 2002 11:30            

So no doubt things are even more complicated than all of this. Perhaps there are frequency bands (low and high) over which these various effects are more pronounced or less pronounced.

Ernst in Germany was pointing out the importance of some spectral components below 700 Hz in order to help reconstruct speech in our minds.