Almost 20 years of effort, to the very date… I finally found another research group that obtained essentially the same EarSpring equation that I found through my exhaustive laboratory measurements.

https://arxiv.org/pdf/1408.2085.pdf

Not precisely the same equation, at the top of page 24. But it means exactly the same thing. They found the gamma dependence that I found. To them, it represents a cubic term in a Taylor series expansion of a nonlinearity term. But to me it represents an increasing stiffness with power of vibration.

So happy to see this. You have no idea how uncertain it feels when you are stumbling in the dark, never sure what the next step means, nor if it is even correct.

The researchers used formal mathematical physics to derive the equation at the top of page 24. They resorted to nonlinear differential equation analysis, using tools that were developed in the 1960’s and 1970’s to analyze nonlinear systems.

So for them, it represents a formal expansion term without any specific apriori reason to assume the existence of 3rd order expansion terms, other than knowing already that our hearing exhibits nonlinearity. They show that second order terms cancel, and the first remaining nonlinearity comes from a sole 3rd order term. And they expound further down the page to show that this implies cube-root compression and results in the IMD products for 3rd order intermodulation distortion.

*[3rd order IMD products – hit the ear with two roughly equal loudness signals at frequencies f1 and f2. Our hearing will produce two new tones located at 2f1-f2 and 2f2-f1. Those are located below f1 and above f2, assuming f1 < f2. They are much less loud, but grow as the cube of the level of the test tones. If the test tones increase by 3 dB, the IMD product tones increase by 9 dB. A sure sign that these are 3rd order IMD product tones.]*

These results match what we already know and observe – that Steven’s law shows cube-root compression in the Sones scale, and that 3rd order IMD products really are generated inside our hearing. But more than that, they show that the equation produces exactly cube-root compression, as long as you ignore the higher order nonlinearities. We also know that the compression is not precisely cube root, and becomes slightly more compressive at very loud levels.

So EarSpring is the equation to 3rd order. And that covers most of the audible range of daily loudness experience, certainly all that matters for listening to music.

They show that IMD products will be generated if there are 3rd order expansion terms in the equation. I found the EarSpring equation by measuring those 3rd order IMD tones – called “phantom tones”. So we derived the same results from opposite approaches. They used theory to predict the results, and I used lab measurements of the results to derive theory.

*[The very existence of phantom tones located at 2f1-f2 and 2f2-f1, and its cubic growth with loudness nails the case for 3rd order IMD products. (and implies a cubic term in the equation). But I measured the 3rd order IMD Intercept Point – the loudness at which the IMD products would theoretically become as loud as the test signals. That tells us not only that they exist, but what their growth factor is – the gamma value in EarSpring. And I saw that the intercept point moves by a tiny bit as loudness increases, so that it is not a fixed point – that our hearing has a cubic term, but also has something more.*

*… and, I found that the 3rd order IMD intercept is right up around 85 dBSPL – which means that when listening to really loud music, our hearing will be swimming in IMD product tones along with the music. These product tones are not harmonically related to their sources, but we must have just become accustomed to them.*

*… and furthermore, when you use perceptual coding of audio, as with MP3, where you leave out the stuff that is supposed to be inaudible as a result of loudness masking – you also remove those sources of potential 3rd order IMD products. Which means that the music won’t sound the same, if those IMD products would have been audible.*

*Hint for Mixing and Mastering Engineers – if you work by listening at loud levels, as in dBX Theater Standard levels, then you might be hearing audible IMD tones. And by dropping the sources of the IMD tones by as little as a couple of dB, those bothersome extra tones will decrease by 6 dB or more. *

*And since it can be difficult to track down those source frequencies, this really just means lowering the overall volume by a few dB. If you try a narrow dip EQ at the bothersome frequency, you should find that it has no effect on the phantom. You need to dip at the source frequencies, generally somewhere above or below the bothersome tone.*

*That could also be a sure sign that you are listening at too loud a level. What you are doing is chasing phantoms that aren’t really in the music.]*

There are no specific publication dates in the paper, but they refer to some other results that were published in 2006. So the paper was written around the same time that I was investigating and discovering EarSpring. So in terms of human history, even though that was more than 10 years ago for both of us, this information is “hot off the press” for humanity.

Hurray!!

- DM

*PS: I also found early on, that my crunchy sounds are sometimes a form of 3rd order IMD product. When playing with a graphic equalizer against high string sections or female chorus sections, I located the crunches at around 1.7 kHz. Yet, by dipping at around 1.1 kHz by just a few dB, I could make the crunchy sounds disappear entirely. At the time, it seemed weird. By now, I pretty much understand them. *

*But that 1.7 kHz is also the location of my hyper-recruitment. And I can use a 1/3 octave (narrow) dip EQ by a few dB ahead of Crescendo to remove objectionable crunchy artifacts when listening to music (not high string sections or female chorus sections, but bothersome musical tones). If the crunchy sounds were solely generated by IMD then a narrow dip EQ at 1.7 kHz would have no effect.*

*So there are really two things going on here. By using a narrow boost EQ I could locate the frequency of maximum crunchy sound. But that boost has no effect on IMD product tone levels. I’m being bothered by hyper-recruitment in this case. So yes, dipping at 1.7 kHz helps, and it seems to take about 6-8 dB of dip across a 1/3 octave band.*

*But then, when the strings or vocals come online, the crunchy sounds reappear at loud listening levels and the EQ has no effect on them. That’s when they come from 3rd order IMD products. Just stop listening at such loud levels and they disappear. A tiny drop in volume overall, causes an enormous drop in the crunchy sounds.*